The groups of interest in Zimmer's conjecture are those involving special "higher-rank" lattices, which are lattices in certain higher-dimensional spaces.
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The Nagaoka-Thouless picture also only applies to simple lattices: two-dimensional lattices of squares or triangles, or a three-dimensional cubic lattice.
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Physicists knew from experiments that the vortices form triangular lattices, called Abrikosov lattices, and so the question was to prove why they form these patterns.
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Even when you allow the lattices to transform a space in very irregular ways — by shearing, bending, stretching — the lattices are still tightly restricted in where they can act.
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Lattices can exist in spaces of any number of dimensions.
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For a system of electrons and a single hole, they proved that the connectivity condition is satisfied for nearly all lattices, including common structures like the two-dimensional honeycomb and the three-dimensional diamond lattices.
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It also encompasses the surface lattices used by Caspar and Klug.
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It starts with the list of spaces on which higher-rank lattices can act — the list that Margulis found — and asks whether this list expands when you allow the lattices to act in less rigid ways.
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The next and even more ambitious phase of work is to focus on just those spaces in which the lattices do appear — and then to classify all the different ways in which those lattices transform those spaces.
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Zimmer's conjecture concerns special kinds of symmetries known as higher-rank lattices.
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Spaces equipped with lattices have an infinite number of different lattice symmetries.
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Diamonds are lattices of carbon atoms where each carbon connects to four others.
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"Untitled" (2016) and "Untitled" (2016), for example, are both lattices of thick brushwork.
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Instead, they see point clouds or lattices that might some amorphous form about them.
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Others have structural color: nanoscopic bubbles, lattices, and granules that scatter and refract light.
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If the Nagaoka-Thouless theorem really explains ferromagnetism, then it should apply to all lattices.
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Untitled (DDCC) and Untitled (Speculative Lover), two gridlike lattices, engage the language of geometric abstraction.
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Then the graphene is torn in two to produce two flakes with perfectly lined-up lattices.
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Between these lattices he'd inscribed legions of Bosch-like tiny figures engaged in odd, feverish interactions.
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You posted open source instructions explaining how you created 3D-printed lattices for cell culture online.
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The research team studies the kinds of bio ink lattices they could build at various extrusion pressures.
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The cubes' have elaborate structures and are made up of numerous layered lattices that make them ultra-durable.
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Diamonds are just lattices in which every carbon atom bonds to four neighboring carbon atoms, making a strong network.
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The technology hides data within complex math problems known as lattices, which even quantum computers find difficult to solve.
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Here, the lattices expand and contract in relation to the fluctuations of population as expressed in the objects' contours.
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But band gaps were once thought to be unique to crystal lattices and direction-dependent, aligning with the crystal's symmetry axes.
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That enables users of Carbon to create lattices, which can give objects like drones or vehicles structural integrity without unnecessary weight.
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The answer: Opposing straps whose complementary lattices interlock like Lego bricks, allowing each to fasten quickly and securely to the other.
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The famous and rare blue diamonds, like the notorious Hope diamond, have varying quantities of boron atoms in their crystal lattices.
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They tracked the way that these progressed from abstract shapes—spirals, lattices, tunnels—to recognizable objects, just as Werner Stoll now did.
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Their first series remained their best known: densely wrought compositions of convoluted golden horns whose intricacies could evoke lattices, weapons or bodies.
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And every once in a while, he would secretly tap fingertips with other inmates, through the metal lattices on their cellblock doors.
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To fix this, researchers led by MIT's Jeehwan Kim, used crystalline forms of silicon and germanium that resemble lattices at the microscopic level.
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Margulis gave a complete description of which kinds of spaces can be transformed by these higher-rank lattices when you permit only rigid transformations.
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Local materials such as teak and red brick help to create jalees (lattices) and beras (perforated screens), which provide cheap shading, shelter and ventilation.
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Perhaps in the summer roses would grow on the lattices up here, but now, the blizzard beginning to pick up again, they are bare.
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Qubits can be stored in a lot of different ways via many different properties of particles: electron spin, atomic energy levels, atomic lattices, nuclear spin.
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Power lines supported by two wooden poles rise above a scrubby terrain in the United States; lattices of metal parade over the hills of Australia.
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In regular crystals, atoms arrange themselves in lattices, so you could say that they prefer to be located in a certain point of space—space crystals.
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Cryptographers are debating the relative merits of such mathematical curiosities as supersingular isogenies, structured and unstructured lattices, and multivariate polynomials as foundations for quantum-proof cryptography.
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The special higher-rank lattices involved in his conjecture were first studied in the 1960s by Grigory Margulis, who won the Fields Medal for his work.
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By answering the conjecture, the coauthors of the new work have placed a rough-cut restriction on the spaces in which higher-rank lattices can act.
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Unlike traditional 3D printers that build layer-by-layer, Branch's machines will create lattices, which they will then fill with liquid foam and concrete that hardens.
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The pies and tarts, after all, are meant to the be our focus: They're pretty spectacular, some with gobsmackingly intricate lattices, others studded with goldenberries and pomegranate seeds.
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But when one of the layers was twisted slightly compared to the other, the rotational misalignment of the two lattices produces a repeating "moiré pattern" stretching across many atoms.
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Bouguereau was influenced by the Italian Renaissance painter Raphael but many of his compositions feel Mannerist (Veronese, Parmigianino, Bronzino) with armatures of ascending, interlocking lattices of arms and hands.
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Wrapped in plastic or covered in beige or green siding, more than a dozen homes float above their neighbors, supported by thick metal I-beams and lattices of wood.
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For 50 years, Nohl constantly tinkered with the art, adding lattices of concrete faces and glass that caught the light, wind chimes in the trees, and whimsical mosaic creatures.
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This absence led artist Johnathan Payne to create Constructions, an ongoing series of intricate lattices made from old comic books including Thor, Coneheads, and Deathstroke the Terminator, among others.
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Near-boiling pools of acidic water bubble between odd formations of rocks and minerals: white beehive-shaped mounds of salt, yolk-colored lattices of sulfuric crust, purplish-red crumbles.
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The green of emeralds and the red of rubies, for instance, are both caused by chromium atoms, but the atoms in question are bonded into their respective lattices in different ways.
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It turns out that back in 1974, a mathematician named Richard Wilson, now at the California Institute of Technology, had figured it out, generalizing and solving the 15-puzzle for all lattices.
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Drifting about the richly hued atmospheres of his paintings are flat silhouettes suggestive of prayer beads; the Buddhist temples called stupas; fans; lattices; medallions; and bulbous cartoony shapes, sometimes with glowing auras.
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Viviano uses a glass-making method called reticello, in which clusters of glass canes are laid crosswise and fused, then sculpted to create forms that are embedded with fluid, diagonal grids or lattices.
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In two small watercolors of gilded honeycomb lattices, it's the surface itself that becomes charged with ambiguity — clearly but not crudely sexual and transmitting a vivid sense of motion though it's obviously static.
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From the medieval towers of Notre Dame to the 19th-century iron lattices of the Eiffel Tower, the riverside offers a stunning survey, complemented by postmodern structures like the Institut du Monde Arabe.
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Carbon's printers work with resins, elastomers and "continuous liquid interface printing" technology, meaning they can form objects with the same kind of strength you'd see in traditional thermoplastics, or can form flexible lattices.
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But other than the square and triangle lattices and the three-dimensional cubic, it wasn't clear whether the connectivity condition would be satisfied in other cases — and thus whether the theorem applies more generally.
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More than 16,000 plants line Salesforce Park, which features geometric metal beams and lattices, a winding pedestrian path and motion-activated fountains that turn on when buses pass through the Transbay Transit Center underneath.
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Lattices of bamboo and the sporadic advertisement, such as the arches of a McDonald's sign, are often the only interruptions in the colorful drapery that has consumed the curves and angles of a skyscraper.
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Evoking the up-cycled metal lattices of Ghanian artist El Anatsui, Padernos has composed an intricate postcolonial work that touches on one's attachment to everyday objects and domestic materials that denote the concept of home.
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His trademark lattices of glass were devised to admit as much light as possible, sometimes by angling the thousands of crystalline facets, sometimes by connecting them with rods so thin they were more like a spiderweb.
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Researchers from different companies (including Google and Microsoft, as well as IBM) have experimented with all sorts of strange machinery to try and master the qubit, including optical lattices, nuclear magnetic devices, and even diamond-based systems.
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The lattices were recently on display in the group exhibition, Viewpoints, at Jenkins Johnson Gallery in San Francisco, and were inspired by a collaboration with Payne's photographer friend D'Angelo Williams, explains the artist to The Creators Project.
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Canyon Bridge supports Lattice's decision and believes "President Trump will recognize the benefits this investment will provide — to keep and grow jobs in the U.S. as well as expand Lattices product portfolio," it said in a statement.
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Ng told him about Project Marvin, an internal effort (named after the celebrated A.I. pioneer Marvin Minsky) he had recently helped establish to experiment with "neural networks," pliant digital lattices based loosely on the architecture of the brain.
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Tokyo's Olympic stadium will be constructed around an unusual set of wooden lattices - a design conceived by architect Kengo Kuma to harmonize with a forest of oak and camphor trees surrounding the nearby Meiji shrine in the Japanese capital.
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For this article, however, I purposely kicked the psychedelia up a notch and turned some already beautiful brain pictures into some seriously meta-psychedelic GIFs, with captions explaining the meaning behind all the rainbows tubes and convoluted mesh lattices.
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Tokyo's Olympic stadium will be constructed around an unusual set of wooden lattices - a design conceived by architect Kengo Kuma to harmonise with a forest of oak and camphor trees surrounding the nearby Meiji shrine in the Japanese capital.
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Composed with impossibly dense lines ranging from minuscule to thin, interwoven lattices resembling Celtic knots wobble and writhe inside gaping swaths of empty paper, each cluster pulled, as if by some strange magnetism, away from the center of the page.
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The spacing between the atoms in their crystal lattices is different from that in silicon, so adding a layer of them to the silicon substrate from which all chips are made causes stress that can have the effect of cracking the chip.
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As part of his proof, he showed that for almost all nonseparable lattices (which are those whose vertices remain linked even after removing one vertex), you could slide the tiles around and get any configuration you want, so long as you make an even number of moves.
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When I rounded one last cluster of trees and finally saw it full on, I was even more bemused — by an intricate network of lattices, taking the form of a rough cube and rising some 40 feet into the air above a slope clad in wildflowers.
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Working with projections on prints, the works will synthesize Murphy's digital architectural spaces with Ratté's glitchy abstractions to depict "an impalpable reality where architectures and interlocking lattices are continuously evolving, transforming and morphing into new environments, thus unfolding a constantly shifting perspective between dimensions," according to the gallery.
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To settle on the final list, Ms. McDowell baked more than a hundred pies, incorporating ideas from an ongoing email chain and a 20-page Google document filled with text and sketches of over a dozen desserts, with suggestions for various architectural features — such as crimp styles, lattices and crust designs.
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The individuality of each body of work is energized by the other; the knobs, loops, and lattices of Mendelson's objects underscore the sculptural presence of Hackett's clay protrusions, while the tortoise-shell solidity of Hackett's surfaces seem to turn Mendelson's plastic animals and non-functioning vessels into clouds and columns of smoke.
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There were a whole number of things that we didn't have, and a lot of things that we did have were completely insufficient ... Our perimeter security is nonexistent, we have walls with lattices that somebody can shoot through; we have walls with footholds people can climb over; we have a 4 foot wall back here; we have no lighting.
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The black-white Bravais lattices characterize the translational symmetry of the structure like the typical Bravais lattices, but also contain additional symmetry elements. For black-white Bravais lattices, the number of black and white sites is always equal. There are 14 traditional Bravais lattices, 14 grey lattices, and 22 black-white Bravais lattices, for a total of 50 two-color lattices in three dimensions.
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Modular lattices are sometimes called Dedekind lattices after Richard Dedekind, who discovered the modular identity in several motivating examples.
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Optimal sampling lattices have been studied in higher dimensions. Generally, optimal sphere packing lattices are ideal for sampling smooth stochastic processes while optimal sphere covering latticesJ. H. Conway, N. J. A. Sloane. Sphere packings, lattices and groups.
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Ideal lattices are a new concept, but similar lattice classes have been used for a long time. For example cyclic lattices, a special case of ideal lattices, are used in NTRUEncrypt and NTRUSign. Ideal lattices also form the basis for quantum computer attack resistant cryptography based on the Ring Learning with Errors. These cryptosystems are provably secure under the assumption that the shortest vector problem (SVP) is hard in these ideal lattices.
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There are 24 even unimodular lattices of dimension 24, called the Niemeier lattices. The mass formula for them is checked in .
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In the mathematics of matroids and lattices, a geometric lattice is a finite atomistic semimodular lattice, and a matroid lattice is an atomistic semimodular lattice without the assumptions of finiteness. Geometric lattices and matroid lattices, respectively, form the lattices of flats of finite and infinite matroids, and every geometric or matroid lattice comes from a matroid in this way.
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Since the lattices of flats of matroids are exactly the geometric lattices, this implies that the lattice of partitions is also geometric..
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Hans-Volker Niemeier is a German mathematician who in 1973 classified the Niemeier lattices, the even positive definite unimodular lattices in 24 dimensions.
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Regular planar lattices like square lattices, triangular lattices, honeycomb lattices, etc., are the simplest example of the cellular structure in which every cell has exactly the same size and the same coordination number. The planar Voronoi diagram, on the other hand, has neither a fixed cell size nor a fixed coordination number. Its coordination number distribution is rather Poissonian in nature.
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In four dimensions, there are 64 Bravais lattices. Of these, 23 are primitive and 41 are centered. Ten Bravais lattices split into enantiomorphic pairs.
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Bravais lattices, also referred to as space lattices, describe the geometric arrangement of the lattice points, and therefore the translational symmetry of the crystal. The three dimensions of space afford 14 distinct Bravais lattices describing the translational symmetry. All crystalline materials recognized today, not including quasicrystals, fit in one of these arrangements. The fourteen three-dimensional lattices, classified by lattice system, are shown above.
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It has also been reported that triangular lattices yield more accurate structures than other lattice shapes when compared to crystallographic data. To combat the parity problem, several researchers have suggested using triangular lattices when possible, as well as a square matrix with diagonals for theoretical applications where the square matrix may be more appropriate. Hexagonal lattices were introduced to alleviate sharp turns of adjacent residues in triangular lattices. Hexagonal lattices with diagonals have also been suggested as a way to combat the parity problem.
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In discrete mathematics, ideal lattices are a special class of lattices and a generalization of cyclic lattices. Vadim Lyubashevsky. Lattice-Based Identification Schemes Secure Under Active Attacks. In Proceedings of the Practice and theory in public key cryptography , 11th international conference on Public key cryptography, 2008.
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Thus, Birkhoff's representation theorem extends to the case of infinite (bounded) distributive lattices in at least three different ways, summed up in duality theory for distributive lattices.
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In this sense, there are 14 possible Bravais lattices in three- dimensional space. The 14 possible symmetry groups of Bravais lattices are 14 of the 230 space groups. In the context of the space group classification, the Bravais lattices are also called Bravais classes, Bravais arithmetic classes, or Bravais flocks.
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Birkhoff's theorem states that in fact all finite distributive lattices can be obtained this way, and later generalizations of Birkhoff's theorem state a similar thing for infinite distributive lattices.
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It is not known for any r > 1 whether there are any r-differential lattices other than those that arise by taking products of the Young–Fibonacci lattices and Young's lattice.
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By commutativity and associativity one can think of join and meet as operations on non-empty finite sets, rather than on pairs of elements. In a bounded lattice the join and meet of the empty set can also be defined (as 0 and 1, respectively). This makes bounded lattices somewhat more natural than general lattices, and many authors require all lattices to be bounded. The algebraic interpretation of lattices plays an essential role in universal algebra.
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Ideal lattices naturally occur in many parts of number theory, but also in other areas. In particular, they have a significant place in cryptography. Micciancio defined a generalization of cyclic lattices as ideal lattices. They can be used in cryptosystems to decrease by a square root the number of parameters necessary to describe a lattice, making them more efficient.
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Reciprocal lattices for the cubic crystal system are as follows.
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Open VOGEL relies on an object-oriented multi-threading calculation core (CC). The aerodynamic part of the CC (the ACC) is based in hierarchical structure of classes that begins with the definition of vortex rings, which in turn, can be assembled together as bricks to form a lattice. Lattices can be of two types: "bounded" or "free". Bounded lattices represent all solid boundaries where boundary conditions have to be imposed, while free lattices represent the wakes shed by the bounded lattices.
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Bethe lattices also occur as the discrete subgroups of certain hyperbolic Lie groups, such as the Fuchsian groups. As such, they are also lattices in the sense of a lattice in a Lie group.
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A structure that satisfies all axioms for Boolean algebras except the two distributivity axioms is called an orthocomplemented lattice. Orthocomplemented lattices arise naturally in quantum logic as lattices of closed subspaces for separable Hilbert spaces.
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In mathematics, specifically in order theory and functional analysis, a Fréchet lattice is a topological vector lattice that is also a Fréchet space space. Fréchet lattices are important in the theory of topological vector lattices.
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A lattice is unimodular if and only if its dual lattice is integral. Unimodular lattices are equal to their dual lattices, and for this reason, unimodular lattices are also known as self- dual. Given a pair (m,n) of nonnegative integers, an even unimodular lattice of signature (m,n) exists if and only if m-n is divisible by 8, but an odd unimodular lattice of signature (m,n) always exists. In particular, even unimodular definite lattices only exist in dimension divisible by 8.
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Every lattice satisfying Mac Lane's condition is semimodular. The converse is true for lattices of finite length, and more generally for relatively atomic lattices. Moreover, every upper continuous lattice satisfying Mac Lane's condition is M-symmetric.
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When lattices with more structure are considered, the morphisms should "respect" the extra structure, too. In particular, a bounded-lattice homomorphism (usually called just "lattice homomorphism") f between two bounded lattices L and M should also have the following property: : f(0L) = 0M , and : f(1L) = 1M . In the order-theoretic formulation, these conditions just state that a homomorphism of lattices is a function preserving binary meets and joins. For bounded lattices, preservation of least and greatest elements is just preservation of join and meet of the empty set.
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In terms of open elements x ∈ N(y) if and only if there is an open element z such that y ≤ z ≤ x. Neighbourhood functions may be defined more generally on (meet)-semilattices producing the structures known as neighbourhood (semi)lattices. Interior algebras may thus be viewed as precisely the Boolean neighbourhood lattices i.e. those neighbourhood lattices whose underlying semilattice forms a Boolean algebra.
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More specific complete lattices are complete Boolean algebras and complete Heyting algebras (locales).
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The identity mapping obviously has these two properties. Thus all complete lattices occur.
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Ryan's work ranges from quantum chromodynamics, and lattices, to particle collisions and muons.
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Some authors refer to M-symmetric lattices as semimodular lattices.For instance Fofanova (2001).
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Pálfy and Pudlák. Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups. Algebra Universalis 11(1), 22–27 (1980). DOI For an overview of the group theoretic approach to the problem, see Pálfy (1993)Péter Pál Pálfy.
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Birkhoff's representation theorem has also been called the fundamental theorem for finite distributive lattices..
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For orthorhombic, tetragonal and cubic lattices with as well, then :V = a b c .
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In mathematics, specifically in order theory and functional analysis, a normed lattice is a topological vector lattice that is also a normed space space whose unit ball is a solid set. Normed lattices are important in the theory of topological vector lattices.
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In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). Complete lattices appear in many applications in mathematics and computer science. Being a special instance of lattices, they are studied both in order theory and universal algebra. Complete lattices must not be confused with complete partial orders (cpos), which constitute a strictly more general class of partially ordered sets.
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More special versions of both are continuous and algebraic cpos. Adding even further completeness properties one obtains continuous lattices and algebraic lattices, which are just complete lattices with the respective properties. For the algebraic case, one finds broader classes of posets that are still worth studying: historically, the Scott domains were the first structures to be studied in domain theory. Still wider classes of domains are constituted by SFP-domains, L-domains, and bifinite domains.
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Hex-splines are the generalization of B-splines for 2-D hexagonal lattices. Similarly, in 3-D and higher dimensions, Voronoi splines provide a generalization of B-splines that can be used to design non- separable FIR filters which are geometrically tailored for any lattice, including optimal lattices. Explicit construction of ideal low-pass filters (i.e., sinc functions) generalized to optimal lattices is possible by studying the geometric properties of Brillouin zones (i.e.
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The mass in this case is large, more than 40 million. This implies that there are more than 80 million even unimodular lattices of dimension 32, as each has automorphism group of order at least 2 so contributes at most 1/2 to the mass. By refining this argument, showed that there are more than a billion such lattices. In higher dimensions the mass, and hence the number of lattices, increases very rapidly.
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Lattices in semisimple Lie groups are always finitely presented. For uniform lattices this is a direct consequence of cocompactness. In the non-uniform case this can be proved using reduction theory. However a much faster proof is by using Kazhdan's property (T) when possible.
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In this case local rigidity never holds. For cocompact lattices a small deformation remains a cocompact lattice but it may not be conjugated to the original one (see Teichmüller space for more detail). Non-cocompact lattices are virtually free and hence have non-lattice deformations.
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Wave propagation in Periodic Structures: Electric Filters and Crystal Lattices, McGraw–Hill, New York, p. 2.
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Pic. 9: Monotonic map f between lattices that preserves neither joins nor meets, since and . The appropriate notion of a morphism between two lattices flows easily from the above algebraic definition. Given two lattices and , a lattice homomorphism from L to M is a function such that for all : : f(a ∨L b) = f(a) ∨M f(b), and : f(a ∧L b) = f(a) ∧M f(b). Thus f is a homomorphism of the two underlying semilattices.
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Also, box splines can be used to compute the volume of polytopes. In the context of multidimensional signal processing, box splines can provide multivariate interpolation kernels (reconstruction filters) tailored to non- Cartesian sampling lattices,Entezari, Alireza. Optimal sampling lattices and trivariate box splines. [Vancouver, BC.]: Simon Fraser University, 2007. .
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In a digression he introduced and studied lattices formally in a general context. He observed that the lattice of submodules of a module satisfies the modular identity. He called such lattices dual groups of module type (). He also proved that the modular identity and its dual are equivalent.
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Ralph McKenzie. Finite forbidden lattices. In: Universal algebra and lattice theory (Puebla, 1982), Lecture Notes in Math., vol.
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For further details on these connections, see the book Sphere Packings, Lattices and Groups by Conway and Sloane.
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Some of the Niemeier lattices are related to sporadic simple groups. The Leech lattice is acted on by a double cover of the Conway group, and the lattices A124 and A212 are acted on by the Mathieu groups M24 and M12. The Niemeier lattices, other than the Leech lattice, correspond to the deep holes of the Leech lattice. This implies that the affine Dynkin diagrams of the Niemeier lattices can be seen inside the Leech lattice, when two points of the Leech lattice are joined by no lines when they have distance \sqrt 4, by 1 line if they have distance \sqrt 6, and by a double line if they have distance \sqrt 8.
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His current projects include: BEC on a chip, magnetic lattices, quantum coherence, molecular BEC, high harmonic generation, ultrafast spectroscopy.
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Most of the building architectural style follows Qing Dynasty style with arched bricks arcades and curved pattern window lattices.
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Unlike older lattice based cryptographic algorithms, the RLWE-KEX is provably reducible to a known hard problem in lattices.
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Other examples of mathematical objects obeying axioms of countability include sigma-finite measure spaces, and lattices of countable type.
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This allows for the easy construction of groups without lattices, for example the group of invertible upper triangular matrices or the affine groups. It is also not very hard to find unimodular groups without lattices, for example certain nilpotent Lie groups as explained below. A stronger condition than unimodularity is simplicity. This is sufficient to imply the existence of a lattice in a Lie group, but in the more general setting of locally compact groups there exists simple groups without lattices, for example the "Neretin groups".
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Therefore, the modular law can also be stated as ;Modular law (variant): x ≤ b implies x ∨ (a ∧ b) ≥ (x ∨ a) ∧ b. By substituting x with x ∧ b, the modular law can be expressed as an equation that is required to hold unconditionally, as follows: ;Modular identity: (x ∧ b) ∨ (a ∧ b) = [(x ∧ b) ∨ a] ∧ b. This shows that, using terminology from universal algebra, the modular lattices form a subvariety of the variety of lattices. Therefore, all homomorphic images, sublattices and direct products of modular lattices are again modular.
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Simple cubic lattices are achieved in a similar way by slicing cubic facets into spheres. This allows for the assembly of simple cubic lattices. A bcc crystal is achieved by faceting a sphere octahedrally. The faceting amount, α, is used in the emergent valence self-assembly to determine what crystal structure will form.
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Different lattice types and algorithms were used to study protein folding with HP model. Efforts were made to obtain higher approximation ratios using approximation algorithms in 2 dimensional and 3 dimensional, square and triangular lattices. Alternative to approximation algorithms, some genetic algorithms were also exploited with square, triangular, and face-centered-cubic lattices.
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Felix Bloch made fundamental theoretical contributions to the understanding of electron behavior in crystal lattices, ferromagnetism, and nuclear magnetic resonance.
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In particular there is a wealth of rigidity results in this setting, and a celebrated theorem of Grigory Margulis states that in most cases all lattices are obtained as arithmetic groups. Lattices are also well-studied in some other classes of groups, in particular groups associated to Kac–Moody algebras and automorphisms groups of regular trees (the latter are known as tree lattices). Lattices are of interest in many areas of mathematics: geometric group theory (as particularly nice examples of discrete groups), in differential geometry (through the construction of locally homogeneous manifolds), in number theory (through arithmetic groups), in ergodic theory (through the study of homogeneous flows on the quotient spaces) and in combinatorics (through the construction of expanding Cayley graphs and other combinatorial objects).
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The Priestley space is called the Priestley dual of . Each Priestley space is isomorphic to the Priestley dual of some bounded distributive lattice.Priestley (1970) Let Dist denote the category of bounded distributive lattices and bounded lattice homomorphisms. Then the above three representations of bounded distributive lattices can be extended to dual equivalenceBezhanishvili et al.
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For this process, elements of the poset are mapped to (Dedekind-) cuts, which can then be mapped to the underlying posets of arbitrary complete lattices in much the same way as done for sets and free complete (semi-) lattices above. The aforementioned result that free complete lattices do not exist entails that an according free construction from a poset is not possible either. This is easily seen by considering posets with a discrete order, where every element only relates to itself. These are exactly the free posets on an underlying set.
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The technique of fuzzy concept lattices is increasingly used in programming for the formatting, relating and analysis of fuzzy data sets.
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Intervals in subgroup lattices of finite groups. In Groups ’93 Galway/St. Andrews, Vol. 2, volume 212 of London Math. Soc.
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Viewed as a lattice the chain with three elements is not congruence-permutable and hence neither is the variety of lattices.
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Stehle, Steinfeld, Tanaka and Xagawa Damien Stehlé, Ron Steinfeld, Keisuke Tanaka and Keita Xagawa. Efficient public key encryption based on ideal lattices. In Lecture Notes in Computer Science, 2009. defined a structured variant of LWE problem (Ideal-LWE) to describe an efficient public key encryption scheme based on the worst case hardness of the approximate SVP in ideal lattices.
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The precise description of which lattices can occur as Picard lattices of K3 surfaces is complicated. One clear statement, due to Viacheslav Nikulin and David Morrison, is that every even lattice of signature (1,\rho-1) with \rho\leq 11 is the Picard lattice of some complex projective K3 surface.Huybrechts (2016), Corollary 14.3.1 and Remark 14.3.7.
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In mathematics, a Niemeier lattice is one of the 24 positive definite even unimodular lattices of rank 24, which were classified by . gave a simplified proof of the classification. has a sentence mentioning that he found more than 10 such lattices, but gives no further details. One example of a Niemeier lattice is the Leech lattice.
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An atomistic (hence algebraic) semimodular bounded lattice is called a matroid lattice because such lattices are equivalent to (simple) matroids. An atomistic semimodular bounded lattice of finite length is called a geometric lattice and corresponds to a matroid of finite rank.These definitions follow Stern (1999). Some authors use the term geometric lattice for the more general matroid lattices.
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Similarly, time-space fractional diffusion equations can be considered as CTRWs with continuously distributed jumps or continuum approximations of CTRWs on lattices.
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Based on this structure, the face lattices of simple polytopes can be reconstructed from their graphs in polynomial time using linear programming .
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67, no. 6, pp. 930 – 949, June 1979. As a consequence, hexagonal lattices are preferred for sampling isotropic fields in \Re^2.
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We now introduce a number of important properties that lead to interesting special classes of lattices. One, boundedness, has already been discussed.
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For finite lattices this means that the previous conditions hold with ∨ and ∧ exchanged, "covers" exchanged with "is covered by", and inequalities reversed.
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Shreddies cereal pieces A former Shreddies box in the United Kingdom Shreddies are a breakfast cereal made from lattices of wholegrain wheat.
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Would there be a free construction of complete lattices from posets, then both constructions could be composed, which contradicts the negative result above.
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This is among the oldest unsolved problems in universal algebra.Joel Berman. Congruence lattices of finite universal algebras. PhD thesis, University of Washington, 1970.
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Both the lattices over the arches and the wall enclosing the central arch were re-used from Visigothic origins in the 7th century.
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This notion includes mapping class groups via their actions on curve complexes. Lattices in higher-rank Lie groups are (still!) not acylindrically hyperbolic.
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The situation for complete lattices with complete homomorphisms obviously is more intricate. In fact, free complete lattices do generally not exist. Of course, one can formulate a word problem similar to the one for the case of lattices, but the collection of all possible words (or "terms") in this case would be a proper class, because arbitrary meets and joins comprise operations for argument-sets of every cardinality. This property in itself is not a problem: as the case of free complete semilattices above shows, it can well be that the solution of the word problem leaves only a set of equivalence classes.
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Thus we immediately find that every complete lattice is represented by Birkhoff's method, up to isomorphism. The construction is utilized in formal concept analysis, where one represents real-word data by binary relations (called formal contexts) and uses the associated complete lattices (called concept lattices) for data analysis. The mathematics behind formal concept analysis therefore is the theory of complete lattices. Another representation is obtained as follows: A subset of a complete lattice is itself a complete lattice (when ordered with the induced order) if and only if it is the image of an increasing and idempotent (but not necessarily extensive) self-map.
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Exact thresholds are only known for certain two-dimensional lattices that can be broken up into a self-dual array, such that under a triangle-triangle transformation, the system remains the same. Studies using numerical methods have led to numerous improvements in algorithms and several theoretical discoveries. Simply duality in two dimensions implies that all fully triangulated lattices (e.g., the triangular, union jack, cross dual, martini dual and asanoha or 3-12 dual, and the Delaunay triangulation) all have site thresholds of 1/2, and self-dual lattices (square, martini-B) have bond thresholds of 1/2.
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This was followed by a DNA truncated octahedron. It soon became clear that these structures, polygonal shapes with flexible junctions as their vertices, were not rigid enough to form extended three-dimensional lattices. Seeman developed the more rigid double-crossover (DX) structural motif, and in 1998, in collaboration with Erik Winfree, published the creation of two-dimensional lattices of DX tiles.
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The behaviour of different processes on discrete regular lattices have been studied quite extensively. They show a rich diversity of behaviour, including a non-trivial dependence on the dimension of the regular lattice. In recent years the study has been extended from regular lattices to complex networks. The shortcut model has been used in studying several processes and their dependence on dimension.
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The cylindrical assemblies possess internal helical order and self- organize into columnar liquid crystalline lattices. When inserted into vesicular membranes, the porous cylindrical assemblies mediate transport of protons across the membrane. Self-assembly of dendrons generates arrays of nanowires. Electron donor-acceptor complexes comprise the core of the cylindrical supramolecular assemblies, which further self-organize into two- dimensional columnar liquid crystaline lattices.
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Exchange interaction can occur in clusters and infinite lattices, resulting in ferromagnetism, antiferromagnetism or ferrimagnetism depending on the relative orientations of the individual spins.
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Therefore, they can possess effective properties which are not found in nature and may not be achieved with larger-scale lattices of the same geometry.
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Because ontology charts have a root that all affordances (realisations/things) are ultimately dependent upon for their existence, they are graphical representation of semi-lattices.
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The Fibonacci cube may be defined in terms of Fibonacci codes and Hamming distance, independent sets of vertices in path graphs, or via distributive lattices.
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In universal algebra, within mathematics, a majority term, sometimes called a Jónsson term, is a term t with exactly three free variables that satisfies the equations t(x, x, y) = t(x, y, x) = t(y, x, x) = x.R. Padmanabhan, Axioms for Lattices and Boolean Algebras, World Scientific Publishing Company (2008) For example for lattices, the term (x ∧ y) ∨ (y ∧ z) ∨ (z ∧ x) is a Jónsson term.
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The criterion for nilpotent Lie groups to have a lattice given above does not apply to more general solvable Lie groups. It remains true that any lattice in a solvable Lie group is uniform and that lattices in solvable groups are finitely presented. Not all finitely generated solvable groups are lattices in a Lie group. An algebraic criterion is that the group be polycyclic.
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The triclinic lattice is the least symmetric of the 14 three-dimensional Bravais lattices. It has (itself) the minimum symmetry all lattices have: points of inversion at each lattice point and at 7 more points for each lattice point: at the midpoints of the edges and the faces, and at the center points. It is the only lattice type that itself has no mirror planes.
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Related to this result is the interesting phenomenon that there are various competing notions of homomorphism for this class of posets, depending on whether they are seen as complete lattices, complete join-semilattices, complete meet-semilattices, or as join-complete or meet-complete lattices. Note that "partial lattice" is not the opposite of "complete lattice" – rather, "partial lattice", "lattice", and "complete lattice" are increasingly restrictive definitions.
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A lattice is sometimes called weakly semimodular if it satisfies the following condition due to Garrett Birkhoff: ;Birkhoff's condition: If a ∧ b <: a and a ∧ b <: b, :then a <: a ∨ b and b <: a ∨ b. Every semimodular lattice is weakly semimodular. The converse is true for lattices of finite length, and more generally for upper continuous (meets distribute over joins of chains) relatively atomic lattices.
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The version of the theorem in Hedlund's paper applied only to one- dimensional finite automata, but a generalization to higher dimensional integer lattices was soon afterwards published by ,. and it can be even further generalized from lattices to discrete groups. One important consequence of the theorem is that, for reversible cellular automata, the reverse dynamics of the automaton can also be described by a cellular automaton.
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8, above left) contain minimal internal energy, retaining only that due to the ever-present background of zero-point energy. No other crystal structure can exceed the 74.048% packing density of a closest-packed arrangement. The two regular crystal lattices found in nature that have this density are hexagonal close packed (HCP) and face-centered cubic (FCC). These regular lattices are at the lowest possible energy state.
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We define two Z-lattices L and M in a quadratic space V over Q to be spinor equivalent if there exists a transformation g in the proper orthogonal group O+(V) and for every prime p there exists a local transformation fp of Vp of spinor norm 1 such that M = g fpLp. A spinor genus is an equivalence class for this equivalence relation. Properly equivalent lattices are in the same spinor genus, and lattices in the same spinor genus are in the same genus. The number of spinor genera in a genus is a power of two, and can be determined effectively.
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Correlation and causation. Journal of agricultural research, 20(7), pp. 557–585 He is, however, best remembered for his work on Bravais lattices, particularly his 1848 discovery that there are 14 unique lattices in three-dimensional crystalline systems, correcting the previous scheme, with 15 lattices, conceived by Frankenheim three years before.Bravais, A.: Mémoire sur les systèmes formés par des points distribués regulièrement sur un plan ou dans l'espace, Journal de l'Ecole Polytechnique 19: 1-128; en translation Allemande par C. et E. Blasius: Abhandlung über die Systeme von regelmässig auf einer Ebene oder im Raum vertheilten Punkten, Leipzig: Engelmann, 1897 (= Ostwalds Klassiker der exakten Wissenschaften, 90).
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Moreover, any inequality x ≤ y can be expressed equivalently as an equation, either x∧y = x or x∨y = y. This along with the equations axiomatizing lattices and monoids then yields a purely equational definition of residuated lattices, provided the requisite operations are adjoined to the signature (L, ≤, •, I) thereby expanding it to (L, ∧, ∨, •, I, /, \\). When thus organized, residuated lattices form an equational class or variety, whose homomorphisms respect the residuals as well as the lattice and monoid operations. Note that distributivity x•(y ∨ z) = (x•y) ∨ (x•z) and x•0 = 0 are consequences of these axioms and so do not need to be made part of the definition.
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Ergodic theory has fruitful connections with harmonic analysis, Lie theory (representation theory, lattices in algebraic groups), and number theory (the theory of diophantine approximations, L-functions).
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Tilman Esslinger is a German experimental physicist. He is Professor at ETH Zurich, Switzerland, and works in the field of ultracold quantum gases and optical lattices.
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The theorem can be given in a geometric formulation (pertaining to finite-volume, complete manifolds), and in an algebraic formulation (pertaining to lattices in Lie groups).
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The concept of a lattice arises in order theory, a branch of mathematics. The Hasse diagram below depicts the inclusion relationships among some important subclasses of lattices.
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MCPs share similar structure and signalling mechanism. MCPs form dimers. Three dimers of MCP spontaneously form trimers. Trimers are complexed by CheA and CheW into hexagonal lattices.
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In mathematics, duality theory for distributive lattices provides three different (but closely related) representations of bounded distributive lattices via Priestley spaces, spectral spaces, and pairwise Stone spaces. This duality, which is originally also due to Marshall H. Stone Stone (1938), generalizes the well-known Stone duality between Stone spaces and Boolean algebras. Let be a bounded distributive lattice, and let denote the set of prime filters of . For each , let }.
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There are 24 orbits of primitive norm 0 vectors, corresponding to the 24 Niemeier lattices. The correspondence is given as follows: if z is a norm 0 vector, then the lattice z⊥/z is a 24-dimensional even unimodular lattice and is therefore one of the Niemeier lattices. The Niemeier lattice corresponding to the norm 0 Weyl vector of the reflection group of II25,1 is the Leech lattice.
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Solid foams, both open-cell and closed-cell, are considered as a sub-class of cellular structures. They often have lower nodal connectivity as compared to other cellular structures like honeycombs and truss lattices, and thus, their failure mechanism is dominated by bending of members. Low nodal connectivity and the resulting failure mechanism ultimately lead to their lower mechanical strength and stiffness compared to honeycombs and truss lattices.
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Maternal cells also synthesize and contribute a store of ribosomes that are required for the translation of proteins before the zygotic genome is activated. In mammalian oocytes, maternally derived ribosomes and some mRNAs are stored in a structure called cytoplasmic lattices. These cytoplasmic lattices, a network of fibrils, protein, and RNAs, have been observed to increase in density as the number of ribosomes decrease within a growing oocyte.
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Active fluids have been shown to organize into regular and irregular lattices in a variety of settings. These include irregular hexagonal lattices by microtubules and regular vortex lattice by sperm cells. From topological considerations, it can be seen that the constituent element in quasi stationary states of active fluids should necessarily be vortices. But very less is known, for instance, about the length scale selection in such systems.
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Artificial lattice is a term encompassing every atomic-scale structures designed and controlled to confine electrons onto a chosen lattice. Research has been done on multiple geometries and one of the most notable being what is called molecular graphene (in order to mimic graphene structure). Molecular graphene is a part of two-dimensional artificial lattices. Artificial lattices can be studied to test theoretical topology predictions or for their engineered electronic proprieties.
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An alternative phrasing, exchanging the roles of and , instead emphasizes that modular lattices form a variety in the sense of universal algebra. Modular lattices arise naturally in algebra and in many other areas of mathematics. In these scenarios, modularity is an abstraction of the 2nd Isomorphism Theorem. For example, the subspaces of a vector space (and more generally the submodules of a module over a ring) form a modular lattice.
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Sachdev predicted density wave order and 'magnetic' quantum criticality in tilted lattices of ultracold atoms. This was subsequently observed in experiments. The modeling of tilted lattices inspired a more general model of interacting bosons in which a coherent external source can create and annihilate bosons on each site. This model exhibits density waves of multiple periods, along with gapless incommensurate phases, and has been realized in experiments on trapped Rydberg atoms.
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A B2 intermetallic compound has equal numbers of atoms of two metals such as aluminium and iron, arranged as two interpenetrating simple cubic lattices of the component metals.
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3 "Forbidden Structures for Lattices", pp. 210–212; . The Hasse diagram of a lattice is planar if and only if its order dimension is at most two., pp.
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The Banach spaces L^p\left( \mu \right) (1 \leq p \leq \infty) are Banach lattices under their canonical orderings. These spaces are order complete for p < \infty.
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Therefore, these results of Margulis pave a way for classification of lattices. Arithmeticity turned out to be closely related to another remarkable property of lattices discovered by Margulis. Superrigidity for a lattice Γ in G roughly means that any homomorphism of Γ into the group of real invertible n × n matrices extends to the whole G. The name derives from the following variant: : If G and G' are semisimple algebraic groups over a local field without compact factors and whose split rank is at least two and Γ and Γ' are irreducible lattices in them, then any homomorphism f: Γ → Γ' between the lattices agrees on a finite index subgroup of Γ with a homomorphism between the algebraic groups themselves. (The case when f is an isomorphism is known as the strong rigidity.) While certain rigidity phenomena had already been known, the approach of Margulis was at the same time novel, powerful, and very elegant.
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The geometric lattices are cryptomorphic to (finite, simple) matroids, and the matroid lattices are cryptomorphic to simple matroids without the assumption of finiteness. Like geometric lattices, matroids are endowed with rank functions, but these functions map sets of elements to numbers rather than taking individual elements as arguments. The rank function of a matroid must be monotonic (adding an element to a set can never decrease its rank) and they must be submodular functions, meaning that they obey an inequality similar to the one for semimodular lattices: :r(X)+r(Y)\ge r(X\cap Y)+r(X\cup Y). \, The maximal sets of a given rank are called flats. The intersection of two flats is again a flat, defining a greatest lower bound operation on pairs of flats; one can also define a least upper bound of a pair of flats to be the (unique) maximal superset of their union that has the same rank as their union.
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On the other hand, contrary to the most of public key cryptography, lattice-based cryptography allows security against subexponential quantum attacks. Most of the cryptosystems based on general lattices rely on the average-case hardness of the Learning with errors (LWE). Their scheme is based on a structured variant of LWE, that they call Ideal-LWE. They needed to introduce some techniques to circumvent two main difficulties that arise from the restriction to ideal lattices. Firstly, the previous cryptosystems based on unstructured lattices all make use of Regev’s worst-case to average-case classical reduction from Bounded Distance Decoding problem (BDD) to LWE (this is the classical step in the quantum reduction from SVP to LWE).
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Margulis's early work dealt with Kazhdan's property (T) and the questions of rigidity and arithmeticity of lattices in semisimple algebraic groups of higher rank over a local field. It had been known since the 1950s (Borel, Harish-Chandra) that a certain simple-minded way of constructing subgroups of semisimple Lie groups produces examples of lattices, called arithmetic lattices. It is analogous to considering the subgroup SL(n,Z) of the real special linear group SL(n,R) that consists of matrices with integer entries. Margulis proved that under suitable assumptions on G (no compact factors and split rank greater or equal than two), any (irreducible) lattice Γ in it is arithmetic, i.e.
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The façade has pre-Turkish carvings and patterns. The windows have marble lattices. Surface decoration consists of interweaved floral tendrils and is repeated with a symmetry on three doorways.
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Similarly, a lattice endomorphism is a lattice homomorphism from a lattice to itself, and a lattice automorphism is a bijective lattice endomorphism. Lattices and their homomorphisms form a category.
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Property (T) has important applications to group representation theory, lattices in algebraic groups over local fields, ergodic theory, geometric group theory, expanders, operator algebras and the theory of networks.
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A spectral map f: X → Y between spectral spaces X and Y is a continuous map such that the preimage of every open and compact subset of Y under f is again compact. The category of spectral spaces, which has spectral maps as morphisms, is dually equivalent to the category of bounded distributive lattices (together with morphisms of such lattices). In this anti-equivalence, a spectral space X corresponds to the lattice K\circ(X).
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In the same paper, Dedekind also investigated the following stronger form of the modular identity, which is also self-dual: : (x ∧ b) ∨ (a ∧ b) = [x ∨ a] ∧ b. He called lattices that satisfy this identity dual groups of ideal type (). In modern literature, they are more commonly referred to as distributive lattices. He gave examples of a lattice that is not modular and of a modular lattice that is not of ideal type.
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For many cryptographic primitives, the only known constructions are based on lattices or closely related objects. These primitives include fully homomorphic encryption, indistinguishability obfuscation, cryptographic multilinear maps, and functional encryption.
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A paper published by Dedekind in 1900 had lattices as its central topic: He described the free modular lattice generated by three elements, a lattice with 28 elements (see picture).
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E.T. Schmidt, Zur Charakterisierung der Kongruenzverbände der Verbände, Mat. Casopis Sloven. Akad. Vied. 18 (1968), 3--20. F. Wehrung, A uniform refinement property for congruence lattices, Proc. Amer. Math. Soc.
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As Hilary Putnam writes, von Neumann replaced classical logic with a logic constructed in orthomodular lattices (isomorphic to the lattice of subspaces of the Hilbert space of a given physical system).
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For contact process on all integer lattices, a major breakthrough came in 1990 when Bezuidenhout and Grimmett showed that the contact process also dies out almost surely at the critical value.
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As further observes, the Young–Fibonacci lattice is modular. incorrectly claims that it is distributive; however, the sublattice formed by the strings {21,22,121,211,221} forms a diamond sublattice, forbidden in distributive lattices.
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Several important recent results include the realization of a Mott insulator in a driven- dissipative Bose-Hubbard system and studies of phase transitions in lattices of superconducting resonators coupled to qubits.
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In recent years, the concept of the FFLO state was taken up in the field of atomic physics and experiments to detect the FFLO state in atomic ensembles in optical lattices.
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Niemeier lattices also correspond to the 24 orbits of primitive norm zero vectors w of the even unimodular Lorentzian lattice II25,1, where the Niemeier lattice corresponding to w is w⊥/w.
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A portion of the discrete Heisenberg group, a discrete subgroup of the continuous Heisenberg Lie group. (The coloring and edges are only for visual aid.) In Lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has finite invariant measure. In the special case of subgroups of Rn, this amounts to the usual geometric notion of a lattice as a periodic subset of points, and both the algebraic structure of lattices and the geometry of the space of all lattices are relatively well understood. The theory is particularly rich for lattices in semisimple Lie groups or more generally in semisimple algebraic groups over local fields.
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Its advantage is good shear strength at a much lower weight than a tower of solid construction would have as well as lower wind resistance. In structural engineering the term lattice tower is used for a freestanding structure, while a lattice mast is a guyed mast supported by guy lines. Lattices of triangular (3-sided) cross-section are most common, particularly in North America. Square lattices(4-sided) are also widely used and are most common in Eurasia.
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Skolem was among the first to write on lattices. In 1912, he was the first to describe a free distributive lattice generated by n elements. In 1919, he showed that every implicative lattice (now also called a Skolem lattice) is distributive and, as a partial converse, that every finite distributive lattice is implicative. After these results were rediscovered by others, Skolem published a 1936 paper in German, "Über gewisse 'Verbände' oder 'Lattices'", surveying his earlier work in lattice theory.
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Reversibly assembled cellular composite materials (RCCM) are three-dimensional lattices of modular structures that can be partially disassembled to enable repairs or other modifications. Each cell incorporates structural material and a reversible interlock, allowing lattices of arbitrary size and shape. RCCM display three-dimensional symmetry derived from the geometry as linked. The discrete construction of reversibly assembled cellular composites introduces a new degree of freedom that determines global functional properties from the local placement of heterogeneous components.
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The following result, proved by Ploščica, Tůma, and Wehrung in 1998, is more striking, because it shows examples of representable semilattices that do not satisfy Schmidt's Condition. We denote by FV(Ω) the free lattice on Ω in V, for any variety V of lattices. Theorem (Ploščica, Tůma, and Wehrung 1998). The semilattice Conc FV(Ω) does not satisfy WURP, for any set Ω of cardinality at least ℵ2 and any non- distributive variety V of lattices.
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In natural environment, crystal lattices of quartz and/or feldspar are bombarded with radiation released from radiogenic source such as in -situ radioactive decay. As the crystals are irradiated, charges are stored up in their crystallographic defects. The charge trapping process involves atomic-scale ionic substitution of both electron and hole within the crystal lattices of quartz and feldspar. The electron diffusion happens in response to ionizing radiation as the minerals cools below their closure temperature.
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Cockayne was educated at the University of Melbourne and the University of Oxford where he was awarded a Doctor of Philosophy degree in 1970 for research into crystal lattices using electron microscopy.
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The lattice structure compresses up to 85% of its original thickness and can recover to its original form. These lattices are stabilized into triangles with cross-members for structural integrity and flexibility.
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Springer, 1999. are ideal for sampling rough stochastic processes. Since optimal lattices, in general, are non-separable, designing interpolation and reconstruction filters requires non-tensor-product (i.e., non-separable) filter design mechanisms.
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L.Bunimovich, H.Spohn, Viscosity for Periodic Two-Disk Fluid, Comm. Math. Phys. v.76 (1996) 661-680 12.L.Bunimovich, Ya.G.Sinai, Space-Time Chaos in Coupled Map Lattices, Nonlinearity v.1 (1988) 491-516 13.
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300px Fig. 5: Spatial aliasing in the form of a Moiré pattern. Fig. 6: Properly sampled image of brick wall. The theorem gives conditions on sampling lattices for perfect reconstruction of the sampled.
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EC-14, no. 3, pp. 399–403, 1965. In terms of the theory of lattices, the C-element is a semimodular distributive circuit, whose operation in time is described by a Hasse diagram.
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Parker-Rhodes also co-authored papers with Needham on the "theory of clumps" in relation to information retrieval and computational linguistics. He wrote a book on language structure and the logic of descriptions, Inferential Semantics, published in 1978.Inferential Semantics, Humanities Press (1978) The work analyzes sentences and longer passages into mathematical lattices (the kind in Lattice Theory, not crystal lattices) which are semantic networks. These are inferred not only from sentence syntax but also from grammatical focus and sometimes prosody.
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Shortly afterwards the finiteness of covolume was proven in full generality by Borel and Harish-Chandra. Meanwhile, there was progress on the general theory of lattices in Lie groups by Atle Selberg, Grigori Margulis, David Kazhdan, M. S. Raghunathan and others. The state of the art after this period was essentially fixed in Raghunathan's treatise, published in 1972. In the seventies Margulis revolutionised the topic by proving that in "most" cases the arithmetic constructions account for all lattices in a given Lie group.
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A transgranular fracture is a fracture that follows the edges of lattices in a granular material, ignoring the grains in the individual lattices. This results in a fairly smooth looking fracture with fewer sharp edges than one that follows the changing grains. This can be visualized as several wooden jigsaw puzzle pieces with the grains showing, but with each piece having grains running in a different direction. A transgranular fracture follows the grains in the wood, not the edges of the puzzle pieces.
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Ewald, was having trouble subtracting out of his calculations the field of the test dipole. The solution was provided by Sommerfeld’s assistant and former doctoral student, Peter Debye, in a discussion that took no more than 15 minutes. Ewald’s paper has been widely cited in the literature as well as scientific books, such as Dynamical Theory of Crystal Lattices,Max Born and Kun Huang Dynamical Theory of Crystal Lattices (Oxford, Clarendon Press, 1954) by Max Born and Kun Huang.Ewald – University of Pennsylvania.
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Below is a mind map showing how some of the exceptional objects in mathematics and mathematical physics are related. 600px The connections can partly be explained by thinking of the algebras as a tower of lattice vertex operator algebras. It just so happens that the vertex algebras at the bottom are so simple that they are isomorphic to familiar non-vertex algebras. Thus the connections can be seen simply as the consequence of some lattices being sub-lattices of others.
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The notation such as (4,82) comes from Grünbaum and Shephard, and indicates that around a given vertex, going in the clockwise direction, one encounters first a square and then two octagons. Besides the eleven Archimedean lattices composed of regular polygons with every site equivalent, many other more complicated lattices with sites of different classes have been studied. Error bars in the last digit or digits are shown by numbers in parentheses. Thus, 0.729724(3) signifies 0.729724 ± 0.000003, and 0.74042195(80) signifies 0.74042195 ± 0.00000080.
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But most authors only deal with the finite case, in which both definitions are equivalent to semimodular and atomistic. Semimodular lattices are also known as upper semimodular lattices; the dual notion is that of a lower semimodular lattice. A finite lattice is modular if and only if it is both upper and lower semimodular. A finite lattice, or more generally a lattice satisfying the ascending chain condition or the descending chain condition, is semimodular if and only if it is M-symmetric.
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In principle one can find explicit examples: for example, even just picking a few "random" lattices will work with high probability. The problem is that testing these lattices to see if they are solutions requires finding their shortest vectors, and the number of cases to check grows very fast with the dimension, so this could take a very long time. This result was stated without proof by and proved by . The result is related to a linear lower bound for the Hermite constant.
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In mathematics, the Chabauty topology is a certain topological structure introduced in 1950 by Claude Chabauty, on the set of all closed subgroups of a locally compact group G. The intuitive idea may be seen in the case of the set of all lattices in a Euclidean space E. There these are only certain of the closed subgroups: others can be found by in a sense taking limiting cases or degenerating a certain sequence of lattices. One can find linear subspaces or discrete groups that are lattices in a subspace, depending on how one takes a limit. This phenomenon suggests that the set of all closed subgroups carries a useful topology. This topology can be derived from the Vietoris topology construction, a topological structure on all non-empty subsets of a space.
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If L is an odd unimodular lattice of dimension 8n and M its sublattice of even vectors, then M is contained in exactly 3 unimodular lattices, one of which is L and the other two of which are even. (If L has a norm 1 vector then the two even lattices are isomorphic.) The Kneser neighborhood graph in 8n dimensions has a point for each even lattice, and a line joining two points for each odd 8n dimensional lattice with no norm 1 vectors, where the vertices of each line are the two even lattices associated to the odd lattice. There may be several lines between the same pair of vertices, and there may be lines from a vertex to itself. Kneser proved that this graph is always connected.
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Geometric phase analysis is a digital signal processing method used with Fast Fourier transform algorithms in high-resolution transmission electron microscopy images to quantify displacement and strain fields in crystalline lattices at nanoscale resolution.
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Chazottes, Jean-René, and Bastien Fernandez. Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems. Springer, 2004. pgs 1-4 Studied systems include populations, chemical reactions, convection, fluid flow and biological networks.
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BDC has also found uses in discrete event dynamic systems (DEDS) in digital network communication protocols. Meanwhile, BDC has seen extensions to multi-valued variables and functions as well as to lattices of Boolean functions.
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In mathematics, umbral moonshine is a mysterious connection between Niemeier lattices and Ramanujan's mock theta functions. It is a generalization of the Mathieu moonshine phenomenon connecting representations of the Mathieu group M24 with K3 surfaces.
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More recently, CMLs have been applied to computational networks Xu, Jian. Wang, Xioa Fan. " Cascading failures in scale-free coupled map lattices." IEEE International Symposium on Circuits and Systems “ ISCAS Volume 4, (2005): 3395–3398.
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The factors of the chosen temperature and applied pressure depend on the diffusion rate. The diffusion occurs between the crystal lattices by lattice vibration. Atoms can not leap over free space, i.e. contamination or vacancies.
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This result catapulted upconversion research in lattices doped with rare-earth metals. One of the first examples of efficient lanthanide doping, the Yb/Er-doped fluoride lattice, was achieved in 1972 by Menyuk et al.
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Recently several models solvable by Bethe ansatz were realized experimentally in solid states and optical lattices. An important role in the theoretical description of these experiments was played by Jean-Sébastien Caux and Alexei Tsvelik.
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Granular Matter, 21(2), 26 . Both the face-centred cubic (fcc) and hexagonal close packed (hcp) crystal lattices have maximum densities equal to this upper limit, which can occur through the process of granular crystallisation.
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This reduction exploits the unstructured-ness of the considered lattices, and does not seem to carry over to the structured lattices involved in Ideal-LWE. In particular, the probabilistic independence of the rows of the LWE matrices allows to consider a single row. Secondly, the other ingredient used in previous cryptosystems, namely Regev’s reduction from the computational variant of LWE to its decisional variant, also seems to fail for Ideal-LWE: it relies on the probabilistic independence of the columns of the LWE matrices.
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In combinatorial mathematics, an Eulerian poset is a graded poset in which every nontrivial interval has the same number of elements of even rank as of odd rank. An Eulerian poset which is a lattice is an Eulerian lattice. These objects are named after Leonhard Euler. Eulerian lattices generalize face lattices of convex polytopes and much recent research has been devoted to extending known results from polyhedral combinatorics, such as various restrictions on f-vectors of convex simplicial polytopes, to this more general setting.
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Galois connections also provide an interesting class of mappings between posets which can be used to obtain categories of posets. Especially, it is possible to compose Galois connections: given Galois connections between posets and and between and , the composite is also a Galois connection. When considering categories of complete lattices, this can be simplified to considering just mappings preserving all suprema (or, alternatively, infima). Mapping complete lattices to their duals, these categories display auto duality, that are quite fundamental for obtaining other duality theorems.
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The number of distinct sites visited by a single random walker S(t) has been studied extensively for square and cubic lattices and for fractals. This quantity is useful for the analysis of problems of trapping and kinetic reactions. It is also related to the vibrational density of states, diffusion reactions processes, and spread of populations in ecology. The generalization of this problem to the number of distinct sites visited by N random walkers, S_N(t), has recently been studied for d-dimensional Euclidean lattices.
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They are also promising candidates for quantum information processing. The best atomic clocks in the world use atoms trapped in optical lattices, to obtain narrow spectral lines that are unaffected by the Doppler effect and recoil.
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Craig Gentry, using lattice-based cryptography, described the first plausible construction for a fully homomorphic encryption scheme. Craig Gentry. Fully Homomorphic Encryption Using Ideal Lattices. In the 41st ACM Symposium on Theory of Computing (STOC), 2009.
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Hilbert's eighteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by mathematician David Hilbert. It asks three separate questions about lattices and sphere packing in Euclidean space.
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Box splines provide a flexible framework for designing such non-separable reconstruction FIR filters that can be geometrically tailored for each lattice.A. Entezari. Optimal sampling lattices and trivariate box splines. [Vancouver, BC.]: Simon Fraser University, 2007. .
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Using the properties listed above, the dual of a lattice can be efficiently calculated, by hand or computer. Certain lattices with importance in mathematics and computer science are dual to each other, and we list some here.
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In mathematics, specifically in order theory and functional analysis, a locally convex vector lattice (LCVL) is a topological vector lattice that is also a locally convex space. LCVLs are important in the theory of topological vector lattices.
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Nowadays, the term "complete semilattice" has no generally accepted meaning, and various inconsistent definitions exist. If completeness is taken to require the existence of all infinite joins, or all infinite meets, whichever the case may be, as well as finite ones, this immediately leads to partial orders that are in fact complete lattices. For why the existence of all possible infinite joins entails the existence of all possible infinite meets (and vice versa), see the entry completeness (order theory). Nevertheless, the literature on occasion still takes complete join- or meet- semilattices to be complete lattices.
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In order theory, arbitrary meets can be expressed in terms of arbitrary joins and vice versa (for details, see completeness (order theory)). In effect, this means that it is sufficient to require the existence of either all meets or all joins to obtain the class of all complete lattices. As a consequence, some authors use the terms complete meet-semilattice or complete join-semilattice as another way to refer to complete lattices. Though similar on objects, the terms entail different notions of homomorphism, as will be explained in the below section on morphisms.
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When further subdivided according to their symmetry groups, there are 22 forms of the parallelohedra. For each form, the centers of its copies in its honeycomb form the points of one of the 14 Bravais lattices. Because there are fewer Bravais lattices than symmetric forms of parallelohedra, certain pairs of parallelohedra map to the same Bravais lattice. By placing one endpoint of each generating line segment of a parallelohedron at the origin of three-dimensional space, the generators may be represented as three-dimensional vectors, the positions of their opposite endpoints.
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All high-order lattice networks can be replaced by a cascade of simpler lattices, provided their characteristic impedances are all equal to that of the original and the sum of their propagation functions equals the original. In the particular case of all-pass networks (networks which modify the phase characteristic only), any given network can always be replaced by a cascade of second-order lattices together with, possibly, one single first order lattice. Whatever the filter requirements being considered, the reduction process results in simpler filter structures, with less stringent demands on component tolerances.
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Probably the most common type of distributivity is the one defined for lattices, where the formation of binary suprema and infima provide the total operations of join (\vee) and meet (\wedge). Distributivity of these two operations is then expressed by requiring that the identity : x \wedge (y \vee z) = (x \wedge y) \vee (x \wedge z) hold for all elements x, y, and z. This distributivity law defines the class of distributive lattices. Note that this requirement can be rephrased by saying that binary meets preserve binary joins.
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They also showed that four other (then) recently announced sporadic groups, namely, Higman- Sims, Suzuki, McLaughlin, and the Janko group J2 could be found inside the Conway groups using the geometry of the Leech lattice. (Ronan, p. 155) , has a single rather cryptic sentence mentioning that he found more than 10 even unimodular lattices in 24 dimensions without giving further details. stated that he found 9 of these lattices earlier in 1938, and found two more, the Niemeier lattice with A root system and the Leech lattice (and also the odd Leech lattice), in 1940.
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The aperiodic structures obtained by the cut-and-project method are made diffractive by choosing a suitable orientation for the construction; this is a geometric approach that has also a great appeal for physicists. Classical theory of crystals reduces crystals to point lattices where each point is the center of mass of one of the identical units of the crystal. The structure of crystals can be analyzed by defining an associated group. Quasicrystals, on the other hand, are composed of more than one type of unit, so, instead of lattices, quasilattices must be used.
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In 8 dimensions it has one point and no lines, in 16 dimensions it has two points joined by one line, and in 24 dimensions it is the following graph: Image:Neighborhood graph of Niemeier lattices.svg Each point represents one of the 24 Niemeier lattices, and the lines joining them represent the 24 dimensional odd unimodular lattices with no norm 1 vectors. (Thick lines represent multiple lines.) The number on the right is the Coxeter number of the Niemeier lattice. In 32 dimensions the neighborhood graph has more than a billion vertices.
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He retired in 1966 and moved to Utica, N.Y., where he continued his research. Nikodym worked in a wide range of areas, but his best-known early work was his contribution to the development of the Lebesgue–Radon–Nikodym integral (see Radon–Nikodym theorem). His work in measure theory led him to an interest in abstract Boolean lattices. His work after coming to the United States centered on the theory of operators in Hilbert space, based on Boolean lattices, culminating in his The Mathematical Apparatus for Quantum-Theories.
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The 14 lattice types in 3D are called Bravais lattices. They are characterized by their space group. 3D patterns with translational symmetry of a particular type cannot have more, but may have less symmetry than the lattice itself.
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The Leech lattice can be constructed in a variety of ways. As with all lattices, it can be constructed by taking the integral span of the columns of its generator matrix, a 24×24 matrix with determinant 1.
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This effect can provide information on the orientation of molecules with a single crystal or material. The spectral information arising from this analysis is often used to understand macro- molecular orientation in crystal lattices, liquid crystals or polymer samples.
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The AKLT model has been solved on lattices of higher dimension, even in quasicrystals . The model has also been constructed for higher Lie algebras including SU(n), SO(n), Sp(n) and extended to the quantum groups SUq(n).
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In mathematics, a Priestley space is an ordered topological space with special properties. Priestley spaces are named after Hilary Priestley who introduced and investigated them. Priestley, (1970). Priestley spaces play a fundamental role in the study of distributive lattices.
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Physical models of phyllotaxis date back to Airy's experiment of packing hard spheres. Gerrit van Iterson diagrammed grids imagined on a cylinder (Rhombic Lattices). Douady et al. showed that phyllotactic patterns emerge as self- organizing processes in dynamic systems.
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Of note are the timber lattices, timber slat blinds and hedge plantings that provide shade in a sub tropical climate. The impressive entrance porch, decorative iron balustrading, roof ventilator, and acroterions also enhance the visual impact of the residence.
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The main difference, when compared to the one-dimensional wavelets, is that multi-dimensional sampling requires the use of lattices (e.g., the quincunx lattice). The wavelet filters themselves can be separable or non-separable regardless of the sampling lattice.
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20, p. 776. HF5 has been applied towards the analysis of lattices and other macromolecules. HF5 was the first form of flow FFF to be developed in 1974. Flat membranes soon outperformed hollow fibers and forced HF5 into obscurity.
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This result holds more generally for modular lattices, see Exercise 4, p. 50.Birkhoff (1961), Corollary IX.1, p. 134 The lattice of subspaces of a vector space provide an example of a complemented lattice that is not, in general, distributive.
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Besides, unlike regular lattices, the sizes of its cells are not equal; rather, the distribution of the area size of its blocks obeys dynamic scaling, whose coordination number distribution follows a power-law. A snapshot of the weighted stochastic lattice.
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The Scott topology was first defined by Dana Scott for complete lattices and later defined for arbitrary partially ordered sets. Scott-continuous functions show up in the study of models for lambda calculi and the denotational semantics of computer programs.
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There are 121 orbits of vectors v of norm –2, corresponding to the 121 isomorphism classes of 25-dimensional even lattices L of determinant 2. In this correspondence, the lattice L is isomorphic to the orthogonal complement of the vector v.
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The Z_N model is a simplified statistical mechanical spin model. It is a generalization of the Ising model. Although it can be defined on an arbitrary graph, it is integrable only on one and two-dimensional lattices, in several special cases.
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There are two such lattices: Γ8⊕Γ8 and Γ16 (constructed in a fashion analogous to that of Γ8). These lead to two version of the heterotic string known as the E8×E8 heterotic string and the SO(32) heterotic string.
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Therefore, ring can be overlapped by a set of hexagons instead of one. JWST telescope array consists from 18 hexagons. Sampling on 18 shifted lattices is possible for 2-d Fourier transform of the array signal (i. e. for emitted signal).
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An important question in statistical mechanics is the dependence of model behaviour on the dimension of the system. The shortcut model was introduced in the course of studying this dependence. The model interpolates between discrete regular lattices of integer dimension.
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Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilattices include lattices, which in turn include Heyting and Boolean algebras. These "lattice-like" structures all admit order-theoretic as well as algebraic descriptions.
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The model has since been studied on the infinite lattice, on other (non-square) lattices, and on arbitrary graphs (including directed multigraphs). It is closely related to the dollar game, a variant of the chip- firing game introduced by Biggs.
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One-dimensional lattices may be studied using density matrix renormalization group (DMRG) and related techniques such as time-evolving block decimation (TEBD). This includes to calculate the ground state of the Hamiltonian for systems of thousands of particles on thousands of lattice sites, and simulate its dynamics governed by the Time-dependent Schrödinger equation. Recently, two dimensional lattices have also been studied using Projected Entangled Pair States which is a generalization of Matrix Product States in higher dimensions, both for the ground state as well as finite temperature. Higher dimensions are significantly more difficult due to the quick growth of entanglement.
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That is, the distribution is peaked about the mean where it is almost impossible to find cells which have significantly higher or fewer coordination number than the mean. Recently, Hassan et al proposed a lattice, namely the weighted planar stochastic lattice. For instance, unlike a network or a graph, it has properties of lattices as its sites are spatially embedded. On the other hand, unlike lattices, its dual (obtained by considering the center of each block of the lattice as a node and the common border between blocks as links) display the property of networks as its degree distribution follows a power law.
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At this stage Kidner began to be interested in the spaces between the lines and crisscross wavy lines began to emerge in his work, culminating in grids and lattices. These were sometimes in phase creating identical spaces in between and then sometimes out of phase so the spaces in between didn't repeat. Kidner used this structure as a basis for creating many variations of this principle and observed that " the endless number of linear intersections both offer and resist any sort of visual resolution." Continuing with his investigation of grids and lattices Kidner experimented with various materials.
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Quadratic forms over the ring of integers are called integral quadratic forms, whereas the corresponding modules are quadratic lattices (sometimes, simply lattices). They play an important role in number theory and topology. An integral quadratic form has integer coefficients, such as ; equivalently, given a lattice Λ in a vector space V (over a field with characteristic 0, such as Q or R), a quadratic form Q is integral with respect to Λ if and only if it is integer-valued on Λ, meaning if . This is the current use of the term; in the past it was sometimes used differently, as detailed below.
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It has been shown that for most proteins the coordination number of the lattice used should fall between 3 and 20, although most commonly used lattices have coordination numbers at the lower end of this range. Lattice shape is an important factor in the accuracy of lattice protein models. Changing lattice shape can dramatically alter the shape of the energetically favorable conformations. It can also add unrealistic constraints to the protein structure such as in the case of the parity problem where in square and cubic lattices residues of the same parity (odd or even numbered) cannot make hydrophobic contact.
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In the theory of lattices, the dual lattice is a construction analogous to that of a dual vector space. In certain respects, the geometry of the dual lattice of a lattice L is the reciprocal of the geometry of L , a perspective which underlies many of its uses. Dual lattices have many applications inside of lattice theory, theoretical computer science, cryptography and mathematics more broadly. For instance, it is used in the statement of the Poisson summation formula, transference theorems provide connections between the geometry of a lattice and that of its dual, and many lattice algorithms exploit the dual lattice.
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For nilpotent groups the theory simplifies much from the general case, and stays similar to the case of Abelian groups. All lattices in a nilpotent Lie group are uniform, and if N is a connected simply connected nilpotent Lie group (equivalently it does not contain a nontrivial compact subgroup) then a discrete subgroup is a lattice if and only if it is not contained in a proper connected subgroup (this generalises the fact that a discrete subgroup in a vector space is a lattice if and only if it spans the vector space). A nilpotent Lie group contains a lattice if and only if it can be defined over the rationals, that is if and only if its structure constants are rational numbers. More precisely, in a nilpotent group satisfying this condition lattices correspond via the exponential map to lattices (in the more elementary sense of Lattice (group)) in the Lie algebra.
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Define the polynomial as the product of the differences over the conjugate sublattices. As a polynomial in , has coefficients that are polynomials over in . On the conjugate lattices, the modular group acts as . It follows that has Galois group isomorphic to over .
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Comparison of fcc and hcp lattices, explaining the formation of stacking faults in close-packed crystals. In crystallography, a stacking fault is a type of defect which characterizes the disordering of crystallographic planes. It is thus considered a planar defect.Fine, Morris E. (1921).
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Centre for Mathematical Sciences, Thiruvananthapuram, India. 1989. #"G-lattices". Proceedings of the Monash Conference on Semigroup Theory in honour of G. B. Preston held in July 1990 : 224-241. World Scientific Publishing Co. 1991. #(with E. Krishnan) "The semigroup of Fredholm operators".
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In two dimensions there are two orthorhombic Bravais lattices: primitive rectangular and centered rectangular. The primitive rectangular lattice can also be described by a centered rhombic unit cell, while the centered rectangular lattice can also be described by a primitive rhombic unit cell.
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Tetsuji Shioda, Oberwolfach 2005 Tetsuji Shioda (塩田 徹治) is a Japanese mathematician who introduced Shioda modular surfaces and who used Mordell–Weil lattices to give examples of dense sphere packings. He was an invited speaker at the ICM in 1990.
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The A lattice (also called A) is the union of nine A8 lattices, and has the vertex arrangement of the dual honeycomb to the omnitruncated 8-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 8-simplex : ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ = dual of .
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The A lattice (also called A) is the union of eight A7 lattices, and has the vertex arrangement to the dual honeycomb of the omnitruncated 7-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 7-simplex. : ∪ ∪ ∪ ∪ ∪ ∪ ∪ = dual of .
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In crystals screw rotations and/or glide reflections are additionally possible. These are rotations or reflections together with partial translation. These operations may change based on the dimensions of the crystal lattice. The Bravais lattices may be considered as representing translational symmetry operations.
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Cluster states have been realized experimentally. They have been obtained in photonic experiments using parametric downconversion. In such systems, the horizontal and vertical polarizations of the photons code the qubit. Cluster states have been created also in optical lattices of cold atoms.
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An example of a Moiré interference pattern. The offset of the two lattices creates a dipole of retinal ganglion cells. This dipole is orientated in various directions that correspond to a specific orientation. A highly debatedSchottdorf M., Eglen S. J., Wolf F. & Keil W. (2014).
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The mass spectrum of the model and the scattering matrix was explicitly evaluated by Bethe Ansatz. An explicit formula for the correlations is not known. J. I. Cirac, P. Maraner and J. K. Pachos applied the massive Thirring model to the description of optical lattices.
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In 2018, chemists from Brown University announced the successful creation of a self-constructing lattice structure based on a strangely shaped quantum dot. While single-component quasicrystal lattices have been previously predicted mathematically and in computer simulations, they had not been demonstrated prior to this.
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T.S. Blyth, Lattices and Ordered Algebraic Structures, Springer, 2005, , p. 11 He also wrote on algebraic geometry, number theory, and integral equations. At Chicago, Moore supervised 31 doctoral dissertations, including those of George Birkhoff, Leonard Dickson, Robert Lee Moore (no relation), and Oswald Veblen.
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More recently, a different method of real-time control of the lattice periodicity was demonstrated, in which the center fringe moved less than 2.7 μm while the lattice periodicity was changed from 0.96 to 11.2 μm. Keeping atoms (or other particles) trapped while changing the lattice periodicity remains to be tested more thoroughly experimentally. Such accordion lattices are useful for controlling ultracold atoms in optical lattices, where small spacing is essential for quantum tunneling, and large spacing enables single- site manipulation and spatially resolved detection. Site-resolved detection of the occupancy of lattice sites of both bosons and fermions within a high tunneling regime is regularly performed in quantum gas microscopes.
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In the study of discrete subgroups of Lie groups, the quotient space of cosets is often a candidate for more subtle compactification to preserve structure at a richer level than just topological. For example, modular curves are compactified by the addition of single points for each cusp, making them Riemann surfaces (and so, since they are compact, algebraic curves). Here the cusps are there for a good reason: the curves parametrize a space of lattices, and those lattices can degenerate ('go off to infinity'), often in a number of ways (taking into account some auxiliary structure of level). The cusps stand in for those different 'directions to infinity'.
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Cold atoms in optical lattices are used as quantum simulators, that is, they act as controllable systems that can model behavior of more complicated systems, such as frustrated magnets. In particular, they are used to engineer one-, two- and three-dimensional lattices for a Hubbard model with pre-specified parameters, and to study phase transitions for antiferromagnetic and spin liquid ordering. In 1995, a gas of rubidium atoms cooled down to a temperature of 170 nK was used to experimentally realize the Bose–Einstein condensate, a novel state of matter originally predicted by S. N. Bose and Albert Einstein, wherein a large number of atoms occupy one quantum state.
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Removing the requirement of existence of a unit from the axioms of Boolean algebra yields "generalized Boolean algebras". Formally, a distributive lattice B is a generalized Boolean lattice, if it has a smallest element 0 and for any elements a and b in B such that a ≤ b, there exists an element x such that a ∧ x = 0 and a ∨ x = b. Defining a ∖ b as the unique x such that (a ∧ b) ∨ x = a and (a ∧ b) ∧ x = 0, we say that the structure (B,∧,∨,∖,0) is a generalized Boolean algebra, while (B,∨,0) is a generalized Boolean semilattice. Generalized Boolean lattices are exactly the ideals of Boolean lattices.
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Free modular lattice generated by three elements {x,y,z} The definition of modularity is due to Richard Dedekind, who published most of the relevant papers after his retirement. In a paper published in 1894 he studied lattices, which he called dual groups () as part of his "algebra of modules" and observed that ideals satisfy what we now call the modular law. He also observed that for lattices in general, the modular law is equivalent to its dual. In another paper in 1897, Dedekind studied the lattice of divisors with gcd and lcm as operations, so that the lattice order is given by divisibility.
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In mathematics, the congruence lattice problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of some other lattice. The problem was posed by Robert P. Dilworth, and for many years it was one of the most famous and long-standing open problems in lattice theory; it had a deep impact on the development of lattice theory itself. The conjecture that every distributive lattice is a congruence lattice is true for all distributive lattices with at most ℵ1 compact elements, but F. Wehrung provided a counterexample for distributive lattices with ℵ2 compact elements using a construction based on Kuratowski's free set theorem.
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The following two conditions are equivalent to each other for all lattices. They were found by Saunders Mac Lane, who was looking for a condition that is equivalent to semimodularity for finite lattices, but does not involve the covering relation. ;Mac Lane's condition 1: For any a, b, c such that b ∧ c < a < c < b ∨ a, :there is an element d such that b ∧ c < d ≤ b and a = (a ∨ d) ∧ c. ;Mac Lane's condition 2: For any a, b, c such that b ∧ c < a < c < b ∨ c, :there is an element d such that b ∧ c < d ≤ b and a = (a ∨ d) ∧ c.
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There are 665 orbits of vectors v of norm –4, corresponding to the 665 isomorphism classes of 25-dimensional unimodular lattices L. In this correspondence, the index 2 sublattice of the even vectors of the lattice L is isomorphic to the orthogonal complement of the vector v.
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The lattice of noncrossing partitions of a finite set has recently taken on importance because of its role in free probability theory. These form a subset of the lattice of all partitions, but not a sublattice, since the join operations of the two lattices do not agree.
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Every algebraic lattice is isomorphic to the congruence lattice of some algebra. The lattice Sub V of all subspaces of a vector space V is certainly an algebraic lattice. As the next result shows, these algebraic lattices are difficult to represent. Theorem (Freese, Lampe, and Taylor 1979).
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Topological spaces in fact lead to very special topoi called locales. The set of open subsets of a topological space determines a lattice. The axioms for a topological space cause these lattices to be complete Heyting algebras. The theory of locales takes this as its starting point.
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On the south side, instead of fences, gates were set up using wooden lattices for the introduction of horses into the building. The miller's apartment consists of three rooms: a sitting room, a kitchen, and a cellar. These rooms are located next to the mill area.
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Basic fuzzy Logic (or shortly BL), the logic of the continuous t-norms, is one of t-norm fuzzy logics. It belongs to the broader class of substructural logics, or logics of residuated lattices;Ono (2003). it extends the logic of all left-continuous t-norms MTL.
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The size of the shell varies between 4 mm and 9 mm. The shell has a discoidal shape, with a flattened spire. The periphery shows two prominent ribs, connected by lattices which a subspinously project. The surface contains clathrate ridges, the interstices of which are finely striated.
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Recent restoration work has been redone on the panels of inlay and has also reproduced the gilded pattern on one of the pillars fronting the hall. In the riverbed below the hall and the connected buildings was the space known as zer-jharokha, or "beneath the lattices".
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The spectral properties of this Hamiltonian can be studied with Stone's theorem; this is a consequence of the duality between posets and Boolean algebras. On regular lattices, the operator typically has both traveling-wave as well as Anderson localization solutions, depending on whether the potential is periodic or random.
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In mathematics, the trace field of a linear group is the field generated by the traces of its elements. It is mostly studied for Kleinian and Fuchsian groups, though related objects are used in the theory of lattices in Lie groups, often under the name field of definition.
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A nanosheet is a two-dimensional nanostructure with thickness in a scale ranging from 1 to 100 nm. A typical example of a nanosheet is graphene, the thinnest two-dimensional material (0.34 nm) in the world. It consists of a single layer of carbon atoms with hexagonal lattices.
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The Wigner–Seitz cell always has the same point symmetry as the underlying Bravais lattice. For example, the cube, truncated octahedron, and rhombic dodecahedron have point symmetry Oh, since the respetive Bravais lattices used to generate them all belong to the cubic lattice system, which has Oh point symmetry.
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Therefore, every modular graph is a bipartite graph. The modular graphs contain as a special case the median graphs, in which every triple of vertices has a unique median; median graphs are related to distributive lattices in the same way that modular graphs are related to modular lattices. However, the modular graphs also include other graphs such as the complete bipartite graphs where the medians are not unique: when the three vertices , , and all belong to one side of the bipartition of a complete bipartite graph, every vertex on the other side is a median. Every chordal bipartite graph (a class of graphs that includes the complete bipartite graphs and the bipartite distance-hereditary graphs) is modular.
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A congruence subgroup is (roughly) a subgroup of an arithmetic group defined by taking all matrices satisfying certain equations modulo an integer, for example the group of 2 by 2 integer matrices with diagonal (respectively off-diagonal) coefficients congruent to 1 (respectively 0) modulo a positive integer. These are always finite-index subgroups and the congruence subgroup problem roughly asks whether all subgroups are obtained in this way. The conjecture (usually attributed to Jean-Pierre Serre) is that this is true for (irreducible) arithmetic lattices in higher-rank groups and false in rank-one groups. It is still open in this generality but there are many results establishing it for specific lattices (in both its positive and negative cases).
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Every finite distributive lattice is isomorphic to the congruence lattice of some finite lattice. It is important to observe that the solution lattice found in Grätzer and Schmidt's proof is sectionally complemented, that is, it has a least element (true for any finite lattice) and for all elements a ≤ b there exists an element x with a ∨ x = b and a ∧ x = 0. It is also in that paper that CLP is first stated in published form, although it seems that the earliest attempts at CLP were made by Dilworth himself. Congruence lattices of finite lattices have been given an enormous amount of attention, for which a reference is Grätzer's 2005 monograph.
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Initially there were also two small forged lattices with monograms at the front door, but to our time has remained only one. The first floor walls are decorated with rust. Window openings are decorated by various platbands. On the second floor over side gate narrow windows are pulled together on three.
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Fully Homomorphic Encryption Using Ideal Lattices. In the 41st ACM Symposium on Theory of Computing (STOC), 2009. In 2009, his dissertation, in which he constructed the first Fully Homomorphic Encryption scheme, won the ACM Doctoral Dissertation Award. In 2010 he won the ACM Grace Murray Hopper Award for the same work.
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This chapter departs slightly from the simultaneous treatment of number fields and function fields. In the number field setting, lattices (that is, fractional ideals) are defined, and the Haar measure volume of a fundamental domain for a lattice is found. This is used to study the discriminant of an extension.
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A lattice is called algebraic if it is complete and compactly generated. In 1963, Grätzer and Schmidt proved that every algebraic lattice is isomorphic to the congruence lattice of some algebra.G. Grätzer and E. T. Schmidt, Characterizations of congruence lattices of abstract algebras, Acta Sci. Math. (Szeged) 24 (1963), 34–59.
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In the mathematical area of order theory, there are various notions of the common concept of distributivity, applied to the formation of suprema and infima. Most of these apply to partially ordered sets that are at least lattices, but the concept can in fact reasonably be generalized to semilattices as well.
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A zincblende unit cell Hg1−xCdxTe has a zincblende structure with two interpenetrating face-centered cubic lattices offset by (1/4,1/4,1/4)ao in the primitive cell. The cations Cd are Hg statistically mixed on the yellow sublattice while the Te anions form the grey sublattice in the image.
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Distributivity is most commonly found in rings and distributive lattices. A ring has two binary operations, commonly denoted + and ∗, and one of the requirements of a ring is that ∗ must distribute over +. Most kinds of numbers form rings. A lattice is another kind of algebraic structure with two binary operations, ∧ and ∨.
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Dilation (usually represented by ⊕) is one of the basic operations in mathematical morphology. Originally developed for binary images, it has been expanded first to grayscale images, and then to complete lattices. The dilation operation usually uses a structuring element for probing and expanding the shapes contained in the input image.
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On August 1, 1984 he retired from the university at the age of 68. Between 1937 and 1988 he published 54 scientific articles. In 1988 he published his last (with Th Ruygrok) in the Journal of Statistical Physics, titled On the energy per particle in three- and four- dimensional Wigner lattices '.
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The seven-direction box spline has been used for modelling surfaces and can be used for interpolation of data on the Cartesian lattice as well as the body centered cubic lattice. Generalization of the four- and six-direction box splines to higher dimensionsKim, Minho. Symmetric Box-Splines on Root Lattices. [Gainesville, Fla.
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Let P and Q be combinatorially equivalent 3-dimensional convex polytopes; that is, they are convex polytopes with isomorphic face lattices. Suppose further that each pair of corresponding faces from P and Q are congruent to each other, i.e. equal up to a rigid motion. Then P and Q are themselves congruent.
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A core enhancement of copula models are dynamic copulas, introduced by Albanese et al. (2005) and (2007). The "dynamic conditioning" approach models the evolution of multi-factor super-lattices, which correlate the return processes of each entity at each time step. Binomial dynamic copulas apply combinatorial methods to avoid Monte Carlo simulations.
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The first gluten free products were launched with the autumn/winter menu in 2016 and included a range of cakes, brownies and crispy rolls. Their Halloween product line includes fairy buns, cakes, biscuits, lattices and gingerbread kits. Their Christmas menu includes bakes, rolls, soups, toasties, baguettes, biscuits, muffins, buns and mince pies.
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Eva Bayer-Fluckiger (born 25 June 1951) is a Hungarian and Swiss mathematician. She is an Emmy Noether Professor Emeritus at École Polytechnique Fédérale de Lausanne. She has worked on several topics in topology, algebra and number theory, e.g. on the theory of knots, on lattices, on quadratic forms and on Galois cohomology.
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7, p. 722 In 2012 he became a fellow of the American Mathematical Society.List of Fellows of the American Mathematical Society, retrieved 2013-02-02. In 2005, Margulis received the Wolf Prize for his contributions to theory of lattices and applications to ergodic theory, representation theory, number theory, combinatorics, and measure theory.
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There are wide range of cellular and intracellular elements which form active fluids. This include systems of microtubule, bacteria, sperm cells as well as inanimate microswimmers. It is known that these systems form a variety of structures such as regular and irregular lattices as well as seemingly random states in two dimensions.
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Anderson localization is a well-known phenomenon that occurs when electrons become trapped in a disordered metallic structure, and this metal goes through a phase transition from conductor to insulator.P. W. Anderson 'Absence of diffusion in certain random lattices' Phys. Rev. 109, 1492 (1958) . These electrons are said to be Anderson-localized.
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Poul S. Jessen holds the position of Professor of Optical Sciences with a joint appointment in Physics at the University of Arizona. He is a founding member of the Center for Quantum Information and Control. He has done experimental research in the areas of optical lattices, quantum information, quantum chaos, and quantum optics.
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In mathematics, specifically in functional analysis and order theory, a topological vector lattice is a Hausdorff topological vector space (TVS) X that has a partial order ≤ making it into vector lattice that is possesses a neighborhood base at the origin consisting of solid sets. Ordered vector lattices have important applications in spectral theory.
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Mai Gehrke (born 10 May 1964) is a Danish mathematician who studies the theory of lattices and their applications to mathematical logic. She is a director of research for the French Centre national de la recherche scientifique (CNRS), affiliated with the Laboratoire J. A. Dieudonné (LJAD) at the University of Nice Sophia Antipolis.
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Hänsch will remain professor at the LMU and director at the MPQ at least until 2016. One research focus of Prof. Bloch is the investigation of ultracold quantum gases in optical lattices. These systems may help to model solid states and to get a better understanding of special properties such as e.g.
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Every distributive lattice with zero satisfies Schmidt's Condition; thus it is representable. This result has been improved further as follows, via a very long and technical proof, using forcing and Boolean-valued models. Theorem (Wehrung 2003). Every direct limit of a countable sequence of distributive lattices with zero and (∨,0)-homomorphisms is representable.
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The spaces described in this article are pitch class spaces which model the relationships between pitch classes in some musical system. These models are often graphs, groups or lattices. Closely related to pitch class space is pitch space, which represents pitches rather than pitch classes, and chordal space, which models relationships between chords.
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He worked on Diophantine approximation and geometry of numbers, where he used both classical and p-adic analytic methods.special issue of Annales de l'Institut Fourier (vol. XXIX, Fasc. 1), March 1979, for Chabauty's retirement He introduced the Chabauty topology to generalise Mahler's compactness theorem from Euclidean lattices to more general discrete subgroups.
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An orthomodular lattice is therefore defined as an orthocomplemented lattice such that for any two elements the implication ::if a ≤ c, then a ∨ (a⊥ ∧ c) = c holds. Lattices of this form are of crucial importance for the study of quantum logic, since they are part of the axiomisation of the Hilbert space formulation of quantum mechanics. Garrett Birkhoff and John von Neumann observed that the propositional calculus in quantum logic is "formally indistinguishable from the calculus of linear subspaces [of a Hilbert space] with respect to set products, linear sums and orthogonal complements" corresponding to the roles of and, or and not in Boolean lattices. This remark has spurred interest in the closed subspaces of a Hilbert space, which form an orthomodular lattice.
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A partially ordered set (L, ≤) is a complete lattice if every subset A of L has both a greatest lower bound (the infimum, also called the meet) and a least upper bound (the supremum, also called the join) in (L, ≤). The meet is denoted by \bigwedge A, and the join by \bigvee A. Note that in the special case where A is the empty set, the meet of A will be the greatest element of L. Likewise, the join of the empty set yields the least element. Since the definition also assures the existence of binary meets and joins, complete lattices thus form a special class of bounded lattices. More implications of the above definition are discussed in the article on completeness properties in order theory.
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Peng was granted his PhD in 1940 and DSc in 1945. Recommended by Born, Peng worked at Dublin Institute for Advanced Studies in Ireland as a postdoctoral scholar from 1941-1943 and later as an assistant professor from 1945-1947. While at DIAS Peng worked with another one of Born's students Sheila Tinney to produce important work on crystal lattices.MacTutor biography of Sheila Christina Power Tinney"On the stability of crystal lattices VIII. Stability of rhombohedral Bravais lattices" (with S. C. Power), Mathematical Proceedings of the Cambridge Philosophical Society, Vol 38, Issue 1, January 1942, pp. 67-81 From August 1941 to July 1943, Peng collaborated with Walter Heitler and James Hamilton to study cosmic ray, and developed HHP theory.
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Many years later, Leverett is contacted by a descendant of a famous horror author, H. Kenneth Allard (supposedly based on H.P. Lovecraft), who hires him to illustrate a volume of Allard's previously unpublished stories. When Leverett decides to base the illustrations on his old sketches of the stick lattices, he is unwittingly drawn into a supernatural conspiracy of potentially apocalyptic magnitude. The mysterious lattices of twigs were inspired by the work of Weird Tales artist Lee Brown Coye, who illustrated two Carcosa Press volumes which Wagner edited: Manly Wade Wellman's Worse Things Waiting and Hugh B. Cave's Murgunstrumm and Others (the latter volume appeared some years after "Sticks" was written). "Sticks" was also the inspiration for the lattice stick structures in the HBO show "True Detective".
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In an infinite distributive lattice, it may not be the case that the lower sets of the join-irreducible elements are in one-to-one correspondence with lattice elements. Indeed, there may be no join-irreducibles at all. This happens, for instance, in the lattice of all natural numbers, ordered with the reverse of the usual divisibility ordering (so x ≤ y when y divides x): any number x can be expressed as the join of numbers xp and xq where p and q are distinct prime numbers. However, elements in infinite distributive lattices may still be represented as sets via Stone's representation theorem for distributive lattices, a form of Stone duality in which each lattice element corresponds to a compact open set in a certain topological space.
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Girard (1987) to model linear logic), while Chu spaces over K realize any category of vector spaces over a field whose cardinality is at most that of K. This was extended by Vaughan Pratt (1995) to the realization of k-ary relational structures by Chu spaces over 2k. For example, the category Grp of groups and their homomorphisms is realized by Chu(Set, 8) since the group multiplication can be organized as a ternary relation. Chu(Set, 2) realizes a wide range of ``logical`` structures such as semilattices, distributive lattices, complete and completely distributive lattices, Boolean algebras, complete atomic Boolean algebras, etc. Further information on this and other aspects of Chu spaces, including their application to the modeling of concurrent behavior, may be found at Chu Spaces.
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A proof of this statement is given by Johnstone;P. T. Johnstone, Stone Spaces, Cambridge University Press, 1982; (see paragraph 4.7) the original argument is attributed to Alfred W. Hales;A. W. Hales, On the non-existence of free complete Boolean algebras, Fundamenta Mathematicae 54: pp.45-66. see also the article on free lattices.
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Two typical applications that require a large number of pure qubits are quantum error correction (QEC) and ensemble computing. In realizations of quantum computing (implementing and applying the algorithms on actual qubits), algorithmic cooling was involved in realizations in optical lattices. In addition, algorithmic cooling can be applied to in vivo magnetic resonance spectroscopy.
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Hasse diagram of the powerset of {x,y,z} ordered by inclusion. Order theory is the study of partially ordered sets, both finite and infinite. Various examples of partial orders appear in algebra, geometry, number theory and throughout combinatorics and graph theory. Notable classes and examples of partial orders include lattices and Boolean algebras.
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The lattice angles and the lengths of the lattice vectors are all the same for both the cubic and rhombohedral lattice systems. The lattice angles for simple cubic, face-centered cubic, and body-centered cubic lattices are /2 radians, /3 radians, and radians, respectively. A rhombohedral lattice will result from lattice angles other than these.
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Depending on the complexity of the relationships under consideration, the models may be multidimensional. Models of pitch space are often graphs, groups, lattices, or geometrical figures such as helixes. Pitch spaces distinguish octave-related pitches. When octave-related pitches are not distinguished, we have instead pitch class spaces, which represent relationships between pitch classes.
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Of the sixteen Solomonic columns, only the two half columns of the main altar are original. The remaining belong to the 2016 restoration. The stone pillars are covered with stucco but remain unpainted. The second level includes a platform with eight semi-circular arches and a set of balconies with lattices on the supporting pillars.
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Thus the central operations of lattices are binary suprema \vee and infima It is in this context that the terms meet for \wedge and join for \vee are most common. A poset in which only non-empty finite suprema are known to exist is therefore called a join-semilattice. The dual notion is meet-semilattice.
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The Metamath website also hosts a few older databases which are not maintained anymore, such as the "Hilbert Space Explorer", which presents theorems pertaining to Hilbert space theory which have now been merged into the Metamath Proof Explorer, and the "Quantum Logic Explorer", which develops quantum logic starting with the theory of orthomodular lattices.
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I. J. Vaughn, A. S. Alenin, and J. S. Tyo, "Focal plane filter array engineering I: rectangular lattices", Optics Express 25: 10 (2017). While snapshot instruments are featured prominently in the research literature, none of these instruments have seen wide adoption in commercial use (i.e. outside the professional astronomical community) due to manufacturing limitations.
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S. P. Efimov from Bauman Moscow State Technical University in 1978 y. found an approach to ease the restrictions for spectrum domain. He considered N identical sampling lattices to be shifted arbitrarily to each other. Optimal sampling is valid for spectrum domain that shifted versions of is close-packed N times on reciprocal lattice.
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Often all the factors (both the lattices and the measures) are identical, i.e., μ is the probability distribution of i.i.d. random variables. The FKG inequality for the case of a product measure is known also as the Harris inequality after Harris , who found and used it in his study of percolation in the plane.
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When talking about solid materials, the discussion is mainly around crystals – periodic lattices. Here we will discuss a 1D lattice of positive ions. Assuming the spacing between two ions is , the potential in the lattice will look something like this: Image:Potential-actual.PNG The mathematical representation of the potential is a periodic function with a period .
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One reason for interest in Dowling lattices is that the characteristic polynomial is very simple. If L is the Dowling lattice of rank n of a finite group G having m elements, then :p_L(y) = (y-1)(y-m-1)\cdots(y-[n-1]m-1) , an exceptionally simple formula for any geometric lattice.
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Nanostructures contain small length scales, and this results in a large surface to volume ratio. Surface defects have hence been the primary focus of research into defects of ZnO nanostructures. Deep level emissions also occur, affecting material characteristics. ZnO can occupy multiple types of lattices, but is often found in a hexagonal wurtzite structure.
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The opal in this bracelet contains a natural periodic microstructure responsible for its iridescent color. It is essentially a natural photonic crystal. Wings of some butterflies contain photonic crystals. A photonic crystal is a periodic optical nanostructure that affects the motion of photons in much the same way that ionic lattices affect electrons in solids.
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No remains of traced streets were found, as the buildings around collapsed and ruined so narrow and crooked traces. Unbaked bricks on wooden framework, with clay molding were used to build the houses. Some of them had windows with gypsum lattices set with small pieces of glass. There were found a lot of coins.
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In February 2012, the start of a major restoration to the windmill began. The first stage saw the removal of the wooden lattices that make up the sails. This work was in preparation for the replacement of the windshaft which had reached the end of its natural life. The work was completed in November 2014.
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An orthocomplementation on a complemented lattice is an involution which is order-reversing and maps each element to a complement. An orthocomplemented lattice satisfying a weak form of the modular law is called an orthomodular lattice. In distributive lattices, complements are unique. Every complemented distributive lattice has a unique orthocomplementation and is in fact a Boolean algebra.
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His father József Grätzer was famous in Hungary as the "Puzzle King" ("rejtvénykirály"). George Grätzer received his PhD from Eötvös Loránd University in 1960 under the supervision of László Fuchs. In 1963 Grätzer and Schmidt published their theorem on the characterization of congruence lattices of algebras. In 1963 Grätzer left Hungary and became a professor at Pennsylvania State University.
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That is all for lattices in the plane. In n-dimensional Euclidean space the same questions can be posed, for example about SO(n)\SLn(R)/SLn(Z). This is harder to compactify. There are a variety of compactifications, such as the Borel–Serre compactification, the reductive Borel-Serre compactification, and the Satake compactifications, that can be formed.
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The G-1 experiment performed at the HWA testing station, under the direction of Kurt Diebner, had lattices of 6,800 uranium oxide cubes (about 25 tons), in the nuclear moderator paraffin.Hentschel and Hentschel, 1996, 369 and 373, Appendix F (see the entry for Nikolaus Riehl and Kurt Diebner), and Appendix D (see the entry for Auergesellschaft).
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According to Paul Halmos, a subset of a locally compact Hausdorff topological space is called a Borel set if it belongs to the smallest σ–ring containing all compact sets. Norberg and Vervaat Tommy Norberg and Wim Vervaat, Capacities on non-Hausdorff spaces, in: Probability and Lattices, in: CWI Tract, vol. 110, Math. Centrum Centrum Wisk. Inform.
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This allows phase transformation to absorb and disperse the point defects that typically accumulate in more rigid lattices. This extends the life of the alloy through making vacancy and interstitial creation less successful as constant neutron excitement in the form of displacement cascades transform the SRO phase, while the SRO reforms in the bulk solid solution.
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Another result analogous to Birkhoff's representation theorem, but applying to a broader class of lattices, is the theorem of that any finite join-distributive lattice may be represented as an antimatroid, a family of sets closed under unions but in which closure under intersections has been replaced by the property that each nonempty set has a removable element.
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Every interval of a geometric lattice (the subset of the lattice between given lower and upper bound elements) is itself geometric; taking an interval of a geometric lattice corresponds to forming a minor of the associated matroid. Geometric lattices are complemented, and because of the interval property they are also relatively complemented., pp. 55, 65–67.
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He was the first ever winner of the Canadian Mathematical Olympiad in 1969. He continued his education at Harvard University and the Massachusetts Institute of Technology in Cambridge, receiving a PhD from the latter institution in 1977. His dissertation was entitled "Witt Theorems for Lattices over Discrete Valuation Rings". He worked as a corporate planner and financial analyst.
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Half-Heusler (HH) alloys have a great potential for high-temperature power generation applications. Examples of these alloys include NbFeSb, NbCoSn and VFeSb. They have a cubic MgAgAs-type structure formed by three interpenetrating face-centered-cubic (fcc) lattices. The ability to substitute any of these three sublattices opens the door for wide variety of compounds to be synthesized.
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In universal algebra, a congruence-permutable algebra is an algebra whose congruences commute under composition. This symmetry has several equivalent characterizations, which lend to the analysis of such algebras. Many familiar varieties of algebras, such as the variety of groups, consist of congruence- permutable algebras, but some, like the variety of lattices, have members that are not congruence-permutable.
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In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set (poset). The most familiar example is the completeness of the real numbers. A special use of the term refers to complete partial orders or complete lattices. However, many other interesting notions of completeness exist.
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Many of the capitals are also reused, those developed specifically for the building following a Corinthian tradition of drawing the characteristic necking stranded (such as wreathed Asturias). Ornamental architecture did not reach a great development. Eaves were decorated, and the openings of the windows filled with beautiful stone lattices. Many times the doors and windows were framed with alfiz.
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The Great Buddha of Gifu is unique due to the method of its construction. First, a central pillar 1.8 meters in circumference was formed from ginkgo tree wood. The Buddha's shape was then formed using bamboo lattices. The bamboo was covered with clay to add shape and many Buddhist scriptures were then placed upon the clay.
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The Young-Fibonacci lattice has a natural r-differential analogue for every positive integer r. These posets are lattices, and can be constructed by a variation of the reflection construction. In addition, the product of an r-differential and s-differential poset is always an (r + s)-differential poset. This construction also preserves the lattice property.
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The term abstract data type can also be regarded as a generalized approach of a number of algebraic structures, such as lattices, groups, and rings., Chapter 7, section 40. The notion of abstract data types is related to the concept of data abstraction, important in object-oriented programming and design by contract methodologies for software development.
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For a distributive lattice, the complement of x, when it exists, is unique. In the case the complement is unique, we write and equivalently, . The corresponding unary operation over L, called complementation, introduces an analogue of logical negation into lattice theory. Heyting algebras are an example of distributive lattices where some members might be lacking complements.
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In the optically actuated sorting process, the cells are flowed through into an optical landscape i.e. 2D or 3D optical lattices. Without any induced electrical charge, the cells would sort based on their intrinsic refractive index properties and can be re-configurability for dynamic sorting. An optical lattice can be created using diffractive optics and optical elements.
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An example of the tetragonal crystals, wulfenite In crystallography, the tetragonal crystal system is one of the 7 crystal systems. Tetragonal crystal lattices result from stretching a cubic lattice along one of its lattice vectors, so that the cube becomes a rectangular prism with a square base (a by a) and height (c, which is different from a).
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In mathematics, specifically in order theory and functional analysis, an ordered vector space X is said to be regularly ordered and its order is called regular if X is Archimedean ordered and the order dual of X distinguishes points in X. Being a regularly ordered vector space is an important property in the theory of topological vector lattices.
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Miroshnichenko A, Vasiliev A, Dmitriev S. Solitons and Soliton Collisions. Standing breathers correspond to localized solutions whose amplitude vary in time (they are sometimes called oscillons). A necessary condition for the existence of breathers in discrete lattices is that the breather main frequency and all its multipliers are located outside of the phonon spectrum of the lattice.
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The crystal structure consists of the same group of atoms, the basis, positioned around each and every lattice point. This group of atoms therefore repeats indefinitely in three dimensions according to the arrangement of one of the Bravais lattices. The characteristic rotation and mirror symmetries of the unit cell is described by its crystallographic point group.
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A Dowling lattice is the geometric lattice of flats associated with a Dowling geometry. The lattice and the geometry are mathematically equivalent: knowing either one determines the other. Dowling lattices, and by implication Dowling geometries, were introduced by Dowling (1973a,b). A Dowling lattice or geometry of rank n of a group G is often denoted Qn(G).
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There are only five topologically distinct polyhedra which tile three-dimensional space, . These are referred to as the parallelohedra. They are the subject of mathematical interest, such as in higher dimensions. These five paralellohedra can be used to classify the three dimensional lattices using the concept of a projective plane, as suggested by John Horton Conway and Neil Sloane.
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The above describes magnetic domain structure in a perfect crystal lattice, such as would be found in a single crystal of iron. However most magnetic materials are polycrystalline, composed of microscopic crystalline grains. These grains are not the same as domains. Each grain is a little crystal, with the crystal lattices of separate grains oriented in random directions.
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In 1504, cheval de frise were installed in Moscow, under which the guards, drawn from the local population, were stationed. The city was divided into areas, between which gates with lattices were built. It was forbidden to move around the city at night or without lighting. Subsequently, the Grand Prince Ivan IV established patrols around Moscow for increased security.
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In another direction one can weaken the assumption about curvature in the examples above: a CAT(0) group is a group admitting a geometric action on a CAT(0) space. This includes Euclidean crystallographic groups and uniform lattices in higher-rank Lie groups. It is not known whether there exists a hyperbolic group which is not CAT(0).
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Goldstine graduated summa cum laude from Amherst College in 1993. She completed a Ph.D. in mathematics at Harvard University in 1998. Her dissertation, Spin Representations and Lattices, was supervised by Benedict Gross. After postdoctoral and visiting assistant professorships at McMaster University, Ohio State University, and Amherst College, she joined the St. Mary's College faculty in 2004.
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Complex metal hydrides are salts wherein the anions contain hydrides. In the older chemical literature as well as contemporary materials science textbooks, a "metal hydride" is assumed to be nonmolecular, i.e. three-dimensional lattices of atomic ions. In such systems, hydrides are often interstitial and nonstoichiometric, and the bonding between the metal and hydrogen atoms is significantly ionic.
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A number of natural branched DNA structures were known at the time, including the DNA replication fork and the mobile Holliday junction, but Seeman's insight was that immobile nucleic acid junctions could be created by properly designing the strand sequences to remove symmetry in the assembled molecule, and that these immobile junctions could in principle be combined into rigid crystalline lattices. The first theoretical paper proposing this scheme was published in 1982, and the first experimental demonstration of an immobile DNA junction was published the following year. Seeman developed the more rigid double-crossover (DX) motif, suitable for forming two-dimensional lattices, demonstrated in 1998 by him and Erik Winfree. In 2006, Paul Rothemund first demonstrated the DNA origami technique for easily and robustly creating folded DNA structures of arbitrary shape.
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The Lattice paintings are typically based on a flexible, constantly varied system of usually diagonal, interlaced bands of colour, within a square canvas. "The basic over and under pattern which holds the forms meshed together on the picture surface is common to all the paintings of the Lattices." Early examples from 1976 to 1977, such as Lattice No. 11 (May 1977, Christchurch Art Gallery), have white horizontal and vertical as well as diagonal bands, their edges marked by black crayon rubbed along the edges of masking tape, with brightly coloured bands seeming to form a grid behind the white bands. From 1978 onwards, the crayon lines were eliminated in favour of clean, masked edges of abutted colour, and with the exception of the Asymmetrical Lattices the square canvas became the standard support.
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She discovered the Si4− butterfly anion in Ba3Si4 during her work on her dissertation. Furthermore, she and Herbert Schäfer extended the definition of Zintl phases: In their definition, Zintl phases are intermetallic compounds with a pronounced heteropolar bonding contribution. Furthermore, their anion partial lattices should obey the (8-N) rule. The latter was newly introduced to the definition of Zintl phases.
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Surprisingly, there is a notion of "distributivity" applicable to semilattices, even though distributivity conventionally requires the interaction of two binary operations. This notion requires but a single operation, and generalizes the distributivity condition for lattices. A join- semilattice is distributive if for all a, b, and x with there exist and such that x = a' ∨ b' . Distributive meet-semilattices are defined dually.
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While the lattice problem is in general intractable, the impurity model is usually solvable through various schemes. The mapping in itself does not constitute an approximation. The only approximation made in ordinary DMFT schemes is to assume the lattice self-energy to be a momentum-independent (local) quantity. This approximation becomes exact in the limit of lattices with an infinite coordination.
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In general, Jónsson terms, more formally, a sequence of Jónsson terms, is a sequence of ternary terms satisfying certain related idenitities. One of the earliest Maltsev condition, a variety is congruence distributive if and only if it has a sequence of Jónsson terms. Originally proved in B. Jónsson, Algebras whose congruence lattices are distributive. Math. Scand., 21:110-121, 1967.
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The isoperimetric problem, a recurring concept in convex geometry, was studied by the Greeks as well, including Zenodorus. Archimedes, Plato, Euclid, and later Kepler and Coxeter all studied convex polytopes and their properties. From the 19th century on, mathematicians have studied other areas of convex mathematics, including higher-dimensional polytopes, volume and surface area of convex bodies, Gaussian curvature, algorithms, tilings and lattices.
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Finite-dimensional vector spaces over local fields and division algebras under the topology uniquely determined by the field's topology are studied, and lattices are defined topologically, an analogue of Minkowski's theorem is proved in this context, and the main theorems about character groups of these vector spaces, which in the commutative one-dimensional case reduces to `self duality’ for local fields, are shown.
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The Beamless Brick Hall was built in 1600 by Wanli Emperor to congratulate the birthday of his mother. It is high, wide and long. Inspired by techniques and styles of India and Myanmar with not a single piece of wood was used. Walls of the hall are decorated with patterns of wood-like structures like circular arches, vertical columns, window lattices, etc.
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The electron and hole may have either parallel or anti-parallel spins. The spins are coupled by the exchange interaction, giving rise to exciton fine structure. In periodic lattices, the properties of an exciton show momentum (k-vector) dependence. The concept of excitons was first proposed by Yakov Frenkel in 1931, when he described the excitation of atoms in a lattice of insulators.
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Structured cellular materials can be remarkably strong despite very low density. Reversibly assembled cellular composite materials enable tailorable composite materials properties, to the ideal linear specific stiffness scaling regime. Using projection microstereolithography, octet microlattices have also been fabricated from polymers, metals, and ceramics. The design of the high performing lattices mean that the individual struts making up the materials do not bend.
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Nature Neuroscience, 14(7), 919-925. The ideal case takes two layers of perfect hexagonal lattices of the on-center and off-center receptive fields of the RGCs. These two layers are superimposed on each other with an angled offset that produces a periodic interference pattern. This pattern produces dipoles of these RGCs that have a preferred orientation scattered throughout the visual field.
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The tetrahedral-angled hydrogen-bonded hexagonal rings are also the mechanism that causes liquid water to be densest at 4 °C. Close to 0 °C, tiny hexagonal ice Ih-like lattices form in liquid water, with greater frequency closer to 0 °C. This effect decreases the density of the water, causing it to be densest at 4 °C when the structures form infrequently.
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A parallelogon is constructed by two or three pairs of parallel line segments. The vertices and edges on the interior of the hexagon are suppressed. Bravais lattices in two dimensions, related to the parallelogon tessellations by their five symmetry variations. A parallelogon is a polygon such that images of the polygon will tile the plane when fitted together along entire sides, without rotation.
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The nine-story building's most notable feature is its translucent, corrugated sunscreens. Supported by aluminum lattices throughout the building's facade, the sunscreens are made of Rohm and Haas's principal product, Plexiglas. In 2007 the Rohm and Haas Corporate Headquarters was listed on the National Register of Historic Places and today is considered one of the best examples of the International style.
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Each cell includes aligned fiber composite beams and looped fiber load-bearing holes that reversibly chain together to form volume-filling lattices. Mass-produced cells can be assembled to fill arbitrary structural shapes, with a resolution prescribed by the part scale that matches the variability of an application's boundary stress. The periodic nature of assemblies simplifies behavior analysis and prediction.
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Crystalline materials (mainly metals and alloys, but also stoichiometric salts and other materials) are made up of solid regions of ordered matter (atoms placed in one of a number of ordered formations called Bravais lattices). These regions are known as crystals. A perfect crystal is a crystal that contains no point, line, or planar defects. There are a wide variety of crystallographic defects.
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Due to the dimension bound on spaces of modular forms, the minimum norm of a nonzero vector of an even unimodular lattice is no greater than ⎣n/24⎦ + 1. An even unimodular lattice that achieves this bound is called extremal. Extremal even unimodular lattices are known in relevant dimensions up to 80, and their non-existence has been proven for dimensions above 163,264.
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Lattices have some connections to the family of group-like algebraic structures. Because meet and join both commute and associate, a lattice can be viewed as consisting of two commutative semigroups having the same domain. For a bounded lattice, these semigroups are in fact commutative monoids. The absorption law is the only defining identity that is peculiar to lattice theory.
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Optical fields transmitted through nonlinear Kerr media can also display pattern formation owing to the nonlinear medium amplifying spatial and temporal noise. The effect is referred to as optical modulation instability. This has been observed both in photo-refractive, photonic lattices, as well as photo-reactive systems. In the latter case, optical nonlinearity is afforded by reaction-induced increases in refractive index.
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As a graduate student, Ashkin contributed to one paper in astrophysics. and two papers in statistical mechanics He collaborated with Lamb in writing the first of the two papers on statistical mechanics and with Teller in writing the second. This second paper, "Statistics of Two- Dimensional Lattices with Four Components" has since been frequently cited. He received his Ph.D. in Physics in 1943.
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One-way quantum computation has been demonstrated by running the 2 qubit Grover's algorithm on a 2x2 cluster state of photons. A linear optics quantum computer based on one-way computation has been proposed. Cluster states have also been created in optical lattices, but were not used for computation as the atom qubits were too close together to measure individually.
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These contain diamond- based crystal lattices oriented in all directions to give a brilliant green coloration that hardly varies with angle. The scales are effectively divided into pixels about a micrometre wide. Each such pixel is a single crystal and reflects light in a direction different from its neighbours.The Photonic Beetle: Nature Builds Diamond-like Crystals for Future Optical Computers .
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In mathematics, the Siegel–Weil formula, introduced by as an extension of the results of , expresses an Eisenstein series as a weighted average of theta series of lattices in a genus, where the weights are proportional to the inverse of the order of the automorphism group of the lattice. For the constant terms this is essentially the Smith–Minkowski–Siegel mass formula.
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Institut (1967) 38–39. Conway has also suggested a system of notation dedicated to describing polyhedra called Conway polyhedron notation. In the theory of tessellations, he devised the Conway criterion which is a fast way to identify many prototiles that tile the plane. He investigated lattices in higher dimensions and was the first to determine the symmetry group of the Leech lattice.
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The MacNeille completion theorem (1937) (that any partial order may be embedded in a complete lattice) is cited in a 2013 survey article "Decomposition of relations on concept lattices".R. Berghammer & M. Winter (2013) "Decomposition of relations on concept lattices", Fundamenta Informaticae 126(1): 37–82 The decomposition is :R \ = \ f \ E \ g^T , where f and g are functions, called mappings or left-total, univalent relations in this context. The "induced concept lattice is isomorphic to the cut completion of the partial order E that belongs to the minimal decomposition (f, g, E) of the relation R." Particular cases are considered below: E total order corresponds to Ferrers type, and E identity corresponds to difunctional, a generalization of equivalence relation on a set. Relations may be ranked by the Schein rank which counts the number of concepts necessary to cover a relation.
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This generalized representation theorem can be expressed as a category-theoretic duality between distributive lattices and spectral spaces (sometimes called coherent spaces, but not the same as the coherent spaces in linear logic), topological spaces in which the compact open sets are closed under intersection and form a base for the topology.. Hilary Priestley showed that Stone's representation theorem could be interpreted as an extension of the idea of representing lattice elements by lower sets of a partial order, using Nachbin's idea of ordered topological spaces. Stone spaces with an additional partial order linked with the topology via Priestley separation axiom can also be used to represent bounded distributive lattices. Such spaces are known as Priestley spaces. Further, certain bitopological spaces, namely pairwise Stone spaces, generalize Stone's original approach by utilizing two topologies on a set to represent an abstract distributive lattice.
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Besides trapping cold atoms, optical lattices have been widely used in creating gratings and photonic crystals. They are also useful for sorting microscopic particles, and may be useful for assembling cell arrays. Atoms in an optical lattice provide an ideal quantum system where all parameters can be controlled. Thus they can be used to study effects that are difficult to observe in real crystals.
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Springer, 1999. are useful for sampling multivariate functions in 2-D, 3-D and higher dimensions. In the 2-D setting the three- direction box spline is used for interpolation of hexagonally sampled images. In the 3-D setting, four-direction and six-direction box splines are used for interpolation of data sampled on the (optimal) body-centered cubic and face- centered cubic lattices respectively.
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However, on non-bipartite lattices, RVB liquid phases possessing topological order and fractionalized spinons also appear. The discovery of topological order in quantum dimer models (more than a decade after the models were introduced) has led to new interest in these models. Classical dimer models have been studied previously in statistical physics, in particular by P. W. Kasteleyn (1961) and M. E. Fisher (1961).
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When ordered by the subset relation ⊆, they are complete lattices. A special instance of Birkhoff's construction starts from an arbitrary poset (P,≤) and constructs the Galois connection from the order relation ≤ between P and itself. The resulting complete lattice is the Dedekind-MacNeille completion. When this completion is applied to a poset that already is a complete lattice, then the result is isomorphic to the original one.
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The G-1 experiment performed at the HWA testing station had lattices of 6,800 uranium oxide cubes (about 25 tons) in the neutron moderator paraffin. Their work verified Höcker's calculations that cubes were better than rods, and rods were better than plates.Walker, 1993, 94-104.Hentschel and Hentschel, 1996, 373 and Appendix F; see the appendix entries for Pose, Abraham Esau, and Kurt Diebner.
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Chapter V, "The flow of steam through nozzles", pages 90 to 99 The study of supersaturation is also relevant to atmospheric studies. Since the 1940s, the presence of supersaturation in the atmosphere has been known. When water is supersaturated in the troposphere, the formation of ice lattices is frequently observed. In a state of saturation, the water particles will not form ice under tropospheric conditions.
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Ultralight materials are solids with a density of less than 10 mg/cm3. Ultralight material is defined by its cellular arrangement and its stiffness and strength that make up its solid constituent. They include silica aerogels, carbon nanotube aerogels, aero graphite, metallic foams, polymeric foams, and metallic micro lattices. Ultralight materials are produced to have the strength of bulk-scaled properties at a micro-size.
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Haüy was not the only researcher to observe that calcite crystals could be composed of smaller rhombohedra, but it was he who introduced the idea of triple periodicity in crystals. This idea was fundamental to later developments in the field on crystal lattices. Between 1784 and 1822, Haüy published more than 100 reports discussing his theories and their application to the structure of crystalline substances.
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The Cocks IBE scheme is based on well-studied assumptions (the quadratic residuosity assumption) but encrypts messages one bit at a time with a high degree of ciphertext expansion. Thus it is highly inefficient and impractical for sending all but the shortest messages, such as a session key for use with a symmetric cipher. A third approach to IBE is through the use of lattices.
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They include Molecular Beam Epitaxy and Metal Organic Vapour Phase Epitaxy. The current efficiency record is made with this process but doesn't have exact matching lattice constants. The losses due to this are not as effective because the differences in lattices allows for more optimal band gap material for the first two cells. This type of cell is expected to be able to be 50% efficient.
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Recent advances in the CCD technology has made hexagonal sampling feasible for real life applications. Historically, because of technology constraints, detector arrays were implemented only on 2-dimensional rectangular sampling lattices with rectangular shape detectors. But the super [CCD] detector introduced by Fuji has an octagonal shaped pixel in a hexagonal grid. Theoretically, the performance of the detector was greatly increased by introducing an octagonal pixel.
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Valentin Afraimovich (, 2 April 1945, Kirov, Kirov Oblast, USSR – 21 February 2018, Nizhny Novgorod, Russia) was a Soviet, Russian and Mexican mathematician. He made contributions to dynamical systems theory, qualitative theory of ordinary differential equations, bifurcation theory, concept of attractor, strange attractors, space-time chaos, mathematical models of nonequilibrium media and biological systems, traveling waves in lattices, complexity of orbits and dimension-like characteristics in dynamical systems.
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Polymer crystals have different properties than simple atomic crystals. They possess high density and long range order. They do not possess isotropy, and therefore are anisotopic in nature, which means they show anisotropy and limited conformation space. However, just as atomic crystals have lattices, polymer crystals also exhibit a periodic structure called a lattice, which describes the repetition of the unit cells in the space.
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The above statement is known to be equivalent to its order dual : x \vee (y \wedge z) = (x \vee y) \wedge (x \vee z) such that one of these properties suffices to define distributivity for lattices. Typical examples of distributive lattice are totally ordered sets, Boolean algebras, and Heyting algebras. Every finite distributive lattice is isomorphic to a lattice of sets, ordered by inclusion (Birkhoff's representation theorem).
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Thouless made many important contributions to the theory of many-body problems. For atomic nuclei, he cleared up the concept of 'rearrangement energy' and derived an expression for the moment of inertia of deformed nuclei. In statistical mechanics, he contributed many ideas to the understanding of ordering, including the concept of 'topological ordering'. Other important results relate to localised electron states in disordered lattices.
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MIT is experimenting with trees that grow quickly and develop an interwoven root structure that's soft enough to "train" over the scaffold, but then hardens into a more durable structure. The inside walls would be conventional clay and plaster. An old methodology new to buildings is introduced in this design - pleaching. Pleaching is a method of weaving together tree branches to form living archways, lattices, or screens.
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Theory Ser. A 26 (1979), no. 2, 210—214 The results in these dimensions stem from the existence of highly symmetrical lattices: the E8 lattice and the Leech lattice. If arrangements are restricted to lattice arrangements, in which the centres of the spheres all lie on points in a lattice, then this restricted kissing number is known for n = 1 to 9 and n = 24 dimensions.
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The strongest form of completeness is the existence of all suprema and all infima. The posets with this property are the complete lattices. However, using the given order, one can restrict to further classes of (possibly infinite) subsets, that do not yield this strong completeness at once. If all directed subsets of a poset have a supremum, then the order is a directed-complete partial order (dcpo).
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In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. It is a part of universal algebra, in the sense that it relates to all types of algebraic structure (with finitary operations). It also has a formulation in terms of category theory, although this is in yet more abstract terms. Examples include free groups, tensor algebras, or free lattices.
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The Dedekind–MacNeille completion of a partially ordered set is the smallest complete lattice with embedded in it, in the sense that, if is any lattice completion of , then the Dedekind–MacNeille completion is a partially ordered subset of .; , Theorem 5.3.8, p. 121. When is finite, its completion is also finite, and has the smallest number of elements among all finite complete lattices containing .
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The serpentine group of minerals are polymorphous, meaning that they have the same chemical formulae, but the atoms are arranged into different structures, or crystal lattices. Chrysotile, which has a fiberous habit, is one polymorph of serpentine and is one of the more important asbestos minerals. Other polymorphs in the serpentine group may have a platy habit. Antigorite and lizardite are the polymorphs with platy habit.
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That is, all totally ordered sets are directed sets (contrast partially ordered sets, which need not be directed). Join semilattices (which are partially ordered sets) are directed sets as well, but not conversely. Likewise, lattices are directed sets both upward and downward. In topology, directed sets are used to define nets, which generalize sequences and unite the various notions of limit used in analysis.
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The buildings at UNBC are designed to represent the northern landscape. The Canfor Winter Garden area has a flowing blue staircase below a ceiling of wooden lattices, representing the west coast rain forests. The cafeteria has a lighthouse design that represents the rugged coastline of northern British Columbia. Another structural feature, a pair of triangular glass peaks, represents mountains and functions as skylights above the UNBC Bookstore.
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A conditionally complete lattice is a lattice in which every nonempty subset that has an upper bound has a join (i.e., a least upper bound). Such lattices provide the most direct generalization of the completeness axiom of the real numbers. A conditionally complete lattice is either a complete lattice, or a complete lattice without its maximum element 1, its minimum element 0, or both.
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Butler's 1986 dissertation was Combinatorial Properties of Partially Ordered Sets associated with Partitions and Finite Abelian Groups. She subsequently published some of this research as "A unimodality result in the enumeration of subgroups of a finite abelian group" (Proc. AMS 1987), which concerned applications of algebraic combinatorics in group theory. Her work in this line of research also included her book Subgroup Lattices and Symmetric Functions (Mem.
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Quartz – Japan twin Pyrite iron cross twin Simple twinned crystals may be contact twins or penetration twins. Contact twins share a single composition surface often appearing as mirror images across the boundary. Plagioclase, quartz, gypsum, and spinel often exhibit contact twinning. Merohedral twinning occurs when the lattices of the contact twins superimpose in three dimensions, such as by relative rotation of one twin from the other.
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The MWC model proved very popular in enzymology, and pharmacology, although it has been shown inappropriate in a certain number of cases. The best example of a successful application of the model is the regulation of hemoglobin function. Extension of the model have been proposed for lattices of proteins, for instance by Changeux, Thiery, Tung and Kittel, by Wyman or by Duke, Le Novere and Bray.
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In mathematics, composition operators commonly occur in the study of shift operators, for example, in the Beurling–Lax theorem and the Wold decomposition. Shift operators can be studied as one-dimensional spin lattices. Composition operators appear in the theory of Aleksandrov–Clark measures. The eigenvalue equation of the composition operator is Schröder's equation, and the principal eigenfunction f(x) is often called Schröder's function or Koenigs function.
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For complex lattices such as diamond, however, the rule has to be modified to allow for internal degrees of freedom between the sublattices. The approximation can then be used to obtain bulk properties of crystalline materials such as stress-strain relationship. For crystalline bodies of finite size, the effect of surface stress is also significant. However, the standard Cauchy–Born rule cannot deduce the surface properties.
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The proof of the negative solution for CLP shows that the problem of representing distributive semilattices by compact congruences of lattices already appears for congruence lattices of semilattices. The question whether the structure of partially ordered set would cause similar problems is answered by the following result. Theorem (Wehrung 2008). For any distributive (∨,0)-semilattice S, there are a (∧,0)-semilattice P and a map μ : P × P → S such that the following conditions hold: (1) x ≤ y implies that μ(x,y)=0, for all x, y in P. (2) μ(x,z) ≤ μ(x,y) ∨ μ(y,z), for all x, y, z in P. (3) For all x ≥ y in P and all α, β in S such that μ(x,y) ≤ α ∨ β, there are a positive integer n and elements x=z0 ≥ z1 ≥ ... ≥ z2n=y such that μ(zi,zi+1) ≤ α (resp.
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In 2015 a practical way to create and access magnetic skyrmions under ambient room-temperature conditions was announced. The device used arrays of magnetized cobalt disks as artificial Bloch skyrmion lattices atop a thin film of cobalt and palladium. Asymmetric magnetic nanodots were patterned with controlled circularity on an underlayer with perpendicular magnetic anisotropy (PMA). Polarity is controlled by a tailored magnetic field sequence and demonstrated in magnetometry measurements.
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Dedekind made other contributions to algebra. For instance, around 1900, he wrote the first papers on modular lattices. In 1872, while on holiday in Interlaken, Dedekind met Georg Cantor. Thus began an enduring relationship of mutual respect, and Dedekind became one of the first mathematicians to admire Cantor's work concerning infinite sets, proving a valued ally in Cantor's disputes with Leopold Kronecker, who was philosophically opposed to Cantor's transfinite numbers..
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Maciej Lewenstein, Anna Sanpera, and Veronica Ahufunger, "Ultracold atoms in optical lattices: Simulating quantum many-body systems", Oxford University Press(2012), . In 2004 he became a Fellow of the American Physical Society and in 2007 he has obtained the Alexander von Humboldt Research Prize. In 2008 he became a recipient of the ERC Advanced Grant QUGATUA. In 2010 he obtained the Prize of the Joachim Herz Foundation of University of Hamburg.
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A lattice is called modular if for all elements a, b and c the implication ::if a ≤ c, then a ∨ (b ∧ c) = (a ∨ b) ∧ c holds. This is weaker than distributivity; e.g. the above-shown lattice M3 is modular, but not distributive. A natural further weakening of this condition for orthocomplemented lattices, necessary for applications in quantum logic, is to require it only in the special case b = a⊥.
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The phyllotactic notation describes such structures by a triplet of positive integers (l = m+n, m, n) with l \geq m \geq n. Each number l, m, and n describes a family of spirals in the 3-dimensional packing. They count the number of spirals in each direction until the spiral repeats. This notation, however, only applies to triangular lattices and is therefore restricted to the ordered structures without internal spheres.
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This would require more than one neutron to be emitted per fission on average in order to keep the chain reaction going. By March 1939, they established that about two were being emitted per fission on average. The delay between an atom absorbing a neutron and fission occurring would be the key to controlling a chain reaction. At this point Zinn began working for Fermi, constructing experimental uranium lattices.
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Most alkali metal halides crystallize with the face- centered cubic lattices. In this structure both the metals and halides feature octahedral coordination geometry, in which each ion has a coordination number of six. Caesium chloride, bromide, and iodide crystallize in a body-centered cubic lattice that accommodates coordination number of eight for the larger metal cation (and the anion also).Wells, A.F. (1984) Structural Inorganic Chemistry, Oxford: Clarendon Press. .
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Spin waves are propagating disturbances in the ordering of magnetic materials. These low-lying collective excitations occur in magnetic lattices with continuous symmetry. From the equivalent quasiparticle point of view, spin waves are known as magnons, which are bosonic modes of the spin lattice that correspond roughly to the phonon excitations of the nuclear lattice. As temperature is increased, the thermal excitation of spin waves reduces a ferromagnet's spontaneous magnetization.
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In crystallography, the orthorhombic crystal system is one of the 7 crystal systems. Orthorhombic lattices result from stretching a cubic lattice along two of its orthogonal pairs by two different factors, resulting in a rectangular prism with a rectangular base (a by b) and height (c), such that a, b, and c are distinct. All three bases intersect at 90° angles, so the three lattice vectors remain mutually orthogonal.
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Two-dimensional (thin film) semi- ordered lattices have been studied using an optical microscope, as well as those collected at electrode surfaces. Digital video microscopy has revealed the existence of an equilibrium hexatic phase as well as a strongly first- order liquid-to-hexatic and hexatic-to-solid phase transition. These observations are in agreement with the explanation that melting might proceed via the unbinding of pairs of lattice dislocations.
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Over time, the CPM has evolved from a specific model of cell sorting to a general framework with many extensions, some of which are partially or entirely off-lattice. Various cell behaviours, such as chemotaxis, elongation and haptotaxis can be incorporated by extending either the Hamiltonian, H, or the change in energy \Delta H . Auxiliary sub-lattices may be used to include additional spatial information, such as the concentrations of chemicals.
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A classical example of membrane bending by rigid protein scaffold is clathrin. Clathrin is involved in cellular endocytosis and is sequestrated by specific signaling molecules. Clathrin can attach to adaptor protein complexes on the cellular membrane, and it polymerizes into lattices to drive greater curvature, resulting in endocytosis of a vesicular unit. Coat protein complex I (COP1) and coat protein complex II (COPII) follow similar mechanism in driving membrane curvature.
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This occurs when the lattice spacing of the two crystals do not match, resulting in a misfit of the lattices at the interface. The stress caused by the lattice misfit is released by forming regularly spaced misfit dislocations. Misfit dislocations are edge dislocations with the dislocation line in the interface plane and the Burgers vector in the direction of the interface normal. Interfaces with misfit dislocations may form e.g.
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Mary Wynne Warner (22 June 1932 – 1 April 1998) was a Welsh mathematician, specializing in fuzzy mathematics.M. W. Warner, "Fuzzy topology with respect to continuous lattices," Fuzzy Sets and Systems 35(1)(1990): 85–91. doi:10.1016/0165-0114(90)90020-7M. W. Warner, "Towards a Mathematical Theory of Fuzzy Topology" in R. Lowen and M. R. Roubens, eds., Fuzzy Logic: State of the Art (Springer 1993): 83–94.
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Lattice Boltzmann models can be operated on a number of different lattices, both cubic and triangular, and with or without rest particles in the discrete distribution function. A popular way of classifying the different methods by lattice is the DnQm scheme. Here "Dn" stands for "n dimensions", while "Qm" stands for "m speeds". For example, D3Q15 is a 3-dimensional lattice Boltzmann model on a cubic grid, with rest particles present.
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The solid is diamagnetic. In terms of their coordination spheres, copper centres are 2-coordinated and the oxides are tetrahedral. The structure thus resembles in some sense the main polymorphs of SiO2, and both structures feature interpenetrated lattices. Copper(I) oxide dissolves in concentrated ammonia solution to form the colourless complex [Cu(NH3)2]+, which is easily oxidized in air to the blue [Cu(NH3)4(H2O)2]2+.
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Two polytopes are called combinatorially isomorphic if their face lattices are isomorphic. The polytope graph (polytopal graph, graph of the polytope, 1-skeleton) is the set of vertices and edges of the polytope only, ignoring higher-dimensional faces. For instance, a polyhedral graph is the polytope graph of a three-dimensional polytope. By a result of Whitney the face lattice of a three-dimensional polytope is determined by its graph.
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It is represented by the vertices and edges of the quarter cubic honeycomb, filling space by regular tetrahedra and truncated tetrahedra. It contains four sets of parallel planes of points and lines, each plane being a two dimensional kagome lattice. A second expression in three dimensions has parallel layers of two dimensional lattices and is called an orthorhombic- kagome lattice. The trihexagonal prismatic honeycomb represents its edges and vertices.
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Charger with Charles II in the Boscobel Oak, English, c. 1685. The plate's diameter is 43 cm; such large plates, for display rather than use, take slip-trailing to an extreme, building up lattices of thick trails of slip. Chinese porcelain sugar bowl with combed, slip-marbled decoration, c. 1795 Slipware is pottery decorated by slip placed onto a wet or leather-hard clay body surface by dipping, painting or splashing.
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Not every locally compact group contains a lattice, and there is no general group-theoretical sufficient condition for this. On the other hand, there are plenty of more specific settings where such criteria exist. For example, the existence or non-existence of lattices in Lie groups is a well-understood topic. As we mentioned, a necessary condition for a group to contain a lattice is that the group must be unimodular.
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This family of processes causes a discontinuous or first order phase transition. This has been observed for random networks as well as lattices. Furthermore, for embedded interdependent networks the transition is particularly precipitous without even a critical exponent for p>p_c. Surprisingly, it has been shown that—contrary to the results for single networks—interdependent random networks with broader degree distributions are more vulnerable than those with narrow degree distributions.
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Much of mathematics is grounded in the study of equivalences, and order relations. Lattice theory captures the mathematical structure of order relations. Even though equivalence relations are as ubiquitous in mathematics as order relations, the algebraic structure of equivalences is not as well known as that of orders. The former structure draws primarily on group theory and, to a lesser extent, on the theory of lattices, categories, and groupoids.
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If the lattices are not fine enough to satisfy the Petersen- Middleton condition, then the field cannot be reconstructed exactly from the samples in general. In this case we say that the samples may be aliased. Again, consider the example in which \Omega is a circular disc. If the Petersen-Middleton conditions do not hold, the support of the sampled spectrum will be as shown in Figure 4.
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Any homomorphism of lattices is necessarily monotone with respect to the associated ordering relation; see Limit preserving function. The converse is not true: monotonicity by no means implies the required preservation of meets and joins (see Pic. 9), although an order-preserving bijection is a homomorphism if its inverse is also order-preserving. Given the standard definition of isomorphisms as invertible morphisms, a lattice isomorphism is just a bijective lattice homomorphism.
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Switching sources while still in the growth phase can create compound nanowires with super-lattices of alternating materials. A single-step vapour phase reaction at elevated temperature synthesises inorganic nanowires such as Mo6S9−xIx. From another point of view, such nanowires are cluster polymers. VSS Growth Similar to VLS synthesis, VSS (vapor-solid-solid) synthesis of nanowires (NWs) proceeds through thermolytic decomposition of a silicon precursor (typically phenylsilane).
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High concentrations of etioplasts will cause leaves to appear yellow rather than green. These plant organelles contain prolamellar bodies, which are membrane aggregations of semi-crystalline lattices of branched tubules that carry the precursor pigment for chlorophyll. The prolamellar bodies are often (and presumed always) arranged in geometric patterns. They are converted to chloroplasts via the stimulation of chlorophyll synthesis by the plant hormone cytokinin soon after exposure to light.
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Failure of one of the two distributive laws brings about near-rings and near-fields instead of rings and division rings respectively. The operations are usually configured to have the near-ring or near-field distributive on the right but not on the left. Rings and distributive lattices are both special kinds of rigs, which are generalizations of rings that have the distributive property. For example, natural numbers form a rig.
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Bogart was originally from Cincinnati, and was a 1965 graduate of Marietta College. He earned his Ph.D. in 1968 at the California Institute of Technology. His dissertation, Structure Theorems for Local Noether Lattices, was supervised by Robert P. Dilworth. He joined the faculty of the Dartmouth College mathematics department in 1968, was promoted to full professor in 1980, and was chair of the department from 1989 to 1995.
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Unlike conventional flow fields, the 3D micro-lattices in the complex field, which act as baffles and induce frequent micro-scale interfacial flux between the GDL and flow-fields[53]. Due to this repeating micro-scale convective flow, oxygen transport to catalyst layer (CL) and liquid water removal from GDL is significantly enhanced. The generated water is quickly drawn out through the flow field, preventing accumulation within the pores.
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The Wigner–Seitz cell, named after Eugene Wigner and Frederick Seitz, is a primitive cell which has been constructed by applying Voronoi decomposition to a crystal lattice. It is used in the study of crystalline materials in solid- state physics. Wigner–Seitz primitive cell for different angle parallelogram lattices. The unique property of a crystal is that its atoms are arranged in a regular three-dimensional array called a lattice.
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This necessary distributivity of • over ∨ does not in general entail distributivity of ∧ over ∨, that is, a residuated lattice need not be a distributive lattice. However distributivity of ∧ over ∨ is entailed when • and ∧ are the same operation, a special case of residuated lattices called a Heyting algebra. Alternative notations for x•y include x◦y, x;y (relation algebra), and x⊗y (linear logic). Alternatives for I include e and 1'.
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The permutations of Rubik's Cube form a group, a fundamental concept within abstract algebra. In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras. The term abstract algebra was coined in the early 20th century to distinguish this area of study from the other parts of algebra.
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Early designs were for a two-span bridge supported by girder lattices, that would join with several land approach arches. It was redesigned when abandoned coal workings were discovered at both ends. The revised design was a lattice girder bridge carrying four tracks with four spans over the river. The spans were supported by five sandstone piers decorated with pairs of cutaway arches, three of which are in the river.
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Her dissertation, Lattices of Functions, Measures, and Open Sets, was supervised by Frank Smithies. After completing her doctorate, she took a faculty position at the University of London. Following her husband's dream of living on a farm in Vermont, she moved to Dartmouth College in 1966. By 1972, she was working at the University of Hull and circa 2008 she became a professor at the University of Leeds.
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Upon leaving University he became a teacher who could not forget the many interesting problems he had learned about. Upon reading a book by Max Born, on crystal lattices, he started calculations on double refraction of light. Publications of this work provided him with a Fellowship from the International Education Board which he used to go and work with Max Born at the University of Göttingen in 1926–28.
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Formally system networks correspond to type lattices in formal lattice theory, although they are occasionally erroneously mistaken for flowcharts or directed decision trees. Such directionality is always only a property of particular implementations of the general notion and may be made for performance reasons in, for example, computational modelling. System networks commonly employ multiple inheritance and "simultaneous" systems, or choices, which therefore combine to generate very large descriptive spaces.
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The upper part of the mausoleum was used for accomplishment of cultic rites, but there was a burial vault. There are three small windows with stone shebeke-lattices in the southern, eastern and western facets of the mausoleum. The burial vault also has a window with shebeke- lattice. The whole mausoleum was revetted with narrow and wide rows of tightly urged to each other and finely nigged stones.
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The River Road Covered Bridge is a historic covered bridge, carrying Veilleux Road across the Missisquoi River in Troy, Vermont. Built in 1910, the Town lattice truss is the only surviving covered bridge in Troy from the historic period of covered bridge construction. It also exhibits some distinctive variations in construction from more typical Town lattices. It was listed on the National Register of Historic Places in 1974.
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The Covered Bridge (originally called Red Bridge) in Cedarburg, Ozaukee County, Wisconsin, United States, is one of the last remaining covered bridges in that state, which once had about 40 covered bridges. Built in 1876 to cross Cedar Creek, the bridge is long and is made of pine with oak lattices. It was listed on the National Register of Historic Places in 1973 and is now used only for pedestrian traffic.
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Weil's original proof is by relating deformations of a subgroup \Gamma in G to the first cohomology group of \Gamma with coefficients in the Lie algebra of G, and then showing that this cohomology vanishes for cocompact lattices when G has no simple factor of absolute type A1. A more geometric proof which also work in the non-compact cases uses Charles Ehresmann (and William Thurston's) theory of (G, X) structures.
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Prasad's early work was on discrete subgroups of real and p-adic semi-simple groups. He proved the "strong rigidity" of lattices in real semi-simple groups of rank 1 and also of lattices in p-adic groups, see [1] and [2]. He then tackled group-theoretic and arithmetic questions on semi-simple algebraic groups. He proved the "strong approximation" property for simply connected semi-simple groups over global function fields [3]. In collaboration with M. S. Raghunathan, Prasad determined the topological central extensions of these groups, and computed the "metaplectic kernel" for isotropic groups, see [11], [12] and [10]. Later, together with Andrei Rapinchuk, Prasad gave a precise computation of the metaplectic kernel for all simply connected semi-simple groups, see [14]. Prasad and Raghunathan have also obtained results on the Kneser-Tits problem, [13]. In 1987, Prasad found a formula for the volume of S-arithmetic quotients of semi-simple groups, [4].
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To overcome this issue, a numerical value is assigned to each attribute along a scale, and the results are placed in a table which links each assigned object-value within the given range to a numerical value (a score) denoting a given degree of applicability. This is the basic idea of a "fuzzy concept lattice", which can also be graphed; different fuzzy concept lattices can be connected to each other as well (for example, in "fuzzy conceptual clustering" techniques used to group data, originally invented by Enrique H. Ruspini). Fuzzy concept lattices are a useful programming tool for the exploratory analysis of big data, for example in cases where sets of linked behavioural responses are broadly similar, but can nevertheless vary in important ways, within certain limits. It can help to find out what the structure and dimensions are, of a behaviour that occurs with an important but limited amount of variation in a large population.
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The vortex structure is imprinted into the underlayer's interfacial region via suppressing the PMA by a critical ion- irradiation step. The lattices are identified with polarized neutron reflectometry and have been confirmed by magnetoresistance measurements. A recent paper (2019) demonstrated a way to move skyrmions, purely using electric field (in the absence of electric current). The authors used Co/Ni multilayers with a thickness slope and Dzyaloshinskii-Moriya interaction and demonstrated skyrmions.
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Additive manufacturing file format (AMF) is an open standard for describing objects for additive manufacturing processes such as 3D printing. The official ISO/ASTM 52915:2016 standard is an XML-based format designed to allow any computer-aided design software to describe the shape and composition of any 3D object to be fabricated on any 3D printer. Unlike its predecessor STL format, AMF has native support for color, materials, lattices, and constellations.
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An acoustic metamaterial, sonic crystal, or phononic crystal, is a material designed to control, direct, and manipulate sound waves or phonons in gases, liquids, and solids (crystal lattices). Sound wave control is accomplished through manipulating parameters such as the bulk modulus β, density ρ, and chirality. They can be engineered to either transmit, or trap and amplify sound waves at certain frequencies. In the latter case, the material is an acoustic resonator.
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The upper façades of buildings were decorated with precut stones mosaic-fashion, erected as facing over the core, forming elaborate compositions of long-nosed deities such as the rain god Chaac and the Principal Bird Deity. The motifs also included geometric patterns, lattices and spools, possibly influenced by styles from highland Oaxaca, outside the Maya area. In contrast, the lower façades were left undecorated. Roof combs were relatively uncommon at Puuc sites.
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The Goldreich–Goldwasser–Halevi (GGH) lattice-based cryptosystem is an asymmetric cryptosystem based on lattices. There is also a GGH signature scheme. The Goldreich–Goldwasser–Halevi (GGH) cryptosystem makes use of the fact that the closest vector problem can be a hard problem. This system was published in 1997 by Oded Goldreich, Shafi Goldwasser, and Shai Halevi, and uses a trapdoor one-way function that is relying on the difficulty of lattice reduction.
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Nanoarchitecture: and subsequently by many other groups. In 2006, Rothemund first demonstrated the DNA origami method for easily and robustly forming folded DNA structures of arbitrary shape. Rothemund had conceived of this method as being conceptually intermediate between Seeman's DX lattices, which used many short strands, and William Shih's DNA octahedron, which consisted mostly of one very long strand. Rothemund's DNA origami contains a long strand which folding is assisted by several short strands.
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In principle, if the thickness of the sample is within few micrometers, the primary beam will be completely attenuated by scattering with other electrons or lattices. In fact, the primary beam interaction process could be elastic or inelastic. For the first case, no loss of energy happens, this is known as a backscattered electron. On the other hand, in case of inelastic interaction process, the emitted electron from the sample from eV to 30 KeV.
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Superlattices were discovered early in 1925 by Johansson and Linde after the studies on gold-copper and palladium-copper systems through their special X-ray diffraction patterns. Further experimental observations and theoretical modifications on the field were done by Bradley and Jay, Gorsky, Borelius, Dehlinger and Graf, Bragg and Williams and Bethe. Theories were based on the transition of arrangement of atoms in crystal lattices from disordered state to an ordered state.
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First published in 2004 according to the CRC Press web page for this book. According to the copyright page of this book, accessible via Google Books, it had gone into its tenth printing by sometime in 2005. In addition, some artificial dielectrics may consist of irregular lattices, random mixtures, or a non-uniform concentration of particles. Artificial dielectrics came into use with the radar microwave technologies developed between the 1940s and 1970s.
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The first Bose–Einstein condensate observed in a gas of ultracold rubidium atoms. The blue and white areas represent higher density. Ultracold atom trapping in optical lattices is an experimental tool commonly used in condensed matter physics, and in atomic, molecular, and optical physics. The method involves using optical lasers to form an interference pattern, which acts as a lattice, in which ions or atoms can be placed at very low temperatures.
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Immanuel Bloch (born 16 November 1972, Fulda) is a German experimental physicist. His research is focused on the investigation of quantum many-body systems using ultracold atomic and molecular quantum gases. Bloch is known for his work on ultracold atoms in artificial crystals of light, so called optical lattices and especially the first realization of a quantum phase transition from a weakly interacting superfluid to a strongly interacting Mott insulating state of matter.
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Twin boom designs can trace their history back to the lattices of booms used on many early boxkite aircraft. With the recognition of the tremendous drag these imposed, more compact structures covered in fabric were developed during the World War One. Prime examples include the Caproni series of trimotor bombers. Around the same time, the first wooden monocoque fuselages appeared, and it wasn't long before this technique was applied to provide twin booms.
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The lattices of squares or 'cubes' of paint employed throughout his Divisionist period have not been entirely abandoned, but pushed to another extreme. Where before brushstrokes had become increasingly larger and organized into groups of color, now they were larger still and elongated, seemingly blended together directly on the canvas (rather than on a palette). The treatment of color and composition is globally free, loose, expressive, and thus dynamic. There are no inert tones.
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Hilary Ann Priestley is a British mathematician. She is a professor at the University of Oxford and a Fellow of St Anne's College, Oxford, where she has been Tutor in Mathematics since 1972. Hilary Priestley introduced ordered separable topological spaces; such topological spaces are now usually called Priestley spaces in her honour. The term "Priestley duality" is also used for her application of these spaces in the representation theory of distributive lattices.
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Columnar liquid crystals are grouped by their structural order and the ways of packing of the columns. Nematic columnar liquid crystals have no long-range order and are less organized than other columnar liquid crystals. Other columnar phases with long-range order are classified by their two-dimensional lattices: hexagonal, tetragonal, rectangular, and oblique phases. The discotic nematic phase includes nematic liquid crystals composed of flat-shaped discotic molecules without long-range order.
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This early use of electrospun fibrous lattices for cell culture and tissue engineering showed that various cell types including Human Foreskin Fibroblasts (HFF), transformed Human Carcinoma (HEp-2), and Mink Lung Epithelium (MLE) would adhere to and proliferate upon polycarbonate fibers. It was noted that, as opposed to the flattened morphology typically seen in 2D culture, cells grown on the electrospun fibers exhibited a more histotypic rounded 3-dimensional morphology generally observed in vivo.
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Some minerals, namely jarosites and herbertsmithite, contain two- dimensional layers or three-dimensional kagome lattice arrangement of atoms in their crystal structure. These minerals display novel physical properties connected with geometrically frustrated magnetism. For instance, the spin arrangement of the magnetic ions in Co3V2O8 rests in a kagome lattice which exhibits fascinating magnetic behavior at low temperatures. Quantum magnets realized on Kagome lattices have been discovered to exhibit many unexpected electronic and magnetic phenomena.
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They introduced asymmetric polystyrene/polymethylmethacrylate lattices from seeded emulsion polymerization. One year later, Casagrande and Veyssie reported the synthesis of glass beads that were made hydrophobic on only one hemisphere using octadecyl trichlorosilane, while the other hemisphere was protected with a cellulose varnish. The glass beads were studied for their potential to stabilize emulsification processes. Then several years later, Binks and Fletcher investigated the wettability of Janus beads at the interface between oil and water.
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It is not well understood why the improved thermoelectric properties appear only in certain materials with specific fabrication processes. SrTe nanocrystals can be embedded in a bulk PbTe matrix so that rocksalt lattices of both materials are completely aligned (endotaxy) with optimal molar concentration for SrTe only 2%. This can cause strong phonon scattering but would not affect charge transport. In such case, ZT~1.7 can be achieved at 815 K for p-type material.
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In statistical mechanics, the ice-type models or six-vertex models are a family of vertex models for crystal lattices with hydrogen bonds. The first such model was introduced by Linus Pauling in 1935 to account for the residual entropy of water ice. Variants have been proposed as models of certain ferroelectric and antiferroelectric crystals. In 1967, Elliott H. Lieb found the exact solution to a two-dimensional ice model known as "square ice".
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ScanIP is capable of generating robust STL files for 3D printing. Files created using ScanIP feature guaranteed watertight triangulations and correct norms, as well as options for volume and topology preserving smoothing. STL files are generated with conforming interfaces, enabling multi-material printing. Internal structures, otherwise known as lattices, can also be added to 3D models of parts in order to reduce weight prior to additive manufacturing.Young, P., Raymont, D., Hao, L, Cotton, R., 2010.
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Experimental phase diagram of MnSi1-xGex alloys, revealing magnetic skyrmion, tetrahedral and cubic hedgehog spin arrangements at different compositions x. At low temperatures, MnGe and its relative MnSi exhibit unusual spatial arrangements of electron spin, which were named magnetic skyrmion, tetrahedral and cubic hedgehog lattices. Their structure can be controlled not only by the Si/Ge ratio, but also by temperature and magnetic field. This property has potential application in ultrahigh-density magnetic storage devices.
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An example of an infinite diophantine equation is: :, which can be expressed as "How many ways can a given integer be written as the sum of a square plus twice a square plus thrice a square and so on?" The number of ways this can be done for each forms an integer sequence. Infinite Diophantine equations are related to theta functions and infinite dimensional lattices. This equation always has a solution for any positive .
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Boolean algebra emerged in the 1860s, in papers written by William Jevons and Charles Sanders Peirce. The first systematic presentation of Boolean algebra and distributive lattices is owed to the 1890 Vorlesungen of Ernst Schröder. The first extensive treatment of Boolean algebra in English is A. N. Whitehead's 1898 Universal Algebra. Boolean algebra as an axiomatic algebraic structure in the modern axiomatic sense begins with a 1904 paper by Edward V. Huntington.
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A zincblende unit cell The space group of the Zincblende structure is called F3m (in Hermann–Mauguin notation), or 216.Birkbeck College, University of London The Strukturbericht designation is "B3".The Zincblende (B3) Structure The Zincblende structure (also written "zinc blende") is named after the mineral zincblende (sphalerite), one form of zinc sulfide (β-ZnS). As in the rock-salt structure, the two atom types form two interpenetrating face-centered cubic lattices.
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As already mentioned, the methods and formalisms of universal algebra are an important tool for many order theoretic considerations. Beside formalizing orders in terms of algebraic structures that satisfy certain identities, one can also establish other connections to algebra. An example is given by the correspondence between Boolean algebras and Boolean rings. Other issues are concerned with the existence of free constructions, such as free lattices based on a given set of generators.
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Another approach, advocated by Bub and Pitowsky, argues that quantum states are information about propositions within event spaces that form non-Boolean lattices. On occasion, the proposals of Bub and Pitowsky are also called "quantum Bayesianism". Zeilinger and Brukner have also proposed an interpretation of quantum mechanics in which "information" is a fundamental concept, and in which quantum states are epistemic quantities. Unlike QBism, the Brukner-Zeilinger interpretation treats some probabilities as objectively fixed.
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An alternative to the unit cell, for every Bravais lattice there is another kind of primitive cell called the Wigner–Seitz cell. In the Wigner–Seitz cell, the lattice point is at the center of the cell, and for most Bravais lattices, the shape is not a parallelogram or parallelepiped. This is a type of Voronoi cell. The Wigner–Seitz cell of the reciprocal lattice in momentum space is called the Brillouin zone.
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Technologies such as quantum well devices, super-lattices, and lasers are possible with MBE. Epitaxial films are useful due to their ability to be produced with electrical properties different from those of the substrate, either higher purity, or fewer defects or with a different concentration of electrically active impurities as desired. Varying the composition of the material alters the band gap due to bonding of different atoms with differing energy level gaps.
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Another important property of Galois connections is that lower adjoints preserve all suprema that exist within their domain. Dually, upper adjoints preserve all existing infima. From these properties, one can also conclude monotonicity of the adjoints immediately. The adjoint functor theorem for order theory states that the converse implication is also valid in certain cases: especially, any mapping between complete lattices that preserves all suprema is the lower adjoint of a Galois connection.
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In this situation, an important feature of Galois connections is that one adjoint uniquely determines the other. Hence one can strengthen the above statement to guarantee that any supremum-preserving map between complete lattices is the lower adjoint of a unique Galois connection. The main property to derive this uniqueness is the following: For every in , is the least element of such that . Dually, for every in , is the greatest in such that .
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Dynamical Theory of Crystal Lattices is a book in solid state physics, authored collaboratively by Max Born and Kun Huang. The book was originally started by Born in c. 1940, and was finished in the 1950s by Huang in consultation with Born. The text is considered a classical treatise on the subject of lattice dynamics, phonon theory, and elasticity in crystalline solids, but excluding metals and other complex solids with order/disorder phenomena.
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Gradient noise is a type of noise commonly used as a procedural texture primitive in computer graphics. It is conceptually different, and often confused with value noise. This method consists of a creation of a lattice of random (or typically pseudorandom) gradients, dot products of which are then interpolated to obtain values in between the lattices. An artifact of some implementations of this noise is that the returned value at the lattice points is 0.
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In two dimensions, holes may be drilled in a substrate that is transparent to the wavelength of radiation that the bandgap is designed to block. Triangular and square lattices of holes have been successfully employed. The Holey fiber or photonic crystal fiber can be made by taking cylindrical rods of glass in hexagonal lattice, and then heating and stretching them, the triangle-like airgaps between the glass rods become the holes that confine the modes.
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Lattices are structures formed of arrays of uniformly sized cells. Ceramic lattice nanostructures have been formed using hollow tubes of titanium nitride (TiN). Using vertex-connected, tessellated octahedra with 7-nm hollow struts with elliptical cross-sections and wall thickness of 75-nm produced approximately cubic cells 100-nm on a side at a scale of up to 1 cubic millimeter. The material's relative density was of the order of 0.013 (similar to aerogels).
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Free abelian groups have properties which make them similar to vector spaces. They have applications in algebraic topology, where they are used to define chain groups, and in algebraic geometry, where they are used to define divisors. Integer lattices also form examples of free abelian groups, and lattice theory studies free abelian subgroups of real vector spaces. The elements of a free abelian group with basis B may be described in several equivalent ways.
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The purpose of the cavity is to select optical frequencies (e.g. to suppress the Stokes process) that resonantly enhance the light intensity and to enhance the sensitivity to the mechanical vibrations. The setup displays features of a true two-way interaction between light and mechanics, which is in contrast to optical tweezers, optical lattices, or vibrational spectroscopy, where the light field controls the mechanics (or vice versa) but the loop is not closed.
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Thus, the Dedekind numbers count the number of elements in free distributive lattices.; ; . The Dedekind numbers also count (one more than) the number of abstract simplicial complexes on n elements, families of sets with the property that any subset of a set in the family also belongs to the family. Any antichain determines a simplicial complex, the family of subsets of antichain members, and conversely the maximal simplices in a complex form an antichain..
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Formally, an elliptic function is a function meromorphic on for which there exist two non-zero complex numbers and with , such that and for all . Denoting the "lattice of periods" by , this can be rephrased as requiring that for all . In terms of complex geometry, an elliptic function consists of a genus-one Riemann surface and a holomorphic mapping . From this perspective, one is treating two lattices and as equivalent if there is a nonzero complex number with .
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Precipitation occurs with most antigens because the antigen is multivalent (i.e. has several antigenic determinants per molecule to which antibodies can bind). Antibodies have at least two antigen binding sites (and in the case of Immunoglobulin M there is a multimeric complex with up to 10 antigen binding sites), thus large aggregates or gel-like lattices of antigen and antibody are formed. Experimentally, an increasing amount of antigen is added to a constant amount of antibody in solution.
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The ceiling is composed of seven groin vaults, each of which has an ornamental bronze chandelier. The first two vaults, as viewed from leaving Grand Central, are painted with cumulus clouds, while the third contains a 1927 mural by Edward Trumbull depicting American transportation. The Graybar Passage, as well as the central portal of the building, connect to an elevator lobby used by tenants. Above the central portal is a flagpole with multicolored lattices at its base.
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Constantin Piron (1932, Paris - 9 May 2012, Lausanne) was a Belgian physicist who worked for most of his career in Switzerland. In 1963 Piron earned his doctor of science degree from the University of Lausanne under the direction of Ernst Stueckelberg and Josef-Maria Jauch with a thesis on quantum logic, "Axiomatique quantique". He developed Jauch's methods (called the Geneva approach) for the foundations of quantum mechanics. Piron's Theorem (1964) is a famous representation theorem for quantum lattices.
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Such complex network features include a heavy tail in the degree distribution, a high clustering coefficient, assortativity or disassortativity among vertices, community structure (see stochastic block model), and hierarchical structure. In the case of agency-directed networks these features also include reciprocity, triad significance profile (TSP, see network motif), and other features. In contrast, many of the mathematical models of networks that have been studied in the past, such as lattices and random graphs, do not show these features.
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In the mathematical fields of order and domain theory, a Scott domain is an algebraic, bounded-complete cpo. They are named in honour of Dana S. Scott, who was the first to study these structures at the advent of domain theory. Scott domains are very closely related to algebraic lattices, being different only in possibly lacking a greatest element. They are also closely related to Scott information systems, which constitute a "syntactic" representation of Scott domains.
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Bloch studied physics at the University of Bonn in 1995, followed by a one-year research visit to Stanford University. He obtained his PhD in 2000 working under Theodor W. Hänsch at the Ludwig- Maximilians University in Munich. As a junior group leader, he continued in Munich starting his work on ultracold quantum gases in optical lattices. In 2003 he moved to a full professor position in experimental physics at the University of Mainz, where he stayed until 2009.
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An extension of the notion of integrability is also applicable to discrete systems such as lattices. This definition can be adapted to describe evolution equations that either are systems of differential equations or finite difference equations. The distinction between integrable and nonintegrable dynamical systems thus has the qualitative implication of regular motion vs. chaotic motion and hence is an intrinsic property, not just a matter of whether a system can be explicitly integrated in exact form.
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The reciprocal lattices (dots) and corresponding first Brillouin zones of (a) square lattice and (b) hexagonal lattice. In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice is broken up into Brillouin zones. The boundaries of this cell are given by planes related to points on the reciprocal lattice.
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Other cases of interest relate to uniform integrability, and the concept of normal family in complex analysis. Mahler's compactness theorem in the geometry of numbers characterizes relatively compact subsets in certain non-compact homogeneous spaces (specifically spaces of lattices). The definition of an almost periodic function at a conceptual level has to do with the translates of being a relatively compact set. This needs to be made precise in terms of the topology used, in a particular theory.
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The Musamman Burj is made of delicate marble lattices with ornamental niches so that the ladies of the court could gaze out unseen. The decoration of the walls is pietra dura. The chamber has a marble dome on top and is surrounded by a verandah with a beautiful carved fountain in the center. The tower looks out over the River Yamuna and is traditionally considered to have one of the most poignant views of the Taj Mahal.
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Many ultracold atom experiments are examples of quantum simulators. These include experiments studying bosons or fermions in optical lattices, the unitary Fermi gas, Rydberg atom arrays in optical tweezers. A common thread for these experiments is the capability of realizing generic Hamiltonians, such as the Hubbard or transverse-field Ising Hamiltonian. Major aims of these experiments include identifying low-temperature phases or tracking out-of-equilibrium dynamics for various models, problems which are theoretically and numerically intractable.
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The ground-floor columns are of Doric style and the keystones of each of the arches bear medallions with images of apostles or Mercedarian friars. The columns of the upper floor, built later, have more decoration than the lower. These columns are decorated with lattices intertwined with foliage and fruit. The spaces between the pilasters are decorated in Baroque style, with sculpted images of Mercedarian friars in the triangles that extend from the arches to the ceiling.
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In music, 41 equal temperament, abbreviated 41-TET, 41-EDO, or 41-ET, is the tempered scale derived by dividing the octave into 41 equally sized steps (equal frequency ratios). Each step represents a frequency ratio of 21/41, or 29.27 cents (), an interval close in size to the septimal comma. 41-ET can be seen as a tuning of the schismatic,"Schismic Temperaments ", Intonation Information. magic and miracle"Lattices with Decimal Notation", Intonation Information. temperaments.
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Mario Romañach 1951. Among the buildings in Havana are: Peletería "California", built in 1951 and located on Calle Galiano between San José and Barcelona, with a modern interior and ample support spaces; the apartment of Josefina Odoardo from 1953, located in 7th. Between 62 and 66, Miramar, with use of ceramic lattices, outdoor staircases, and roofless terraces, the latter with rectangular wooden rails; the building of Oswaldo Pardo, of 1954, located in 98 between 5ta. Avenida, F and 7th.
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Let X be a topological space. Each of the following properties are equivalent to the property of X being spectral: #X is homeomorphic to a projective limit of finite T0-spaces. #X is homeomorphic to the spectrum of a bounded distributive lattice L. In this case, L is isomorphic (as a bounded lattice) to the lattice K\circ(X) (this is called Stone representation of distributive lattices). #X is homeomorphic to the spectrum of a commutative ring.
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Zimmer's conjecture is a statement in mathematics "which has to do with the circumstances under which geometric spaces exhibit certain kinds of symmetries." It was named after the mathematician Robert Zimmer. The conjecture states that there can exist symmetries (specifically higher-rank lattices) in a higher dimension that cannot exist in lower dimensions. In 2017, the conjecture was proven by Aaron Brown and Sebastián Hurtado-Salazar of the University of Chicago and David Fisher of Indiana University.
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DNA is thus used as a structural material rather than as a carrier of biological information. This has led to the creation of two-dimensional periodic lattices (both tile-based and using the DNA origami method) and three-dimensional structures in the shapes of polyhedra. Nanomechanical devices and algorithmic self-assembly have also been demonstrated, and these DNA structures have been used to template the arrangement of other molecules such as gold nanoparticles and streptavidin proteins.
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When simple systems like the lattices of elements or simple compounds are studied it is often not very difficult to calculate eigenstates in high-symmetry points analytically. So the tight-binding model can provide nice examples for those who want to learn more about group theory. The tight-binding model has a long history and has been applied in many ways and with many different purposes and different outcomes. The model doesn't stand on its own.
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In addition, they performed research on skyrmion lattices in the quantum Hall effect and collective modes in supersolid 4He. Below the results obtained from the study of degenerate Fermi gases are briefly summarized. Already in 1996 they predicted that an atomic gas of 6Li (a fermionic isotope of lithium) becomes a Bardeen-Cooper- Schrieffer (BCS) superfluid at experimentally obtainable temperatures. They have also performed a detailed study of the superfluid behaviour of this gas below the critical temperature.
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Consequently, this also shows that Poisson's ratio for wood is time- dependent during constant loading, meaning that the strain in the axial and transverse direction do not increase in the same rate. Media with engineered microstructure may exhibit negative Poisson's ratio. In a simple case auxeticity is obtained removing material and creating a periodic porous media. Lattices can reach lower values of Poisson's ratio, which can be indefinitely close to the limiting value −1 in the isotropic case.
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His introduction of the theory of prequantization has led to the theory of quantum Toda lattices. The Kostant partition function is named after him. With Gerhard Hochschild and Alex F. T. W. Rosenberg, he is one of the namesakes of the Hochschild–Kostant–Rosenberg theorem which describes the Hochschild homology of some algebras.. His students include James Harris Simons, James Lepowsky, Moss Sweedler, David Vogan, and Birgit Speh. At present he has more than 100 mathematical descendants.
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Already at his lycée, Riguet was impressed by the power of geometric reasoning. He studied Louis Couturat and Bourbaki, who made contributions to logic and set theory.Stephane Dugowson and others Hommage a Jacques Riguet at Google Sites Riguet studied higher mathematics with Albert Châtelet and was introduced to lattices. In 1948 he published "Relations binaires, fermetures, correspondances de Galois"Bulletin de la Société Mathématique de France 76: 114–55 which revived the calculus of binary relations.
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It was built around 1914 CE (1333 AH) and is built of adobe. This minaret is designed by an architect Awad Salman Afif al-Tirmi, who had already carried out many designs and constructions of clay lattices and domes, and the maintenance and supervision was conducted by Abu Bakr bin Shihab (d. 1345 AH). It is considered as one of the most important architectural sites and destinations for visitors and researchers of the city of Tarim.
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The binary Golay code, independently developed in 1949, is an application in coding theory. More specifically, it is an error-correcting code capable of correcting up to three errors in each 24-bit word, and detecting a fourth. It was used to communicate with the Voyager probes, as it is much more compact than the previously-used Hadamard code. Quantizers, or analog-to-digital converters, can use lattices to minimise the average root-mean-square error.
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The erosion of the dark-blue square by a disk, resulting in the light-blue square. Erosion (usually represented by ⊖) is one of two fundamental operations (the other being dilation) in morphological image processing from which all other morphological operations are based. It was originally defined for binary images, later being extended to grayscale images, and subsequently to complete lattices. The erosion operation usually uses a structuring element for probing and reducing the shapes contained in the input image.
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Space groups and crystals are divided into seven crystal systems according to their point groups, and into seven lattice systems according to their Bravais lattices. Five of the crystal systems are essentially the same as five of the lattice systems, but the hexagonal and trigonal crystal systems differ from the hexagonal and rhombohedral lattice systems. The six crystal families are formed by combining the hexagonal and trigonal crystal systems into one hexagonal family, in order to eliminate this confusion.
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A residuated semilattice is defined almost identically for residuated lattices, omitting just the meet operation ∧. Thus it is an algebraic structure L = (L, ∨, •, 1, /, \\) satisfying all the residuated lattice equations as specified above except those containing an occurrence of the symbol ∧. The option of defining x ≤ y as x∧y = x is then not available, leaving only the other option x∨y = y (or any equivalent thereof). Any residuated lattice can be made a residuated semilattice simply by omitting ∧.
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Main entrance and Shridharani Art Gallery facade on the left. Triveni was one of the first buildings by noted American architect, Joseph Allen Stein (1957-1977) in India, who also designed several important building in New Delhi, like India International Centre and India Habitat Centre, Lodhi Road. Designed in modern architecture style, the complex is noted for its " multiple spaces for multiple purposes" and use of jali work (stone lattices), which was to become Stein's hallmark.
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Among many other topics, he has made substantial contributions to the development of reflexive and reductive operator algebras and to the study of lattices of invariant subspaces, composition operators on the Hardy-Hilbert space and linear operator equations. His publications include many with his long-time collaborator Heydar Radjavi, including the book "Invariant subspaces" (Springer-Verlag, 1973; second edition 2003). Rosenthal has supervised the Ph.D. theses of fifteen students and the research work of a number of post- doctoral fellows.
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Gentry's Ph.D. thesis provides additional details. The Gentry-Halevi implementation of Gentry's original cryptosystem reported timing of about 30 minutes per basic bit operation. Extensive design and implementation work in subsequent years have improved upon these early implementations by many orders of magnitude runtime performance. In 2010, Marten van Dijk, Craig Gentry, Shai Halevi and Vinod Vaikuntanathan presented a second fully homomorphic encryption scheme, which uses many of the tools of Gentry's construction, but which does not require ideal lattices.
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Besides the previous representation results, there are some other statements that can be made about complete lattices, or that take a particularly simple form in this case. An example is the Knaster–Tarski theorem, which states that the set of fixed points of a monotone function on a complete lattice is again a complete lattice. This is easily seen to be a generalization of the above observation about the images of increasing and idempotent functions, since these are instances of the theorem.
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The Theatre Royal consisted of a classical proscenium stage, pit, boxes, a middle and upper gallery, lattices (which were a type of box peculiar to Dublin) and a music/orchestra loft above the stage, also the acoustics were said to be excellent. The pit had backless benches and a raked floor that rose toward the back of the audience to help sightlines. Mostly single men sat here, and it was the noisiest, rowdiest area in the theatre. Boxes sat upper-class aristocrats.
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An interesting situation occurs if a function preserves all suprema (or infima). More accurately, this is expressed by saying that a function preserves all existing suprema (or infima), and it may well be that the posets under consideration are not complete lattices. For example, (monotone) Galois connections have this property. Conversely, by the order theoretical Adjoint Functor Theorem, mappings that preserve all suprema/infima can be guaranteed to be part of a unique Galois connection as long as some additional requirements are met.
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Chromium hydrides are compounds of chromium and hydrogen, and possibly other elements. Intermetallic compounds with not-quite-stoichometric quantities of hydrogen exist, as well as highly reactive molecules. When present at low concentrations, hydrogen and certain other elements alloyed with chromium act as softening agents that enables the movement of dislocations that otherwise not occur in the crystal lattices of chromium atoms. The hydrogen in typical chromium hydride alloys may contribute only a few hundred parts per million in weight at ambient temperatures.
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Thus there is essentially no restriction on the shape of a congruence lattice of an algebra. The finite lattice representation problem asks whether the same is true for finite lattices and finite algebras. That is, does every finite lattice occur as the congruence lattice of a finite algebra? In 1980, Pálfy and Pudlák proved that this problem is equivalent to the problem of deciding whether every finite lattice occurs as an interval in the subgroup lattice of a finite group.
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The bridge was constructed with a Town lattice, patented by Ithiel Town who is said to have supervised the construction of the bridge. If accurate, Town would have supervised the construction of the current bridge in the years preceding his death. The bridge has vertical planking and the seams are covered by battens and the roof of the structure has wood shingles. A lot of the joinery has been preserved, though the queen-post trusses around the Town lattices are not original.
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The generalization is considered to be a good candidate for formulating a theory of non-extensive thermodynamics. The resulting theory is not intended to replace Boltzmann–Gibbs statistics, but rather supplement it, such as in the case of anomalous systems characterised by non-ergodicity or metastable states. One experimental verification of the predictions of Tsallis statistics concerned cold atoms in dissipative optical lattices. Eric Lutz made an analytical prediction in 2003 which was verified in 2006 by a London team.
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Physics Dr. Krishna Kant Shukla's PhD work is titled "Calculations of Electron Effective Mass based on 3-D Kroenig Penny Model with Application to Solids". A pioneering mathematical model to predict the behaviour of electrons in simple crystal lattices. This work was later published in the book "Quantum Statistical theory of Superconductivity" edited by Dr. Van der Meswe; Reidel Kluwer Book co. After his Doctorate, Dr. Shukla joined Hartwick College, Oneonta, New York, as an assistant professor in Physics and Astronomy, in 1991.
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The isotopes of ytterbium range in atomic weight from 147.967 u (148Yb) to 180.9562 u (181Yb). The primary decay mode before the most abundant stable isotope, 174Yb is electron capture, and the primary mode after is beta emission. The primary decay products before 174Yb are isotopes of thulium, and the primary products after are isotopes of lutetium. Of interest to modern quantum optics, the different ytterbium isotopes follow either Bose–Einstein statistics or Fermi–Dirac statistics, leading to interesting behavior in optical lattices.
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Then it became a book. The public interest in this work was such that The New York Times ran a front-page story. In this book, von Neumann declared that economic theory needed to use functional analysis, especially convex sets and the topological fixed-point theorem, rather than the traditional differential calculus, because the maximum-operator did not preserve differentiable functions. Independently, Leonid Kantorovich's functional analytic work on mathematical economics also focused attention on optimization theory, non-differentiability, and vector lattices.
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This same basic structure is found in many other compounds and is commonly known as the halite or rock-salt crystal structure. It can be represented as a face-centered cubic (fcc) lattice with a two-atom basis or as two interpenetrating face centered cubic lattices. The first atom is located at each lattice point, and the second atom is located halfway between lattice points along the fcc unit cell edge. Solid sodium chloride has a melting point of 801 °C.
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They can be made with much sharper transition bands than can be achieved with conventional LC filters. Mason invented a new type of mechanical filter, the quartz crystal filter consisting of lattices of crystals, which became the standard form of filtering on these systems. Mason showed that the efficiency and bandwidth of acoustic transducers, such as those used in sonar, could be massively improved through mechanical-electrical analogies and applying electrical network theory, in particular filter theory. A modern distributed-element circuit.
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A coupled map lattice (CML) is a dynamical system that models the behavior of non-linear systems (especially partial differential equations). They are predominantly used to qualitatively study the chaotic dynamics of spatially extended systems. This includes the dynamics of spatiotemporal chaos where the number of effective degrees of freedom diverges as the size of the system increases. Features of the CML are discrete time dynamics, discrete underlying spaces (lattices or networks), and real (number or vector), local, continuous state variables.
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On the original construction, in order to prevent damage from high water, most of the lattices were covered and only the top sections exposed for light. Two lights have been placed inside the bridge for nighttime illumination. The reconstruction project and final landscaping took about six months to complete at a cost of roughly $60,000. On August 14, 2007, a ribbon-cutting ceremony was held at Opelika Municipal Park to re-open the Salem-Shotwell Covered Bridge to the public.
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However, lattices in which every two elements have a supremum and an infimum, and in particular total orders, are F-algebras. This is because they can equivalently be defined in terms of the algebraic operations: x∨y = inf(x,y) and x∧y = sup(x,y), subject to certain axioms (commutativity, associativity, absorption and idempotency). Thus they are F-algebras of signature P x P + P x P. It is often said that lattice theory draws on both order theory and universal algebra.
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In the context of network theory, a complex network is a graph (network) with non-trivial topological features--features that do not occur in simple networks such as lattices or random graphs but often occur in networks representing real systems. The study of complex networks is a young and active area of scientific research (since 2000) inspired largely by empirical findings of real-world networks such as computer networks, biological networks, technological networks, brain networks, climate networks and social networks.
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In many-body physics, most commonly within condensed-matter physics, a gapped Hamiltonian is a Hamiltonian for an infinitely large many-body system where there is a finite energy gap separating the (possibly degenerate) ground space from the first excited states. A Hamiltonian that is not gapped is called gapless. The property of being gapped or gapless is formally defined through a sequence of Hamiltonians on finite lattices in the thermodynamic limit. An example is the BCS Hamiltonian in the theory of superconductivity.
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A Fibonacci crystal or quasicrystal is a model used to study systems with aperiodic structure. Both names are acceptable as a 'Fibonacci crystal' denotes a quasicrystal and a 'Fibonacci' quasicrystal is a specific type of quasicrystal. Fibonacci 'chains' or 'lattices' are closely related terms, depending on the dimension of the model.Searching database of physical papers reveals that 'Fibonacci chain' is the most frequently used; for instance more than 150 such items are found in Arxiv and just a few dozen other appelations.
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The braid group is the universal central extension of the modular group. The braid group is the universal central extension of the modular group, with these sitting as lattices inside the (topological) universal covering group . Further, the modular group has a trivial center, and thus the modular group is isomorphic to the quotient group of modulo its center; equivalently, to the group of inner automorphisms of . The braid group in turn is isomorphic to the knot group of the trefoil knot.
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Among other things his work reveals "dimension nature" of many of the well-known thermodynamics invariants such as metric and topological entropies and topological pressure. It also provides a unified approach to describe various dimension spectra and related multi-fractal formalism (see ). 5) Pesin's work in Mathematical Physics includes the study of Coupled Map Lattices associated with infinite chains of hyperbolic systems as well as the ones generated by some diffusion-type PDEs such as FitzHu-Nagumo and Belousov- Zhabotinsky equations.
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With another scholarship from the University of London, she went to the University of Edinburgh for graduate study with Max Born, beginning in 1941; her dissertation was The Statistical Thermodynamics of Crystal Lattices. She taught briefly at Edinburgh and the University of Dundee before returning to Royal Holloway as an instructor in 1945. She remained at Royal Holloway through its 1965 transition from a women's college to a coeducational one (a change that she supported), until her retirement in 1980.
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The rock-salt crystal structure. Each atom has six nearest neighbors, with octahedral geometry. The space group of the rock-salt (NaCl) structure is called Fmm (in Hermann–Mauguin notation), or "225" (in the International Tables for Crystallography). The Strukturbericht designation is "B1".The NaCl (B1) Structure In the rock-salt or sodium chloride (halite) structure, each of the two atom types forms a separate face-centered cubic lattice, with the two lattices interpenetrating so as to form a 3D checkerboard pattern.
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A finite lattice is modular if and only if it is both upper and lower semimodular. For a graded lattice, (upper) semimodularity is equivalent to the following condition on the rank function r: :r(x) + r(y) ≥ r(x ∧ y) + r(x ∨ y). Another equivalent (for graded lattices) condition is Birkhoff's condition: : for each x and y in L, if x and y both cover , then covers both x and y. A lattice is called lower semimodular if its dual is semimodular.
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In gases there is a lot more spatial uncertainty because most of their volume is merely empty space. We can regard the mixing process as allowing the contents of the two originally separate contents to expand into the combined volume of the two conjoined containers. The two lattices that allow us to conceptually localize molecular centers of mass also join. The total number of empty cells is the sum of the numbers of empty cells in the two components prior to mixing.
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Recent advances in both AI and quantum information theory have given rise to the concept of quantum neural networks. These hold promise in quantum information processing, which is challenging to classical networks, but can also find application in solving classical problems. In 2018, a physical realization of a quantum reservoir computing architecture was demonstrated in the form of nuclear spins within a molecular solid. In 2019, another possible implementation of quantum reservoir processors was proposed in the form of two-dimensional fermionic lattices.
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Hypothetical (H2O)100 icosahedral water cluster and the underlying structure. In chemistry, a water cluster is a discrete hydrogen bonded assembly or cluster of molecules of water. Many such clusters have been predicted by theoretical models (in silico), and some have been detected experimentally, in various contexts such as ice, and bulk liquid water, in the gas phase, in dilute mixtures with non-polar solvents, and as water of hydration in crystal lattices. The simplest example is the water dimer (H2O)2.
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Further interesting examples for Galois connections are described in the article on completeness properties. Roughly speaking, it turns out that the usual functions ∨ and ∧ are lower and upper adjoints to the diagonal map . The least and greatest elements of a partial order are given by lower and upper adjoints to the unique function Going further, even complete lattices can be characterized by the existence of suitable adjoints. These considerations give some impression of the ubiquity of Galois connections in order theory.
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To this end, only a small amount of Sulfur Dioxide would be used during the winemaking process. This would only kill off the unwanted wild yeasts but allow the wine yeasts to survive on the grapes. For the fermentation process, Bennion used a system of wooden lattices to keep the cap submerged in the must. This technique, which was used widely in France and elsewhere in Europe in the 19th century, proved extremely successful and would later be adopted by many California wineries.
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The first discussion of the RVB state on square lattice using the RVB picture only consider nearest neighbour bonds that connect different sub-lattices. The constructed RVB state is an equal amplitude superposition of all the nearest- neighbour bond configurations. Such a RVB state is believed to contain emergent gapless U(1) gauge field which may confine the spinons etc. So the equal-amplitude nearest-neighbour RVB state on square lattice is unstable and does not corresponds to a quantum spin phase.
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The concrete formulae above are consequences of a more general mathematical duality called Pontryagin duality. Applied to the group R, it yields the classical Fourier transform; an application in physics are reciprocal lattices, where the underlying group is a finite-dimensional real vector space endowed with the additional datum of a lattice encoding positions of atoms in crystals. Fourier series are used to solve boundary value problems in partial differential equations. In 1822, Fourier first used this technique to solve the heat equation.
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Mg2Cl3+ ). The movement of the magnesium ion into cathode host lattices is also (as of 2014) problematically slow. In 2018 a chloride free electrolyte together with a quinone based polymer cathode demonstrated promising performance, with up to per kg energy density, up to 3.4 kW/kg power density, and up to 87% retention at 2,500 cycles. The absence of chloride in the electrolyte was claimed to improve ion kinetics, and so reduce the amount of electrolyte used, increasing performance density figures.
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The theatre's auditorium had boxes, lattices, an orchestra pit and first and second floor galleries. However, the theatre had been built hastily causing concerns about its safety, forcing the manager to issue a statement that he would obtain the necessary certificates from Master Builders. This theatre closed in 1749 and the building was appropriated for other uses.John C. Greene, Gladys L. H. Clark, The Dublin Stage, 1720-1745: A Calendar of Plays, Entertainments, and Afterpieces, Leigh University Press (1993) - Google Books pg.
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Such systems are called non self-averaging. Thus unlike the self-averaging scenario, numerical simulations cannot lead to an improved picture in larger lattices (large N), even if the critical point is exactly known. In summary, various types of self-averaging can be indexed with the help of the asymptotic size dependence of a quantity like RX. If RX falls off to zero with size, it is self-averaging whereas if RX approaches a constant as N → ∞, the system is non-self- averaging.
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Prasad earned his bachelor's degree with honors in Mathematics from Magadh University in 1963. Two years later, in 1965, he received his masters in Mathematics from Patna University. After a brief stay at the Indian Institute of Technology Kanpur in their Ph.D. program for Mathematics, Prasad entered the Ph.D. program at the Tata Institute of Fundamental Research (TIFR) in 1966. There he began a long and extensive collaboration with his advisor M. S. Raghunathan on several topics including the study of lattices in semi-simple Lie groups.
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The Ephraim Atwood House stands in the Dana Hill area of Cambridge, between Central Square and Harvard Square north of Massachusetts Avenue. It is set on the east side of Harvard Street, opposite its junction with Centre Street, and is oriented facing south. It is a 1-1/2 story wood frame structure, with a gabled roof and clapboarded exterior. The roof extends over a shallow front porch for its full five-bay width, supported by Doric columns which have trellis lattices between some pairs.
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In the Yokaichi and Gokoku districts is a street of merchant houses, which have solemn white or cream coloured plaster walls, lattices, decorative walls and old-style Japanese desks. This traditional street is about 600 meters long and around 90 of these historical houses are still lived in. The campaign to preserve this old town area has been ongoing since 1975 and the national government designated this area as an "Important Traditional Construction Preservation Area" in 1982. The Nobel laureate Kenzaburō Ōe was born in Uchiko.
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Physicists often use various lattices to apply their favorite models in them. For instance, the most favorite lattice is perhaps the square lattice. There are 14 Bravais space lattice where every cell has exactly the same number of nearest, next nearest, nearest of next nearest etc neighbors and hence they are called regular lattice. Often physicists and mathematicians study phenomena which require disordered lattice where each cell do not have exactly the same number of neighbors rather the number of neighbors can vary wildly.
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The reduced amplified spontaneous emission noise in the radiation of Ti:sapphire lasers lends great strength in their application as optical lattices for the operation of state- of-the-art atomic clocks. Apart from fundamental science applications in the laboratory, this laser has found biological applications such as deep-tissue multiphoton imaging and industrial applications cold micromachining. When operated in the chirped pulse amplification mode, they can be used to generate extremely high peak powers in the terawatt range, which finds use in nuclear fusion research.
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Coding with fuzzy lattices can be useful, for instance, in the psephological analysis of big data about voter behaviour, where researchers want to explore the characteristics and associations involved in "somewhat vague" opinions; gradations in voter attitudes; and variability in voter behaviour (or personal characteristics) within a set of parameters.Daniel Kreiss, Prototype Politics: Technology-Intensive Campaigning and the Data of Democracy. Oxford University Press, 2016. The basic programming techniques for this kind of fuzzy concept mapping and deep learning are by now well-establishedE.g.
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The application of epitaxial growth of silicon on sapphire substrates for fabricating MOS devices involves a silicon purification process that mitigates crystal defects which result from a mismatch between sapphire and silicon lattices. For example, Peregrine Semiconductor's SP4T switch is formed on an SOS substrate where the final thickness of silicon is approximately 95 nm. Silicon is recessed in regions outside the polysilicon gate stack by poly oxidation and further recessed by the sidewall spacer formation process to a thickness of approximately 78 nm.
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In many specialized areas of order theory, one restricts to classes of partially ordered sets that are complete with respect to certain limit constructions. For example, in lattice theory, one is interested in orders where all finite non-empty sets have both a least upper bound and a greatest lower bound. In domain theory, on the other hand, one focuses on partially ordered sets in which every directed subset has a supremum. Complete lattices and orders with a least element (the "empty supremum") provide further examples.
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To produce nanolattice materials, polymer templates are manufactured by high- resolution 3D printing processes, such as multiphoton lithography, or by self- assembly techniques. Ceramic, metal or composite material nanolattices are formed by post-treatment of the polymer templates with techniques including pyrolysis, atomic layer deposition, electroplating and electroless plating. Pyrolysis, which additionally shrinks the lattices by up to 90%, creates the smallest-size structures, whereby the polymeric template material transforms into carbon, or other ceramics and metals, through thermal decomposition in inert atmosphere or vacuum.
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In metallurgy, materials science and structural geology, subgrain rotation recrystallization is recognized as an important mechanism for dynamic recrystallisation. It involves the rotation of initially low-angle sub-grain boundaries until the mismatch between the crystal lattices across the boundary is sufficient for them to be regarded as grain boundaries. This mechanism has been recognized in many minerals (including quartz, calcite, olivine, pyroxenes, micas, feldspars, halite, garnets and zircons) and in metals (various magnesium, aluminium and nickel alloys).Microtectonics by C.W.Passchier and R.A.J.Trouw, 2nd rev.
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Gady Kozma (right), Hugo Duminil-Copin (left), Oberwolfach 2012 Gady Kozma is an Israeli mathematician. Kozma obtained his Ph.D. in 2001 at the University of Tel Aviv with Alexander Olevskii. The Mathematics Genealogy Project (North Dakota State University Department of Mathematics / American Mathematical Society) He is a scientist at the Weizmann Institute. In 2005, he demonstrated the existence of the scaling limit value (that is, for increasingly finer lattices) of the Loop Erased Random Walk (LERW) in three dimensions and its invariance under rotations and dilations.
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Kot (1992) "Discrete-time travelling waves: ecological examples", J. Math. Biol. 30: 413-436. DOI:10.1007/BF00173295 coupled map latticesM. D. S. Herrera, J. S. Martin (2009) "An analytical study in coupled map lattices of synchronized states and traveling waves, and of their period-doubling cascades", Chaos, Solitons & Fractals 42: 901–910.DOI:10.1016/j.chaos.2009.02.040 and cellular automataJ. A. Sherratt (1996) "Periodic travelling waves in a family of deterministic cellular automata", Physica D 95: 319–335. DOI:10.1016/0167-2789(96)00070-XM.
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The lattices are offset so that the lithium atoms are farthest from the cobalt atoms, and the structure repeats in the direction perpendicular to the planes every three cobalt (or lithium) layers. The point group symmetry is R\bar 3m in Hermann-Mauguin notation, signifying a unit cell with threefold improper rotational symmetry and a mirror plane. The threefold rotational axis (which is normal to the layers) is termed improper because the triangles of oxygen (being on opposite sides of each octahedron) are anti- aligned.
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RCCM lattices behave as an elastic solid in both tension and compression. They offer both a linear regime and a nonlinear super-elastic deformation mode a modulus an order of magnitude breater than for an ultralight material (12.3 megapascals at a density of 7.2 mg per cubic centimeter). Bulk properties can be predicted from component measurements and deformation modes determined by the placement of part types. Site locations are locally constrained, yielding structures that merge desirable features of carbon fiber composites, cellular materials and additive manufacturing.
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Cellular composites extend stretch-dominated lattices to the ultralight regime (below ten milligrams per cubic centimeter). Performance depends positively on the framework rigidity of the lattice, node connectivity, slenderness of strut members and the scaling of the density cost of mechanical connections. Conventional fiber composites make truss cores and structural frames, with bonded assembly of substructures or continuous fiber winding. Examples of such truss cores have been reported with continuous two-dimensional (2D) geometric symmetry and nearly ideal but highly anisotropic specific modulus scaling.
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Rutile epitaxial on hematite nearly 6 cm long. Bahia, Brazil In mineralogy, epitaxy is the overgrowth of one mineral on another in an orderly way, such that certain crystal directions of the two minerals are aligned. This occurs when some planes in the lattices of the overgrowth and the substrate have similar spacings between atoms. If the crystals of both minerals are well formed so that the directions of the crystallographic axes are clear then the epitaxic relationship can be deduced just by a visual inspection.
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For random walks in n-dimensional integer lattices, George Pólya published in 1919 and 1921 work, where he studied the probability of a symmetric random walk returning to a previous position in the lattice. Pólya showed that a symmetric random walk, which has an equal probability to advance in any direction in the lattice, will return to a previous position in the lattice an infinite number of times with probability one in one and two dimensions, but with probability zero in three or higher dimensions.
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Copper sulfides can be classified into three groups: Monosulfides, 1.6 ≤ Cu/S ≤ 2: their crystal structures consist of isolated sulfide anions that are closely related to either hcp or fcc lattices, without any direct S-S bonds. The copper ions are distributed in a complicated manner over interstitial sites with both trigonal as well as distorted tetrahedral coordination and are rather mobile. Therefore, this group of copper sulfides shows ionic conductivity at slightly elevated temperatures. In addition, the majority of its members are semiconductors.
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His theory was successful in predicting many generally observed features of the fine structure, including similar structure from similar lattices, inverse r2 dependence, correct r versus T dependence and increasing energy separation of the fine structure features with energy from the edge. The equation, which was re-derived in a more quantitative way in 1932 was simple to apply and interpret. Every experimenter found approximate agreement with the theory. There were always some absorption features close to that predicted by the possible lattice planes.
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The Young–Fibonacci graph, the Hasse diagram of the Young–Fibonacci lattice. The canonical examples of differential posets are Young's lattice, the poset of integer partitions ordered by inclusion, and the Young–Fibonacci lattice. Stanley's initial paper established that Young's lattice is the only 1-differential distributive lattice, while showed that these are the only 1-differential lattices. There is a canonical construction (called "reflection") of a differential poset given a finite poset that obeys all of the defining axioms below its top rank.
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In smaller systems the coordination number for the surface is more significant and the magnetic moments have a stronger effect on the system. Although fluctuations in particles can be minuscule, they are heavily dependent on the structure of crystal lattices as they react with their nearest neighbouring particles. Fluctuations are also affected by the exchange interaction as parallel facing magnetic moments are favoured and therefore have less disturbance and disorder, therefore a tighter structure influences a stronger magnetism and therefore a higher Curie temperature.
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Yttrium is one of the elements that was used to make the red color in CRT televisions The red component of color television cathode ray tubes is typically emitted from a yttria () or yttrium oxide sulfide () host lattice doped with europium (III) cation (Eu3+) phosphors. The red color itself is emitted from the europium while the yttrium collects energy from the electron gun and passes it to the phosphor.Daane 1968, p. 818 Yttrium compounds can serve as host lattices for doping with different lanthanide cations.
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One of the main objectives of modeling is the calculation of the atomistic displacements u caused by an applied force f. The displacements, in principle, are given by Eq. (8). However, it involves inversion of the matrix K which is 3N x 3N. For any calculation of practical interest N ~ 10,000 but preferably a million for more realistic simulations. Inversion of such a large matrix is computationally extensive and special techniques are needed for the calculation of u’s. For regular periodic lattices, LSGF is one such technique.
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A modular lattice of order dimension 2. As with all finite 2-dimensional lattices, its Hasse diagram is an st-planar graph. In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self-dual condition: ;Modular law: implies for every , where ≤ is the partial order, and ∨ and ∧ (called join and meet respectively) are the operations of the lattice. This phrasing emphasizes an interpretation in terms of projection onto the sublattice , a fact known as the diamond isomorphism theorem.
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His collaborators include Jean Bourgain, Alex Eskin, Elon Lindenstrauss, Gregory Margulis, and Hee Oh. In 2000 Mozes received the Erdős Prize. In 1998 he was an invited speaker with talk Products of trees, lattices and simple groups at the International Congress of Mathematicians (ICM) in Berlin. He was a plenary speaker at the ICM Satellite Conference on "Geometry Topology and Dynamics in Negative Curvature" held at the Raman Research Institute of the International Centre for Theoretical Sciences (ICTS) from August 2 to August 7, 2010.
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One usually thinks of dimension for a set which is dense, like the points on a line, for example. Dimension makes sense in a discrete setting, like for graphs, only in the large system limit, as the size tends to infinity. For example, in Statistical Mechanics, one considers discrete points which are located on regular lattices of different dimensions. Such studies have been extended to arbitrary networks, and it is interesting to consider how the definition of dimension can be extended to cover these cases.
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The Leech lattice is also a 12-dimensional lattice over the Eisenstein integers. This is known as the complex Leech lattice, and is isomorphic to the 24-dimensional real Leech lattice. In the complex construction of the Leech lattice, the binary Golay code is replaced with the ternary Golay code, and the Mathieu group M24 is replaced with the Mathieu group M12. The E6 lattice, E8 lattice and Coxeter-Todd lattice also have constructions as complex lattices, over either the Eisenstein or Gaussian integers.
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For composite lattices, (crystals which have more than one vector in their basis) each single lattice point represents multiple atoms. We can break apart each Wigner–Seitz cell into subcells by further Voronoi decomposition according to the closest atom, instead of the closest lattice point. For example, the diamond crystal structure contains a two atom basis. In diamond, carbon atoms have tetraheral sp3 bonding, but since tetrahedra do not tile space, the voronoi decomposition of the diamond crystal structure is actually the triakis truncated tetrahedral honeycomb.
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Semën Samsonovich Kutateladze (born October 2, 1945 in Leningrad, now St. Petersburg) is a mathematician. He is known for contributions to functional analysis and its applications to vector lattices and optimization. In particular, he has made contributions to the calculus of subdifferentials for vector-lattice valued functions, to whose study he introduced methods of Boolean-valued models and infinitesimals. He is professor of mathematics at Novosibirsk State University,About Kutateladze in Russian where he has continued and enriched the scientific tradition of Leonid Kantorovich.
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The equivalence between preorders and finite topologies can be interpreted as a version of Birkhoff's representation theorem, an equivalence between finite distributive lattices (the lattice of open sets of the topology) and partial orders (the partial order of equivalence classes of the preorder). This correspondence also works for a larger class of spaces called finitely generated spaces. Finitely generated spaces can be characterized as the spaces in which an arbitrary intersection of open sets is open. Finite topological spaces are a special class of finitely generated spaces.
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Using pulsed EPR it is possible to measure electron spin relaxation, which can help uncover the dynamics of molecules. examined the frequency dependence of electron spin relaxation in fluid solution for several important types of radicals. She has studied the mechanisms of spin and spin-lattice relaxation in rigid lattices and fluid solutions; identifying the role of methyl groups and nuclear spins in spin echo dephasing. Her systematic studies in both glassy solvents and fluid solutions can be used to establish the relationship between structure and relaxation.
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Nucleic acid design is used in DNA nanotechnology to design strands which will self-assemble into a desired target structure. These include examples such as DNA machines, periodic two- and three-dimensional lattices, polyhedra, and DNA origami. It can also be used to create sets of nucleic acid strands which are "orthogonal", or non- interacting with each other, so as to minimize or eliminate spurious interactions. This is useful in DNA computing, as well as for molecular barcoding applications in chemical biology and biotechnology.
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The above model can be extended for variable (but deterministic) rates and volatilities. The model may also be used to value European options on instruments paying dividends. In this case, closed-form solutions are available if the dividend is a known proportion of the stock price. American options and options on stocks paying a known cash dividend (in the short term, more realistic than a proportional dividend) are more difficult to value, and a choice of solution techniques is available (for example lattices and grids).
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Ions in crystal lattices of purely ionic compounds are spherical; however, if the positive ion is small and/or highly charged, it will distort the electron cloud of the negative ion, an effect summarised in Fajans' rules. This polarization of the negative ion leads to a build-up of extra charge density between the two nuclei, that is, to partial covalency. Larger negative ions are more easily polarized, but the effect is usually important only when positive ions with charges of 3+ (e.g., Al3+) are involved.
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The lattice describes the repeat pattern; for diamond cubic crystals this lattice is "decorated" with a motif of two tetrahedrally bonded atoms in each primitive cell, separated by of the width of the unit cell in each dimension.. The diamond lattice can be viewed as a pair of intersecting face-centered cubic lattices, with each separated by of the width of the unit cell in each dimension. Many compound semiconductors such as gallium arsenide, β-silicon carbide, and indium antimonide adopt the analogous zincblende structure, where each atom has nearest neighbors of an unlike element. Zincblende's space group is F3m, but many of its structural properties are quite similar to the diamond structure.. The atomic packing factor of the diamond cubic structure (the proportion of space that would be filled by spheres that are centered on the vertices of the structure and are as large as possible without overlapping) is ≈ 0.34,. significantly smaller (indicating a less dense structure) than the packing factors for the face- centered and body-centered cubic lattices.. Zincblende structures have higher packing factors than 0.34 depending on the relative sizes of their two component atoms.
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The family of all subsets of a set S, ordered by set inclusion, forms a lattice in which the meet is represented by the set-theoretic intersection and the join is represented by the set-theoretic union; a lattice formed in this way is called a Boolean lattice. The lattice-theoretic version of Frankl's conjecture is that in any finite lattice there exists an element x that is not the join of any two smaller elements, and such that the number of elements greater than or equal to x totals at most half the lattice, with equality only if the lattice is a Boolean lattice. As shows, this statement about lattices is equivalent to the Frankl conjecture for union-closed sets: each lattice can be translated into a union-closed set family, and each union-closed set family can be translated into a lattice, such that the truth of the Frankl conjecture for the translated object implies the truth of the conjecture for the original object. This lattice-theoretic version of the conjecture is known to be true for several natural subclasses of lattices; ; but remains open in the general case.
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Having the full spectroscopic information available in every measurement spot has the advantage that several components can be mapped at the same time, including chemically similar and even polymorphic forms, which cannot be distinguished by detecting only one single wavenumber. Furthermore, material properties such as stress and strain, crystal orientation, crystallinity and incorporation of foreign ions into crystal lattices (e.g., doping, solid solution series) can be determined from hyperspectral maps. Taking the cell culture example, a hyperspectral image could show the distribution of cholesterol, as well as proteins, nucleic acids, and fatty acids.
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The strictest form of order in a solid is lattice periodicity: a certain pattern (the arrangement of atoms in a unit cell) is repeated again and again to form a translationally invariant tiling of space. This is the defining property of a crystal. Possible symmetries have been classified in 14 Bravais lattices and 230 space groups. Lattice periodicity implies long-range order: if only one unit cell is known, then by virtue of the translational symmetry it is possible to accurately predict all atomic positions at arbitrary distances.
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Lewenstein conducted successful research in above mentioned areas and is author of over 480 publications, including over 80 Phys. Rev. Lett., feature and invited articles in Science and Nature group journals, and several reviews.Web of Science, Accession Number: WOS:A1994NA92000074 (Web of Science). Lewenstein is an author of two books: Polish Jazz Recordings and Beyond Maciej Lewenstein Warszawska Firma Wydawnicza (2015)Maciej Lewenstein, "Polish Jazz Recordings and Beyond", Warszawska Firma Wydawnicza (2015), Ultracold atoms in optical lattices: Simulating quantum many-body systems Maciej Lewenstein, Anna Sanpera, and Veronica Ahufunger, Oxford University Press(2012).
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The X-Ionizer is a molecular hardening technology invented by the specialists of Project Atom, introduced in the Cary Bates run of the DC Comics published Captain Atom. Because of the nearly invulnerable nature of the alien metal discovered, the scientists needed some way to cut it in order to perform experiments. Doctor Heinrich Megala, one of the lead researchers of the Project; developed a device that would make the molecular lattices of an object knit together in such a way that it became superdense and compact. This, in effect, making the object nearly indestructible.
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In mathematics, the Hermite constant, named after Charles Hermite, determines how short an element of a lattice in Euclidean space can be. The constant γn for integers n > 0 is defined as follows. For a lattice L in Euclidean space Rn unit covolume, i.e. vol(Rn/L) = 1, let λ1(L) denote the least length of a nonzero element of L. Then is the maximum of λ1(L) over all such lattices L. The square root in the definition of the Hermite constant is a matter of historical convention.
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His doctoral thesis established the quantum theory of solids, using waves to describe electrons in periodic lattices. On March 14, 1940, Bloch married Lore Clara Misch (1911–1996), a fellow physicist working on X-ray crystallography, whom he had met at an American Physical Society meeting.Former Fellows of The Royal Society of Edinburgh 1783 – 2002. royalsoced.org.uk They had four children, twins George Jacob Bloch and Daniel Arthur Bloch (born January 15, 1941), son Frank Samuel Bloch (born January 16, 1945), and daughter Ruth Hedy Bloch Alexander (born September 15, 1949).
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The electromagnetic nature of photons and phonons are studied which show that the oscillations of electromagnetic fields and of crystal lattices have much in common. Waves form a basis for both, provided one incorporates quantum theory. Other topics studied in thermal physics include: chemical potential, the quantum nature of an ideal gas, i.e. in terms of fermions and bosons, Bose–Einstein condensation, Gibbs free energy, Helmholtz free energy, chemical equilibrium, phase equilibrium, the equipartition theorem, entropy at absolute zero, and transport processes as mean free path, viscosity, and conduction.
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In 1883, Groth compiled a monumental five-volume collection entitled Chemische Kristallographie, which contained crystalline morphology and physical property data on thousands of substances. By Groth's time, Dalton's atomic theory was already well established. In 1888, Groth was the first to suggest the possibility that spherical atoms reside at equivalent positions of space lattices, which gave a physical significance to this still somewhat abstract idea of the regular and symmetric partitioning of space. The German physicist Leonhard Sohncke (1842–1897) had previously derived the 65 chiral space groups (i.e.
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Complex metallic alloys (CMAs) are intermetallic compounds characterized by large unit cells comprising some tens up to thousands of atoms; the presence of well-defined clusters of atoms (frequently with icosahedral symmetry); and partial disorder within their crystalline lattices. They are composed of two or more metallic elements, sometimes with metalloids or chalcogenides added. They include, for example, NaCd2, with 348 sodium atoms and 768 cadmium atoms in the unit cell. Linus Pauling attempted to describe the structure of NaCd2 in 1923, but did not succeed until 1955.
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DX arrays: DX arrays: Two-dimensional arrays have been made from other motifs as well, including the Holliday junction rhombus lattice,Other arrays: and various DX-based arrays making use of a double- cohesion scheme.Other arrays: Other arrays: The top two images at right show examples of tile-based periodic lattices. Two-dimensional arrays can be made to exhibit aperiodic structures whose assembly implements a specific algorithm, exhibiting one form of DNA computing. The DX tiles can have their sticky end sequences chosen so that they act as Wang tiles, allowing them to perform computation.
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This eventually led the way to the creation of the U.S. Atomic Energy Commission in 1947. From 1946 to 1957, he worked as a senior physicist at the Brookhaven National Laboratory on Long Island, and wrote a textbook titled Nuclear Physics. Kaplan visited MIT in 1957, and became a professor in 1958 to participate in the new department. He participated in various projects such as the research on lattices of partially enriched uranium rods in heavy water, and development of graduate and undergraduate courses such as the history of science and classical Greek.
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E3 ubiquitin-protein ligase TRIM63, also known as "MuRF1" is an enzyme that in humans is encoded by the TRIM63 gene. This gene encodes a member of the RING zinc finger protein family found in striated muscle and iris. The product of this gene is localized to the Z-line and M-line lattices of myofibrils, where titin's N-terminal and C-terminal regions respectively bind to the sarcomere. In vitro binding studies have shown that this protein also binds directly to titin near the region of titin containing kinase activity.
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Smaller scale works are assembled in metal stands or hung in such a way as to reflect and capture the light. In her large-scale work, the viewer can enter the piece and experience the play of light, observing how this transforms a sculptural interior into a spiritual one. The effect is described in the essay by Albert Ruetz in the catalogue accompanying the Cairo Biennial XII. “Light and the passage of time turn the delicate lattices into translucent assemblies that are constantly created anew, evoking a most varied range of emotions.
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Lattice theoretic information about the lattice of subgroups can sometimes be used to infer information about the original group, an idea that goes back to the work of . For instance, as Ore proved, a group is locally cyclic if and only if its lattice of subgroups is distributive. If additionally the lattice satisfies the ascending chain condition, then the group is cyclic. The groups whose lattice of subgroups is a complemented lattice are called complemented groups , and the groups whose lattice of subgroups are modular lattices are called Iwasawa groups or modular groups .
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For example, the group PSL2(R) is not a group of 2×2 matrices, but it has a faithful representation as 3×3 matrices (the adjoint representation), which can be used in the general case. Many Lie groups are linear but not all of them. The universal cover of SL2(R) is not linear, as are many solvable groups, for instance the quotient of the Heisenberg group by a central cyclic subgroup. Discrete subgroups of classical Lie groups (for example lattices or thin groups) are also examples of interesting linear groups.
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In condensed matter physics, the resonating valence bond theory (RVB) is a theoretical model that attempts to describe high temperature superconductivity, and in particular the superconductivity in cuprate compounds. It was first proposed by an American physicist P. W. Anderson and Indian theoretical physicist Ganapathy Baskaran in 1987. The theory states that in copper oxide lattices, electrons from neighboring copper atoms interact to form a valence bond, which locks them in place. However, with doping, these electrons can act as mobile Cooper pairs and are able to superconduct.
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But then x ≤ y ∨ z, and x is not less than or equal to either y or z, showing that it is not join-prime. There exist lattices in which the join-prime elements form a proper subset of the join- irreducible elements, but in a distributive lattice the two types of elements coincide. For, suppose that x is join-irreducible, and that x ≤ y ∨ z. This inequality is equivalent to the statement that x = x ∧ (y ∨ z), and by the distributive law x = (x ∧ y) ∨ (x ∧ z).
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Noises based on lattices, such as simulation noise and Perlin noise, are often calculated at different frequencies and summed together to form band-limited fractal signals. Other approaches developed later that use vector calculus identities to produce divergence free fields, such as "Curl-Noise" as suggested by Robert Bridson, and "Divergence-Free Noise" due to Ivan DeWolf. These often require calculation of lattice noise gradients, which sometimes are not readily available. A naive implementation would call a lattice noise function several times to calculate its gradient, resulting in more computation than is strictly necessary.
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The G-1 experiment had lattices of 6,800 uranium oxide cubes (about 25 tons) in the nuclear moderator paraffin. The work verified Karl Heinz Höcker's calculations that cubes were better than rods, and rods were better than plates. The G-III experiment was a small-scale design, but it generated an exceptionally high rate of neutron production. The G-III model was superior to nuclear fission chain reaction experiments that had been conducted at the KWIP in Berlin- Dahem, the University of Heidelberg, or the University of Leipzig.
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The same is true for simple polytopes of arbitrary dimension (Blind & Mani-Levitska 1987, proving a conjecture of Micha Perles).. Kalai (1988). gives a simple proof based on unique sink orientations. Because these polytopes' face lattices are determined by their graphs, the problem of deciding whether two three-dimensional or simple convex polytopes are combinatorially isomorphic can be formulated equivalently as a special case of the graph isomorphism problem. However, it is also possible to translate these problems in the opposite direction, showing that polytope isomorphism testing is graph-isomorphism complete.
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When shined through by UV light, the mask will transmit the image of the planes onto a lens which subsequently project it onto a photosensitive polymer resin such as 1,6-hexanediol diacrylate (HDDA) causing the liquid to cure in the light exposed areas. These process is repeated for each layer and assembled together to form a 3-D system. Non-polymer lattices can also be created from this process by additional processing. For instance, metallic structures can be created by electroless plating onto the base structure followed by removal of the polymer through thermal heating.
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H. N. Tsao, H. J. Räder, W. Pisula, A. Rouhanipour, K. Müllen: Novel organic semiconductors and processing techniques for organic field- effect transistors, physica status solidi, 2008, 205, 421–429. The considered two-dimensional benzene ring structures are examples of subunits of graphene lattices (graphene nanostructures). The graphene-like structures synthesized and investigated by Müllen include two-dimensional bands of less than 50 nanometers width with jagged edges. Of interest here are the electronic conduction properties and spintronics properties with a view to future replacement of silicon-semiconductor technology.
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The applications of Er2O3 are varied due to their electrical, optical and photoluminescence properties. Nanoscale materials doped with Er3+ are of much interest because they have special particle-size-dependent optical and electrical properties. Erbium oxide doped nanoparticle materials can be dispersed in glass or plastic for display purposes, such as display monitors. The spectroscopy of Er3+ electronic transitions in host crystals lattices of nanoparticles combined with ultrasonically formed geometries in aqueous solution of carbon nanotubes is of great interest for synthesis of photoluminescence nanoparticles in 'green' chemistry.
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By definition, metamaterials are realized as lattices whose periodicity is assumed to be much smaller than the wavelength. However, it is important that, though small, the periodicity is not negligible with respect to the wavelength. For this reason, if one formally introduces constitutive parameters for such regime, they will not be measurable response functions, and it will not be possible to use them for a sample of other dimensions or for a sample excited in another way. In other words, such formally introduced material parameters cannot satisfy the conditions of locality.
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Nachna Vaishnavism themed ruin with four Vishnu Avatar representations; above: left is man-lion Narasimha, right is boar-headed Varaha. Three Jali windows, which let little light into the dark sanctum, are among the temple's attractions. Its multi-layered composition and decorative figures are significantly more elaborate than the Jalis at Parvati Temple and more representative of the local adornments artistically-speaking. The actual window panel consists of two shells with rich profiled—reminiscent of wooden model—lattices inside and three small arcades in the exterior, formed as horseshoe arches.
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Erik Winfree (born September 26, 1969Erik Winfree resume) is an American applied computer scientist, bioengineer, and professor at California Institute of Technology.Erik Winfree's homepage He is a leading researcher into DNA computing and DNA nanotechnology.Technology Review's 1999 TR35 In 1998, Winfree in collaboration with Nadrian Seeman published the creation of two- dimensional lattices of DNA tiles using the "double crossover" motif. These tile-based structures provided the capability to implement DNA computing, which was demonstrated by Winfree and Paul Rothemund in 2004, and for which they shared the 2006 Feynman Prize in Nanotechnology.
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Most social, biological, and technological networks display substantial non-trivial topological features, with patterns of connection between their elements that are neither purely regular nor purely random. Such features include a heavy tail in the degree distribution, a high clustering coefficient, assortativity or disassortativity among vertices, community structure, and hierarchical structure. In the case of directed networks these features also include reciprocity, triad significance profile and other features. In contrast, many of the mathematical models of networks that have been studied in the past, such as lattices and random graphs, do not show these features.
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The illustrator of Murgunstrumm and Worse Things Waiting was the noted Weird Tales artist Lee Brown Coye. Coye's macabre designs, incorporating mysterious lattices of twigs, were the inspiration for Wagner's British Fantasy Award-winning story "Sticks". A connoisseur of rare horror stories, Wagner perspicaciously edited many horror and fantasy anthologies; perhaps his greatest achievement of this topic was the annual anthology series The Year's Best Horror Stories (DAW Books), which he edited for fourteen years from volume VIII (1980) until volume XXII (1994). The series was canceled after Wagner's death.
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Above this pressure, the VII–VIII transition temperature drops rapidly, reaching 0 K at ~60 GPa.. Thus, ice VII has the largest stability field of all of the molecular phases of ice. The cubic oxygen sub- lattices that form the backbone of the ice VII structure persist to pressures of at least 128 GPa;. this pressure is substantially higher than that at which water loses its molecular character entirely, forming ice X. In high pressure ices, protonic diffusion (movement of protons around the oxygen lattice) dominates molecular diffusion, an effect which has been measured directly.
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It is also possible to restrict the classes of point sets that may be Danzer sets in other ways than by their densities. In particular, they cannot be the union of finitely many lattices, they cannot be generated by choosing a point in each tile of a substitution tiling (in the same position for each tile of the same type), and they cannot be generated by the cut-and-project method for constructing aperiodic tilings. Therefore, the vertices of the pinwheel tiling and Penrose tiling are not Danzer sets.
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In crystallography, a lattice plane of a given Bravais lattice is a plane (or family of parallel planes) whose intersections with the lattice (or any crystalline structure of that lattice) are periodic (i.e. are described by 2d Bravais lattices) and intersect the Bravais lattice; equivalently, a lattice plane is any plane containing at least three noncollinear Bravais lattice points.Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt: New York, 1976). All lattice planes can be described by a set of integer Miller indices, and vice versa (all integer Miller indices define lattice planes).
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The atoms in each grain are organized into one of seven 3d stacking arrangements or crystal lattices (cubic, tetrahedral, hexagonal, monoclinic, triclinic, rhombohedral and orthorhombic). The direction of alignment of the matrices differ between adjacent crystals, leading to variance in the reflectivity of each presented face of the interlocked grains on the galvanized surface. The average grain size can be controlled by processing conditions and composition, and most alloys consist of much smaller grains not visible to the naked eye. This is to increase the strength of the material (see Hall-Petch Strengthening).
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There are two distinct types of interactions that can occur between the subunits of lateral protofilaments within the microtubule called the A-type and B-type lattices. In the A-type lattice, the lateral associations of protofilaments occur between adjacent α and β-tubulin subunits (i.e. an α-tubulin subunit from one protofilament interacts with a β-tubulin subunit from an adjacent protofilament). In the B-type lattice, the α and β-tubulin subunits from one protofilament interact with the α and β-tubulin subunits from an adjacent protofilament, respectively.
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On both sides of the main building there are approximately symmetrical, single storey extensions. The platform canopy in front of the entrance building is built as lattices on thin columns with small capitals, built in cast iron. The building is listed by the Hessian heritage office as an early railway station of outstanding historical significance. Heppenheim station was modernised and equipped for the disabled in preparation for Hessentag (a festival devised to promote unity in the state of Hesse, which was created in 1945) in the summer of 2004.
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His contributions in the subsequent 35 years of his career include transfer grammar, string analysis (adjunction grammar), elementary sentence- differences (and decomposition lattices), algebraic structures in language, operator grammar, sublanguage grammar, a theory of linguistic information, and a principled account of the nature and origin of language.Harris's account of the nature and origin of language, and its learnability, is in the final chapters of Language and information (1988) and A theory of language and information (1991), and in number four of the Bampton Lectures at Columbia in 1986, on which the former was based.
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Thornton Willis "Conversion" 2008, Oil on Canvas, 97x70 inches. Shortly after the exhibition entitled Painting: 40 Years,” a retrospective at the Sideshow Gallery in 2007, Willis returned to a rectilinear format. Combining the early "Slat" paintings, with exploration of form and field in his “Wedge” series, he created a body of work he entitled “Lattices” where lines appear to weave forward and back. Michael Feldman documented the transition to this new work in a film, in 2008-09, Portrait of an American Painter Portrait of an American Painter.
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A significant international community of researchers follow Wille's work on Formal concept analysis, the main forums being the International Conferences on Formal Concept Analysis (ICFCA), Conceptual Structures (see also Conceptual graphs) (ICCS) and Concept Lattices and their Application (CLA) conferences. The first two are published in the Lecture Notes in Computer Science and the latter is a multi-stage conference that produces journal papers. A leader, inter-disciplinarian, peace activist and prolific mentor, Wille oversaw more than 100 German "Diplom- und Staatsexamenarbeiten" in Mathematics, 51 PhD dissertations, and 8 Postdoctoral "habilitation" qualifications.
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Many systems, such as crystal lattices, have a unique ground state, and (since ) this means that they have zero entropy at absolute zero. Other systems have more than one state with the same, lowest energy, and have a non-vanishing "zero-point entropy". For instance, ordinary ice has a zero-point entropy of , because its underlying crystal structure possesses multiple configurations with the same energy (a phenomenon known as geometrical frustration). The third law of thermodynamics states that the entropy of a perfect crystal at absolute zero, or 0 kelvin is zero.
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The testing station is where Pose and Ernst Rexer compared the effectiveness of neutron production in a paraffin-moderated reactor using uranium plates, rods, and cubes. Internal reports (See section below: Internal Reports.) on their activities were classified Top Secret and had limited distribution. The G-1 experiment performed at the HWA testing station had lattices of 6,800 uranium oxide cubes (about 25 tons) in the neutron moderator paraffin. Their work verified Karl Heinz Höcker's calculations that cubes were better than rods, and rods were better than plates.
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The tetrameric streptavidin has also been used as a hub around which other proteins may be arranged, either by an affinity tag such as Strep-tag or AviTag or by genetic fusion to SpyTag. Fusion to SpyTag allowed generation of assemblies with 8 or 20 streptavidin subunits. As well as a molecular force probe for atomic force microscopy studies, novel materials such as 3D crystalline lattices have also been created. Streptavidin has a mildly acidic isoelectric point (pI) of ~5, but a recombinant form of streptavidin with a near-neutral pI is also commercially available.
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Concrete is a mixture of coarse (stone or brick chips) and fine (generally sand or crushed stone) aggregates with a paste of binder material (usually Portland cement) and water. When cement is mixed with a small amount of water, it hydrates to form microscopic opaque crystal lattices encapsulating and locking the aggregate into a rigid structure. The aggregates used for making concrete should be free from harmful substances like organic impurities, silt, clay, lignite etc. Typical concrete mixes have high resistance to compressive stresses (about ); however, any appreciable tension (e.g.
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To achieve this, Litchinitser designed crystal lattices with carefully controlled geometries, which allow light to travel perfectly across their surfaces but block it from travelling through the interior. The ability for light to travel around corners is essential for photonic-based microchips, which will be essential for future data transmission. Litchinitser delivered a plenary lecture at the 2018 SPIE Optics and Photonics conference, where she discussed the interaction of structured light and nanostructured media. At the 2020 SPIE Optics and Photonics conference Litchinitser chaired the session on Nanoscience and Engineering.
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In the context of forcing, authors will sometimes also omit the "strong" term and merely refer to antichains. To resolve ambiguities in this case, the weaker type of antichain is called a weak antichain. If (P, ≤) is a partial order and there exist distinct x, y ∈ P such that {x, y} is a strong antichain, then (P, ≤) cannot be a lattice (or even a meet semilattice), since by definition, every two elements in a lattice (or meet semilattice) must have a common lower bound. Thus lattices have only trivial strong antichains (i.e.
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Although most lattice field theories are not exactly solvable, they are of tremendous appeal because they can be studied by simulation on a computer. One hopes that, by performing simulations on larger and larger lattices, while making the lattice spacing smaller and smaller, one will be able to recover the behaviour of the continuum theory. Just as in all lattice models, numerical simulation gives access to field configurations that are not accessible to perturbation theory, such as solitons. Likewise, non-trivial vacuum states can be discovered and probed.
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Suppose and are arbitrary sets and a binary relation over and is given. For any subset of , we define Similarly, for any subset of , define Then and yield an antitone Galois connection between the power sets of and , both ordered by inclusion ⊆. Birkhoff, 1st edition (1940): §32, 3rd edition (1967): Ch. V, §7 and §8 Up to isomorphism all antitone Galois connections between power sets arise in this way. This follows from the "Basic Theorem on Concept Lattices" Ganter, B. and Wille, R. Formal Concept Analysis -- Mathematical Foundations, Springer (1999), .
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A low-temperature process for creating and arranging such fibers, inspired by sponges, could offer more control over the optical properties of the fibers. These nano-structures are also potentially useful for the creation of more efficient, low-cost solar cells. Furthermore, its skeletal structure has inspired a new type of structural lattice with a higher strength to weight ratio than other diagonally reinforced square lattices used in engineering applications. These sponges skeletons have complex geometric configurations, which have been extensively studied for their stiffness, yield strength, and minimal crack propagation.
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A. I. Mal'cev, On a class of homogeneous spaces, AMS Translation No. 39 (1951). Such a subgroup \Gamma as above is called a lattice in N. It is well known that a nilpotent Lie group admits a lattice if and only if its Lie algebra admits a basis with rational structure constants: this is Malcev's criterion. Not all nilpotent Lie groups admit lattices; for more details, see also M. S. Raghunathan. A compact Riemannian nilmanifold is a compact Riemannian manifold which is locally isometric to a nilpotent Lie group with left-invariant metric.
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The fruit of that inspiration is non-Frege logic, one of the key achievements of the post-war Polish logic. One student of Suszko and a colleague of Wolniewicz was a logician Mieczysław Omyła, who continued both logical and ontological work of Suszko and referenced Wolniewicz's situational ontology more than once. Situational ontology also inspires mathematical papers on conditionally distributive lattices by Jan Zygmunt and Jacek Hawranek. Wolniewicz's papers were also met with interest from philosophers and logicians abroad, as evidenced by multiple reviews of his English papers in "Mathematical Review".
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The tool supports Moses-based servers able to provide an enhanced CAT-MT communication. In particular, the GT API is augmented with feedback information provided to the MT engine every time a segment is post-edited as well as enriched MT output, including confidence scores, word lattices, etc. The developed MT server supports multi-threading to serve multiple translators, handles text segments including tags and adapts from the post-edits performed by each user Nicola Bertoldi, Mauro Cettolo, and Marcello Federico. 2013. Cache-based Online Adaptation for Machine Translation Enhanced Computer Assisted Translation.
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This separation, which appears in other Asturian churches, is not repeated in any other with a similar structure. Both the lattices over the arches and the wall enclosing the central arch were re-used from Visigothic origins in the 7th century. On the outside of the church, it is worth noting the large number of buttresses (32) which seem in some cases to have a merely aesthetic function. Nearby this church is the Asturian Pre-Romanesque Information Centre, located in the old Norte de la Cobertoria Railway Station.
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April 1937 survey drawing legend by report by Clarence W. Jahn, Historic American Buildings Survey team Interior It was built using oak and pine timbers, wood that was prepared at a site near Baraboo, Wisconsin. The lattices are very large, made of oak, and were assembled without the use of bolts or nails. Instead, the trusses, consisting of planks, are secured at the joints using oak pegs measuring in diameter, and having rounded ends. The boards used at the sides and ends measure , and have joints covered by battens.
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The deepest point of the tunnel lies 20.32 metres below sea- level. The tunnel was built of sections of rectangular concrete constructions of 24.8 x 8.75 metres, subdivided into two tubes for automobile traffic and in between tubes for cables and pipes. Ventilators, located in two ventilation buildings on the banks of the IJ, blow clean air into the traffic areas via tubes under the surface of the road and openings in the tunnel walls, and suck polluted air out. At the entrance on the north side there are sun-blocking lattices over the road.
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These lattices are missing on the south side, where the Nemobuilding is built on top of the tunnel. The traffic in the tunnel is monitored by 22 closed-circuit cameras. A heating system prevents the forming of ice on the surface of the road, and a computer regulates the intensity of light at the beginning and the end of the tunnel, so that a gradual transition from tunnel light to daylight takes place. The route through the IJtunnel is an urban avenue, formed out of two divided tunnels, each with two lanes of traffic.
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My pen fails to describe in detail the other arts and > rare inventions employed in decorating the woodwork of this palace. Probably > nowhere else in the whole world can wooden houses be built with such > decoration and figure-carving as by the people of this country. The sides of > this palace have been partitioned into wooden lattices of various designs > carved in relief, and adorned, both with and outside, with mirrors of brass, > polished so finely that when sunbeams fall on them, the eye is dazzled by > the flashing back on light. This mansion was completed by 12,000 men working > for one year.
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Rosenthal graduated from Queens College, City University of New York with a B.S. in Mathematics in 1962. In 1963 he obtained an MA in Mathematics and in 1967 a Ph.D. in Mathematics from the University of Michigan; his Ph.D. thesis advisor was Paul Halmos. His thesis, "On lattices of invariant subspaces" concerns operators on Hilbert space, and most of his subsequent research has been in operator theory and related fields. Much of his work has been related to the invariant subspace problem, the still-unsolved problem of the existence of invariant subspaces for bounded linear operators on Hilbert space.
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107 The tractor motors, neatly cowled, projected well clear of the leading edge; the pusher pair were placed in deep cut-outs in the trailing edge. The cowlings of the latter pair were removed after the first few flights to improve cooling. The wing centre section, between the engines, had an all-metal internal structure and the forward part of the skin was also metal, replaced by fabric further aft. The outer wing sections were slightly tapered on the leading edge only, with elliptical tips; they were entirely fabric-covered over a largely wooden structure, though the principal ribs were metal lattices.
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In 2012, Gannon proved that all but the first of the multiplicities are non-negative integral combinations of representations of M24, and Gaberdiel–Persson–Ronellenfitsch–Volpato computed all analogues of generalized moonshine functions, strongly suggesting that some analogue of a holomorphic conformal field theory lies behind Mathieu moonshine. Also in 2012, Cheng, Duncan, and Harvey amassed numerical evidence of an umbral moonshine phenomenon where families of mock modular forms appear to be attached to Niemeier lattices. The special case of the A124 lattice yields Mathieu Moonshine, but in general the phenomenon does not yet have an interpretation in terms of geometry.
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Gases have a negative entropy of solution, due to the decrease in gaseous volume as gas dissolves. Since their enthalpy of solution does not decrease too much with temperature, and their entropy of solution is negative and does not vary appreciably with temperature, most gases are less soluble at higher temperatures. Enthalpy of solvation can help explain why solvation occurs with some ionic lattices but not with others. The difference in energy between that which is necessary to release an ion from its lattice and the energy given off when it combines with a solvent molecule is called the enthalpy change of solution.
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Periodic ordered lattices behave as linear viscoelastic solids when subjected to small amplitude mechanical deformations. Okano's group experimentally correlated the shear modulus to the frequency of standing shear modes using mechanical resonance techniques in the ultrasonic range (40 to 70 kHz). In oscillatory experiments at lower frequencies (< 40 Hz), the fundamental mode of vibration as well as several higher frequency partial overtones (or harmonics) have been observed. Structurally, most systems exhibit a clear instability toward the formation of periodic domains of relatively short-range order Above a critical amplitude of oscillation, plastic deformation is the primary mode of structural rearrangement.
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Left, a model of a DNA tile used to make another two-dimensional periodic lattice. Right, an atomic force micrograph of the assembled lattice.Other arrays: Small nucleic acid complexes can be equipped with sticky ends and combined into larger two-dimensional periodic lattices containing a specific tessellated pattern of the individual molecular tiles. The earliest example of this used double-crossover (DX) complexes as the basic tiles, each containing four sticky ends designed with sequences that caused the DX units to combine into periodic two-dimensional flat sheets that are essentially rigid two- dimensional crystals of DNA.
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Translated from the 1977 French original by John Stillwell. Springer-Verlag, Berlin-New York, 1980. . derives structural algebraic information about groups by studying group actions on simplicial trees. External precursors of geometric group theory include the study of lattices in Lie Groups, especially Mostow rigidity theorem, the study of Kleinian groups, and the progress achieved in low-dimensional topology and hyperbolic geometry in the 1970s and early 1980s, spurred, in particular, by William Thurston's Geometrization program. The emergence of geometric group theory as a distinct area of mathematics is usually traced to the late 1980s and early 1990s.
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His fellow physicists had never stopped nominating him. Franck and Fermi had nominated him in 1947 and 1948 for his work on crystal lattices, and over the years, he had also been nominated for his work on solid state physics, quantum mechanics and other topics. In 1954, he received the prize for "fundamental research in Quantum Mechanics, especially in the statistical interpretation of the wave function"—something that he had worked on alone. In his Nobel lecture he reflected on the philosophical implications of his work: In retirement, he continued scientific work, and produced new editions of his books.
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Kupiainen works on constructive quantum field theory and statistical mechanics. In the 1980s he developed, with Krzysztof Gawedzki, a renormalization group method (RG) for mathematical analysis of field theories and phase transitions for spin systems on lattices. In addition in the 1980s he and Gawedzki did research on conformal field theories, in particular the WZW (Wess-Zumino-Witten) model. Then he was involved in applications of the RG method to other problems in probability theory, the theory of partial differential equations (for example, pattern formation, blow up, and moving fronts in asymptotic solutions of nonlinear parabolic differential equations), and dynamical systems (e.g.
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In 1906, Pierre Weiss introduced the concept of magnetic domains to explain the main properties of ferromagnets. The first attempt at a microscopic description of magnetism was by Wilhelm Lenz and Ernst Ising through the Ising model that described magnetic materials as consisting of a periodic lattice of spins that collectively acquired magnetization. The Ising model was solved exactly to show that spontaneous magnetization cannot occur in one dimension but is possible in higher- dimensional lattices. Further research such as by Bloch on spin waves and Néel on antiferromagnetism led to developing new magnetic materials with applications to magnetic storage devices.
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Another major achievement working with bosons was his realization of matter wave solitons in a BEC. Hulet has also performed pioneering experiments with degenerate Fermi gases. He achieved the first observation of a polarized degenerate Fermi gas, realized a degenerate Bose-Fermi mixture, and studied spin-imbalanced Fermi gases, including a possible realization of the FFLO state in a 1D system. His group has investigated fermions in optical lattices as a model of systems of solid-state physics, and observed short-range antiferromagnetism in a Hubbard system, similar to physics also observed in the cuprate high-temperature superconductors.
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If lattices are incommensurable, friction was not observed, however, if the surfaces are commensurable, friction force is present. At the atomic level, these tribological properties are directly connected with superlubricity. An example of this is given by solid lubricants, such as graphite, MoS2 and Ti3SiC2: this can be explained with the low resistance to shear between layers due to the stratified structure of these solids. Even if at the macroscopic scale friction involves multiple microcontacts with different size and orientation, basing on these experiments one can speculate that a large fraction of contacts will be in superlubric regime.
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His superrigidity theorem from 1975 clarified an area of classical conjectures about the characterisation of arithmetic groups amongst lattices in Lie groups. He was awarded the Fields Medal in 1978, but was not permitted to travel to Helsinki to accept it in person, allegedly due to anti-semitism against Jewish mathematicians in the Soviet Union. His position improved, and in 1979 he visited Bonn, and was later able to travel freely, though he still worked in the Institute of Problems of Information Transmission, a research institute rather than a university. In 1991, Margulis accepted a professorial position at Yale University.
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Many lattices can be defined in such a way that the elements of the lattice are represented by sets, the join operation of the lattice is represented by set union, and the meet operation of the lattice is represented by set intersection. For instance, the Boolean lattice defined from the family of all subsets of a finite set has this property. More generally any finite topological space has a lattice of sets as its family of open sets. Because set unions and intersections obey the distributive law, any lattice defined in this way is a distributive lattice.
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Birkhoff's representation theorem may also be generalized to finite structures other than distributive lattices. In a distributive lattice, the self-dual median operation. :m(x,y,z)=(x\vee y)\wedge(x\vee z)\wedge(y\vee z)=(x\wedge y)\vee(x\wedge z)\vee(y\wedge z) gives rise to a median algebra, and the covering relation of the lattice forms a median graph. Finite median algebras and median graphs have a dual structure as the set of solutions of a 2-satisfiability instance; formulate this structure equivalently as the family of initial stable sets in a mixed graph.
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At ultralight densities a further reduced cubic scaling law E∝ρ3 is common, such as with aerogels and aerogel composites. The dependence of scaling on geometry is seen in periodic lattice-based materials that have nearly ideal E∝ρ scaling, with high node connectedness relative to stochastic foams. These structures have previously been implemented only in relatively dense engineered materials. For the ultralight regime the E∝ρ2 scaling seen in denser stochastic cellular materials applies to electroplated tubular nickel micro-lattices, as well as carbon-based open-cell stochastic foams, including carbon microtube aerographite and graphene cork.
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Montroll had an exceptionally varied career: was a Sterling Research Fellow at Yale University where his work on the Ising model of a ferromagnet led him to solve certain Markov chain problems. Following this he was a Research Associate at Cornell University in 1941–42 where he began his studies of the problem of finding the frequency spectrum of elastic vibrations in crystal lattices. During 1942–43 Montroll was an instructor in physics at Princeton University. In 1943, Montroll was appointed as Head of the Mathematics Research Group at the Kellex Corporation in New York, working on programs associated with the Manhattan Project.
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There are not many other elements which appear to be promising candidates for supporting biological systems and processes as fundamentally as carbon does, for example, processes such as metabolism. The most frequently suggested alternative is silicon. Silicon shares a group in the periodic table with carbon, can also form four valence bonds, and also bonds to itself readily, though generally in the form of crystal lattices rather than long chains. Despite these similarities, silicon is considerably more electropositive than carbon, and silicon compounds do not readily recombine into different permutations in a manner that would plausibly support lifelike processes.
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Considering the sets of local norms and phases as discrete compact groups, spatially distributed in a square lattice, the gradient moments have the basic property of being globally invariant (for rotation and modulation). The primary research on gradient lattices applied to characterize weak wave turbulence from X-ray images of solar active regions was developed in the Department of Astronomy at University of Maryland, College Park, USA. A key line of research on GPA's algorithms and applications has been developed at Lab for Computing and Applied Mathematics (LAC) at National Institute for Space Research (INPE) in Brazil.
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It can be shown that finding collisions in SWIFFT is at least as difficult as finding short vectors in cyclic/ideal lattices in the worst case. By giving a security reduction to the worst-case scenario of a difficult mathematical problem, SWIFFT gives a much stronger security guarantee than most other cryptographic hash functions. Unlike many other provably secure hash functions, the algorithm is quite fast, yielding a throughput of 40Mbit/s on a 3.2 GHz Intel Pentium 4. Although SWIFFT satisfies many desirable cryptographic and statistical properties, it was not designed to be an "all- purpose" cryptographic hash function.
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For points in Euclidean space, a set X is a Meyer set if it is relatively dense and its difference set X − X is uniformly discrete. Equivalently, X is a Meyer set if both X and X − X are Delone. Meyer sets are named after Yves Meyer, who introduced them (with a different but equivalent definition based on harmonic analysis) as a mathematical model for quasicrystals. They include the point sets of lattices, Penrose tilings, and the Minkowski sums of these sets with finite sets.. The Voronoi cells of symmetric Delone sets form space-filling polyhedra called plesiohedra..
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Another way to say this is that empty space in the frame actually denotes space, not (as in many text editors) just the absence of content. Frames being fixed in size scrolling as a form of interaction is eliminated (as the designers felt that scrolling is suboptimal) opting instead for larger aggregates such as documents and programs to be structured as hierarchies (or more generally, lattices) of hypermedia nodes. This flexibility makes it possible to create a document, search, run programs from a tree of frames starting at any frame. In KMS, links are one way and are embedded in frames.
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Wulfenite crystallizes in the tetragonal system and possesses nearly equal axial ratios; as a result, it is considered to be crystallographically similar to scheelite(CaWO4). Wulfenite is classed by a pyramidal-hemihedral (tetragonal dipyramidal) (C4h) crystal symmetry. Therefore, the unit cell is formed by placing points at the vertices and centers of the faces of rhomboids with square bases and the crystallographic axes coincide in directions with the edges of the rhomboids. Two of these lattices interpenetrate such that a point on the first is diagonal to the second and one quarter the distance between the two seconds.
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The term is used widely with this definition that focuses on suprema and there is no common name for the dual property. However, bounded completeness can be expressed in terms of other completeness conditions that are easily dualized (see below). Although concepts with the names "complete" and "bounded" were already defined, confusion is unlikely to occur since one would rarely speak of a "bounded complete poset" when meaning a "bounded cpo" (which is just a "cpo with greatest element"). Likewise, "bounded complete lattice" is almost unambiguous, since one would not state the boundedness property for complete lattices, where it is implied anyway.
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The Dedekind–MacNeille completion is self-dual: the completion of the dual of a partial order is the same as the dual of the completion.. The Dedekind–MacNeille completion of has the same order dimension as does itself.This result is frequently attributed to an unpublished 1961 Harvard University honors thesis by K. A. Baker, "Dimension, join-independence and breadth in partially ordered sets". It was published by . In the category of partially ordered sets and monotonic functions between partially ordered sets, the complete lattices form the injective objects for order-embeddings, and the Dedekind–MacNeille completion of is the injective hull of ..
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Unit cell definition using parallelopiped with lengths a, b, c and angles between the sides given by α, β, γ The lattice constant, or lattice parameter, refers to the physical dimension of unit cells in a crystal lattice. Lattices in three dimensions generally have three lattice constants, referred to as a, b, and c. However, in the special case of cubic crystal structures, all of the constants are equal and are referred to as a. Similarly, in hexagonal crystal structures, the a and b constants are equal, and we only refer to the a and c constants.
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In addition to the question of whether there are other differential lattices, there are several long-standing open problems relating to the rank growth of differential posets. It was conjectured in that if P is a differential poset with vertices at rank n, then : p(n) \le r_n \le F_n, where p(n) is the number of integer partitions of n and is the nth Fibonacci number. In other words, the conjecture states that at every rank, every differential poset has a number of vertices lying between the numbers for Young's lattice and the Young-Fibonacci lattice. The upper bound was proved in .
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In agriculture, the correct experimental design is the root of a good study and the arrangement of treatments within the study is essential because environment largely affects the plots (plants, livestock, microorganisms). These main arrangements can be found in the literature under the names of “lattices”, “incomplete blocks”, “split plot”, “augmented blocks”, and many others. All of the designs might include control plots, determined by the researcher, to provide an error estimation during inference. In clinical studies, the samples are usually smaller than in other biological studies, and in most cases, the environment effect can be controlled or measured.
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However, it differs from rock-salt structure in how the two lattices are positioned relative to one another. The zincblende structure has tetrahedral coordination: Each atom's nearest neighbors consist of four atoms of the opposite type, positioned like the four vertices of a regular tetrahedron. Altogether, the arrangement of atoms in zincblende structure is the same as diamond cubic structure, but with alternating types of atoms at the different lattice sites. Examples of compounds with this structure include zincblende itself, lead(II) nitrate, many compound semiconductors (such as gallium arsenide and cadmium telluride), and a wide array of other binary compounds.
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Historically, research on electrospun fibrous scaffolds dates back to at least the late 1980s when Simon showed that electrospinning could be used to produced nano- and submicron-scale fibrous scaffolds from polymer solutions specifically intended for use as in vitro cell and tissue substrates. This early use of electrospun lattices for cell culture and tissue engineering showed that various cell types would adhere to and proliferate upon polycarbonate fibers. It was noted that as opposed to the flattened morphology typically seen in 2D culture, cells grown on the electrospun fibers exhibited a more rounded 3-dimensional morphology generally observed of tissues in vivo.
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In addition to the compounds where a noble gas atom is involved in a covalent bond, noble gases also form non-covalent compounds. The clathrates, first described in 1949, consist of a noble gas atom trapped within cavities of crystal lattices of certain organic and inorganic substances. The essential condition for their formation is that the guest (noble gas) atoms must be of appropriate size to fit in the cavities of the host crystal lattice. For instance, argon, krypton, and xenon form clathrates with hydroquinone, but helium and neon do not because they are too small or insufficiently polarizable to be retained.
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Each side of the unit cell is 3.57 angstroms in length. A diamond cubic lattice can be thought of as two interpenetrating face-centered cubic lattices with one displaced by 1/4 of the diagonal along a cubic cell, or as one lattice with two atoms associated with each lattice point. Looked at from a crystallographic direction, it is formed of layers stacked in a repeating ABCABC ... pattern. Diamonds can also form an ABAB ... structure, which is known as hexagonal diamond or lonsdaleite, but this is far less common and is formed under different conditions from cubic carbon.
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A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every two elements have a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). An example is given by the natural numbers, partially ordered by divisibility, for which the unique supremum is the least common multiple and the unique infimum is the greatest common divisor. Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities.
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Sarvis was born on September 15, 1976 in Fairfax, Virginia to a father of English and Irish descent and a mother of Chinese descent. Growing up in West Springfield, he attended Thomas Jefferson High School for Science and Technology, a public magnet school and one of Virginia's "Governor's schools". In his senior year at Thomas Jefferson, Sarvis placed fourth in the 1994 Westinghouse Science Talent Search for a theoretical math project studying lattices, winning a $15,000 scholarship. Upon graduating from high school, Sarvis attended Harvard University, pursuing a bachelor's degree in mathematics and graduated in 1998.
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It was mainly a court architecture, and five stages are distinguished: a first period (737–791) from the reign of the king Fáfila to Vermudo I, a second stage comprises the reign of Alfonso II (791–842), entering a stage of stylistic definition. These two first stages receive the name of 'Pre-Ramirense'. The most important example is the church San Julián de los Prados in Oviedo, with an interesting volume system and a complex iconographic fresco program, related narrowly to the Roman mural paintings. Lattices and trifoliate windows in the apse appear for the first time at this stage.
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Conversely, the Cathedral Bridge was a lattice truss bridge with spans up to and a collective width of . It might, however, be mentioned that the Britannia Bridge successfully took increasingly heavy railway trains across the Menai Strait from its opening in 1850 until it was seriously damaged by fire in 1970. Designed by Robert Stephenson as a tubular bridge, the longest spans of the Britannia Bridge measured , with a width of . The wrought iron latticework of the Cathedral Bridge was designed by hydraulic engineer Hermann Lohse and formed an intricate network of diagonal lattices both on the inside and outside of the bridge.
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Most quantizers are based on the one-dimensional integer lattice, but using multi-dimensional lattices reduces the RMS error. The Leech lattice is a good solution to this problem, as the Voronoi cells have a low second moment. The vertex algebra of the two- dimensional conformal field theory describing bosonic string theory, compactified on the 24-dimensional quotient torus R24/Λ24 and orbifolded by a two-element reflection group, provides an explicit construction of the Griess algebra that has the monster group as its automorphism group. This monster vertex algebra was also used to prove the monstrous moonshine conjectures.
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In 1953 National Titanium Pigments Ltd (or Laporte Titanium Ltd) established a titanium dioxide manufacturing plant on the site of the former gun battery. The plant became known as the Battery works. Through the latter part of the 20th century the plant was expanded and modernised, later becoming part of SCM Corporation (1983), Hanson plc (1986), Millennium Chemicals (1996), and Cristal (2007).See Laporte "Battery Works", Stallingborough In the 1960s a number of companies (Doverstrand, Revertex, Harco) developed chemical plants producing synthetic lattices and resins at a site south-east of the Battery Works, also on the estuary foreshore.
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Springer-Verlag, Berlin-New York, 1979. xi+258 pp. In algebra, an ordered semigroup is a semigroup (S,•) together with a partial order ≤ that is compatible with the semigroup operation, meaning that x ≤ y implies z•x ≤ z•y and x•z ≤ y•z for all x, y, z in S. If S is a group and it is ordered as a semigroup, one obtains the notion of ordered group, and similarly if S is a monoid it may be called ordered monoid. Partially ordered vector spaces and vector lattices are important in optimization with multiple objectives.
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Kreindler was born on 15 October 1931 in Brăila, Romania into a Jewish family. She obtained in 1951 a fellowship and spent the next four years in the USSR studying mathematics at the Ural State University, located in Sverdlovsk (nowadays Yekaterinburg). In 1955, she completed a master thesis on "Multiplicative Lattices with Additive Basis" under the supervision of Petr Grigor'evich Kontorovich, before returning to Bucharest to join the faculty of Mathematics at the Polytechnic Institute of Bucharest. Next to her duties as assistant professor, she continued with her research in the field of functional analysis under the guidance of Grigore Moisil.
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In mathematics and probability theory, continuum percolation theory is a branch of mathematics that extends discrete percolation theory to continuous space (often Euclidean space ). More specifically, the underlying points of discrete percolation form types of lattices whereas the underlying points of continuum percolation are often randomly positioned in some continuous space and form a type of point process. For each point, a random shape is frequently placed on it and the shapes overlap each with other to form clumps or components. As in discrete percolation, a common research focus of continuum percolation is studying the conditions of occurrence for infinite or giant components.
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Slender surfaces are bounded lattices specifically intended to simulate wings. The actual model can be viewed as a paper sheet, but their behavior approaches very well to that of thin wings. Open VOGEL employs this kind of surfaces because they require less than a half of the number of panels required to generate a full 3D model, while they achieve very good results in the prediction of the main air-loads (lift, induced drag and parasitic drag through an experimental polar curve). Open VOGEL lets designers construct wings by shaping a set of adjacent quadrilateral macro-panels.
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Lubotzky holds a Maurice and Clara Weil Chair in mathematics at the Einstein Institute of Mathematics of the Hebrew University of Jerusalem. He is known for contributions to geometric group theory, the study of lattices in Lie groups, representation theory of discrete groups and Kazhdan's property (T), the study of subgroup growth and applications of group theory to combinatorics and computer science (expander graphs) and error correcting codes. Lubotzky received the Erdős Prize in 1990.About the Institute Einstein Institute of Mathematics in the years 1994–1996 Lubotzky was the chairman of Einstein Institute of Mathematics at the Hebrew University of Jerusalem.
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The original interest behind the model stemmed from the fact that in simulations on lattices, it is attracted to its critical state, at which point the correlation length of the system and the correlation time of the system go to infinity, without any fine tuning of a system parameter. This contrasts with earlier examples of critical phenomena, such as the phase transitions between solid and liquid, or liquid and gas, where the critical point can only be reached by precise tuning (e.g., of temperature). Hence, in the sandpile model we can say that the criticality is self-organized.
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Jar, Giyan IV type, Western Iran, 2500-2000 BC, earthenware with slip-painted decoration Charger with Charles II in the Boscobel Oak, English, c. 1685. Such large plates, for display rather than use, take slip-trailing to an extreme, building up lattices of thick trails of slip. Slipware is pottery identified by its primary decorating process where slip is placed onto the leather-hard clay body surface before firing by dipping, painting or splashing. Slip is an aqueous suspension of a clay body, which is a mixture of clays and other minerals such as quartz, feldspar and mica.
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In applied mathematics, a weighted planar stochastic lattice (WPSL) is a structure that has properties in common with those of lattices and those of graphs. In general, space-filling planar cellular structures can be useful in a wide variety of seemingly disparate physical and biological systems. Examples include grain in polycrystalline structures, cell texture and tissues in biology, acicular texture in martensite growth, tessellated pavement on ocean shores, soap froths and agricultural land division according to ownership etc. The question of how these structures appear and the understanding of their topological and geometrical properties have always been an interesting proposition among scientists in general and physicists in particular.
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In other words, it is possible that proper classes of the class of all terms have the same meaning and are thus identified in the free construction. However, the equivalence classes for the word problem of complete lattices are "too small", such that the free complete lattice would still be a proper class, which is not allowed. Now one might still hope that there are some useful cases where the set of generators is sufficiently small for a free complete lattice to exist. Unfortunately, the size limit is very low and we have the following theorem: : The free complete lattice on three generators does not exist; it is a proper class.
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His proof observes that the idempotents used in the proofs of Remak and Schmidt can be restricted to module homomorphisms; the remaining details of the proof are largely unchanged. O. Ore unified the proofs from various categories include finite groups, abelian operator groups, rings and algebras by proving the exchange theorem of Wedderburn holds for modular lattices with descending and ascending chain conditions. This proof makes no use of idempotents and does not reprove the transitivity of Remak's theorems. Kurosh's The Theory of Groups and Zassenhaus' The Theory of Groups include the proofs of Schmidt and Ore under the name of Remak–Schmidt but acknowledge Wedderburn and Ore.
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CRR formulae In finance, a lattice model is a technique applied to the valuation of derivatives, where a discrete time model is required. For equity options, a typical example would be pricing an American option, where a decision as to option exercise is required at "all" times (any time) before and including maturity. A continuous model, on the other hand, such as Black–Scholes, would only allow for the valuation of European options, where exercise is on the option's maturity date. For interest rate derivatives lattices are additionally useful in that they address many of the issues encountered with continuous models, such as pull to par.
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Lattice-based cryptography is the generic term for constructions of cryptographic primitives that involve lattices, either in the construction itself or in the security proof. Lattice-based constructions are currently important candidates for post-quantum cryptography. Unlike more widely used and known public-key schemes such as the RSA, Diffie-Hellman or elliptic-curve cryptosystems, which could, theoretically, be easily attacked by a quantum computer, some lattice-based constructions appear to be resistant to attack by both classical and quantum computers. Furthermore, many lattice-based constructions are considered to be secure under the assumption that certain well-studied computational lattice problems cannot be solved efficiently.
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Creutz's research spans a wide variety of topics in particle physics and mathematical physics, but he is best known for his work on lattice QCD. His 1983 textbook Quarks, Gluons, and Lattices was the first full-length textbook on lattice QCD and is considered a classic in the field. Creutz is a fellow of the American Physical Society and was the 2000 recipient of the Aneesur Rahman Prize for Computational Physics "for first demonstrating that properties of QCD could be computed numerically on the lattice through Monte Carlo methods, and for numerous contributions to the field thereafter." In 2009 he received a Humboldt Research Award.
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Pyramids of Guimar Plaza de España) Auditorio de Tenerife, icon of architecture in Canary Islands Tenerife is characterized by an architecture whose best representatives are the local manor houses and also the most humble and common dwellings. This style, while influenced by those of Andalusia and Portugal, nevertheless had a very particular and native character. Of the manor houses, the best examples can be found in La Orotava and in La Laguna, characterized by their balconies and by the existence of interior patios and the widespread use of the wood known as pino tea ("pitch pine"). These houses are characterized by simple façades and wooden lattices with little ornamentation.
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Through marriage, Born is related to jurists Victor Ehrenberg, his father-in-law, and Rudolf von Jhering, his wife's maternal grandfather, as well as Hans Ehrenberg, and is a great uncle of British comedian Ben Elton. By the end of 1913, Born had published 27 papers, including important work on relativity and the dynamics of crystal lattices (3 with Theodore von Karman), which became a book. In 1914, received a letter from Max Planck explaining that a new professor extraordinarius chair of theoretical physics had been created at the University of Berlin. The chair had been offered to Max von Laue, but he had turned it down.
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The Library absorbed smaller lending libraries and outgrew its rooms, renting larger space on the second floor of the new Carpenters' Company hall in 1773. "The Books (inclosed within Wire Lattices) are kept in one large Room," Franklin was informed in London, "and in another handsome Apartment the [scientific] Apparatus is deposited and the Directors meet." On September 5, 1774, the First Continental Congress met on the first floor of Carpenters' Hall, and the Library Company extended members' privileges to all the delegates. The offer was renewed when the Second Continental Congress met the following spring, and again when the delegates to the Constitutional Convention met in 1787.
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DNA nanotechnology is an area of current research that uses the bottom-up, self-assembly approach for nanotechnological goals. DNA nanotechnology uses the unique molecular recognition properties of DNA and other nucleic acids to create self-assembling branched DNA complexes with useful properties. DNA is thus used as a structural material rather than as a carrier of biological information, to make structures such as complex 2D and 3D lattices (both tile-based as well as using the "DNA origami" method) and three-dimensional structures in the shapes of polyhedra. These DNA structures have also been used as templates in the assembly of other molecules such as gold nanoparticles and streptavidin proteins.
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Theoretical understanding of condensed matter physics is closely related to the notion of emergence, wherein complex assemblies of particles behave in ways dramatically different from their individual constituents. For example, a range of phenomena related to high temperature superconductivity are understood poorly, although the microscopic physics of individual electrons and lattices is well known. Similarly, models of condensed matter systems have been studied where collective excitations behave like photons and electrons, thereby describing electromagnetism as an emergent phenomenon. Emergent properties can also occur at the interface between materials: one example is the lanthanum aluminate-strontium titanate interface, where two non-magnetic insulators are joined to create conductivity, superconductivity, and ferromagnetism.
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George Daniel Mostow (July 4, 1923 – April 4, 2017) was an American mathematician, renowned for his contributions to Lie theory. He was the Henry Ford II (emeritus) Professor of Mathematics at Yale University, a member of the National Academy of Sciences, the 49th President of the American Mathematical Society (1987–1988), and former Trustee of the Institute for Advanced Study in Princeton, New Jersey. The rigidity phenomenon for lattices in Lie groups he discovered and explored is known as Mostow rigidity. His work on rigidity played an essential role in the work of three Fields medalists, namely Grigori Margulis, William Thurston, and Grigori Perelman.
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In 2009,Dejter I. J. "Quasiperfect domination in triangular lattices" Discussiones Mathematicae Graph Theory, 29(1) (2009), 179-198. Dejter defined a vertex subset S of a graph G as a quasiperfect dominating set in G if each vertex v of G not in S is adjacent to dv ∈{1,2} vertices of S, and then investigated perfect and quasiperfect dominating sets in the regular tessellation graph of Schläfli symbol {3,6} and in its toroidal quotient graphs, yielding the classification of their perfect dominating sets and most of their quasiperfect dominating sets S with induced components of the form Kν, where ν ∈{1,2,3} depends only on S.
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In the US, the U.S. Geological Survey estimates that the median groundwater concentration is 1 μg/L or less, although some groundwater aquifers, particularly in the western United States, can contain much higher levels. For example, median levels in Nevada were about 8 μg/L but levels of naturally occurring arsenic as high as 1000 μg/L have been measured in the United States in drinking water. Geothermally active zones occur at hotspots where mantle-derived plumes ascend, such as in Hawaii and Yellowstone National Park, USA. Arsenic is an incompatible element (does not fit easily into the lattices of common rock-forming minerals).
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Reconstruction of Salem-Shotwell Covered Bridge eventually began on February 2, 2007 over a section of Rocky Brook which had already been cleared, graded and landscaped for foundation settlement. Due to the amount of usable original materials and in order to fit the new span, the bridge was shortened from 76 feet to 43 feet in length. The Town Lattice truss setup was kept, using many of the pieces which were recovered from Wacoochee Creek. A change was made to the exterior wooden sides covering the lattices, now just draping the bottom section so park visitors can view the brook and other surroundings from inside the bridge.
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In the 1970s Perkins was one of a small group of female artisan glass blowers. In the 1980s Perkins built her own glass studio in Pojoaque, New Mexico where she worked on learning Venetian glassblowing techniques. She further refined her glass making technique while studying with the Murano glass master Lino Tagliapietra While living in New Mexico in the 1980s and 1990s she studied botanical forms including cacti, flower buds and bouquets and began incorporating those forms into her work. In the 1990s she began incorporating bronze, steel bars and iron into her glass work to create larger works including lattices, wreaths, swags and bouquets.
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An increase in alloying agents or carbon content causes an increase in retained austenite. Austenite has much higher stacking-fault energy than martensite or pearlite, lowering the wear resistance and increasing the chances of galling, although some or most of the retained austenite can be transformed into martensite by cold and cryogenic treatments prior to tempering. The martensite forms during a diffusionless transformation, in which the transformation occurs due to shear-stresses created in the crystal lattices rather than by chemical changes that occur during precipitation. The shear-stresses create many defects, or "dislocations," between the crystals, providing less-stressful areas for the carbon atoms to relocate.
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At any vertex, there are six configurations of the arrows which satisfy the ice rule (justifying the name "six-vertex model"). The valid configurations for the (two-dimensional) square lattice are the following: :500px The energy of a state is understood to be a function of the configurations at each vertex. For square lattices, one assumes that the total energy E is given by : E = n_1\epsilon_1 + n_2\epsilon_2 + \ldots + n_6\epsilon_6, for some constants \epsilon_1,\ldots,\epsilon_6, where n_i here denotes the number of vertices with the ith configuration from the above figure. The value \epsilon_i is the energy associated with vertex configuration number i.
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The synagogue is a rectangular, symmetrical building oriented west to east oriented towards Jerusalem, with an ample interior spaces, a posterior support spaces and an exterior towards an elevated area. The principal facade of the synagogue has its foundations, corners and decorative capstones in stone, with rectangular windows framed in geometric lattices. The three registers are divided between second and third floor a by a frieze, that divide rectangular windows from three rounded windows in the pediment. The main doorway includes a Gothic-like portico and doorway flanked by pilasters, used by male members of the congregation, while on the left lateral facade includes another access female members.
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Unlike many other thermococci, T. litoralis is non- motile. Its cell wall consists only of a single S-layer that does not form hexagonal lattices. Additionally, while many thermococcales obligately use sulfur as an electron acceptor in metabolism, T. litoralis only needs sulfur to help stimulate growth, and can live without it. T. litoralis has recently been popularized by the scientific community for its ability to produce an alternative DNA polymerase to the commonly used Taq polymerase. The T. litoralis polymerase, dubbed the vent polymerase, has been shown to have a lower error rate than Taq but due to its proofreading 3’-5’ exonuclease abilities.
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Assuming one atom per lattice point, in a primitive cubic lattice with cube side length a, the sphere radius would be and the atomic packing factor turns out to be about 0.524 (which is quite low). Similarly, in a bcc lattice, the atomic packing factor is 0.680, and in fcc it is 0.740. The fcc value is the highest theoretically possible value for any lattice, although there are other lattices which also achieve the same value, such as hexagonal close packed (hcp) and one version of tetrahedral bcc. As a rule, since atoms in a solid attract each other, the more tightly packed arrangements of atoms tend to be more common.
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In the United States and Europe, solid state became a prominent field through its investigations into semiconductors, superconductivity, nuclear magnetic resonance, and diverse other phenomena. During the early Cold War, research in solid state physics was often not restricted to solids, which led some physicists in the 1970s and 1980s to found the field of condensed matter physics, which organized around common techniques used to investigate solids, liquids, plasmas, and other complex matter. Today, solid-state physics is broadly considered to be the subfield of condensed matter physics, often referred to as hard condensed matter, that focuses on the properties of solids with regular crystal lattices.
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In projective geometry, the Veblen–Young theorem states that a projective geometry of dimension at least 3 is isomorphic to the projective geometry of a vector space over a division ring. This can be restated as saying that the subspaces in the projective geometry correspond to the principal right ideals of a matrix algebra over a division ring. Neumann generalized this to continuous geometries, and more generally to complemented modular lattices, as follows . His theorem states that if a complemented modular lattice L has order at least 4, then the elements of L correspond to the principal right ideals of a von Neumann regular ring.
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Subsequently, the roles of zitterbewegung, antiparticles and the Dirac sea in the chessboard model have been elucidated, and the implications for the Schrödinger equation considered through the non- relativistic limit. Further extensions of the original 2-dimensional spacetime model include features such as improved summation rules and generalized lattices. There has been no consensus on an optimal extension of the chessboard model to a fully four-dimensional space-time. Two distinct classes of extensions exist, those working with a fixed underlying lattice Frank D. Smith, HyperDiamond Feynman Checkerboard in 4-dimensional Spacetime, 1995, arXiv:quant-ph/9503015 and those that embed the two-dimensional case in higher dimension.
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Ooids with radial crystals (such as the aragonitic ooids in the Great Salt Lake, Utah, US) grow by ions extending the lattices of the radial crystals. The mode of growth of ooids with tangential (usually minute needle-like) crystals is less clear. They may be accumulated in a "snowball" fashion from tiny crystals in the sediment or water, or they may crystallize in place on the ooid surface. A hypothesis of growth by accretion (like a snowball) from the polymineralic sediment of fine aragonite, high-magnesium calcite (HMC) and low-magnesium calcite (LMC), must explain how only aragonite needles are added to the ooid cortex.
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Niggli used morphological methods to account for internal structure and, in his 1928 Kristallographische und Strukturtheoretische Grundbegriffe, he took up what is essentially the reverse process, the task of establishing the connection between space lattices and external crystal morphology. The great aim of his life was to integrate the whole field of Earth sciences. In 1920, Niggli became the lead scientist at the ETH's Institut für Mineralogie und Petrographie, where he brought his systematic approach to the study of crystal morphologies using X-ray diffraction. In 1935, Niggli and his doctoral student Werner Nowacki (1909–1988) determined the 73 three-dimensional arithmetic crystal classes (symmorphic space groups).
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Gill's research is concerned with laser frequency stabilisation techniques for very high resolution spectroscopy, and the development of leading-edge optical atomic clocks that look to form the basis of a future redefinition of the SI base unit second. These include optical clocks based on laser-cooled single ions confined in radiofrequency traps and neutral atoms held in optical lattices, and which now reach uncertainties below that of the caesium fountain primary frequency standard. Additionally, he has developed a range of stable lasers and optical metrology instrumentation with application to high technology sectors such as precision engineering and manufacture, space science, satellite navigation, Earth observation, defence and security and optical telecommunications.
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In mathematics, the Barnes–Wall lattice Λ16, discovered by Eric Stephen Barnes and G. E. (Tim) Wall (), is the 16-dimensional positive-definite even integral lattice of discriminant 28 with no norm-2 vectors. It is the sublattice of the Leech lattice fixed by a certain automorphism of order 2, and is analogous to the Coxeter-Todd lattice. The automorphism group of the Barnes-Wall lattice has order 89181388800 = 221 35 52 7 and has structure 21+8 PSO8+(F2). The genus of the Barnes-Wall lattice was described by and contains 24 lattices; all the elements other than the Barnes-Wall lattice have root system of maximal rank 16.
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In string theory, the left-moving and the right-moving excitations are completely decoupled, and it is possible to construct a string theory whose left-moving (counter-clockwise) excitations are treated as a bosonic string propagating in D = 26 dimensions, while the right-moving (clockwise) excitations are treated as a superstring in D = 10 dimensions. The mismatched 16 dimensions must be compactified on an even, self-dual lattice (a discrete subgroup of a linear space). There are two possible even self-dual lattices in 16 dimensions, and it leads to two types of the heterotic string. They differ by the gauge group in 10 dimensions.
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It is one of the first rigidity statements in dynamical systems. In the last two decades Katok has been working on other rigidity phenomena, and in collaboration with several colleagues, made contributions to smooth rigidity and geometric rigidity, to differential and cohomological rigidity of smooth actions of higher-rank abelian groups and of lattices in Lie groups of higher rank, to measure rigidity for group actions and to nonuniformly hyperbolic actions of higher-rank abelian groups. Katok's works on topological properties of nonuniformly hyperbolic dynamical systems. It includes density of periodic points and lower bounds on their number as well as exhaustion of topological entropy by horseshoes.
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The fundamental polygon of the torus, on which the cars move The cars are typically placed on a square lattice that is topologically equivalent to a torus: that is, cars that move off the right edge would reappear on the left edge; and cars that move off the bottom edge would reappear on the top edge. There has also been research in rectangular lattices instead of square ones. For rectangles with coprime dimensions, the intermediate states are self-organized bands of jams and free-flow with detailed geometric structure, that repeat periodically in time. In non-coprime rectangles, the intermediate states are typically disordered rather than periodic.
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She did her Master's at UCD in 1939, and was subsequently awarded a National University of Ireland travelling studentship, which enabled her to undertake research at the University of Edinburgh in Scotland. Two years later, in 1941, she earned her doctorate under the supervision of the celebrated physicist Max Born on the stability of crystal lattices. Returning to Dublin, she became an assistant lecturer at University College Dublin, and was also one of the first three scholars appointed to the brand new Dublin Institute for Advanced Studies (DIAS), in October 1941. While at the DIAS she worked with Paul Dirac, Arthur Eddington and Erwin Schrödinger.
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Markus Greiner is a German physicist and Professor of Physics at Harvard University. Greiner studied under the Nobel Laureate Theodor Hänsch at the Ludwig-Maximilians University and at the Max-Planck-Institute of Quantum Optics, where he received his diploma and PhD in physics for experimental work on Bose-Einstein condensates and bosons in optical lattices. He was involved in the first realization of the quantum phase transition from a superfluid to Mott insulator in a Bose-Hubbard system. He then moved to the United States and conducted postdoctoral research at JILA under Deborah Jin, working on the creation of a fermionic condensate of ultracold atoms.
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The cloister with twisted columns The two-storey cloister of the monastery, which has large capitals with carved scenes, and also relief panels, is considered a masterpiece of Romanesque art, and has been written about extensively, notably by Meyer Schapiro in his Romanesque Art (1977).Schapiro, Meyer, From Mozarabic to Romanesque in Silos, in Selected Papers, volume 2, Romanesque Art, 1977, Chatto & Windus, London, The capitals in the lower cloister are decorated with dragons, centaurs, lattices, and mermaids. There is also an important Romanesque free-standing enthroned Madonna and Child. The cloisters are the only surviving part of the monastery that hasn't changed since its inception.
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The ACC will impose the boundary conditions by solving a system of linear equations. In a rigid model-problem, it will build a unique matrix of coefficients, generate the LU decomposition, and reuse that same matrices to find the circulation in the rings at every time step. For aeroelastic simulations, however, because the relative position of the rings is permanently changing due to the deformation of the lattices, the CC will recalculate the influence matrix and LU decomposition at each time step. The right hand side of the system of equations containing the free-velocity and source-potential terms, is only updated at each time for unsteady problems.
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The family of all monotone Boolean functions on n inputs, together with these two operations, forms a distributive lattice, the lattice given by Birkhoff's representation theorem from the partially ordered set of subsets of the n variables with set inclusion as the partial order. This construction produces the free distributive lattice with n generators.The definition of free distributive lattices used here allows as lattice operations any finite meet and join, including the empty meet and empty join. For the free distributive lattice in which only pairwise meets and joins are allowed, one should eliminate the top and bottom lattice elements and subtract two from the Dedekind numbers.
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Ultracold atoms in optical lattices are considered a standard realization of the Bose–Hubbard model. The ability to tune parameters of the model using simple experimental techniques and the lack of the lattice dynamics which are present in solid-state electronic systems mean that ultracold atoms offer a very clean, controllable realisation of the Bose–Hubbard model. The biggest downside with optical lattice technology is the trap lifetime, with atoms typically only being trapped for a few tens of seconds. To see why ultracold atoms offer such a convenient realisation of Bose-Hubbard physics, we can derive the Bose-Hubbard Hamiltonian starting from the second quantized Hamiltonian which describes a gas of ultracold atoms in the optical lattice potential.
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The crystallographic restriction theorem in its basic form was based on the observation that the rotational symmetries of a crystal are usually limited to 2-fold, 3-fold, 4-fold, and 6-fold. However, quasicrystals can occur with other diffraction pattern symmetries, such as 5-fold; these were not discovered until 1982 by Dan Shechtman.Shechtman et al (1982) Crystals are modeled as discrete lattices, generated by a list of independent finite translations . Because discreteness requires that the spacings between lattice points have a lower bound, the group of rotational symmetries of the lattice at any point must be a finite group (alternatively, the point is the only system allowing for infinite rotational symmetry).
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During diagenesis, pore water is squeezed out of the sediments and, as burial continues and heat increases, water is liberated from clay minerals as the peripheral hydroxyl bonds are broken. As the rock enters the submetamorphic field, generally zeolite facies metamorphism, clay minerals begin to recrystallize into low-temperature metamorphic phyllite minerals such as chlorite, prehnite, pumpellyite, glauconite and so forth. This liberates not only water but incompatible elements attached to the mineral and trapped within crystal lattices. Metals liberated from clay and carbonate minerals as they are changed from clays and low-pressure disordered carbonate forms enters the remaining pore fluid which by this time has become concentrated into what is known as a deep formation brine.
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As describes, Hermann Minkowski was led to a special case of the cube-tiling conjecture from a problem in diophantine approximation. One consequence of Minkowski's theorem is that any lattice (normalized to have determinant one) must contain a nonzero point whose Chebyshev distance to the origin is at most one. The lattices that do not contain a nonzero point with Chebyshev distance strictly less than one are called critical, and the points of a critical lattice form the centers of the cubes in a cube tiling. Minkowski conjectured in 1900 that, whenever a cube tiling has its cubes centered at lattice points in this way, it must contain two cubes that meet face to face.
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Finally, he shows that any bootstrappable somewhat homomorphic encryption scheme can be converted into a fully homomorphic encryption through a recursive self- embedding. For Gentry's "noisy" scheme, the bootstrapping procedure effectively "refreshes" the ciphertext by applying to it the decryption procedure homomorphically, thereby obtaining a new ciphertext that encrypts the same value as before but has lower noise. By "refreshing" the ciphertext periodically whenever the noise grows too large, it is possible to compute an arbitrary number of additions and multiplications without increasing the noise too much. Gentry based the security of his scheme on the assumed hardness of two problems: certain worst-case problems over ideal lattices, and the sparse (or low-weight) subset sum problem.
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Graphite is made of two-dimensional layers in which each carbon is covalently bonded to three other carbons; atoms in other layers are further away and are not nearest neighbours, giving a coordination number of 3. bcc structure Ions with coordination number six comprise the highly symmetrical "rock salt structure". For chemical compounds with regular lattices such as sodium chloride and caesium chloride, a count of the nearest neighbors gives a good picture of the environment of the ions. In sodium chloride each sodium ion has 6 chloride ions as nearest neighbours (at 276 pm) at the corners of an octahedron and each chloride ion has 6 sodium atoms (also at 276 pm) at the corners of an octahedron.
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The study, published in the journal of the Geochemical and Meteoritic Society, used more advanced high resolution electron microscopy than was possible in 1996. A serious difficulty with the claims for a biogenic origin of the magnetites is that the majority of them exhibit topotactic crystallographic relationships with the host carbonates (i.e., there are 3D orientation relationships between the magnetite and carbonate lattices), which is strongly indicative that the magnetites have grown in-situ by a physico-chemical mechanism. While water is no indication of life, many of the meteorites found on Earth have shown water, including NWA 7034 which is a more rare meteorite from the Amazonian period of Martian history.
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It is most commonly used on the street side of the building; however, it may also be used internally on the sahn (courtyard) side. The term "" is sometimes used of similar lattices elsewhere, for instance in a takhtabush. There is a difference between Mashrabiya and Rawshan as recorded in many studies. This research extends Hasan Fathy’s (1986) principle of vernacular architecture by focusing on the Rawshan through an investigation of two criteria: aesthetics and energy efficiency. The paper discusses the views of both the Saudi public and key decision-makers on reviving vernacular architecture in the context of Saudi Arabia’s rapidly developing economy, characterized by relatively high rates of energy consumption and CO2 emissions.
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In mathematics, the genus is a classification of quadratic forms and lattices over the ring of integers. An integral quadratic form is a quadratic form on Zn, or equivalently a free Z-module of finite rank. Two such forms are in the same genus if they are equivalent over the local rings Zp for each prime p and also equivalent over R. Equivalent forms are in the same genus, but the converse does not hold. For example, x2 \+ 82y2 and 2x2 \+ 41y2 are in the same genus but not equivalent over Z. Forms in the same genus have equal discriminant and hence there are only finitely many equivalence classes in a genus.
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Algorithmic self-assembly: DX arrays have been made to form hollow nanotubes 4–20 nm in diameter, essentially two-dimensional lattices which curve back upon themselves. These DNA nanotubes are somewhat similar in size and shape to carbon nanotubes, and while they lack the electrical conductance of carbon nanotubes, DNA nanotubes are more easily modified and connected to other structures. One of many schemes for constructing DNA nanotubes uses a lattice of curved DX tiles that curls around itself and closes into a tube.DNA nanotubes: In an alternative method that allows the circumference to be specified in a simple, modular fashion using single-stranded tiles, the rigidity of the tube is an emergent property.
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The woodcut Depth (pictured) by M. C. Escher reportedly inspired Nadrian Seeman to consider using three-dimensional lattices of DNA to orient hard-to- crystallize molecules. This led to the beginning of the field of DNA nanotechnology. The conceptual foundation for DNA nanotechnology was first laid out by Nadrian Seeman in the early 1980s.History: Seeman's original motivation was to create a three-dimensional DNA lattice for orienting other large molecules, which would simplify their crystallographic study by eliminating the difficult process of obtaining pure crystals. This idea had reportedly come to him in late 1980, after realizing the similarity between the woodcut Depth by M. C. Escher and an array of DNA six-arm junctions.
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Prior to 1962, the only isolated compounds of noble gases were clathrates (including clathrate hydrates); other compounds such as coordination compounds were observed only by spectroscopic means. Clathrates (also known as cage compounds) are compounds of noble gases in which they are trapped within cavities of crystal lattices of certain organic and inorganic substances. The essential condition for their formation is that the guest (noble gas) atoms should be of appropriate size to fit in the cavities of the host crystal lattice; for instance, Ar, Kr, and Xe can form clathrates with crystalline β-quinol, but He and Ne cannot fit because they are too small. As well, Kr and Xe can appear as guests in crystals of melanophlogite.
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Arborist standing next to a redwood named Screaming Titans, described in a chapter of the book – The Lost Valley – within section 4: Love in Zeus. This redwood tree is located in the Grove of Titans The Wild Trees: A Story of Passion and Daring is a non-fiction book by Richard Preston about California's coastal redwoods (Sequoia sempervirens) and the recreational climbers who climbed them. It is a narrative-style collection of stories from climbers who pioneered redwood climbing, including botanist Steve Sillett, lichenologist Marie Antoine, and Michael Taylor. They inadvertently discovered a thriving ecosystem hidden among the tree tops, 60-90 meters (200-300 ft) above, of redwood lattices, berry bushes, bonsai trees, epiphytes, lichens, voles, and salamanders.
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This configuration means that the integration over the whole domain of the Brillouin zone can be reduced to a 48-th part of the whole Brillouin zone. As a crystal structure periodic table shows, there are many elements with a FCC crystal structure, like diamond, silicon and platinum and their Brillouin zones and dispersion relations have this 48-fold symmetry. Two other familiar crystal structures are the body- centered cubic lattice (BCC) and hexagonal closed packed structures (HCP) with cubic and hexagonal lattices, respectively. The BCC structure has the 24-fold pyritohedral symmetry of the point group Th. The HCP structure has the 12-fold prismatic dihedral symmetry of the point group D3h.
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In 1912, The structure of crystalline solids was studied by Max von Laue and Paul Knipping, when they observed the X-ray diffraction pattern of crystals, and concluded that crystals get their structure from periodic lattices of atoms. In 1928, Swiss physicist Felix Bloch provided a wave function solution to the Schrödinger equation with a periodic potential, known as Bloch's theorem. Calculating electronic properties of metals by solving the many-body wavefunction is often computationally hard, and hence, approximation methods are needed to obtain meaningful predictions. The Thomas–Fermi theory, developed in the 1920s, was used to estimate system energy and electronic density by treating the local electron density as a variational parameter.
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She and her research teams developed knowledge about spatial dynamical systems (differential equations, coupled-map lattices, cellular automata and individual based models) to study the behaviour of a wide range of ecosystems. The ideas and theories developed led to significant insights into the role of evolutionarily stable attractors, invasion exponents and phenotype dynamics in a wide range of ecosystems, including the African savannah, marine communities, annual and perennial plants and forests and red grouse. From 1998-2000, she was Director of NERC's Center for Coastal and Marine Science. She also served as a Board Member of the Environment Agency England and Wales and as a member of the Advisory Council for the Campaign for Science and Engineering.
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Different versions of lattice proteins may adopt different types of lattice (typically square and triangular ones), in two or three dimensions, but it has been shown that generic lattices can be used and handled via a uniform approach. Lattice proteins are made to resemble real proteins by introducing an energy function, a set of conditions which specify the interaction energy between beads occupying adjacent lattice sites. The energy function mimics the interactions between amino acids in real proteins, which include steric, hydrophobic and hydrogen bonding effects. The beads are divided into types, and the energy function specifies the interactions depending on the bead type, just as different types of amino acids interact differently.
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Auguste Bravais (; 23 August 1811, Annonay, Ardèche – 30 March 1863, Le Chesnay, France) was a French physicist known for his work in crystallography, the conception of Bravais lattices, and the formulation of Bravais law. Bravais also studied magnetism, the northern lights, meteorology, geobotany, phyllotaxis, astronomy, statistics and hydrography. He studied at the Collège Stanislas in Paris before joining the École Polytechnique in 1829, where he was a classmate of groundbreaking mathematician Évariste Galois, whom Bravais actually beat in a scholastic mathematics competition.Toti Rigatelli, L.: "Evariste Galois 1811–1832" , page 41 Towards the end of his studies he became a naval officer, and sailed on the Finistere in 1832 as well as the Loiret afterwards.
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Dextral slickenside of pyrite The mechanisms of shearing depend on the pressure and temperature of the rock and on the rate of shear which the rock is subjected to. The response of the rock to these conditions determines how it accommodates the deformation. Shear zones which occur in more brittle rheological conditions (cooler, less confining pressure) or at high rates of strain, tend to fail by brittle failure; breaking of minerals, which are ground up into a breccia with a milled texture. Shear zones which occur under brittle-ductile conditions can accommodate much deformation by enacting a series of mechanisms which rely less on fracture of the rock and occur within the minerals and the mineral lattices themselves.
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In contrast, networks with a single well-defined scale are somewhat similar to a lattice in that every node has (roughly) the same degree. Examples of networks with a single scale include the Erdős–Rényi (ER) random graph, random regular graphs, regular lattices, and hypercubes. Some models of growing networks that produce scale-invariant degree distributions are the Barabási–Albert model and the fitness model. In a network with a scale-free degree distribution, some vertices have a degree that is orders of magnitude larger than the average - these vertices are often called "hubs", although this language is misleading as, by definition, there is no inherent threshold above which a node can be viewed as a hub.
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Early iron structures using a Town-type lattice replicated this appearance, leading to the instantly recognisable lattice-work shown in the bridges in Part A of this list. However, design considerations required that an iron (as opposed to a wooden) structure required many of the latticed bars to be stiffened in the third dimension. Thus, on closer examination, the delicate appearance of these early iron lattices is belied by this much more complex stiffening in the thickness or third dimension. This complex stiffening is itself also sometimes described as a ‘lattice girder’, being composed of (typically) two or four parallel flat or angled steel bars, closely spaced but linked by lattice work.
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The use of set membership as a key components of category theory can be generalized to fuzzy sets. This approach which initiated in 1968 shortly after the introduction of fuzzy set theoryJ. A. Goguen "Categories of fuzzy sets : applications of non- Cantorian set theory" PhD Thesis University of California, Berkeley, 1968 led to the development of Goguen categories in the 21st century.Michael Winter "Goguen Categories:A Categorical Approach to L-fuzzy Relations" 2007 Springer Michael Winter "Representation theory of Goguen categories" Fuzzy Sets and Systems Volume 138, Issue 1, 16 August 2003, Pages 85–126 In these categories, rather than using two valued set membership, more general intervals are used, and may be lattices as in L-fuzzy sets.
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No tiling admitted by such a set of tiles can be periodic, simply because no single translation can leave the entire hierarchical structure invariant. Consider Robinson's 1971 tiles: The Robinson Tiles Any tiling by these tiles can only exhibit a hierarchy of square lattices: each orange square is at the corner of a larger orange square, ad infinitum. Any translation must be smaller than some size of square, and so cannot leave any such tiling invariant. A portion of tiling by the Robinson tiles Robinson proves these tiles must form this structure inductively; in effect, the tiles must form blocks which themselves fit together as larger versions of the original tiles, and so on.
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He essentially invented the Chirikov standard map, described its chaotic properties, established its ubiquity, and found a variety of applications. The quantum version of this map provides canonical kicked rotator model and demonstrates the phenomenon of dynamical localization of quantum chaos, which has been observed, for example, in experiments with hydrogen and Rydberg atoms in a microwave field and cold atoms and Bose–Einstein condensates in kicked optical lattices. Boris Chirikov was one of the first teachers in Novosibirsk State University.The photo of B. Chirikov The physical theory of deterministic chaos developed by Boris Chirikov has found applications in solar system dynamics, particle dynamics in accelerators and plasma magnetic traps, and numerous other systems.
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While the difference in price between a call to a land-line and a call to a mobile may not seem much at 9 UK pence, the volumes can be one hundred thousand minutes a day or more, leading to losses of over £250,000 a month. When a carrier's dial code table can contain three thousand items, comparing codes is a critical and complex part of the process. The theory of dial code relationships actually involves the mathematical theory of lattices and code comparisons have to be done with computer software. Number plan management monitors changes in suppliers' dial codes and adds or removes codes from the company's own code tables to improve costs.
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Bronze bust of Huang Kun at Physics department, Peking U Huang Kun (; September 2, 1919 – July 6, 2005) was a Chinese physicist and an academician of the Chinese Academy of Sciences. He was awarded the State Preeminent Science and Technology Award (the highest science award in China) by President Jiang Zemin in 2001. Born in Beijing, China, in 1919, Huang graduated from Yenching University with a degree in physics. In 1948, he earned his PhD from the H. H. Wills Physics Lab of Bristol University in England and continued his postdoctoral studies at Liverpool University, where he coauthored the book Dynamical Theory of Crystal Lattices with Max Born between 1949 and 1951.
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In cryptography, a public key exchange algorithm is a cryptographic algorithm which allows two parties to create and share a secret key, which they can use to encrypt messages between themselves. The ring learning with errors key exchange (RLWE-KEX) is one of a new class of public key exchange algorithms that are designed to be secure against an adversary that possesses a quantum computer. This is important because some public key algorithms in use today will be easily broken by a quantum computer if and when such computers are implemented. RLWE-KEX is one of a set of post-quantum cryptographic algorithms which are based on the difficulty of solving certain mathematical problems involving lattices.
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The network is extended by taking each node in turn and, with probability \textstyle p, adding an edge to a new location \textstyle m nodes distant. The rewiring process allows the model to interpolate between a one- dimensional regular lattice and a two-dimensional regular lattice. When the rewiring probability \textstyle p=0, we have a one-dimensional regular lattice of size \textstyle N. When \textstyle p=1, every node is connected to a new location and the graph is essentially a two-dimensional lattice with \textstyle m and \textstyle N/m nodes in each direction. For \textstyle p between \textstyle 0 and \textstyle 1, we have a graph which interpolates between the one and two dimensional regular lattices.
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Siegel gave a more general formula that counts the weighted number of representations of one quadratic form by forms in some genus; the Smith–Minkowski–Siegel mass formula is the special case when one form is the zero form. Tamagawa showed that the mass formula was equivalent to the statement that the Tamagawa number of the orthogonal group is 2, which is equivalent to saying that the Tamagawa number of its simply connected cover the spin group is 1. André Weil conjectured more generally that the Tamagawa number of any simply connected semisimple group is 1, and this conjecture was proved by Kottwitz in 1988. gave a mass formula for unimodular lattices without roots (or with given root system).
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One finds linenfold there too, a Gothic motif par excellence, but also, on the door stop, a salamander, an emblem of François I. Above, Louis de la Saussaye had engraved, in Greek, the sentence "Small is the house, but oh how much happiness, if it is filled with friends." ascribed to the general Themistocles. Le marmouset The tower dates from the 19th century. Modeled on the towers of the Louis XII wing of the Château de Blois, it shares their most notable characteristic: lattices of red and black bricks. Above a François I shell, a small marmouset commemorates the construction with a banner in Latin: "united by friendship, Louis de la Saussaye wanted, Jules de la Morandière realized".
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This theorem has been generalized by to a tight bound on the dimension of the height-three partially ordered sets formed analogously from the vertices, edges and faces of a convex polyhedron, or more generally of an embedded planar graph: in both cases, the order dimension of the poset is at most four. However, this result cannot be generalized to higher-dimensional convex polytopes, as there exist four-dimensional polytopes whose face lattices have unbounded order dimension. Even more generally, for abstract simplicial complexes, the order dimension of the face poset of the complex is at most , where is the minimum dimension of a Euclidean space in which the complex has a geometric realization .
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Thus any continuous function f from a space X to a space Y defines an inverse mapping f −1 from Ω(Y) to Ω(X). Furthermore, it is easy to check that f −1 (like any inverse image map) preserves finite intersections and arbitrary unions and therefore is a morphism of frames. If we define Ω(f) = f −1 then Ω becomes a contravariant functor from the category Top to the category Frm of frames and frame morphisms. Using the tools of category theory, the task of finding a characterization of topological spaces in terms of their open set lattices is equivalent to finding a functor from Frm to Top which is adjoint to Ω.
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Eric Simon, in a 1988 NIH SBIR grant report, showed that electrospinning could be used to produced nano- and submicron-scale polymeric fibrous scaffolds specifically intended for use as in vitro cell and tissue substrates. This early use of electrospun fibrous lattices for cell culture and tissue engineering showed that various cell types would adhere to and proliferate upon polycarbonate fibers. It was noted that as opposed to the flattened morphology typically seen in 2D culture, cells grown on the electrospun fibers exhibited a more rounded 3-dimensional morphology generally observed of tissues in vivo. Plant tissue culture in particular is concerned with the growing of entire plants from small pieces of plant tissue, cultured in medium.
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In graph theory, a voltage graph is a directed graph whose edges are labelled invertibly by elements of a group. It is formally identical to a gain graph, but it is generally used in topological graph theory as a concise way to specify another graph called the derived graph of the voltage graph. Typical choices of the groups used for voltage graphs include the two-element group ℤ2 (for defining the bipartite double cover of a graph), free groups (for defining the universal cover of a graph), d-dimensional integer lattices ℤd (viewed as a group under vector addition, for defining periodic structures in d-dimensional Euclidean space),; ; . and finite cyclic groups ℤn for n > 2\.
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The spin ice model is only one subdivision of frustrated systems. The word frustration was initially introduced to describe a system's inability to simultaneously minimize the competing interaction energy between its components. In general frustration is caused either by competing interactions due to site disorder (see also the Villain model or by lattice structure such as in the triangular, face-centered cubic (fcc), hexagonal-close-packed, tetrahedron, pyrochlore and kagome lattices with antiferromagnetic interaction. So frustration is divided into two categories: the first corresponds to the spin glass, which has both disorder in structure and frustration in spin; the second is the geometrical frustration with an ordered lattice structure and frustration of spin.
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In the theory of random surfaces, it is also called the harmonic crystal. It is also the starting point for many constructions in quantum field theory, where it is called the Euclidean bosonic massless free field. A key property of the 2-dimensional GFF is conformal invariance, which relates it in several ways to the Schramm-Loewner Evolution, see and . Similarly to Brownian motion, which is the scaling limit of a wide range of discrete random walk models (see Donsker's theorem), the continuum GFF is the scaling limit of not only the discrete GFF on lattices, but of many random height function models, such as the height function of uniform random planar domino tilings, see .
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It turns out that every solution of the Yang–Baxter equation with spectral parameters in a tensor product of vector spaces V\otimes V yields an exactly-solvable vertex model. right Although the model can be applied to various geometries in any number of dimensions, with any number of possible states for a given bond, the most fundamental examples occur for two dimensional lattices, the simplest being a square lattice where each bond has two possible states. In this model, every particle is connected to four other particles, and each of the four bonds adjacent to the particle has two possible states, indicated by the direction of an arrow on the bond. In this model, each vertex can adopt 2^4 possible configurations.
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"History of Engineering", The Canadian Encyclopedia"Canada's Nuclear Achievement: Technical and Economic Perspectives", IEEE Canada"Canadian Engineering Achievements", University of Waterloo"Candu: The Canadian Nuclear Reactor", CBC Digital Archives ZEEP was used to test reactivity effects and other physics parameters needed for reactor development at Chalk River Laboratories, including fuel lattices for the NRU reactor situated next door. ZEEP was one of the world's first heavy water reactors, and it was also designed to use natural (unenriched) uranium; a feature carried through to the CANDU design. Uranium enrichment is a complex and expensive process; thus, the ability to use unenriched uranium gave ZEEP and its descendants a number of distinct advantages. ZEEP continued to be used for basic research until 1970.
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If a complete lattice is freely generated from a given poset used in place of the set of generators considered above, then one speaks of a completion of the poset. The definition of the result of this operation is similar to the above definition of free objects, where "sets" and "functions" are replaced by "posets" and "monotone mappings". Likewise, one can describe the completion process as a functor from the category of posets with monotone functions to some category of complete lattices with appropriate morphisms that is left adjoint to the forgetful functor in the converse direction. As long as one considers meet- or join-preserving functions as morphisms, this can easily be achieved through the so-called Dedekind–MacNeille completion.
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It was not until four years after in 1808 the same patterns were discovered by Count Alois von Beckh Widmanstätten who was heating iron meteorites when he noticed geometric patterns caused by the differing oxidation rates of kamacite and taenite. Widmanstätten told many of his colleagues about these patterns in correspondence leading to them being referred to as Widmanstätten patterns in most literature. Thomson structures or Widmanstätten patterns are created as the meteorite cools; at high temperatures both iron and nickel have face- centered lattices. When the meteorite is formed it starts out as entirely molten taenite (greater than 1500 °C) and as it cools past 723 °C the primary metastable phase of the alloy changes into taenite and kamacite begins to precipitate out.
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The JTE is usually associated with degeneracies that are well localised in space, like those occurring in a small molecule or associated to an isolated transition metal complex. However, in many periodic high-symmetry solid-state systems, like perovskites, some crystalline sites allow for electronic degeneracy giving rise under adequate compositions to lattices of JT-active centers. This can produce a cooperative JTE, where global distortions of the crystal occur due to local degeneracies. In order to determine the final electronic and geometric structure of a cooperative JT system, it is necessary to take into account both the local distortions and the interaction between the different sites, which will take such form necessary to minimise the global energy of the crystal.
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Bray J., "Innovation and the Communications Revolution", The IEE, London, 2002. George Ashley Campbell was a key contributor to this new filter theory, as was Otto Julius Zobel. They and many colleagues worked at the laboratories of Western Electric and the American Telephone and Telegraph Co., and their work was reported in the early editions of the Bell System Technical Journal. Campbell discussed lattice filters in his article of 1922, while other early workers with an interest in the lattice included JohnsonJohnson K.S., "Lattice type wave filters", US Patent 1,501,667, 1924 and Bartlett.Bartlett A.C., "Lattice Type Filters", British Patent 253,629 Zobel's article on filter theory and design, published at about this time, mentioned lattices only briefly, with his main emphasis on ladder networks.
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Although much of Kuperberg's early mathematical work is in topology, he is best known today for his work in geometry, and in particular on packing and covering problems. His first paper in this area (1982) showed that the ratio of packing density to covering density of any convex body in the plane is at least 3/4. His 1990 paper on double lattices with his son Greg provides the best lower bound known at that time for packing densities of arbitrary two-dimensional convex bodies; with Bezdek (1990) he calculated the exact packing density of the infinite cylinder, which prior to Hales' 1998 solution of the Kepler conjecture was the first nontrivial calculation of the packing density of any three-dimensional convex body.
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Paving matroids were initially studied by , in their equivalent formulation in terms of d-partitions; Hartmanis called them generalized partition lattices. In their 1970 book Combinatorial Geometries, Henry Crapo and Gian-Carlo Rota observed that these structures were matroidal; the name "paving matroid" was introduced by following a suggestion of Rota. The simpler structure of paving matroids, compared to arbitrary matroids, has allowed some facts about them to be proven that remain elusive in the more general case. An example is Rota's basis conjecture, the statement that a set of n disjoint bases in a rank-n matroid can be arranged into an n × n matrix so that the rows of the matrix are the given bases and the columns are also bases.
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Blundy is most noted for advancing the understanding of how magmas are generated in the Earth's crust and mantle and of the processes that occur in volcanoes before they erupt. He undertook his PhD research at the University of Cambridge under the supervision of Professor Robert Stephen John Sparks on the granites of Adamello-Presanella in the Italian Alps. In series of seminal papers with Professor Bernard John Wood in the 1990s Blundy developed a theory of elastic strain to describe the uptake of trace elements into the crystal lattices of igneous minerals. The theory was based on high temperature and pressure experiments on molten rocks, and is now widely used to predict crystal-melt partition coefficients for use in modelling magmatic processes.
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The median of three vertices in a median graph In graph theory, a division of mathematics, a median graph is an undirected graph in which every three vertices a, b, and c have a unique median: a vertex m(a,b,c) that belongs to shortest paths between each pair of a, b, and c. The concept of median graphs has long been studied, for instance by or (more explicitly) by , but the first paper to call them "median graphs" appears to be . As Chung, Graham, and Saks write, "median graphs arise naturally in the study of ordered sets and discrete distributive lattices, and have an extensive literature".. In phylogenetics, the Buneman graph representing all maximum parsimony evolutionary trees is a median graph.; ; .
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Similarly, each edge of the dual corresponds to an edge of the primal, and each face of the dual corresponds to a vertex of the primal. These correspondences are incidence-preserving: if two parts of the primal polyhedron touch each other, so do the corresponding two parts of the dual polyhedron. More generally, using the concept of polar reciprocation, any convex polyhedron, or more generally any convex polytope, corresponds to a dual polyhedron or dual polytope, with an -dimensional feature of an -dimensional polytope corresponding to an -dimensional feature of the dual polytope. The incidence-preserving nature of the duality is reflected in the fact that the face lattices of the primal and dual polyhedra or polytopes are themselves order-theoretic duals.
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Although concepts of degeneracy and coloring number are frequently considered in the context of finite graphs, the original motivation for was the theory of infinite graphs. For an infinite graph G, one may define the coloring number analogously to the definition for finite graphs, as the smallest cardinal number α such that there exists a well-ordering of the vertices of G in which each vertex has fewer than α neighbors that are earlier in the ordering. The inequality between coloring and chromatic numbers holds also in this infinite setting; state that, at the time of publication of their paper, it was already well known. The degeneracy of random subsets of infinite lattices has been studied under the name of bootstrap percolation.
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He said that "most rewarding are the moments when the band breaks loose from their throbbing chords to deliver a crunchy punch to the solar plexus, such as on "Plot Twist" and "Silver Lining." Mostly, though, the effect is of rich lattices of guitar and intermingling swells, lifting and propelling the songs through an omnipresent guitar drone." Michael Petitti of Tucson Weekly, in another retrospective commentary, said the album was "gorgeous [and] buzzing" and "perhaps [the] finest hour by the Susans." In 2014, music journalist Andrew Earles included the album in his book Gimme Indie Rock: 500 Essential American Underground Rock Albums 1981–1996, one of two Band of Susans albums included in the list, alongside Here Comes Success (1995).
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On 18 September 1745, the "Young Pretender" Charles Edward Stuart had his father proclaimed King James VIII of Scotland and himself Regent at the Cross. According to Robert Chambers in his History of the Rebellion of 1745, "The ladies, who viewed the scene from their lofty lattices in the High Street, strained their voices in acclamation, and waved white handkerchiefs in honour of the day", cite but another history claims that "few gentlemen were, however, to be seen in the streets or at the windows, and even among the common people, there were not a few who preserved a stubborn silence". cites Following the Prince's defeat the following year at Culloden, the Jacobite colours captured in the battle were ceremoniously burned at the Cross.
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Prior to the availability of private biometrics, research focused on ensuring the prover's biometric would be protected against misuse by a dishonest verifier through the use of partially homomorphic data or decrypted(plaintext) data coupled with a private verification function intended to shield private data from the verifier. This method introduced a computational and communication overhead which was computationally inexpensive for 1:1 verification but proved infeasible for large 1:many identification requirements. From 1998 to 2018 cryptographic researchers pursued four independent approaches to solve the problem: cancelable biometrics, BioHashing, Biometric Cryptosystems, and two-way partially homomorphic encryption. Yasuda M., Shimoyama T., Kogure J., Yokoyama K., Koshiba T. (2013) Packed Homomorphic Encryption Based on Ideal Lattices and Its Application to Biometrics.
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In the body-centred cubic arrangement, there is an iron atom in the centre of each cube, and in the face-centred cubic, there is one at the center of each of the six faces of the cube. It is the interaction of the allotropes of iron with the alloying elements that gives iron-hydrogen alloy its range of unique properties. In pure iron, the crystal structure has relatively little resistance to the iron atoms slipping past one another, and so pure iron is quite ductile, or soft and easily formed. In iron hydride, small amounts of hydrogen within the iron act as a softening agent that promote the movement of dislocations that are common in the crystal lattices of iron atoms.
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Often, these metaphors are used as a visual shorthand in explanations as they allow one to refer to the Internet as a definite object without having to explain the intricate details of its functioning. Clouds are the most common of abstract metaphors employed for this purpose in cloud computing and have been used since the creation of the Internet. Other abstract metaphors of the Internet draw on the fractal branching of trees and leaves, and the lattices of coral and webs, while others are based on the aesthetics of astronomy such as gas nebulas, and star clusters. Technical methods such as algorithms are often used to create huge, complex graphs or maps of raw data from networks and the topology of connections.
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Each office is divided into two parts—one for the MSP, with a floor space of 15 square metres (160 sq ft) and another part for their staff, which has a floor space of 12 square metres (130 sq ft). The most distinctive feature of the MSP block are the unusual windows which project out from the building onto the western elevation of the parliamentary complex, inspired by a combination of the repeated leaf motif and the traditional Scottish stepped gable. In each office, these bay windows have a seat and shelving and are intended as "contemplation spaces". Constructed from stainless steel and framed in oak, with oak lattices covering the glass, the windows are designed to provide MSPs with privacy and shade from the sun.
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In mathematics, especially in order theory, a partially ordered set with a unique minimal element 0 has the disjunction property of Wallman when for every pair (a, b) of elements of the poset, either b ≤ a or there exists an element c ≤ b such that c ≠ 0 and c has no nontrivial common predecessor with a. That is, in the latter case, the only x with x ≤ a and x ≤ c is x = 0. A version of this property for lattices was introduced by , in a paper showing that the homology theory of a topological space could be defined in terms of its distributive lattice of closed sets. He observed that the inclusion order on the closed sets of a T1 space has the disjunction property.
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Like the majority of ice phases (including ice Ih), the hydrogen atom positions are disordered.. In addition, the oxygen atoms are disordered over multiple sites.... The structure of ice VII comprises a hydrogen bond framework in the form of two interpenetrating (but non-bonded) sublattices. Hydrogen bonds pass through the center of the water hexamers and thus do not connect the two lattices. Ice VII has a density of about 1.65 g cm−3 (at 2.5 GPa and ),D. Eisenberg and W. Kauzmann, The structure and properties of water (Oxford University Press, London, 1969); (b) The dodecahedral interstitial model is described in L. Pauling, The structure of water, In Hydrogen bonding, Ed. D. Hadzi and H. W. Thompson (Pergamon Press Ltd, London, 1959) pp 1–6.
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The Sommer’s sector (CA1) of the hippocampus has been described to be influential in the formation of new memories, as well as, containing inclusion bodies that contribute to a hallmark of Alzheimer’s disease (AD), intellectual deficit. Alzheimer’s neurofibrillary tangles show a preference to form in the CA1, which is one of the major areas in which Hb’s have been observed. There are a larger number of Hb’s found in people with Alzheimer’s disease than those without the disease. Additionally many processes of Alzheimer’s neurofibrillary tangles have been observed to contain Hirano bodies. Hirano bodies are described as cytoplasmic paracrystalline lattices, which are a main form of a pathological feature seen in a broad spectrum of neurodegenerative diseases, such as Alzheimer’s disease (AD).
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In mathematics, in the theory of discrete groups, superrigidity is a concept designed to show how a linear representation ρ of a discrete group Γ inside an algebraic group G can, under some circumstances, be as good as a representation of G itself. That this phenomenon happens for certain broadly defined classes of lattices inside semisimple groups was the discovery of Grigory Margulis, who proved some fundamental results in this direction. There is more than one result that goes by the name of Margulis superrigidity. One simplified statement is this: take G to be a simply connected semisimple real algebraic group in GLn, such that the Lie group of its real points has real rank at least 2 and no compact factors.
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For instance, the square lattice is replaced by a circle,, Self-dual property of the Potts model in one dimension, F. Y. Wu random lattice,, Dirac operator and Ising model on a compact 2D random lattice, L.Bogacz, Z.Burda, J.Jurkiewicz, A.Krzywicki, C.Petersen, B.Petersson nonhomogeneous torus,, Duality of the 2D Nonhomogeneous Ising Model on the Torus, A.I. Bugrij, V.N. Shadura triangular lattice,, Selfduality for coupled Potts models on the triangular lattice, Jean-Francois Richard, Jesper Lykke Jacobsen, Marco Picco labyrinth,, A critical Ising model on the Labyrinth, M. Baake, U. Grimm, R. J. Baxter lattices with twisted boundaries,, Duality and conformal twisted boundaries in the Ising model, Uwe Grimm chiral Potts model,, Duality and Symmetry in Chiral Potts Model, Shi-shyr Roan and many others.
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More generally, a lattice Γ in a Lie group G is a discrete subgroup, such that the quotient G/Γ is of finite measure, for the measure on it inherited from Haar measure on G (left-invariant, or right-invariant—the definition is independent of that choice). That will certainly be the case when G/Γ is compact, but that sufficient condition is not necessary, as is shown by the case of the modular group in SL2(R), which is a lattice but where the quotient isn't compact (it has cusps). There are general results stating the existence of lattices in Lie groups. A lattice is said to be uniform or cocompact if G/Γ is compact; otherwise the lattice is called non-uniform.
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The expanded Bravais lattice concept, including the unit cell, is used to formally define a crystalline arrangement and its (finite) frontiers. A crystal is made up of a periodic arrangement of one or more atoms (the basis or motif) occurring exactly once in each primitive unit cell. The basis may consist of atoms, molecules, or polymer strings of solid matter, Consequently, the crystal looks the same when viewed in any given direction from any equivalent points in two different unit cells (two points in two different unit cells of the same lattice are equivalent if they have the same relative position with respect to their individual unit cell boundaries). Two Bravais lattices are often considered equivalent if they have isomorphic symmetry groups.
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In mathematics, the Smith–Minkowski–Siegel mass formula (or Minkowski–Siegel mass formula) is a formula for the sum of the weights of the lattices (quadratic forms) in a genus, weighted by the reciprocals of the orders of their automorphism groups. The mass formula is often given for integral quadratic forms, though it can be generalized to quadratic forms over any algebraic number field. In 0 and 1 dimensions the mass formula is trivial, in 2 dimensions it is essentially equivalent to Dirichlet's class number formulas for imaginary quadratic fields, and in 3 dimensions some partial results were given by Gotthold Eisenstein. The mass formula in higher dimensions was first given by , though his results were forgotten for many years.
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Re scoring is usually done by trying to minimize the Bayes risk (or an approximation thereof): Instead of taking the source sentence with maximal probability, we try to take the sentence that minimizes the expectancy of a given loss function with regards to all possible transcriptions (i.e., we take the sentence that minimizes the average distance to other possible sentences weighted by their estimated probability). The loss function is usually the Levenshtein distance, though it can be different distances for specific tasks; the set of possible transcriptions is, of course, pruned to maintain tractability. Efficient algorithms have been devised to re score lattices represented as weighted finite state transducers with edit distances represented themselves as a finite state transducer verifying certain assumptions.
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A high salt concentration, which can be achieved by adding or increasing the concentration of ammonium sulfate in a solution, enables protein separation based on a decrease in protein solubility; this separation may be achieved by centrifugation. Precipitation by ammonium sulfate is a result of a reduction in solubility rather than protein denaturation, thus the precipitated protein can be solubilized through the use of standard buffers. Ammonium sulfate precipitation provides a convenient and simple means to fractionate complex protein mixtures. In the analysis of rubber lattices, volatile fatty acids are analyzed by precipitating rubber with a 35% ammonium sulfate solution, which leaves a clear liquid from which volatile fatty acids are regenerated with sulfuric acid and then distilled with steam.
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The green circles show the isoradial constraint on both the original and dual lattices. The yellow polygons highlight the three types of polygons on the lattice, and the pink polygons highlight the two types of polygons on the dual lattice. The lattice has vertex types (1/2)(33,42) + (1/2)(3,4,6,4), while the dual lattice has vertex types (1/15)(46)+(6/15)(42,52)+(2/15)(53)+(6/15)(52,4). The critical point is where the longer bonds (on both the lattice and dual lattice) have occupation probability p = 2 sin (π/18) = 0.347296... which is the bond percolation threshold on a triangular lattice, and the shorter bonds have occupation probability 1 − 2 sin(π/18) = 0.652703..., which is the bond percolation on a hexagonal lattice.
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Additionally, CMM requirements were challenging in achieving high levels of accuracy and lack of deviation in the gap between the curved hood and the curved headlamp housings, where the LS400 featured rectangular lamps that were more traditional. Toyota also felt it was important to blend the bumper into the metal fenders with minimal gaps and differentiation in visual cohesion. Engineers also adopted many influences from the LS400 and Lexus ES300 such as alloy lattices and metallurgical eutectic techniques that led to engines with low thermal expansion coefficient, better fuel atomization, rust corrosion, and flusher body panels. The development of reducing NVH levels in the cabin was the result of hundreds of engineers, which was revealed by Chris Goffey during a Top Gear review.
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Simon, in a 1988 NIH SBIR grant report, showed that electrospinning could be used to produced nano- and submicron-scale polystyrene and polycarbonate fibrous mats specifically intended for use as in vitro cell substrates. This early use of electrospun fibrous lattices for cell culture and tissue engineering showed that Human Foreskin Fibroblasts (HFF), transformed Human Carcinoma (HEp-2), and Mink Lung Epithelium (MLE) would adhere to and proliferate upon the fibers. Nanofiber scaffolds are used in bone tissue engineering to mimic the natural extracellular matrix of the bones. The bone tissue is arranged either in a compact or trabecular pattern and composed of organized structures that vary in length from the centimeter range all the way to the nanometer scale.
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Frank den Hollander studied theoretical physics at Leiden University with undergraduate degree and M.Sc. in 1980 and a Ph.D. in 1985 with thesis advisor Pieter Kasteleyn and thesis Random Walks on Random Lattices. As a postdoc he studied from 1985 to 1989 with Michael Keane at Delft Technical University (TU Delft) and from 1989 to 1991 was at TU Delft on a scholarship from the Royal Netherlands Academy of Arts and Sciences. Den Hollander was from 1991 to 1994 an associate professor at Utrecht University and from 1994 to 2000 a professor of probability and statistics at the Radboud University Nijmegen. He was from 2000 to 2005 a professor at Eindhoven University of Technology (TU Eindhoven) and scientific director of EURANDOM (TU Eindhoven's center for stochastic sciences).
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Vortices in Bose–Einstein condensates are also currently the subject of analogue gravity research, studying the possibility of modeling black holes and their related phenomena in such environments in the laboratory. Experimenters have also realized "optical lattices", where the interference pattern from overlapping lasers provides a periodic potential. These have been used to explore the transition between a superfluid and a Mott insulator, and may be useful in studying Bose–Einstein condensation in fewer than three dimensions, for example the Tonks–Girardeau gas. Further, the sensitivity of the pinning transition of strongly interacting bosons confined in a shallow one- dimensional optical lattice originally observed by Haller has been explored via a tweaking of the primary optical lattice by a secondary weaker one.
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The BCS formalism is applicable without modifications to the description of quark matter with color group SU(2), where Cooper pairs are colorless. The Nambu-Jona-Lasinio model predicts the existence of the superconducting phase of SU(2) color quark matter at high densities . This physical picture is confirmed in the Polyakov-Nambu-Jona-Lasinio model , and also in lattice QCD models , in which the properties of cold quark matter can be described based on the first principles of quantum chromodynamics. The possibility of modeling on the lattices of two-color QCD at finite chemical potentials for even numbers of the quark flavors is associated with the positive-definiteness of the integral measure and the absence of a sign problem.
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Hecke operators can be realized in a number of contexts. The simplest meaning is combinatorial, namely as taking for a given integer n some function f(Λ) defined on the lattices of fixed rank to :\sum f(\Lambda') with the sum taken over all the Λ′ that are subgroups of Λ of index n. For example, with n=2 and two dimensions, there are three such Λ′. Modular forms are particular kinds of functions of a lattice, subject to conditions making them analytic functions and homogeneous with respect to homotheties, as well as moderate growth at infinity; these conditions are preserved by the summation, and so Hecke operators preserve the space of modular forms of a given weight. Another way to express Hecke operators is by means of double cosets in the modular group.
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Overview: History: See for a statement of the problem, and for the proposed solution. Several natural branched DNA structures were known at the time, including the DNA replication fork and the mobile Holliday junction, but Seeman's insight was that immobile nucleic acid junctions could be created by properly designing the strand sequences to remove symmetry in the assembled molecule, and that these immobile junctions could in principle be combined into rigid crystalline lattices. The first theoretical paper proposing this scheme was published in 1982, and the first experimental demonstration of an immobile DNA junction was published the following year.Overview: In 1991, Seeman's laboratory published a report on the synthesis of a cube made of DNA, the first synthetic three-dimensional nucleic acid nanostructure, for which he received the 1995 Feynman Prize in Nanotechnology.
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Animation about the magnetoresistance discovery Graphs Sampathkumaran's early research was focused on intermetallics and oxides, with regard to their solid state properties and his work led to the identification of new fluctuating-valent rare-earth systems. The thermal and transport behaviour of magnetic systems, superconductivity, physics of d- and f-electron systems, Kondo lattices, geometrically frustrated magnetism, spin- chain magnetism, multiferroics and nanomagnetism have been some of the other areas of his work. His studies have been documented by way of a number of articles and the online article repository of the Indian Academy of Sciences has listed 302 of them. Sampathkumaran is a member of the editorial boards of Solid State Communications and Journal of Magnetism and Magnetic Materials of Elsevier and Scientific Reports journal of Nature.
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In 2012, amassed numerical evidence of an extension of Mathieu moonshine, where families of mock modular forms were attached to divisors of 24. After some group-theoretic discussion with Glauberman, found that this earlier extension was a special case (the A-series) of a more natural encoding by Niemeier lattices. For each Niemeier root system X, with corresponding lattice LX, they defined an umbral group GX, given by the quotient of the automorphism group of LX by the subgroup of reflections- these are also known as the stabilizers of deep holes in the Leech lattice. They conjectured that for each X, there is an infinite dimensional graded representation KX of GX, such that the characters of elements are given by a list of vector-valued mock modular forms that they computed.
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In particular, they include the laws of quantum mechanics, electromagnetism and statistical mechanics. More exotic condensed phases include the superconducting phase exhibited by certain materials at low temperature, the ferromagnetic and antiferromagnetic phases of spins on crystal lattices of atoms, and the Bose–Einstein condensate found in ultracold atomic systems. The study of condensed matter physics involves measuring various material properties via experimental probes along with using methods of theoretical physics to develop mathematical models that help in understanding physical behavior. The diversity of systems and phenomena available for study makes condensed matter physics the most active field of contemporary physics: one third of all American physicists self-identify as condensed matter physicists, and the Division of Condensed Matter Physics is the largest division at the American Physical Society.
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The grids recall the lattices of the Arab-Andalusian tradition, and carved wood panels, which enabled women to watch the street without being seen. At the beginning of the twentieth century, each region and often even each village had its own costume. Today, traditional dress is mostly reserved for only weddings and other national or native ceremonies. Costume traditionnel (Portail national de l’artisanat tunisien) On a national level, the jebba has become traditional dress, a wide coat covering the whole body, which differs depending on the quality of its fabric, its colors and its trimmings. The men's slippers are usually the natural color of the leather, Cuir et maroquinerie (Portail national de l’artisanat tunisien) while women's are generally of embroidered silk, cotton, gold and silver with floral patterns.
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Furstenberg gained attention at an early stage in his career for producing an innovative topological proof of the infinitude of prime numbers in 1955. In a series of articles beginning in 1963 with A Poisson Formula for Semi-Simple Lie Groups, he continued to establish himself as a ground-breaking thinker. His work showing that the behavior of random walks on a group is intricately related to the structure of the group - which led to what is now called the Furstenberg boundary – has been hugely influential in the study of lattices and Lie groups. In his 1967 paper, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Furstenberg introduced the notion of ‘disjointness,’ a notion in ergodic systems that is analogous to coprimality for integers.
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Tilman Esslinger received his PhD in physics from the University of Munich and the Max Planck Institute of Quantum Optics, Germany, in 1995. In his doctoral research he worked under the supervision of Theodor Hänsch on subrecoil laser cooling and optical lattices. He then build up his own group in Hänsch’s lab and conducted pioneering work on atom lasers, observed long-range phase coherence in a Bose–Einstein condensate, and realized the superfluid to Mott-insulator transition with a Bose gas in an optical lattice. Following his habilitation, Esslinger was in October 2001 appointed full professor at ETH Zurich, Switzerland, where he pioneered one- dimensional atomic quantum gases, Fermi–Hubbard models with atoms, a quantum- gas analogue of the topological Haldane model and the merger of quantum gas experiments with cavity quantum electrodynamics.
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The towers are usually steel lattices or trusses (wooden structures are used in Canada, Germany, and Scandinavia in some cases) and the insulators are either glass or porcelain discs or composite insulators using silicone rubber or EPDM rubber material assembled in strings or long rods whose lengths are dependent on the line voltage and environmental conditions. Typically, one or two ground wires, also called "guard" wires, are placed on top to intercept lightning and harmlessly divert it to ground. Towers for high- and extra-high voltage are usually designed to carry two or more electric circuits (with very rare exceptions, only one circuit for 500-kV and higher). If a line is constructed using towers designed to carry several circuits, it is not necessary to install all the circuits at the time of construction.
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STM image of surfaces at the edge of a 1 μm thick layer of ta-C "diamond-like" coating on 304 stainless steel after various durations of tumbling in a slurry of 240 mesh SiC abrasive. The first 100 min shows a burnishing away from the coating of an overburden of soft carbons than had been deposited after the last cycle of impacts converted bonds to sp3. On the uncoated part of the sample, about 5 μm of steel were removed during subsequent tumbling while the coating completely protected the part of the sample it covered. Within the "cobblestones", nodules, clusters, or "sponges" (the volumes in which local bonding is sp3) bond angles may be distorted from those found in either pure cubic or hexagonal lattices because of intermixing of the two.
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Also stands out the apartment building for Evangelina Aristigueta de Vidaña for its formal simplicity and its strong Cuban identity, given in a series of details such as the use of traditional materials of the country's architecture, presence of lattices and stained glass windows, is found in the corner of 7th and 60, in Miramar, Playa. Finally, the apartment building of The Goods and Bonds Investment Co., built between 1956-1958 and located in C between 29 and Zapata, El Vedado. It is a work of great secrecy, of a fairly simple formality and its compositional richness is given by the movement of a module in its façade. Inside one can see Romañach's interest in Japanese culture specifically in the design of the stair and in the details of the hand railings.
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The space groups in three dimensions are made from combinations of the 32 crystallographic point groups with the 14 Bravais lattices, each of the latter belonging to one of 7 lattice systems. What this means is that the action of any element of a given space group can be expressed as the action of an element of the appropriate point group followed optionally by a translation. A space group is thus some combination of the translational symmetry of a unit cell (including lattice centering), the point group symmetry operations of reflection, rotation and improper rotation (also called rotoinversion), and the screw axis and glide plane symmetry operations. The combination of all these symmetry operations results in a total of 230 different space groups describing all possible crystal symmetries.
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After the Second World War, the Ollerenshaws moved to Manchester, where Kathleen worked as a part-time lecturer in the mathematics department at Manchester University while raising her children and continued her work on lattices. In 1949, at the age of 37, she received her first effective hearing aid. Kathleen Ollerenshaw, aged 95, at Manchester Astronomical Society in 2007 Outside of academics, Ollerenshaw served as a Conservative Councillor for Rusholme for twenty-six years (1956–1981), Lord Mayor of Manchester (1975–1976), High Sheriff of Greater Manchester from 1978 to 1979, and the prime motivator in the creation of the Royal Northern College of Music. She was made a Freeman of the City of Manchester and was an advisor on educational matters to Margaret Thatcher's government in the 1980s.
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Liu Zhaogan’s Former Residence () Liu Zhaogan, who was nicknamed “White Duck”, earned his certified student title at the age of eight, the first-degree scholar title at the age of thirteen, and became the Grand Preceptor of the then-Prince Jiaqing during the Qianlong Period of the Qing Dynasty. The gate is wide, featuring wide head and narrow tail for the purpose of preventing dissipation of Qi. A pair of drum-shaped bearing stones, which was a typical indication of the architectural class and family status of dignitaries in the Qing Dynasty, is placed apart on both sides. The gate used to carry a plaque of "" () which is missing now. The central room is decorated with a caisson ceiling and lattices on doors and windows are also elaborately designed and carved.
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Velocity-distribution data of a gas of rubidium atoms, confirming the discovery of a new phase of matter, the Bose–Einstein condensate Condensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. In particular, it is concerned with the "condensed" phases that appear whenever the number of particles in a system is extremely large and the interactions between them are strong. The most familiar examples of condensed phases are solids and liquids, which arise from the bonding by way of the electromagnetic force between atoms. More exotic condensed phases include the superfluid and the Bose–Einstein condensate found in certain atomic systems at very low temperature, the superconducting phase exhibited by conduction electrons in certain materials, and the ferromagnetic and antiferromagnetic phases of spins on atomic lattices.
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Hemker's research expertise centers on identifying the underlying atomic, nano and microstructural details that govern the mechanical response, performance and reliability of materials. His group has made key observations and discoveries that have challenged the way the community thinks about and understands materials behavior in: nanostructured materials, materials for MEMS, metallic micro-lattices, structural intermetallics, thermal barrier coatings for gas turbines and satellites, armor ceramics and high temperature aerospace materials. More recent activities include the development of nanotwinned NiMoW thin films with extreme strength (3-4GPa) and an exceptional balance of properties. Having demonstrated that they can be micromachined into micro-cantilevers with requisite dimensional stability, Hemker collaborates with General Electric to promote the use of metal MEMS devices for use in extreme environments and the Internet of Things (IoT).
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This work had to be performed with great care as the steel of the structure had for many years been painted with lead based paint, which had to be carefully removed and contained by workers with extensive protective gear. laced ties (left, eastern span) and bolted box beam retrofit (right, western span) Most of the beams were originally constructed of two plate -beams joined with lattices of flat strip or angle stock, depending upon structural requirements. These have all been reconstructed by replacing the riveted lattice elements with bolted steel plate and so converting the lattice beams into box beams. This replacement included adding face plates to the large diagonal beams joining the faces of the main towers, which now have an improved appearance when viewed from certain angles.
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The congruence lattice Con A of an algebra A is an algebraic lattice. The (∨,0)-semilattice of compact elements of Con A is denoted by Conc A, and it is sometimes called the congruence semilattice of A. Then Con A is isomorphic to the ideal lattice of Conc A. By using the classical equivalence between the category of all (∨,0)-semilattices and the category of all algebraic lattices (with suitable definitions of morphisms), as it is outlined here, we obtain the following semilattice-theoretical formulation of CLP. \---- Semilattice- theoretical formulation of CLP: Is every distributive (∨,0)-semilattice isomorphic to the congruence semilattice of some lattice? \---- Say that a distributive (∨,0)-semilattice is representable, if it is isomorphic to Conc L, for some lattice L. So CLP asks whether every distributive (∨,0)-semilattice is representable.
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It is now accepted that a boundary consists of structural units which depend on both the misorientation of the two grains and the plane of the interface. The types of structural unit that exist can be related to the concept of the coincidence site lattice, in which repeated units are formed from points where the two misoriented lattices happen to coincide. In coincident site lattice (CSL) theory, the degree of fit (Σ) between the structures of the two grains is described by the reciprocal of the ratio of coincidence sites to the total number of sites. In this framework, it is possible to draw the lattice for the 2 grains and count the number of atoms that are shared (coincidence sites), and the total number of atoms on the boundary (total number of site).
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By Birkhoff's representation theorem for finite distributive lattices, the upper sets of any partially ordered set form a finite distributive lattice, and every finite distributive lattice can be represented in this way. The upper sets correspond to the vertices of the order polytope, so the mapping from upper sets to vertices provides a geometric representation of any finite distributive lattice. Under this representation, the edges of the polytope connect comparable elements of the lattice. If two functions p and q both belong to the order polytope of a partially ordered set (S,\le), then the function p\wedge q that maps x to \min(p(x),q(x)), and the function p\vee q that maps x to \max(p(x),q(x)) both also belong to the order polytope.
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Suvorov was the founder of a new branch of function theory concerned with the study of classes of plane and spatial mappings with bounded Dirichlet integral and of a new trend in mathematics at the border of the theory of functions and general topology that deals with the topological aspects of the boundary correspondence in a conformal mapping. Suvorov made major contributions on the theory of topological and metric mappings on 2-dimensional space. Later at Donetsk he extended Lavrent'ev's work on stability and differentiable function theorems, to more general classes of transformations. Suvorov introduced new methods to help in the understanding of metric properties of mappings with bounded Dirichlet integral. Suvorov and Oleg Ivanov collaborated on a number of papers culminating in a joint monograph “Complete lattices of conformally invariant compactifications of a domain”.
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The original motivation of Erdős in studying this problem was to extend from finite to infinite graphs the theorem that, whenever a graph has an orientation with finite maximum out-degree k, it also has a (2k+1)-coloring. For finite graphs this follows because such graphs always have a vertex of degree at most 2k, which can be colored with one of 2k+1 colors after all the remaining vertices are colored recursively. Infinite graphs with such an orientation do not always have a low-degree vertex (for instance, Bethe lattices have k=1 but arbitrarily large minimum degree), so this argument requires the graph to be finite. But the De Bruijn–Erdős theorem shows that a (2k+1)-coloring exists even for infinite graphs.. See in particular p.
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The entire portal is made of hardened plaster, which is the predominant material in the interior of the Aljafería, as Mudéjar craftsmen perpetuate the materials and techniques that are common in Islam. In the same wall, two large windows of triple mixtilineal arc with shutters on their keys are escorted by the entrance, thanks to which the inner space of the royal rooms is illuminated. Once the space of the gallery is crossed, several rooms are arranged that precede to the great Throne Room, that are denominated "rooms of the lost steps". These are three small rooms of square plan communicated to each other by big windows shut with lattices that give to the Patio de San Martín, and that served as waiting rooms for those who were to be received in audience by the kings.
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The Festival architects tried to show by the design and layout of the South Bank Festival what could be achieved by applying modern town planning ideas.A Tonic to the Nation The Festival Style, (also called "Contemporary") combining modernism with whimsy and Englishness, influenced architecture, interior design, product design and typography in the 1950s. William Feaver describes the Festival Style as "Braced legs, indoor plants, lily-of-the valley sprays of lightbulbs, aluminium lattices, Costswold-type walling with picture windows, flying staircases, blond wood, the thorn, the spike, the molecule."William Feaver, "Festival Star", in Mary Banham and Bevis Hillier, A Tonic to the Nation: The Festival of Britain 1951, London, Thames and Hudson, 1976 , p. 54 The influence of the Festival Style was felt in the new towns, coffee bars and office blocks of the fifties.
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He constructed a pair of flat tori of 16 dimension, using arithmetic lattices first studied by Ernst Witt. After this example, many isospectral pairs in dimension two and higher were constructed (for instance, by M. F. Vignéras, A. Ikeda, H. Urakawa, C. Gordon). In particular , based on the Selberg trace formula for PSL(2,R) and PSL(2,C), constructed examples of isospectral, non-isometric closed hyperbolic 2-manifolds and 3-manifolds as quotients of hyperbolic 2-space and 3-space by arithmetic subgroups, constructed using quaternion algebras associated with quadratic extensions of the rationals by class field theory. In this case Selberg's trace formula shows that the spectrum of the Laplacian fully determines the length spectrum, the set of lengths of closed geodesics in each free homotopy class, along with the twist along the geodesic in the 3-dimensional case.
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The book begins with a motivating problem, the coin problem of determining which amounts of money can be represented (and what is the largest non-representable amount of money) for a given system of coin values. Other topics touched on include face lattices of polytopes and the Dehn–Sommerville equations relating numbers of faces; Pick's theorem and the Ehrhart polynomials, both of which relate lattice counting to volume; generating functions, Fourier transforms, and Dedekind sums, different ways of encoding sequences of numbers into mathematical objects; Green's theorem and its discretization; Bernoulli polynomials; the Euler–Maclaurin formula for the difference between a sum and the corresponding integral; special polytopes including zonotopes, the Birkhoff polytope, and permutohedra; and the enumeration of magic squares. In this way, the topics of the book connect together geometry, number theory, and combinatorics.
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First, a distance sequence from a vertex v in a graph is the sequence n1, n2, n3, ..., where ni is the number of vertices of distance i from v. The coordination sequence is the sequence s1, s2, s3, ..., where si is the weighted mean of the i-th entries of the distance sequences of vertices of the (orbits of the) crystal nets, where the weights are the asymptotic proportion of vertices of each orbit. The cumulative sums of the coordination sequence is denoted the topological density, and the sum of the first ten terms (plus 1 for the zero- th term) – often denoted TD10 – is a standard search term in crystal net databases. SeeM. Kotani and T. Sunada "Geometric aspects of large deviations for random walks on crystal lattices" In: Microlocal Analysis and Complex Fourier Analysis (T.
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Generally, the active medium of a solid-state laser consists of a glass or crystalline "host" material, to which is added a "dopant" such as neodymium, chromium, erbium, thulium or ytterbium.Z. Su, J. D. Bradley, N. Li, E. S. Magden, Purnawirman, D. Coleman, N. Fahrenkopf, C. Baiocco, T. Adam, G. Leake, D. Coolbaugh, D. Vermeulen, and M. R. Watts (2016) "Ultra-Compact CMOS- Compatible Ytterbium Microlaser", Integrated Photonics Research, Silicon and Nanophotonics 2016, IW1A.3. Many of the common dopants are rare-earth elements, because the excited states of such ions are not strongly coupled with the thermal vibrations of their crystal lattices (phonons), and their operational thresholds can be reached at relatively low intensities of laser pumping. There are many hundreds of solid-state media in which laser action has been achieved, but relatively few types are in widespread use.
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The De Morgan duals ▷ and ◁ of residuation arise as follows. Among residuated lattices, Boolean algebras are special by virtue of having a complementation operation ¬. This permits an alternative expression of the three inequalities :y ≤ x\z ⇔ x•y ≤ z ⇔ x ≤ z/y in the axiomatization of the two residuals in terms of disjointness, via the equivalence x ≤ y ⇔ x∧¬y = 0. Abbreviating x∧y = 0 to x # y as the expression of their disjointness, and substituting ¬z for z in the axioms, they become with a little Boolean manipulation :¬(x\¬z) # y ⇔ x•y # z ⇔ ¬(¬z/y) # x Now ¬(x\¬z) is reminiscent of De Morgan duality, suggesting that x\ be thought of as a unary operation f, defined by f(y) = x\y, that has a De Morgan dual ¬f(¬y), analogous to ∀xφ(x) = ¬∃x¬φ(x).
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It is the merit of Werner Urland for stating the AOM version for f-type compounds, advocating for it by systematic applications acting as proof for the validity of this approach. The theoretical activity was complemented by involvement in synthetic coordination chemistry, producing new coordination compounds taken as relevant new case studies for ligand field interpretation of magnetic properties. A series consisting in individual octahedral units [LnCl6]3−, is interesting by the intrinsic simplicity of these complexes, once is known that lanthanide complexes are usually adopting higher coordination numbers, the hexa-coordination being enforced mostly by the doping regime, in solid lattices, such as elpasolites (a variety of Halide minerals with ABM2X3 stoichiometry). The magnetic properties of [LnCl6]3− complexes (with pyridinium counter ions) were analysed in the non-trivial details of the causal role of the ligand field effects.
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As shown in Figure 2, k-vectors outside the first Brillouin zone are physically equivalent to vectors inside it and can be mathematically transformed into each other by the addition of a reciprocal lattice vector G. These processes are called Umklapp scattering and change the total phonon momentum. Umklapp scattering is the dominant process for electrical resistivity at low temperatures for low defect crystalsNiel W. Ashcroft and N. David Mermin, (1976) "Solid State Physics", Holt Rinehart and Winston, New York. See pages 523-526 for a discussion of resistivity at high temperatures, and pages 526-528 for the contribution of Umklapp to resistivity at low temperatures. (as opposed to phonon-electron scattering, which dominates at high temperatures, and high-defect lattices which lead to scattering at any temperature.) Umklapp scattering is the dominant process for thermal resistivity at high temperatures for low defect crystals.
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If the even lattice is generated by its "root vectors" (those satisfying (α, α)=2), and any two root vectors are joined by a chain of root vectors with consecutive inner products non-zero then the vertex operator algebra is the unique simple quotient of the vacuum module of the affine Kac–Moody algebra of the corresponding simply laced simple Lie algebra at level one. This is known as the Frenkel–Kac (or Frenkel–Kac–Segal) construction, and is based on the earlier construction by Sergio Fubini and Gabriele Veneziano of the tachyonic vertex operator in the dual resonance model. Among other features, the zero modes of the vertex operators corresponding to root vectors give a construction of the underlying simple Lie algebra, related to a presentation originally due to Jacques Tits. In particular, one obtains a construction of all ADE type Lie groups directly from their root lattices.
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His study of discrete geometric analysis includes a graph-theoretic interpretation of Ihara zeta functions, a discrete analogue of periodic magnetic Schrödinger operators as well as the large time asymptotic behaviors of random walk on crystal lattices. The study of random walk led him to the discovery of a "mathematical twin" of the diamond crystal out of an infinite universe of hypothetical crystals (2005). He named it the K4 crystal due to its mathematical relevance (see the linked article). What was noticed by him is that the K4 crystal has the "strong isotropy property", meaning that for any two vertices x and y of the crystal net, and for any ordering of the edges adjacent to x and any ordering of the edges adjacent to y, there is a net-preserving congruence taking x to y and each x-edge to the similarly ordered y-edge.
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Lattices were woven from willow branches and every year renewed, unless made with juniper poles that had been reinforced with charred oak. Of the planted crops, turnips required the most room, and planted next to these were coleworts, and a path leading to plots of sorrel, arugula, parsley, spinach, beets, and orach, then separated from the greens another path to the root vegetables, leeks, onions, garlic, carrots, and scallions, and so on for edible flowers and winter potherbs like thyme, sage, lavender, rosemary, hyssop, southern wormwood, savoury, lemon balm, basil, costmary, spikenard, chamomile, and pennyroyal. Marigolds could grow perennially in untilled fields, and their juice and flowers were reputed to have many benefits from soothing eye irritation to relieving tooth pain. Strawberry juice and wine were rumored to have similar benefits for the eyes, and, according to Estienne, the berries themselves had "no neede of greate toile or tilling".
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Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time. But some also use the term to refer to processes that change in continuous time, particularly the Wiener process used in finance, which has led to some confusion, resulting in its criticism. There are other various types of random walks, defined so their state spaces can be other mathematical objects, such as lattices and groups, and in general they are highly studied and have many applications in different disciplines. A classic example of a random walk is known as the simple random walk, which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each Bernoulli variable takes either the value positive one or negative one.
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A homomorphism between two Boolean algebras A and B is a function f : A → B such that for all a, b in A: : f(a ∨ b) = f(a) ∨ f(b), : f(a ∧ b) = f(a) ∧ f(b), : f(0) = 0, : f(1) = 1. It then follows that f(¬a) = ¬f(a) for all a in A. The class of all Boolean algebras, together with this notion of morphism, forms a full subcategory of the category of lattices. An isomorphism between two Boolean algebras A and B is a homomorphism f : A → B with an inverse homomorphism, that is, a homomorphism g : B → A such that the composition g ∘ f: A → A is the identity function on A, and the composition f ∘ g: B → B is the identity function on B. A homomorphism of Boolean algebras is an isomorphism if and only if it is bijective.
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A useful feature of the TEBD algorithm is that it can be reliably employed for time evolution simulations of time-dependent Hamiltonians, describing systems that can be realized with cold atoms in optical lattices, or in systems far from equilibrium in quantum transport. From this point of view, TEBD had a certain ascendance over DMRG, a very powerful technique, but until recently not very well suited for simulating time-evolutions. With the Matrix Product States formalism being at the mathematical heart of DMRG, the TEBD scheme was adopted by the DMRG community, thus giving birth to the time dependent DMRG , t-DMRG for short. Around the same time, other groups have developed similar approaches in which quantum information plays a predominant role as, for example, in DMRG implementations for periodic boundary conditions , and for studying mixed-state dynamics in one-dimensional quantum lattice systems,.
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Frankenheim's focus of research was crystallography, particularly studies of crystal structure and the mathematical and theoretical basis of the symmetry of crystals. By 1826, he was already using the integer reciprocals of Weiss' coefficients (the intersection of a plane with the three crystallographic axes) to describe the spatial positions of crystal surfaces, from which the British crystallographer William Hallowes Miller (1801-1880) developed the concept of Miller indices in 1839. By assigning symmetry elements to the crystal systems defined previously by Weiss and Friedrich Mohs (1773-1839), Frankenheim was able, for the first time, to define 32 point groups (crystal classes) and to classify them into four crystal systems (the regular one, the fourfold, the twofold and the sixfold). From his observations he derived 15 lattice types for crystals, which were later reduced by Auguste Bravais (1811-1863) to 14 and today as Bravais lattices describe unit cells of crystal structures.
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Tetrahedral packing: The dihedral angle of a tetrahedron is not commensurable with 2; consequently, a hole remains between two faces of a packing of five tetrahedra with a common edge. A packing of twenty tetrahedra with a common vertex in such a way that the twelve outer vertices form an irregular icosahedron The stability of metals is a longstanding question of solid state physics, which can only be understood in the quantum mechanical framework by properly taking into account the interaction between the positively charged ions and the valence and conduction electrons. It is nevertheless possible to use a very simplified picture of metallic bonding and only keeps an isotropic type of interactions, leading to structures which can be represented as densely packed spheres. And indeed the crystalline simple metal structures are often either close packed face-centered cubic (fcc) or hexagonal close packing (hcp) lattices.
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Another proof of this theorem based on unique sink orientations was given by , and showed how to use this theorem to derive a polynomial time algorithm for reconstructing the face lattices of simple polytopes from their graphs. However, testing whether a given graph or lattice can be realized as the face lattice of a simple polytope is equivalent (by polarity) to realization of simplicial polytopes, which was shown to be complete for the existential theory of the reals by . In the context of the simplex method for linear programming, it is important to understand the diameter of a polytope, the minimum number of edges needed to reach any vertex by a path from any other vertex. The system of linear inequalities of a linear program define facets of a polytope representing all feasible solutions to the program, and the simplex method finds the optimal solution by following a path in this polytope.
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The high field strength and large ionic radii of rare-earths make them incompatible with the crystal lattices of most rock-forming minerals, so REE will undergo strong partitioning into a melt phase if one is present. REE are chemically very similar and have always been difficult to separate, but a gradual decrease in ionic radius from LREE to HREE, called lanthanide contraction, can produce a broad separation between light and heavy REE. The larger ionic radii of LREE make them generally more incompatible than HREE in rock-forming minerals, and will partition more strongly into a melt phase, while HREE may prefer to remain in the crystalline residue, particularly if it contains HREE-compatible minerals like garnet. The result is that all magma formed from partial melting will always have greater concentrations of LREE than HREE, and individual minerals may be dominated by either HREE or LREE, depending on which range of ionic radii best fits the crystal lattice.
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His research contributions in various fields including low angle scattering, thermal diffuse scattering of X-rays from crystals, diffraction of X-rays by liquids "Professor Banerjee's article in journal NATURE on liquid Crystal in Most Prominent Journal Nature". Professor Kedareswar Banerjee article on liquid crystal in journal NATURE., jute fibre and organic polymers, structures of coal and glass, determination of the elastic constants of crystals by X-rays, theoretical modelling of the vibrational spectra of crystal lattices and some topics in crystal optics have received international recognition. Perhaps his most significant contribution to the advancement of science in India was the creation of active schools of research wherever he went leaving behind a band of young, energetic research workers who became the torch bearers of his scientific tradition.“Professor Kedareswar Banerjee – The Crystallographer (15 September 1900 – 30 April 1975)”. Professor Kedareswar Banerjee's works from IUCR web page“Kedareswar Banerjee”. Professor Kedareswar Banerjee's works from IISc Bangore web page“ K Banerjee was fellow of IAS, Bangalore. Brief Biography .
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The Baghdad Koshk was also built at the Topkapı Palace in 1638–39, by Sultan Murad IV. The building is again domed, offering direct views onto the gardens and park of the Palace as well as the architecture of the city of Istanbul. Sultan Ahmed III (1703–1730) also built a glass room of the Sofa Kiosk at the Topkapı Palace incorporating some Western elements, such as the gilded brazier designed by Duplessis père, which was given to the Ottoman ambassador by King Louis XV of France. Morisco Kiosk in Mexico The first English contact with Turkish Kiosk came through Lady Wortley Montagu (1689–1762), the wife of the English ambassador to Istanbul, who in a letter written on 1 April 1717 to Anne Thistlethwayte, mentions a "chiosk" describing it as "raised by 9 or 10 steps and enclosed with gilded lattices".R. Halsband, The complete letters of Lady Mary Wortley Montagu, Clarendon Press, Oxford, 1965 European monarchs adopted the building type.
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Lunar water has two potential origins: water-bearing comets (and other bodies) striking the Moon, and in situ production. It has been theorized that the latter may occur when hydrogen ions (protons) in the solar wind chemically combine with the oxygen atoms present in the lunar minerals (oxides, silicates etc.) to produce small amounts of water trapped in the minerals' crystal lattices or as hydroxyl groups, potential water precursors. (This mineral-bound water, or mineral surface, must not be confused with water ice.) The hydroxyl surface groups (X–OH) formed by the reaction of protons (H+) with oxygen atoms accessible at oxide surface (X=O) could further be converted in water molecules (H2O) adsorbed onto the oxide mineral's surface. The mass balance of a chemical rearrangement supposed at the oxide surface could be schematically written as follows: :2 X–OH → X=O + X + H2O or, :2 X–OH → X–O–X + H2O where "X" represents the oxide surface.
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Alternatively three-dimensional lattices (hydrogel/xerogel) can be used to chemically or physically entrap these (where by chemically entrapped it is meant that the biological element is kept in place by a strong bond, while physically they are kept in place being unable to pass through the pores of the gel matrix). The most commonly used hydrogel is sol-gel, a glassy silica generated by polymerization of silicate monomers (added as tetra alkyl orthosilicates, such as TMOS or TEOS) in the presence of the biological elements (along with other stabilizing polymers, such as PEG) in the case of physical entrapment. Another group of hydrogels, which set under conditions suitable for cells or protein, are acrylate hydrogel, which polymerize upon radical initiation. One type of radical initiator is a peroxide radical, typically generated by combining a persulfate with TEMED (Polyacrylamide gel are also commonly used for protein electrophoresis), alternatively light can be used in combination with a photoinitiator, such as DMPA (2,2-dimethoxy-2-phenylacetophenone).
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For example, in his photography series "Buildings Made of Sky," Wegner reverses urban streetscapes to reveal how skyscrapers shape the open-air spaces between one another into skyscraper-like forms of their own. Chasin described a 2004 piece from the series in these terms: “A magical reversal thereby takes place: the physical buildings read visually as a darkened background offset by architectural contours from startling blue-hued visions of skyscrapers carved from atmosphere. Sky becomes building; building, sky—or to invoke K. Michael Hays in a different context, ‘not architecture but evidence that it exists.’” Wegner has also often pushed the construction of his works in an architectural direction, presenting paintings in the form of leaning columns, complex lattices, and multi-layered scrims. Huldisch noted that “[h]is stacks, grids, and lattice structures reveal both an interest in the forms of Minimalism and a rejection of the stringent doctrine that predicated them.
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The Oranienburg plant provided the uranium sheets and cubes for the Uranmaschine experiments conducted at the KWIP and the Versuchsstelle (testing station) of the Heereswaffenamt (Army Ordnance Office) in Gottow. The G-1 experimentF. Berkei, W. Borrmann, W. Czulius, Kurt Diebner, Georg Hartwig, K. H. Höcker, W. Herrmann, H. Pose, and Ernst Rexer Bericht über einen Würfelversuch mit Uranoxyd und Paraffin G-125 (dated before 26 November 1942). performed at the HWA testing station, under the direction of Kurt Diebner, had lattices of 6,800 uranium oxide cubes (about 25 tons), in the nuclear moderator paraffin.Hentschel and Hentschel, 1996, 369 and 373, Appendix F (see the entry for Nikolaus Riehl and Kurt Diebner), and Appendix D (see the entry for Auergesellschaft). Work of the American Operation Alsos teams, in November 1944, uncovered leads which took them to a company in Paris that handled rare earths and had been taken over by the Auergesellschaft.
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The definition of a congruence depends on the type of algebraic structure under consideration. Particular definitions of congruence can be made for groups, rings, vector spaces, modules, semigroups, lattices, and so forth. The common theme is that a congruence is an equivalence relation on an algebraic object that is compatible with the algebraic structure, in the sense that the operations are well-defined on the equivalence classes. For example, a group is an algebraic object consisting of a set together with a single binary operation, satisfying certain axioms. If G is a group with operation \ast, a congruence relation on G is an equivalence relation \equiv on the elements of G satisfying :g_1 \equiv g_2 \ \ \, and \ \ \, h_1 \equiv h_2 \implies g_1 \ast h_1 \equiv g_2 \ast h_2 for all g_1, g_2, h_1, h_2 \in G. For a congruence on a group, the equivalence class containing the identity element is always a normal subgroup, and the other equivalence classes are the cosets of this subgroup.
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If we do this for half of the holes in a second plane above the first, we create a new compact layer. There are two possible choices for doing this, call them B and C. Suppose that we chose B. Then one half of the hollows of B lies above the centers of the balls in A and one half lies above the hollows of A which were not used for B. Thus the balls of a third layer can be placed either directly above the balls of the first one, yielding a layer of type A, or above the holes of the first layer which were not occupied by the second layer, yielding a layer of type C. Combining layers of types A, B, and C produces various close-packed structures. Two simple arrangements within the close-packed family correspond to regular lattices. One is called cubic close packing (or face-centred cubic, "FCC")—where the layers are alternated in the ABCABC... sequence.
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These graphs are closely related to partially ordered sets and lattices. The Hasse diagram of a partially ordered set is a directed acyclic graph whose vertices are the set elements, with an edge from x to y for each pair x, y of elements for which x ≤ y in the partial order but for which there does not exist z with x ≤ y ≤ z. A partially ordered set forms a complete lattice if and only if every subset of elements has a unique greatest lower bound and a unique least upper bound, and the order dimension of a partially ordered set is the least number of total orders on the same set of elements whose intersection is the given partial order. If the vertices of an st-planar graph are partially ordered by reachability, then this ordering always forms a two-dimensional complete lattice, whose Hasse diagram is the transitive reduction of the given graph.
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In mathematics, specifically in order theory and functional analysis, a subset A of an ordered vector space is said to be order complete in X if for every non-empty subset S of C that is order bounded in A (i.e. is contained in an interval [a, b] := { z ∈ X : a ≤ z and z ≤ b } for some a and b belonging to A), the supremum sup S and the infimum inf S both exist and are elements of A. An ordered vector space is called order complete, Dedekind complete, a complete vector lattice, or a complete Riesz space, if it is order complete as a subset of itself, in which case it is necessarily a vector lattice. An ordered vector space is said to be countably order complete if each countable subset that is bounded above has a supremum. Being an order complete vector space is an important property that is used frequently in the theory of topological vector lattices.
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A residuated Boolean algebra is an algebraic structure (L, ∧, ∨, ¬, 0, 1, •, I, \, /) such that : (i) (L, ∧, ∨, •, I, \, /) is a residuated lattice, and :(ii) (L, ∧, ∨, ¬, 0, 1) is a Boolean algebra. An equivalent signature better suited to the relation algebra application is (L, ∧, ∨, ¬, 0, 1, •, I, ▷, ◁) where the unary operations x\ and x▷ are intertranslatable in the manner of De Morgan's laws via :x\y = ¬(x▷¬y), x▷y = ¬(x\¬y), and dually /y and ◁y as : x/y = ¬(¬x◁y), x◁y = ¬(¬x/y), with the residuation axioms in the residuated lattice article reorganized accordingly (replacing z by ¬z) to read :(x▷z)∧y = 0 ⇔ (x•y)∧z = 0 ⇔ (z◁y)∧x = 0 This De Morgan dual reformulation is motivated and discussed in more detail in the section below on conjugacy. Since residuated lattices and Boolean algebras are each definable with finitely many equations, so are residuated Boolean algebras, whence they form a finitely axiomatizable variety.
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Hemker has been named Fellow of AAAS, ASME, ASM International and TMS Hemker has mentored over 75 postdoctoral fellows, doctoral and masters students since coming to Hopkins, about 20 of whom now have tenured or tenure-track academic positions in major research universities (U Penn, UIUC, UCSB, Tohoku U, U Freiburg, KAIST, TAMU, and others). His group strives to elucidate the underlying atomic-level details that govern the mechanical response, performance and reliability of disparate material systems. They have made key observations and discoveries that have challenged the way the community thinks about and understands materials behavior in: additive manufacturing, nanocrystalline materials, materials for MEMS, metallic micro-lattices, thermal barrier coatings for satellites and gas turbines, armor ceramics, extreme environments, and high temperature structural materials in general. The results of Hemker's research have been disseminated in approximately 250 scientific articles, 4 co-edited books and over 300 invited presentations and plenary lectures.
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Each set of elements has a least upper bound and a greatest lower bound, so that it forms a lattice, and more specifically (for partitions of a finite set) it is a geometric lattice.. The partition lattice of a 4-element set has 15 elements and is depicted in the Hasse diagram on the left. Based on the cryptomorphism between geometric lattices and matroids, this lattice of partitions of a finite set corresponds to a matroid in which the base set of the matroid consists of the atoms of the lattice, namely, the partitions with n-2 singleton sets and one two-element set. These atomic partitions correspond one-for-one with the edges of a complete graph. The matroid closure of a set of atomic partitions is the finest common coarsening of them all; in graph-theoretic terms, it is the partition of the vertices of the complete graph into the connected components of the subgraph formed by the given set of edges.
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For every pair of elements x, y of a GCD domain R, a GCD d of x and y and an LCM m of x and y can be chosen such that , or stated differently, if x and y are nonzero elements and d is any GCD d of x and y, then xy/d is an LCM of x and y, and vice versa. It follows that the operations of GCD and LCM make the quotient R/~ into a distributive lattice, where "~" denotes the equivalence relation of being associate elements. The equivalence between the existence of GCDs and the existence of LCMs is not a corollary of the similar result on complete lattices, as the quotient R/~ need not be a complete lattice for a GCD domain R. If R is a GCD domain, then the polynomial ring R[X1,...,Xn] is also a GCD domain.Robert W. Gilmer, Commutative semigroup rings, University of Chicago Press, 1984, p. 172.
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He was one of the first to become convinced of the conceptual revolution whereby string theory is viewed as a unified theory of all particle interactions, including gravity, rather than simply as a model of hadrons. This was the subject of his plenary talk at the 1974 Rochester conference in London. In 1977, Olive returned to the UK to take up a lectureship at Imperial College, becoming Professor in 1984 and Head of the Theoretical Physics Group in 1988. He had by now begun collaboration with Peter Goddard and together they produced a series of papers on the mathematical foundations of string theory, notably on Virasoro and Kac-Moody algebras and their representations and relation to vertex operators. One outcome of their work on algebras and lattices was the identification of the special role played by the two Lie groups SO(32) and E8 x E8, which would shortly be shown by Michael Green and John Schwarz to exhibit anomaly cancellation that led to the renaissance of string theory in 1984.
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The Book of Mormon describes that the Jaredite people were familiar with the concept of "windows" near the time of the biblical Tower of Babel, and that they specifically avoided crafting windows for lighting in their covered seagoing vessels, because of fears that "they would be dashed in pieces" during the ocean voyage.Ether 2:22–23 Transparent window panes are a more recent invention, dating to the 11th century AD in Germany. FairMormon, citing several such uses in the KJV of the Bible, notes that 'the term "window" originally referred to an opening through which the wind could enter' and which sometimes had doors or shutters (2 Kings 13:17) or lattices (Song of Solomon 2:9) and were not made of glass, so these parts of the window could be what is referred to as being "dashed to pieces." It is also suggested that the warning in Ether may have referred to the entire vessel being "dashed in pieces" if the structure was weakened by additional openings.
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The connections transfer forces through load-bearing surface contacts, requiring that the characteristic dimensions of the connections scale with the cross section of the attached strut members, t2, because this dimension determines the maximum stress transferable through the joint. These definitions give a cubic scaling relation between the relative mass contribution of the joints and the strut's thickness-to-length ratio (ρc/ρs ∝ Cc(t/l)3, where Cc is the connection contribution constant determined by the lattice geometry). The struts' relative density contribution scales quadratically with the thickness-to-length ratio of the struts (ρm/ρs ∝ Cm (t/l)2), which agrees with the literature on classical cellular materials. Mechanical properties (such as modulus and strength) scale with overall relative density, which in turn scales primarily with the strut and not the connection, considering only open cell lattices with slender struts [t/l < 0.1 (7)], given that the geometric constants Cc and Cm are of the same order of magnitude [ρ/ρs ∝ Cc (t/l)3 + Cm (t/l)2].
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For lattices, prime ideals can be characterized as follows: A subset I of a lattice (P, ≤) is a prime ideal, if and only if # I is a proper ideal of P, and # for all elements x and y of P, x\wedgey in I implies that x is in I or y is in I. It is easily checked that this is indeed equivalent to stating that P \ I is a filter (which is then also prime, in the dual sense). For a complete lattice the further notion of a completely prime ideal is meaningful. It is defined to be a proper ideal I with the additional property that, whenever the meet (infimum) of some arbitrary set A is in I, some element of A is also in I. So this is just a specific prime ideal that extends the above conditions to infinite meets. The existence of prime ideals is in general not obvious, and often a satisfactory amount of prime ideals cannot be derived within ZF (Zermelo–Fraenkel set theory without the axiom of choice).
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Over fields of characteristic greater than 3, all pseudo-reductive groups can be obtained from reductive groups by the "standard construction", a generalization of the construction above. The standard construction involves an auxiliary choice of a commutative pseudo-reductive group, which turns out to be a Cartan subgroup of the output of the construction, and the main complication for a general pseudo-reductive group is that the structure of Cartan subgroups (which are always commutative and pseudo-reductive) is mysterious. The commutative pseudo-reductive groups admit no useful classification (in contrast with the connected reductive case, for which they are tori and hence are accessible via Galois lattices), but modulo this one has a useful description of the situation away from characteristics 2 and 3 in terms of reductive groups over some finite (possibly inseparable) extensions of the ground field. Over imperfect fields of characteristics 2 and 3 there are some extra pseudo-reductive groups (called exotic) coming from the existence of exceptional isogenies between groups of types B and C in characteristic 2, between groups of type F₄ in characteristic 2, and between groups of type G₂ in characteristic 3, using a construction analogous to that of the Ree groups.
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