501 Sentences With "topologically" | Random Sentence Generator

The knots are topologically stable: unlike the knots we tie in ropes or shoelaces, a topologically stable knot cannot be untied without cutting the rope, although you can relocate the knots within the rope.

You can describe the number of holes in each shape topologically, he said.

Thus, a coffee cup and a doughnut are what's known as topologically equivalent.

Different kinds of braids would encode different computational tasks, and those structures would be topologically stable.

The mutations affect a newly discovered design feature of the DNA molecule called topologically associating domains, or TADs.

Giving away the money would begin to clean up the gross, topologically complex web of influence trading that Epstein helped weave.

This behavior is topologically protected, meaning that mathematics prevents the edges from losing their wave-trapping behavior, even when the object is deformed.

Such a design would braid qubits into a kind of knot; different kinds of braids would encode different computational tasks, and those structures would be topologically stable.

In the same way, a topological qubit will preserve its contained information as long as it remains in a topologically equivalent state, which means you can deform that qubit "as much as a doughnut is different from a coffee cup, and it still works," says Dowling.

"I thought it was a weird, bizarre hiccup I'd stumbled into," Stoll told me when we first spoke last year, after I called the home number he lists on the very eclectic website for his business selling klein bottles—blown-glass oddities that, topologically speaking, have only one side, with no inside or outside.

Brodmann-1909 regarded area 32 as topologically, but not cytoarchitecturally, homologous to the human dorsal anterior cingulate area 32; area 25 of Walker-1940 is topologically homologous to area 32.

The historically attested instances of the symbol appear in two traditional, topologically distinct, forms. The symbol appears in unicursal form, topologically a trefoil knot also seen in the triquetra. This unicursal form is found, for example, on the Tängelgårda stone. The symbol also appears in tricursal form, consisting of three linked triangles, topologically equivalent to the Borromean rings.

In this interpretation, the triquetra represents the topologically simplest possible knot.

It just reduces the point set until points become topologically distinguishable.

In topology, two points of a topological space X are topologically indistinguishable if they have exactly the same neighborhoods. That is, if x and y are points in X, and Nx is the set of all neighborhoods that contain x, and Ny is the set of all neighborhoods that contain y, then x and y are "topologically indistinguishable" if and only if Nx = Ny. (See Hausdorff's axiomatic neighborhood systems.) Intuitively, two points are topologically indistinguishable if the topology of X is unable to discern between the points. Two points of X are topologically distinguishable if they are not topologically indistinguishable. This means there is an open set containing precisely one of the two points (equivalently, there is a closed set containing precisely one of the two points).

The Szilassi polyhedron is a nonconvex polyhedron, topologically a torus, with seven hexagonal faces.

A topological space is Hausdorff if and only if it is both preregular (i.e. topologically distinguishable points are separated by neighbourhoods) and Kolmogorov (i.e. distinct points are topologically distinguishable). A topological space is preregular if and only if its Kolmogorov quotient is Hausdorff.

Topologically, de Sitter space is (so that if then de Sitter space is simply connected).

A T0 space is a topological space in which every pair of distinct points is topologically distinguishable. That is, for any two different points x and y there is an open set which contains one of these points and not the other. Note that topologically distinguishable points are automatically distinct. On the other hand, if the singleton sets {x} and {y} are separated, then the points x and y must be topologically distinguishable.

Regular maps are typically defined and studied in three ways: topologically, group-theoretically, and graph- theoretically.

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n). This tiling is topologically related as a part of sequence of regular tilings with order-6 vertices with Schläfli symbol {n,6}, and Coxeter diagram , progressing to infinity.

It is topologically equivalent to a quotient space of the hyperbolic order-5 square tiling, by distorting the rhombi into squares. As such, it is topologically a regular polyhedron of index two:The Regular Polyhedra (of index two), David A. Richter 250px Note that the order-5 square tiling is dual to the order-4 pentagonal tiling, and a quotient space of the order-4 pentagonal tiling is topologically equivalent to the dual of the medial rhombic triacontahedron, the dodecadodecahedron.

This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbol {n,3}.

This tiling is topologically related as part of a sequence of regular polyhedra with Schläfli symbol {3,p}.

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n).

This matrix can be represented topologically as a complex network where direct and indirect influences between variables are visualized.

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n).

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (5n).

The tetragonal hosohedron is topologically equivalent to the bicylinder Steinmetz solid, the intersection of two cylinders at right-angles.

2, p. 105 (illustration of colored tilings), Figure 2.5.3(d), p. 83 (topologically equivalent star tiling), and Exercise 4.1.

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (6n).

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (5n).

Statistical law of stream numbers. Journal of Geology 74, 17–37.Shreve, R.L., 1967. Infinite topologically random channel networks.

It is important to distinguish two different forms of the quartic. The closed quartic is what is generally meant in geometry; topologically it has genus 3 and is a compact space. The open or "punctured" quartic is of interest in number theory; topologically it is a genus 3 surface with 24 punctures, and geometrically these punctures are cusps. The open quartic may be obtained (topologically) from the closed quartic by puncturing at the 24 centers of the tiling by regular heptagons, as discussed below.

A topologically ordered state is a state with complicated non-local quantum entanglement. The non-locality means that the quantum entanglement in a topologically ordered state is distributed among many different particles. As a result, the pattern of quantum entanglements cannot be destroyed by local perturbations. This significantly reduces the effect of decoherence.

Topologically, an n-simplex is equivalent to an n-ball. Every n-simplex is an n-dimensional manifold with corners.

In topology, a branch of mathematics, a topologically stratified space is a space X that has been decomposed into pieces called strata; these strata are manifolds and are required to fit together in a certain way. Topologically stratified spaces provide a purely topological setting for the study of singularities analogous to the more differential-geometric theory of Whitney. They were introduced by René Thom, who showed that every Whitney stratified space was also a topologically stratified space, with the same strata. Another proof was given by John Mather in 1970, inspired by Thom's proof.

The uranyl sheets in schoepite/metaschoepite and α-UO2(OH)2 are topologically related via the substitution 2(OH) = O2 + vacancy.

That is, :separated ⇒ topologically distinguishable ⇒ distinct The property of being topologically distinguishable is, in general, stronger than being distinct but weaker than being separated. In a T0 space, the second arrow above reverses; points are distinct if and only if they are distinguishable. This is how the T0 axiom fits in with the rest of the separation axioms.

Similarly to topological order, topological insulators also have gapless boundary states. The boundary states of topological insulators play a key role in the detection and the application of topological insulators. This observation naturally leads to a question: are topological insulators examples of topologically ordered states? In fact topological insulators are different from topologically ordered states defined in this article.

More generally, if X has the initial topology induced by a family of maps f_\alpha : X \to Y_\alpha then :x ≡ y if and only if fα(x) ≡ fα(y) for all α. It follows that two elements in a product space are topologically indistinguishable if and only if each of their components are topologically indistinguishable.

It is possible for networks to be symmetric or antimetric in their electrical properties without being physically or topologically symmetric or antimetric.

Chirality is the 'handedness' of a knot. Topologically speaking, a knot and its mirror image may or may not have knot equivalence.

The limit of this tower yields a topological but not differentiable map, hence surgery works topologically but not differentiably in dimension 4.

This tiling is topologically related as a part of sequence of regular tilings with order-6 vertices with Schläfli symbol {n,6}, and Coxeter diagram , progressing to infinity. This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram , progressing to infinity.

This tiling is topologically related as a part of sequence of regular tilings with order-5 vertices with Schläfli symbol {n,5}, and Coxeter diagram , progressing to infinity. This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram , progressing to infinity.

This tiling is topologically related as a part of sequence of regular polyhedra and tilings, extending into the hyperbolic plane: {4,p}, p=3,4,5... This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.

They also proved the theorem of the three geodesics, that a Riemannian manifold topologically equivalent to a sphere has at least three simple closed geodesics.

His most recent works are on the topologically protected magnetic solitons called skyrmions and on the conversion between charge and spin current by topological insulators.

This tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry.

In the mathematical field of topology, a manifold M is called topologically rigid if every manifold homotopically equivalent to M is also homeomorphic to M.

In mathematics, the Ahlfors conjecture, now a theorem, states that the limit set of a finitely-generated Kleinian group is either the whole Riemann sphere, or has measure 0. The conjecture was introduced by , who proved it in the case that the Kleinian group has a fundamental domain with a finite number of sides. proved the Ahlfors conjecture for topologically tame groups, by showing that a topologically tame Kleinian group is geometrically tame, so the Ahlfors conjecture follows from Marden's tameness conjecture that hyperbolic 3-manifolds with finitely generated fundamental groups are topologically tame (homeomorphic to the interior of compact 3-manifolds). This latter conjecture was proved, independently, by and by .

This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry.

This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (6.n.n), and [n,3] Coxeter group symmetry.

This hyperbolic tiling is topologically related as a part of sequence of uniform snub polyhedra with vertex configurations (3.3.3.3.n), and [n,3] Coxeter group symmetry.

Symplectic field theory provides invariants of Legendrian submanifolds called relative contact homology that can sometimes distinguish distinct Legendrian submanifolds that are topologically identical (i.e. smoothly isotopic).

Topologically enriched categories (sometimes simply topological categories) are categories enriched over some convenient category of topological spaces, e.g. the category of compactly generated Hausdorff topological spaces.

The concept of stratifolds was invented by Matthias Kreck. The basic idea is similar to that of a topologically stratified space, but adapted to differential topology.

This hyperbolic tiling is topologically related as a part of sequence of uniform cantellated polyhedra with vertex configurations (3.4.n.4), and [n,3] Coxeter group symmetry.

Topologically, this projective line is equivalent to a cylinder. Points on this cylinder are in a natural one-to-one correspondence with oriented lines on the plane.

Topologically associating domains (TADs) are megabase-size regions of relatively high DNA interaction frequencies. Mechanistic studies indicate TADs are single insulated neighborhoods or collections of insulated neighborhoods.

Some DNA molecules are circular and are topologically constrained. More recently circular RNA was described as well to be a natural pervasive class of nucleic acids, expressed in many organisms (see CircRNA). A covalently closed, circular DNA (also known as cccDNA) is topologically constrained as the number of times the chains coiled around one other cannot change. This cccDNA can be supercoiled, which is the tertiary structure of DNA.

Cohesin is thought to mediate enhancer-promoter interactions and generate Topologically associating domain. As well as mediating cohesion and regulating DNA architecture the cohesin complex is required for DNA repair by homologous recombination. Given that NIPBL is required for cohesin's association with DNA it is thought that NIPBL is also required for all of these processes. Consistently, inactivation of Nipbl results in the loss topologically associating domains and cohesion.

This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram , progressing to infinity. This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.

Assuming a model of a collection of polyhedra with the boundary of each topologically equivalent to a sphere and with faces topologically equivalent to disks, according to Euler's formula, there are Θ(n) faces. Testing Θ(n2) line segments against Θ(n) faces takes Θ(n3) time in the worst case. Appel's algorithm is also unstable, because an error in visibility will be propagated to subsequent segment endpoints.J. F. Blinn.

Strings can be either open or closed. A closed string is a string that has no end-points, and therefore is topologically equivalent to a circle. An open string, on the other hand, has two end-points and is topologically equivalent to a line interval. Not all string theories contain open strings, but every theory must contain closed strings, as interactions between open strings can always result in closed strings.

This suggests that if we use different quantum entanglements in a topologically ordered state to encode quantum information, the information may last much longer. The quantum information encoded by the topological quantum entanglements can also be manipulated by dragging the topological defects around each other. This process may provide a physical apparatus for performing quantum computations. Therefore, topologically ordered states may provide natural media for both quantum memory and quantum computation.

This open set can then be used to distinguish between the two points. A T0 space is a topological space in which every pair of distinct points is topologically distinguishable. This is the weakest of the separation axioms. Topological indistinguishability defines an equivalence relation on any topological space X. If x and y are points of X we write x ≡ y for "x and y are topologically indistinguishable".

This hyperbolic tiling is topologically related as a part of sequence of uniform quasiregular polyhedra with vertex configurations (3.n.3.n), and [n,3] Coxeter group symmetry.

Topologically, the pinched torus is homotopy equivalent to the wedge of a sphere and a circle. It is homeomorphic to a sphere with two distinct points being identified.

Thinning is the transformation of a digital image into a simplified, but topologically equivalent image. It is a type of topological skeleton, but computed using mathematical morphology operators.

M can be thought of, topologically, as the interior of a compact manifold with toral boundary. Suppose we have chosen a meridian and longitude for each boundary torus, i.e.

A cost of 0 corresponds to a pair of topologically identical nodes v and u, while a cost close to 2 corresponds to a pair of topologically different nodes. GRAAL chooses as the initial seed a pair of nodes (v,u), v\in V and u\in U, that have the smallest cost. Ties are broken randomly. Once the seed is found, GRAAL builds "spheres" of all possible radii around nodes v and u.

GML 3.1.1 contains encoding support for more advanced geometric representations: curves, surfaces, multi- dimensions (time, elevation, multi-band imagery). In addition, GML 3.1.1 includes encoding support for topologically integrated datasets.

Topologically protected Dirac cone surface states have been observed in Bismuth selenide and its insulating derivatives leading to intrinsic topological insulators, which later became the subject of world-wide scientific research.

In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a topologically invariant way.

Linards Reiziņš (14 January 1924 – 1991) was a Latvian mathematician who specialized in the field of differential equations and the conditions under which two systems of differential equations are topologically equivalent.

His early research was focused on molecular recognition, models of serine proteases,Cramer, K. D.; Zimmerman, S. C. "Kinetic effect of a syn- oriented carboxylate on a proximate imidazole in catalysis: a model for the histidine-aspartate couple in enzymes," J. Am. Chem. Soc. 1990, 112, 3680-3682. and topologically novel DNA intercalators.Zimmerman, S. C.; Lamberson, C. R.; Cory, M.; Fairley, T. A. "Topologically constrained bifunctional intercalators: DNA intercalation by a macrocyclic bisacridine," J. Am. Chem. Soc.

This tiling is topologically part of sequence of regular polyhedra and tilings with Schläfli symbol {n,3}. And also is topologically part of sequence of regular tilings with Schläfli symbol {8,n}. From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 10 forms.

Since the conclusion of the Baire category theorem is purely topological, it applies to these spaces as well. Completely metrizable spaces are often called topologically complete. However, the latter term is somewhat arbitrary since metric is not the most general structure on a topological space for which one can talk about completeness (see the section Alternatives and generalizations). Indeed, some authors use the term topologically complete for a wider class of topological spaces, the completely uniformizable spaces.

Wormholes have been defined both geometrically and topologically. From a topological point of view, an intra-universe wormhole (a wormhole between two points in the same universe) is a compact region of spacetime whose boundary is topologically trivial, but whose interior is not simply connected. Formalizing this idea leads to definitions such as the following, taken from Matt Visser's Lorentzian Wormholes (1996). Geometrically, wormholes can be described as regions of spacetime that constrain the incremental deformation of closed surfaces.

This tiling is topologically related as a part of sequence of regular tilings with octagonal faces, starting with the octagonal tiling, with Schläfli symbol {8,n}, and Coxeter diagram , progressing to infinity.

A mug without a handle, i.e., a bowl or a beaker, is topologically equivalent to a saucer, which is quite evident when a raw clay bowl is flattened on a potter's wheel.

This tiling is topologically related as a part of sequence of regular tilings with octagonal faces, starting with the octagonal tiling, with Schläfli symbol {8,n}, and Coxeter diagram , progressing to infinity.

Gram-negative bacteria have two membranes, thus making secretion topologically more complex. There are at least six specialized secretion systems in gram-negative bacteria. Many secreted proteins are particularly important in bacterial pathogenesis.

Cells are defined in a normed space, commonly a two-dimensional Euclidean geometry, like a grid. The cells are not limited to two-dimensional spaces however; they can be defined in an arbitrary number of dimensions and can be square, triangle, hexagonal, or any other spatially invariant arrangement. Topologically, cells can be arranged on an infinite plane or on a toroidal space. Cell interconnect is local, meaning that all connections between cells are within a specified radius (with distance measured topologically).

The double-pointed cofinite topology is the cofinite topology with every point doubled; that is, it is the topological product of the cofinite topology with the indiscrete topology on a two-element set. It is not T0 or T1, since the points of the doublet are topologically indistinguishable. It is, however, R0 since the topologically distinguishable points are separable. An example of a countable double-pointed cofinite topology is the set of even and odd integers, with a topology that groups them together.

Much of his work on stratified sets was developed so as to understand the notion of topologically stable maps, and to eventually prove the result that the set of topologically stable mappings between two smooth manifolds is a dense set. Thom's lectures on the stability of differentiable mappings, given at the University of Bonn in 1960, were written up by Harold Levine and published in the proceedings of a year long symposium on singularities at Liverpool University during 1969-70, edited by C. T. C. Wall. The proof of the density of topologically stable mappings was completed by John Mather in 1970, based on the ideas developed by Thom in the previous ten years. A coherent detailed account was published in 1976 by Christopher Gibson, Klaus Wirthmüller, Andrew du Plessis, and Eduard Looijenga.

A crossed pentagonal antiprism is topologically identical to the pentagonal antiprism, although it can't be made uniform. The sides are isosceles triangles. It has d5d symmetry, order 10. Its vertex configuration is 3.3/2.3.

The model of quasi-categories is, however, better suited to applications than that of topologically enriched categories, though it has been proved by Lurie that the two have natural model structures that are Quillen equivalent.

In RMCNN processors, the cell interconnect is local and topologically invariant, but the weights are used to store previous states and not to control dynamics. The weights of the cells are modified during some learning state creating long-term memory. The topology and dynamics of CNN processors closely resembles that of CA. Like most CNN processors, CA consists of a fixed-number of identical processors that are spatially discrete and topologically uniform. The difference is that most CNN processors are continuous-valued whereas CA have discrete-values.

The three- dimensional associahedron K5 is an enneahedron topologically equivalent to the order-4 truncated triangular bipyramid with nine faces (three squares and six pentagons) and fourteen vertices, and its dual is the triaugmented triangular prism.

It is part of a truncation process between a dodecahedron and icosahedron: This polyhedron is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry.

Circular chess is a chess variant played using the standard set of pieces on a circular board consisting of four rings, each of sixteen squares. This is topologically equivalent to playing on the surface of a cylinder.

Braiding those topological defect leads to topologically protected non-Abelian geometric phase, which can be used to perform topologically protected quantum computation. Topological degeneracy of topological order can be defined on a closed space or an open space with gapped boundaries or gapped domain walls, including both Abelian topological orders and non-Abelian topological orders. The application of these types of systems for quantum computation has been proposed. In certain generalized cases, one can also design the systems with topological interfaces enriched or extended by global or gauge symmetries.

This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram , progressing to infinity. This tiling is topologically related to regular polyhedra with vertex figure n3, as a part of sequence that continues into the hyperbolic plane. It is similarly related to the uniform truncated polyhedra with vertex figure n.6.6. This tiling is also a part of a sequence of truncated rhombic polyhedra and tilings with [n,3] Coxeter group symmetry.

Four-dimensional manifolds are the most unusual: they are not geometrizable (as in lower dimensions), and surgery works topologically, but not differentiably. Since topologically, 4-manifolds are classified by surgery, the differentiable classification question is phrased in terms of "differentiable structures": "which (topological) 4-manifolds admit a differentiable structure, and on those that do, how many differentiable structures are there?" Four-manifolds often admit many unusual differentiable structures, most strikingly the uncountably infinitely many exotic differentiable structures on R4. Similarly, differentiable 4-manifolds is the only remaining open case of the generalized Poincaré conjecture.

In mathematics, a completely metrizable spaceWillard, Definition 24.2 (metrically topologically complete spaceKelley, Problem 6.K, p. 207) is a topological space (X, T) for which there exists at least one metric d on X such that (X, d) is a complete metric space and d induces the topology T. The term topologically complete space is employed by some authors as a synonym for completely metrizable space,e. g. Steen and Seebach, I §5: Complete Metric Spaces but sometimes also used for other classes of topological spaces, like completely uniformizable spacesKelley, Problem 6.

SPT states are short-range entangled while topologically ordered states are long-range entangled. Both intrinsic topological order, and also SPT order, can sometimes have protected gapless boundary excitations. The difference is subtle: the gapless boundary excitations in intrinsic topological order can be robust against any local perturbations, while the gapless boundary excitations in SPT order are robust only against local perturbations that do not break the symmetry. So the gapless boundary excitations in intrinsic topological order are topologically protected, while the gapless boundary excitations in SPT order are symmetry protected.

This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.

This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,6}, and Coxeter diagram , with n progressing to infinity.

This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,5}, and Coxeter diagram , with n progressing to infinity.

That is, a pseudometric is a metric if and only if the topology it generates is T0 (i.e. distinct points are topologically distinguishable). The definitions of Cauchy sequences and metric completion for metric spaces carry over to pseudometric spaces unchanged.

A toroidal polyhedron is a polyhedron whose Euler characteristic is less than or equal to 0, or equivalently whose genus is 1 or greater. Topologically, the surfaces of such polyhedra are torus surfaces having one or more holes through the middle.

RP1 is called the real projective line, which is topologically equivalent to a circle. RP2 is called the real projective plane. This space cannot be embedded in R3. It can however be embedded in R4 and can be immersed in R3.

If two spaces are homeomorphic, they have identical topological properties, and are considered topologically the same. The cube and the sphere are homeomorphic, as are the coffee cup and the doughnut. But the circle is not homeomorphic to the doughnut.

Finite embedding problems characterize profinite groups. The following theorem gives an illustration for this principle. Theorem. Let F be a countably (topologically) generated profinite group. Then # F is projective if and only if any finite embedding problem for F is solvable.

Spherical pentagonal hexecontahedron This polyhedron is topologically related as a part of sequence of polyhedra and tilings of pentagons with face configurations (V3.3.3.3.n). (The sequence progresses into tilings the hyperbolic plane to any n.) These face-transitive figures have (n32) rotational symmetry.

Figure 2. Adding another T-section to the ladders of figure 1 Figure 3. Examples of symmetric (top) and antimetric (bottom) networks which do not exhibit topological symmetry nor antimetry. Symmetric and antimetric networks are often also topologically symmetric and antimetric, respectively.

It has 10 triangular faces, 15 edges, and 6 vertices. It is also related to the nonconvex uniform polyhedron, the tetrahemihexahedron, which could be topologically identical to the hemi- icosahedron if each of the 3 square faces were divided into two triangles.

It is related to the order-4 120-cell honeycomb, {5,3,3,4}, and order-5 120-cell honeycomb, {5,3,3,5}. It is topologically similar to the finite 5-cube, {4,3,3,3}, and 5-simplex, {3,3,3,3}. It is analogous to the 120-cell, {5,3,3}, and dodecahedron, {5,3}.

It is related to the order-5 tesseractic honeycomb, {4,3,3,5}, and order-5 120-cell honeycomb, {5,3,3,5}. It is topologically similar to the finite 5-orthoplex, {3,3,3,4}, and 5-simplex, {3,3,3,3}. It is analogous to the 600-cell, {3,3,5}, and icosahedron, {3,5}.

They also find applications in elliptic curve cryptography (ECC) and integer factorization. An elliptic curve is not an ellipse: see elliptic integral for the origin of the term. Topologically, a complex elliptic curve is a torus, while a complex ellipse is a sphere.

YedZ (TC# 5.B.7) of E. coli has been examined topologically and has 6 transmembrane segments (TMSs) with both the N- and C-termini localized to the cytoplasm. von Rozycki et al. 2004 identified homologues of YedZ in bacteria and animals.

They will be infinite unless the rotations are specially chosen. All the infinite groups mentioned so far are not closed as topological subgroups of O(3). We now discuss topologically closed subgroups of O(3). An unmarked sphere has O(3) symmetry.

Michael A. Levin and Xiao-Gang Wen have also defined string-nets using tensor categories that are objects very similar to spin networks. However the exact connection with spin networks is not clear yet. String-net condensation produces topologically ordered states in condensed matter.

The symmetry will be the product of the symmetry of the two polygons. So a rectangle-rectangle duopyramid would be topologically identical to the uniform 4-4 duopyramid, but a lower symmetry [2,2,2], order 16, possibly doubled to 32 if the two rectangles are identical.

A related, but weaker, notion is that of a preregular space. X is a preregular space if any two topologically distinguishable points can be separated by disjoint neighbourhoods. Preregular spaces are also called R_1 spaces. The relationship between these two conditions is as follows.

Here, a vacuum angle (or 'theta angle') distinguishes between topologically different sectors in the vacuum. These topological sectors correspond to the robustly quantized phases. The quantum Hall transitions can then be understood by looking at the topological excitations (instantons) that occur between those phases.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, all 8 forms are distinct. However treating faces identically, there are only three unique topologically forms: square tiling, truncated square tiling, snub square tiling.

In dimension 5 and above (and 4 dimensions topologically), manifolds are classified by surgery theory. The Whitney trick requires 2+1 dimensions (2 space, 1 time), hence the two Whitney disks of surgery theory require 2+2+1=5 dimensions. The reason for dimension 5 is that the Whitney trick works in the middle dimension in dimension 5 and more: two Whitney disks generically don't intersect in dimension 5 and above, by general position (2+2 < 5). In dimension 4, one can resolve intersections of two Whitney disks via Casson handles, which works topologically but not differentiably; see Geometric topology: Dimension for details on dimension.

For example, compactness and connectedness are topological properties, whereas boundedness and completeness are not. Algebraic topology is the study of topologically invariant abstract algebra constructions on topological spaces. ;Topological space: A topological space (X, T) is a set X equipped with a collection T of subsets of X satisfying the following axioms: :# The empty set and X are in T. :# The union of any collection of sets in T is also in T. :# The intersection of any pair of sets in T is also in T. :The collection T is a topology on X. ;Topological sum: See Coproduct topology. ;Topologically complete: Completely metrizable spaces (i. e.

The third rendering is more commonly known as lattice topology. It is not so obvious that this is topologically equivalent. It can be seen that this is indeed so by visualising the top left node moved to the right of the top right node. Figure 1.9.

His early work was mainly on the theory of 3-manifolds. He dealt mainly with Haken manifolds and Heegaard splitting. Among other things, he proved that, roughly speaking, any homotopy equivalence of Haken manifolds is homotopic to a homeomorphism, i.e. that closed Haken manifolds are topologically rigid.

The automorphisms form the projective linear group PGL(2,R). Topologically, the real projective line is homeomorphic to the circle. The real projective line is the boundary of the hyperbolic plane. Every isometry of the hyperbolic plane induces a unique geometric transformation of the boundary, and vice versa.

In geometry, a simplicial polytope is a polytope whose facets are all simplices. For example, a simplicial polyhedron in three dimensions contains only triangular facesPolyhedra, Peter R. Cromwell, 1997. (p.341) and corresponds via Steinitz's theorem to a maximal planar graph. They are topologically dual to simple polytopes.

A concentric- cylinder Fluidyne pumping engine. Topologically equivalent to a U-tube design. A Fluidyne engine is an alpha or gamma type Stirling engine with one or more liquid pistons. It contains a working gas (often air), and either two liquid pistons or one liquid piston and a displacer.

Multiple levels of mammalian genome organization. Chromosomes occupy discrete territories in the nucleus (left). Topologically associating domains (TADs) are regions of the genome with locally high interaction frequency (center). Insulated neighborhoods are loops formed by the interaction of CTCF/cohesin- bound anchors containing genes and their regulatory elements.

This is called a glitch. Some reactive languages are glitch-free, and prove this property. This is usually achieved by topologically sorting expressions and updating values in topological order. This can, however, have performance implications, such as delaying the delivery of values (due to the order of propagation).

Johnson solid J₇. In geometry, the elongated triangular pyramid is one of the Johnson solids (J7). As the name suggests, it can be constructed by elongating a tetrahedron by attaching a triangular prism to its base. Like any elongated pyramid, the resulting solid is topologically (but not geometrically) self- dual.

Johnson solid J₈. In geometry, the elongated square pyramid is one of the Johnson solids (J8). As the name suggests, it can be constructed by elongating a square pyramid (J1) by attaching a cube to its square base. Like any elongated pyramid, it is topologically (but not geometrically) self-dual.

The Jahn- Teller distorted [Cu(OH)4Cl2] octahedra share the edges to form partially occupied layers parallel to (101). This layer is topologically the same as that in mica. Adjacent layers of octahedra are offset, such that vacant sites in one sheet align with occupied sites in the neighboring sheet.

It also interacts with the C-terminal region of the E1-like atg7 enzyme. Autophagocytosis is a starvation-induced process responsible for transport of cytoplasmic proteins to the lysosome/vacuole. Atg3 is a ubiquitin like modifier that is topologically similar to the canonical E2 enzyme. It catalyses the conjugation of Atg8 and phosphatidylethanolamine.

In the former case, the point is necessarily a codimension 3 metric singularity. The general problem of topologically classifying singularities in polyhedral spaces is largely unresolved (apart from simple statements that e.g. any singularity is locally a cone over a spherical polyhedral space one dimension less and we can study singularities there).

In soft matter physics, plumber's nightmare are structures that are characterized by fully connected, periodic, and topologically nontrivial surfaces. The term plumber's nightmare became widely known through a publicationDavid A. Huse, Stanislas Leibler. Phase behaviour of an ensemble of nonintersecting random fluid films. Journal de Physique, 1988, 49 (4), pp.605-621.

Topologically associating domains within chromosome territories, their borders and interactions A topologically associating domain (TAD) is a self- interacting genomic region, meaning that DNA sequences within a TAD physically interact with each other more frequently than with sequences outside the TAD. The median size of a TAD in mouse cells is 880 kb, and they have similar sizes in non-mammalian species. Boundaries at both side of the these domains are conserved between different mammalian cell types and even across species and are highly enriched with CCCTC-binding factor (CTCF) and cohesin binding sites. In addition, some types of genes (such as transfer RNA genes and housekeeping genes) appear near TAD boundaries more often than would be expected by chance.

Topologically, SO(3) is the real projective space RP3, and it is often desirable to represent rotations by a set of three numbers, known as Euler angles (in numerous variants), both because this is conceptually simple, and because one can build a combination of three gimbals to produce rotations in three dimensions. Topologically this corresponds to a map from the 3-torus T3 of three angles to the real projective space RP3 of rotations, but this map does not have rank 3 at all points (formally because it cannot be a covering map, as the only (non- trivial) covering space is the hypersphere S3), and the phenomenon of the rank dropping to 2 at certain points is referred to in engineering as gimbal lock.

For Euclidean spaces, using the L2 norm gives rise to the Euclidean metric in the product space; however, any other choice of p will lead to a topologically equivalent metric space. In the category of metric spaces (with Lipschitz maps having Lipschitz constant 1), the product (in the category theory sense) uses the sup metric.

Pentagonal stephanoid. This stephanoid has pentagonal dihedral symmetry and has the same vertices as the uniform pentagonal prism. A crown polyhedron or stephanoid is a toroidal polyhedron which is also noble, being both isogonal (equal vertices) and isohedral (equal faces). Crown polyhedra are self-intersecting and topologically self-dual.. See in particular p. 60.

On a ringed space X, an invertible sheaf L is trivial if isomorphic to OX, as an OX-module. If the base X is a complex manifold, then an invertible sheaf is (the sheaf of sections of) a holomorphic line bundle, and trivial means holomorphically equivalent to a trivial bundle, not just topologically equivalent.

Shapes can be drawn precisely or pushed and pulled as the designer chooses. ; Solid modelling: Cobalt exports topologically correct ACIS, Parasolids, and STEP files for tool-path and Gcode generation using external computer-aided manufacturing (CAM) software. Alternatively IGES and DXF files can be used to send surface or profile data to external CAM software.

However in this duality the authors are able to cast the dynamics of the quantum foam in the familiar language of a topologically twisted U(1) gauge theory, whose field strength is linearly related to the Kähler form of the A-model. In particular this suggests that the A-model Kähler form should be quantized.

The structure of a zeolitic imidazolate framework is made through three- dimensional assembly of metal(imidazolate)4 tetrahedra. Zeolitic imidazolate frameworks (ZIFs) are a class of metal-organic frameworks (MOFs) that are topologically isomorphic with zeolites. ZIFs are composed of tetrahedrally- coordinated transition metal ions (e.g. Fe, Co, Cu, Zn) connected by imidazolate linkers.

A regular space is necessarily also preregular, i.e., any two topologically distinguishable points can be separated by neighbourhoods. Since a Hausdorff space is the same as a preregular T0 space, a regular space which is also T0 must be Hausdorff (and thus T3). In fact, a regular Hausdorff space satisfies the slightly stronger condition T2½.

Simplicially enriched categories, or simplicial categories, are categories enriched over simplicial sets. However, when we look at them as a model for (infinity,1)-categories, then many categorical notions (e.g., limits) do not agree with the corresponding notions in the sense of enriched categories. The same for other enriched models like topologically enriched categories.

In philosophy, specifically metaphysics, mereology is the study of parthood relationships. In mathematics and formal logic, wellfoundedness prohibits \cdots for any x. Thus non-wellfounded mereology treats topologically circular, cyclical, repetitive, or other eventual self-containment. More formally, non-wellfounded partial orders may exhibit \cdots for some x whereas well-founded orders prohibit that.

In particle theory, the skyrmion () is a topologically stable field configuration of a certain class of non-linear sigma models. It was originally proposed as a model of the nucleon by Tony Skyrme in 1962.Brown, G. E. (ed.) (1994) Selected Papers, with Commentary, of Tony Hilton Royle Skyrme. World Scientific Series in 20th Century Physics: Volume 3.

There are 2606 topologically distinct convex enneahedra, excluding mirror images. These can be divided into subsets of 8, 74, 296, 633, 768, 558, 219, 50, with 7 to 14 vertices respectively.Counting polyhedra A table of these numbers, together with a detailed description of the nine- vertex enneahedra, was first published in the 1870s by Thomas Kirkman..

100; Randazzo, p. 12), though most rivers in Canada and in parts of the United States drain north (Verdin, K. L. (March 19, 2010). "A System for Topologically Coding Global Drainage Basins and Stream Networks" , ESRI; Gonzalez, M. A. (2003). "Continental Divides in North Dakota and North America", North Dakota Geological Survey Newsletter 30 (1), pp. 1–7; ).

This effect, the quantum anomalous Hall effect has only previously been realised in magnetically doped topological insulators. As well as Dirac/linear SGSs, the other major category of SGS are parabolic spin gapless semiconductors. Electron mobility in such materials is two to four orders of magnitude higher than in classical semiconductors. SGSs are topologically non-trivial.

For vector data it must be made "topologically correct" before it can be used for some advanced analysis. For example, in a road network, lines must connect with nodes at an intersection. Errors such as undershoots and overshoots must also be removed. For scanned maps, blemishes on the source map may need to be removed from the resulting raster.

Take a copy of S^3, the three-dimensional sphere. Now find a compact unknotted solid torus T_1 inside the sphere. (A solid torus is an ordinary three-dimensional doughnut, i.e., a filled-in torus, which is topologically a circle times a disk.) The closed complement of the solid torus inside S^3 is another solid torus.

From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal tiling. It can also be generated from the (4 3 3) hyperbolic tilings: This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (n.6.6), and [n,3] Coxeter group symmetry.

Simple polytopes are topologically dual to simplicial polytopes. The family of polytopes which are both simple and simplicial are simplices or two-dimensional polygons. A simple polyhedron is a three- dimensional polyhedron whose vertices are adjacent to three edges and three faces. The dual to a simple polyhedron is a simplicial polyhedron, in which all faces are triangles.

The 11-cell, discovered independently by H. S. M. Coxeter and Branko Grünbaum, is an abstract 4-polytope. Its facets are hemi-icosahedra. Since its facets are, topologically, projective planes instead of spheres, the 11-cell is not a tessellation of any manifold in the usual sense. Instead, the 11-cell is a locally projective polytope.

However, many details that are not typically provided by common data sources are needed, such as traffic signals, turn lanes, etc. In addition, street network must be topologically appropriate, that is, connections between links must be consistent and representative. Transit network must be compatible with the street network layer. Data usually must be compiled from several independent sources.

Bour's surface crosses itself on three coplanar rays, meeting at equal angles at the origin of the space. The rays partition the surface into six sheets, topologically equivalent to half-planes; three sheets lie in the halfspace above the plane of the rays, and three below. Four of the sheets are mutually tangent along each ray.

The convex hull of two cantellated 24-cells in opposite positions is a nonuniform polychoron composed of 864 cells: 48 cuboctahedra, 144 square antiprisms, 384 octahedra (as triangular antipodiums), 288 tetrahedra (as tetragonal disphenoids), and 576 vertices. Its vertex figure is a shape topologically equivalent to a cube with a triangular prism attached to one of its square faces.

An alternative way to develop two openings from the blastopore during gastrulation, called amphistomy, appears to exist in some animals, such as nematodes.Amphistomy - Contributions to Zoology In humans the perforation of the anus and mouth happen at 8 weeks and 4 weeks respectively. When both the anus and mouth are perforated, the organism becomes topologically a torus.

An example of software which could do this is Vexcel FotoG 5.Vexcel FotoG This software has now been replaced by Vexcel GeoSynth. Another similar software program is Microsoft Photosynth.Photosynth3D data acquisition and object reconstruction using photos A semi-automatic method for acquiring 3D topologically structured data from 2D aerial stereo images has been presented by Sisi Zlatanova.

ZIP was the protocol by which AppleTalk network numbers were associated with zone names. A zone was a subdivision of the network that made sense to humans (for example, "Accounting Department"); but while a network number had to be assigned to a topologically-contiguous section of the network, a zone could include several different discontiguous portions of the network.

This technology has further aided the genetic and epigenetic study of chromosomes both in model organisms and in humans. These methods have revealed large-scale organization of the genome into topologically associating domains (TADs), which correlate with epigenetic markers. Some TADs are transcriptionally active, while others are repressed. Many TADs have been found in D. melanogaster, mouse and human.

Such realizations of quantum memory and quantum computation may potentially be made fault tolerant. Topologically ordered states in general have a special property that they contain non-trivial boundary states. In many cases, those boundary states become perfect conducting channel that can conduct electricity without generating heat. This can be another potential application of topological order in electronic devices.

The final step in the data flow reconstruction is the topological sorting of the association graph. The directed graph created in the previous step is topologically sorted to obtain the order in which the actors have modified the data. This inherit order of the actors defines the data flow of the big data pipeline or task.

In the 2-D XY model, vortices are topologically stable configurations. It is found that the high- temperature disordered phase with exponential correlation decay is a result of the formation of vortices. Vortex generation becomes thermodynamically favorable at the critical temperature T_c of the KT transition. At temperatures below this, vortex generation has a power law correlation.

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.

Image texturization is an application of multidimensional AM-FM modulation. In this method, the image (2-D signal) is expressed into its special frequencies and amplitude estimates. The signal is represented as a product of 2 FM functions (using the independent frequencies), making is separable. Using the instantaneous frequency, the image can be represented topologically to illustrate its texture.

This implies immediately that compact nilmanifolds (except a torus) cannot admit a Kähler structure (see also ). Topologically, all nilmanifolds can be obtained as iterated torus bundles over a torus. This is easily seen from a filtration by ascending central series.Sönke Rollenske, Geometry of nilmanifolds with left-invariant complex structure and deformations in the large, 40 pages, arXiv:0901.3120, Proc.

Adding a point at infinity to the complex plane results in a space that is topologically a sphere. Hence the complex projective line is also known as the Riemann sphere (or sometimes the Gauss sphere). It is in constant use in complex analysis, algebraic geometry and complex manifold theory, as the simplest example of a compact Riemann surface.

Topologically, an -sphere can be constructed as a one-point compactification of -dimensional Euclidean space. Briefly, the -sphere can be described as , which is -dimensional Euclidean space plus a single point representing infinity in all directions. In particular, if a single point is removed from an -sphere, it becomes homeomorphic to . This forms the basis for stereographic projection.

Manifolds differ radically in behavior in high and low dimension. High-dimensional topology refers to manifolds of dimension 5 and above, or in relative terms, embeddings in codimension 3 and above. Low- dimensional topology is concerned with questions in dimensions up to 4, or embeddings in codimension up to 2. Dimension 4 is special, in that in some respects (topologically), dimension 4 is high-dimensional, while in other respects (differentiably), dimension 4 is low-dimensional; this overlap yields phenomena exceptional to dimension 4, such as exotic differentiable structures on R4. Thus the topological classification of 4-manifolds is in principle easy, and the key questions are: does a topological manifold admit a differentiable structure, and if so, how many? Notably, the smooth case of dimension 4 is the last open case of the generalized Poincaré conjecture; see Gluck twists. The distinction is because surgery theory works in dimension 5 and above (in fact, it works topologically in dimension 4, though this is very involved to prove), and thus the behavior of manifolds in dimension 5 and above is controlled algebraically by surgery theory. In dimension 4 and below (topologically, in dimension 3 and below), surgery theory does not work, and other phenomena occur.

Steinmetz solid (intersection of two cylinders) In geometry, a Steinmetz solid is the solid body obtained as the intersection of two or three cylinders of equal radius at right angles. Each of the curves of the intersection of two cylinders is an ellipse. The intersection of two cylinders is called a bicylinder. Topologically, it is equivalent to a square hosohedron.

Topologically, a regular 2-dimensional tessellation may be regarded as similar to a (3-dimensional) polyhedron, but such that the angular defect is zero. Thus, Schläfli symbols may also be defined for regular tessellations of Euclidean or hyperbolic space in a similar way as for polyhedra. The analogy holds for higher dimensions. For example, the hexagonal tiling is represented by {6,3}.

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.

It is also sometimes called a trisoctahedron, or, more fully, trigonal trisoctahedron. Both names reflect the fact that it has three triangular faces for every face of an octahedron. The tetragonal trisoctahedron is another name for the deltoidal icositetrahedron, a different polyhedron with three quadrilateral faces for every face of an octahedron. This convex polyhedron is topologically similar to the concave stellated octahedron.

The house with two rooms is a contractible 2-complex that is not collapsible. Another such example, popularized by E.C. Zeeman, is the dunce hat. The house with two rooms can also be thickened and then triangulated to be unshellable, despite the thickened house topologically being a 3-ball. The house with two rooms shows up in various ways in topology.

Like the uniform polyhedra there are eight uniform tilings that can be based from the regular square tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, all 8 forms are distinct. However treating faces identically, there are only three topologically distinct forms: square tiling, truncated square tiling, snub square tiling.

In 2016 Logan Hoehn and Lex Oversteegen classified all planar homogeneous continua, up to a homeomorphism, as the circle, pseudo-arc and circle of pseudo-arcs. In 2019 Hoehn and Oversteegen showed that the pseudo-arc is topologically the only, other than the arc, hereditarily equivalent planar continuum, thus providing a complete solution to the planar case of Mazurkiewicz's problem from 1921.

Schuldiner is a world expert in the expression, purification, and characterization of membrane proteins. Specifically, he focuses on the multidrug antiporter EmrE. Investigation of its subunit structure led the Schuldiner Group to propose a dimer of topologically parallel subunits. This led to a long-running controversy over EmrE topology that eventually led Schuldiner to demonstrate that both parallel and antiparallel dimers are functional.

Bredon (1997), Theorem II.17.4; Borel (1984), V.3.17. For example, for a compact Hausdorff space X that is locally contractible (in the weak sense discussed above), the sheaf cohomology group Hj(X,Z) is finitely generated for every integer j. One case where the finiteness result applies is that of a constructible sheaf. Let X be a topologically stratified space.

The evolution of the Sierpinski triangle The Sierpinski triangle may be constructed from an equilateral triangle by repeated removal of triangular subsets: # Start with an equilateral triangle. # Subdivide it into four smaller congruent equilateral triangles and remove the central triangle. # Repeat step 2 with each of the remaining smaller triangles infinitely. Each removed triangle (a trema) is topologically an open set.

What if the cut is made from down the real axis to the point at infinity, and from , up the real axis until the cut meets itself? Again a Riemann surface can be constructed, but this time the "hole" is horizontal. Topologically speaking, both versions of this Riemann surface are equivalent – they are orientable two-dimensional surfaces of genus one.

In mathematics, a complex line is a one-dimensional affine subspace of a vector space over the complex numbers.. A common point of confusion is that while a complex line has dimension one over C (hence the term "line"), it has dimension two over the real numbers R, and is topologically equivalent to a real plane, not a real line..

This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbol {n,3}. From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.

It is related to topologically massive gravity and the Cotton tensor. It is a possible UV completion of general relativity. Also, the speed of light goes to infinity at high energies. The novelty of this approach, compared to previous approaches to quantum gravity such as Loop quantum gravity, is that it uses concepts from condensed matter physics such as quantum critical phenomena.

Wraparound, in video games, is a gameplay variation on the single-screen in which space is finite but unbounded; objects leaving one side of the screen immediately reappear on the opposite side, maintaining speed and trajectory. This is referred to as "wraparound", since the top and bottom of the screen wrap around to meet, as do the left and right sides (this is topologically equivalent to a Euclidean 2-torus).The medium of the video game, Mark J. P. Wolf, University of Texas Press, 2001, 203 pp, p. 56, at Google Books Some games wrap around in some directions but not others, such as games of the Civilization series that wrap around left to right, or east and west but the top and bottom remain edges representing the North and South Pole (topologically equivalent to a cylinder).

Topologically, SO(3) is the real projective space RP3, with fundamental group Z/2, and only (non-trivial) covering space the hypersphere S3, which is the group Spin(3), and represented by the unit quaternions. Thus quaternions are a preferred method for representing spatial rotations – see quaternions and spatial rotation. However, it is often desirable to represent rotations by a set of three numbers, known as Euler angles (in numerous variants), both because this is conceptually simpler for someone familiar with planar rotation, and because one can build a combination of three gimbals to produce rotations in three dimensions. Topologically this corresponds to a map from the 3-torus T3 of three angles to the real projective space RP3 of rotations, and the resulting map has imperfections due to this map being unable to be a covering map.

In the end, falls out of all topologically important quantities. In fact, it can be shown that Bose–Einstein condensation exists in the fitness model even without mapping to a Bose gas. A similar gelation can be seen in models with superlinear preferential attachment, however, it is not clear whether this is an accident or a deeper connection lies between this and the fitness model.

A different approach is used for geomipmapping, a popular terrain rendering algorithm because this applies to terrain meshes which are both graphically and topologically different from "object" meshes. Instead of computing an error and simplify the mesh according to this, geomipmapping takes a fixed reduction method, evaluates the error introduced and computes a distance at which the error is acceptable. Although straightforward, the algorithm provides decent performance.

Another view states that BPST- like instantons play an important role in the vacuum structure of QCD. These instantons were discovered in 1975 by Alexander Belavin, Alexander Markovich Polyakov, Albert S. Schwarz and Yu. S. Tyupkin as topologically stable solutions to the Yang-Mills field equations. They represent tunneling transitions from one vacuum state to another. These instantons are indeed found in lattice calculations.

Abelson graduated with an A.B. in mathematics from Princeton University in 1969 after completing a senior thesis, titled "Actions with fixed-point set: a homology sphere", under the supervision of William Browder. He later received a Ph.D. in mathematics from the Massachusetts Institute of Technology in 1973 after completing his doctoral dissertation, titled "Topologically distinct conjugate varieties with finite fundamental group", under the supervision Dennis Sullivan.

Shannon's formula is an analytic expression for calculating the gain of an interconnected set of amplifiers in an analog computer. During World War II, while investigating the functional operation of an analog computer, Claude Shannon developed his formula. Because of wartime restrictions, Shannon's work was not published at that time, and, in 1952, Mason rediscovered the same formula. Happ generalized the Shannon formula for topologically closed systems.

In mathematics, specifically in the field of differential topology, Morse homology is a homology theory defined for any smooth manifold. It is constructed using the smooth structure and an auxiliary metric on the manifold, but turns out to be topologically invariant, and is in fact isomorphic to singular homology. Morse homology also serves as a model for the various infinite-dimensional generalizations known as Floer homology theories.

The conjecture was raised in the form of a question by Albert Marden, who proved that any geometrically finite hyperbolic 3-manifold is topologically tame. The conjecture was also called the Marden conjecture or the tame ends conjecture. There had been steady progress in understanding tameness before the conjecture was resolved. Partial results had been obtained by Thurston, Brock, Bromberg, Canary, Evans, Minsky, Ohshika.

The process involves the manual digitizing of a number of points necessary for automatically reconstructing the 3D objects. Each reconstructed object is validated by superimposition of its wire frame graphics in the stereo model. The topologically structured 3D data is stored in a database and are also used for visualization of the objects. Notable software used for 3D data acquisition using 2D images include e.g.

In mathematics, the Denjoy theorem gives a sufficient condition for a diffeomorphism of the circle to be topologically conjugate to a diffeomorphism of a special kind, namely an irrational rotation. proved the theorem in the course of his topological classification of homeomorphisms of the circle. He also gave an example of a C1 diffeomorphism with an irrational rotation number that is not conjugate to a rotation.

Not every planar graph corresponds to a convex polyhedron in this way: the trees do not, for example. Steinitz's theorem says that the polyhedral graphs formed from convex polyhedra are precisely the finite 3-connected simple planar graphs. More generally, Euler's formula applies to any polyhedron whose faces are simple polygons that form a surface topologically equivalent to a sphere, regardless of its convexity.

These marks promote heterochromatin formation. Analysis of the spread of X chromosome inactivation into autosomal material in one study showed that genes that were subject to (or escaped from) X chromosome inactivation clustered within topologically associating domains, and these genes were more likely to be found in regions that have PRC2 and histone 3 lysine 27 trimethylation marks normally on non-rearranged chromosomes.Portoso M, Cavalli G (2008).

Peripheral cycles have also been called non-separating cycles, but this term is ambiguous, as it has also been used for two related but distinct concepts: simple cycles the removal of which would disconnect the remaining graph,E.g. see . and cycles of a topologically embedded graph such that cutting along the cycle would not disconnect the surface on which the graph is embedded.E.g. see .

The insertion of (viral) DNA into chromosomes and other forms of recombination can also require the action of topoisomerases. Topologically linked circular molecules, aka catenanes, adopt a positive supercoiled form during the process of replication of circular plasmids. The unlinking of catenanes is performed by type IIA topoisomerases, which were recently found to be more efficient unlinking positive supercoiled DNA.The conformational properties of negative vs.

Those excitations turn out to be gauge bosons. The ends of strings are defects which correspond to another type of excitations. Those excitations are the gauge charges and can carry Fermi or fractional statistics. The condensations of other extended objects such as "membranes", "brane-nets", and fractals also lead to topologically ordered phases; Topological Orders and Chern–Simons Theory in strongly correlated quantum liquid.

Descartes' theorem on the "total defect" of a polyhedron states that if the polyhedron is homeomorphic to a sphere (i.e. topologically equivalent to a sphere, so that it may be deformed into a sphere by stretching without tearing), the "total defect", i.e. the sum of the defects of all of the vertices, is two full circles (or 720° or 4π radians). The polyhedron need not be convex.

Due to the three- dimensional curvature of this structure, two-dimensional sections such as shown are commonly seen. Neuroimaging pictures can show a number of different shapes, depending on the angle and location of the cut. Shape of human hippocampus and associated structures. Topologically, the surface of a cerebral hemisphere can be regarded as a sphere with an indentation where it attaches to the midbrain.

Plexcitons were found to emerge from an organic molecular layer (excitons) and a metallic film (plasmons). Dirac cones appeared in the plexcitons' two-dimensional band-structure. An external magnetic field created a gap between the cones when the system was interfaced to a magneto-optical layer. The resulting energy gap became populated with topologically protected one-way modes, which traveled only at the system interface.

Data from high-throughput chromosome conformation capture experiments, such as Hi-C (experiment) and ChIA-PET, can provide information on the spatial proximity of DNA loci. Analysis of these experiments can determine the three-dimensional structure and nuclear organization of chromatin. Bioinformatic challenges in this field include partitioning the genome into domains, such as Topologically Associating Domains (TADs), that are organised together in three-dimensional space.

However, a tree network connected to another tree network is still topologically a tree network, not a distinct network type. A hybrid topology is always produced when two different basic network topologies are connected. A star-ring network consists of two or more ring networks connected using a multistation access unit (MAU) as a centralized hub. Snowflake topology is a star network of star networks.

In 1971, Varchenko proved that a family of complex quasi- projective algebraic sets with an irreducible base form a topologically locally trivial bundle over a Zariski open subset of the base. This statement, conjectured by Oscar Zariski, had filled up a gap in the proof of Zariski's theorem on the fundamental group of the complement to a complex algebraic hypersurface published in 1937. In 1973, Varchenko proved René Thom's conjecture that a germ of a generic smooth map is topologically equivalent to a germ of a polynomial map and has a finite dimensional polynomial topological versal deformation, while the non-generic maps form a subset of infinite codimension in the space of all germs. Varchenko was among creators of the theory of Newton polygons in singularity theory, in particular, he gave a formula, relating Newton polygons and asymptotics of the oscillatory integrals associated with a critical point of a function.

In another direction, every sober space (which may not be T1) is T0; this includes the underlying topological space of any scheme. Given any topological space one can construct a T0 space by identifying topologically indistinguishable points. T0 spaces that are not T1 spaces are exactly those spaces for which the specialization preorder is a nontrivial partial order. Such spaces naturally occur in computer science, specifically in denotational semantics.

42, No. 2, 2001, pp. 301–306. Similarly a stuck open chain is an open polygonal chain such that the segments may not be aligned by moving rigidly its segments. Topologically such a chain can be unknotted, but the limitation of using only rigid motions of the segments can create nontrivial knots in such a chain. Consideration of such "stuck" configurations arises in the study of molecular chains in biochemistry.

Small-scale DLGs are sold in state units and are cast on either the Albers equal-area conic projection system or the geographic coordinate system of latitude and longitude, depending on the distribution format. All DLGs are referenced to the North American Datum of 1927 (NAD27) or the North American Datum of 1983 (NAD83). USGS DLGs are topologically structured for use in mapping and geographic information system (GIS) applications.

The rational numbers form a metric space by using the absolute difference metric and this yields a third topology on Q. All three topologies coincide and turn the rationals into a topological field. The rational numbers are an important example of a space which is not locally compact. The rationals are characterized topologically as the unique countable metrizable space without isolated points. The space is also totally disconnected.

The Egyptian pyramids have been the models used so far for this purpose and this is reflected in pieces like Snefru for accordion and electronics or Nebmaat for saxophone, clarinet and string trio. With regard to painting, its relationship with music is examined in one of its perspective techniques – the anamorphosis – as a procedure to topologically transform music. This is the approach used in the eponymous ensemble piece Anamorfosis.

3D model of a dual uniform pentagrammic bipyramid 3D model of a pentagrammic bipyramid with regular faces In geometry, the pentagrammic bipyramid (or dipyramid) is first of the infinite set of face-transitive star bipyramids containing star polygon arrangement of edges. It has 10 intersecting isosceles triangle faces. It is topologically identical to the pentagonal bipyramid. Each star bipyramid is the dual of a star polygon based uniform prism.

Escher's solid. This image does not depict the stellation, because different visible parts of a single hexagonal face of the stellation have different colors. However, the coloring is consistent with a depiction of the polyhedral compound of three flattened octahedra. Escher's solid is topologically equivalent to the disdyakis dodecahedron, a Catalan solid, which can be seen as a rhombic dodecahedron with shorter rhombic pyramids augumented to each face.

20 stacked pentagonal antiprisms occur in two disjoint rings of 10 antiprisms each. The antiprisms in each ring are joined to each other via their pentagonal faces. The two rings are mutually perpendicular, in a structure similar to a duoprism. The 300 tetrahedra join the two rings to each other, and are laid out in a 2-dimensional arrangement topologically equivalent to the 2-torus and the ridge of the duocylinder.

This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry. From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.

In topology, a branch of mathematics, the ends of a topological space are, roughly speaking, the connected components of the "ideal boundary" of the space. That is, each end represents a topologically distinct way to move to infinity within the space. Adding a point at each end yields a compactification of the original space, known as the end compactification. The notion of an end of a topological space was introduced by .

It has been shown that condensin, a large protein complex that plays a central role in mitotic chromosome assembly, induces positive supercoils in an ATP hydrolysis- dependent manner in vitro. Supercoiling could also play an important role during interphase in the formation and maintenance of topologically associating domains (TADs). Supercoiling is also required for DNA/RNA synthesis. Because DNA must be unwound for DNA/RNA polymerase action, supercoils will result.

Singularities of codimension 2 are of major importance; they are characterized by a single number, the conical angle. The singularities can also studied topologically. Then, for example, there are no topological singularities of codimension 2. In a 3-dimensional polyhedral space without a boundary (faces not glued to other faces) any point has a neighborhood homeomorphic either to an open ball or to a cone over the projective plane.

Wenninger suggested these figures are members of a new class of stellation figures, called stellation to infinity. However, he also suggested that strictly speaking they are not polyhedra because their construction does not conform to the usual definitions. Topologically it is considered to contain seven vertices. The three vertices considered at infinity (the real projective plane at infinity) correspond directionally to the three vertices of the hemi- octahedron, an abstract polyhedron.

A topological polytope is a topological space given along with a specific decomposition into shapes that are topologically equivalent to convex polytopes and that are attached to each other in a regular way. Such a figure is called simplicial if each of its regions is a simplex, i.e. in an n-dimensional space each region has n+1 vertices. The dual of a simplicial polytope is called simple.

MoTe2 exhibits topological Fermi arcs. This is evidence for a new type (type-II) of Weyl fermion that arises due to the breaking of Lorentz invariance, which does not have a counterpart in high-energy physics, which can emerge as topologically protected touching between electron and hole pockets. The topological surface states are confirmed by directly observing the surface states using bulk- and surface-sensitive angle-resolved photoemission spectroscopy.

Sundance Osland Bilson-Thompson is an Australian theoretical particle physicist. He has developed the idea that certain preon models may be represented topologically, rather than by treating preons as pointlike particles. His ideas have attracted interest in the field of loop quantum gravity, as they may represent a way of incorporating the Standard Model into loop quantum gravity. This would make loop quantum gravity a candidate theory of everything.

Alexander's theorem states that the 2-sphere does not smoothly (or PL or tame topologically) knot in the 3-sphere. In the tame topological category, it's known that the -sphere does not knot in the -sphere for all . This is a theorem of Morton Brown, Barry Mazur, and Marston Morse. The Alexander horned sphere is an example of a knotted 2-sphere in the 3-sphere which is not tame.

J. Biol. Chem. 265, 3961-3966 #Yerushalmi, H., Lebendiker, M., and Schuldiner, S. (1995) EmrE, an Escherichia coli 12-kDa multidrug transporter, exchanges toxic cations and H+ and is soluble in organic solvents. J. Biol. Chem. 270, 6856-6863 #Nasie, I., Steiner-Mordoch, S., Gold, A., and Schuldiner, S. (2010) Topologically Random Insertion of Emre Supports a Pathway for Evolution of Inverted Repeats in Ion-Coupled Transporters. J. Biol. Chem.

Circular plot of the bipartite mouse X chromosome, generated by the Epigenome Browser. Image from The different 3C-style experiments produce data with very different structures and statistical properties. As such, specific analysis packages exist for each experiment type. Hi-C data is often used to analyze genome-wide chromatin organization, such as topologically associating domains (TADs), linearly contiguous regions of the genome that are associated in 3-D space.

Crystals that break translation symmetry lead to well defined electronic bands which in turn allow us to make semiconducting devices such as transistors. Different types of topological orders are even richer than different types of symmetry- breaking orders. This suggests their potential for exciting, novel applications. One theorized application would be to use topologically ordered states as media for quantum computing in a technique known as topological quantum computing.

This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry. From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are eight forms.

This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (n.6.6), and [n,3] Coxeter group symmetry. From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.

The disk has circular symmetry. The open disk and the closed disk are not topologically equivalent (that is, they are not homeomorphic), as they have different topological properties from each other. For instance, every closed disk is compact whereas every open disk is not compact.. However from the viewpoint of algebraic topology they share many properties: both of them are contractible. and so are homotopy equivalent to a single point.

The baker's map can be understood as the bilateral shift operator of a bi-infinite two-state lattice model. The baker's map is topologically conjugate to the horseshoe map. In physics, a chain of coupled baker's maps can be used to model deterministic diffusion. As with many deterministic dynamical systems, the baker's map is studied by its action on the space of functions defined on the unit square.

The notion of graph minor can be promoted from a well-quasi-ordering to an equivalence relation if we assume the relation is symmetric. This is the notion of equivalence used for Bratteli diagrams. The major result in this field is that equivalent essentially simple ordered Bratteli diagrams correspond to topologically conjugate pointed dynamical systems. This allows us apply results from the former field into the latter and vice versa.

Such fields themselves exhibit magnetic helicity, reflecting their own topologically nontrivial structure. Much interest attaches to the determination of states of minimum energy, subject to prescribed topology. Many problems of fluid dynamics and magnetohydrodynamics fall within this category. Recent developments in topological fluid dynamics include also applications to magnetic braids in the solar corona, DNA knotting by topoisomerases, polymer entanglement in chemical physics and chaotic behavior in dynamical systems.

The edge, or boundary, of a Möbius strip is homeomorphic (topologically equivalent) to a circle. Under the usual embeddings of the strip in Euclidean space, as above, the boundary is not a true circle. However, it is possible to embed a Möbius strip in three dimensions so that the boundary is a perfect circle lying in some plane. For example, see Figures 307, 308, and 309 of "Geometry and the imagination".

After a lemma of Augustin Cauchy on the impossibility of labeling the edges of a polyhedron by positive and negative signs so that each vertex has at least four sign changes, the remainder of chapter 2 outlines the content of the remaining book. Chapters 3 and 4 prove Alexandrov's uniqueness theorem, characterizing the surface geometry of polyhedra as being exactly the metric spaces that are topologically spherical locally like the Euclidean plane except at a finite set of points of positive angular defect, obeying Descartes' theorem on total angular defect that the total angular defect should be 4\pi. Chapter 5 considers the metric spaces defined in the same way that are topologically a disk rather than a sphere, and studies the flexible polyhedral surfaces that result. Chapters 6 through 8 of the book are related to a theorem of Hermann Minkowski that a convex polyhedron is uniquely determined by the areas and directions of its faces, with a new proof based on invariance of domain.

In mathematics, a stuck unknot is a closed polygonal chain that is topologically equal to the unknot but cannot be deformed to a simple polygon by rigid motions of the segments.G. Aloupis, G. Ewald, and G. T. Toussaint, "More classes of stuck unknotted hexagons," Contributions to Algebra and Geometry, Vol. 45, No. 2, 2004, pp. 429–434.G. T. Toussaint, "A new class of stuck unknots in Pol-6," Contributions to Algebra and Geometry, Vol.

More generally, formal power series can include series with any finite (or countable) number of variables, and with coefficients in an arbitrary ring. In algebraic geometry and commutative algebra, rings of formal power series are especially tractable topologically complete local rings, allowing calculus-like arguments within a purely algebraic framework. They are analogous in many ways to p-adic numbers. Formal power series can be created from Taylor polynomials using formal moduli.

The spatial arrangement of the chromatin within the nucleus is not random - specific regions of the chromatin can be found in certain territories. Territories are, for example, the lamina-associated domains (LADs), and the topologically associating domains (TADs), which are bound together by protein complexes. Currently, polymer models such as the Strings & Binders Switch (SBS) model and the Dynamic Loop (DL) model are used to describe the folding of chromatin within the nucleus.

Topologically, the deltoidal hexecontahedron is identical to the nonconvex rhombic hexecontahedron. The deltoidal hexecontahedron can be derived from a dodecahedron (or icosahedron) by pushing the face centers, edge centers and vertices out to different radii from the body center. The radii are chosen so that the resulting shape has planar kite faces each such that vertices go to degree-3 corners, faces to degree-five corners, and edge centers to degree-four points.

The regular dodecahedron is topologically related to a series of tilings by vertex figure n3. The regular dodecahedron can be transformed by a truncation sequence into its dual, the icosahedron: The regular dodecahedron is a member of a sequence of otherwise non-uniform polyhedra and tilings, composed of pentagons with face configurations (V3.3.3.3.n). (For n > 6, the sequence consists of tilings of the hyperbolic plane.) These face-transitive figures have (n32) rotational symmetry.

Knot theory can be used to determine if a molecule is chiral (has a "handedness") or not. Chemical compounds of different handedness can have drastically differing properties, thalidomide being a notable example of this. More generally, knot theoretic methods have been used in studying topoisomers, topologically different arrangements of the same chemical formula. The closely related theory of tangles have been effectively used in studying the action of certain enzymes on DNA.

Thus one of the main problems in quantum topology has been to interpret quantum invariants topologically. The theory of claspers comes to provide such an interpretation. A clasper, like a framed link, is an embedded topological object in a 3-manifold on which one can perform surgery. In fact, clasper calculus can be thought of as a variant of Kirby calculus on which only certain specific types of framed links are allowed.

The harmonic table note layout is a recently developed musical interface that uses a note layout topologically equivalent to the Tonnetz. A Tonnetz of the syntonic temperament can be derived from a given isomorphic keyboard by connecting lines of successive perfect fifths, lines of successive major thirds, and lines of successive minor thirds. Like a Tonnetz itself, the isomorphic keyboard is tuning invariant. The topology of the syntonic temperament's Tonnetz is generally cylindrical.

Hunter's bend (or rigger's bend) is a knot used to join two lines. It consists of interlocking overhand knots, and can jam under moderate strain. It is topologically similar to the Zeppelin bend. When assessed against other bends In stress tests using paracord, it was found to be "not as strong as the blood knot, similar to the reverse figure of eight and stronger than the fisherman's bend, sheet bend or reef knot".

120px Vertex figure for the omnisnub cubic antiprism Also related is the bialternatosnub octahedral hosochoron, constructed by removing alternating long rectangles from the octagons, but is also not uniform. It has 40 cells: 2 rhombicuboctahedra (with Th symmetry), 6 rectangular trapezoprisms (topologically equivalent to a cube but with D2d symmetry), 8 octahedra (as triangular antiprisms), 24 triangular prisms (as Cs-symmetry wedges) filling the gaps, and 48 vertices. It has [4,(3,2)+] symmetry, order 48.

The Conway Knot. The slice property of the Conway knot was a long-standing unsolved problem in knot theory. The knot was named after its discoverer, English mathematician John Horton Conway, who first wrote about the knot in 1970. The Conway knot was determined to be topologically slice in the 1980s; however the nature of its sliceness, and whether or not it was smoothly slice, eluded mathematicians for decades up until Piccirillo's breakthrough.

Thus, if a first-order stream joins a second-order stream, it remains a second-order stream. It is not until a second-order stream combines with another second-order stream that it becomes a third-order stream. As with mathematical trees, a segment with index i must be fed by at least 2i − 1 different tributaries of index 1. Shreve noted that Horton’s and Strahler’s Laws should be expected from any topologically random distribution.

However, it is possible for a non- discrete finite space to be T0. In general, two points x and y are topologically indistinguishable if and only if x ≤ y and y ≤ x, where ≤ is the specialization preorder on X. It follows that a space X is T0 if and only if the specialization preorder ≤ on X is a partial order. There are numerous partial orders on a finite set. Each defines a unique T0 topology.

Similarly, a space is R0 if and only if the specialization preorder is an equivalence relation. Given any equivalence relation on a finite set X the associated topology is the partition topology on X. The equivalence classes will be the classes of topologically indistinguishable points. Since the partition topology is pseudometrizable, a finite space is R0 if and only if it is completely regular. Non-discrete finite spaces can also be normal.

Epsilon Geminorum is radiating around 8,500 times the luminosity of the Sun from its outer atmosphere at an effective temperature of 4,662 K. It is this temperature that gives it the yellow-hued glow of a G-type star. A surface magnetic field with a strength of has been detected on this star. This topologically complex field is most likely generated by a dynamo formed from the deep convection zone in the star's outer envelope.

The 'plane' formed by a finite field is the cartesian product of all ordered pairs of field elements, with opposite edges identified forming the surface topologically equivalent to a discretized torus. Individual elements correspond to standard 'points' and 'lines' to sets of no more than p points related by incidence (an initial point) plus direction or slope given in lowest terms (say all points '2 over and 1 up') that 'wrap' the plane before repeating.

Immanuel Bomze is an Austrian mathematician. In his Ph.D. thesis, he completely classified all (more than 100 topologically different) possible flows of the generalized Lotka–Volterra dynamics (generalized Lotka–Volterra equation) on the plane, employing equivalence of this dynamics to the 3-type replicator equation.I. Bomze, Lotka–Volterra equation and replicator dynamics: a two-dimensional classification. Biological Cybernetics 48, 201–211 (1983); I. Bomze, Lotka–Volterra equation and replicator dynamics: new issues in classification.

In topology and related branches of mathematics, a topological space X is a T0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of X, at least one of them has a neighborhood not containing the other. In a T0 space, all points are topologically distinguishable. This condition, called the T0 condition, is the weakest of the separation axioms. Nearly all topological spaces normally studied in mathematics are T0 spaces.

USGS logo A Digital Line Graph (DLG) is a cartographic map feature represented in digital vector form that is distributed by the U.S. Geological Survey (USGS). DLGs are collected from USGS maps and are distributed in large-, intermediate- and small-scale with up to nine different categories of features, depending on the scale. They come in optional and Spatial Data Transfer Standard (SDTS) format and are topologically structured for use in mapping and geographic information system (GIS) applications.

Bobfergusonite has a layered crystal structure topologically identical to that of alluaudite and wyllieite but with differences in the ordering of metal cations. The two types of layer alternate along Y. One layer consists of chains of metal cation octahedra cross-linked by phosphate tetrahedra. Within the chains metal cations are ordered M3+–M2+ in a similar fashion to wyllieite. However, the structure of bobfergusonite is distinct by the presence of Al and Fe3+ ordering between chains.

Using linear programming, it is possible to test whether a given polyhedron has an ideal version, in polynomial time. Every two ideal polyhedra with the same number of vertices have the same surface area, and it is possible to calculate the volume of an ideal polyhedron using the Lobachevsky function. The surface of an ideal polyhedron forms a hyperbolic manifold, topologically equivalent to a punctured sphere, and every such manifold forms the surface of a unique ideal polyhedron.

The strong real Jacobian conjecture was that a real polynomial map with a nowhere vanishing Jacobian determinant has a smooth global inverse. That is equivalent to asking whether such a map is topologically a proper map, in which case it is a covering map of a simply connected manifold, hence invertible. constructed two variable counterexamples of total degree 25 and higher. It is well-known that the Dixmier conjecture implies the Jacobian conjecture (see Bass et al. 1982).

Mathematically, a wallpaper group or plane crystallographic group is a type of topologically discrete group of isometries of the Euclidean plane that contains two linearly independent translations. Two such isometry groups are of the same type (of the same wallpaper group) if they are the same up to an affine transformation of the plane. Thus e.g. a translation of the plane (hence a translation of the mirrors and centres of rotation) does not affect the wallpaper group.

There are various methods used to form polymeric macromolecular cages. One synthetic method uses ring opening and multiple click chemistry in the first step to form trefoil and quatrefoil-shaped polymers, which can then be topologically converted into cages using hydrogenolysis. The initiator in this synthesis is azido and hydroxy functionalized p-xylene and the monomer is butylene oxide. The ring opening polymerization and simultaneous click cyclizations of butylene oxide with the initiator is catalyzed by t-Bu-P4.

Church Walk The nearest railway station, Nutfield, is about away in South Nutfield. Bletchingley is architecturally and topologically distinct: the central part of the village is a conservation area with several buildings timber-framed from the late Middle Ages and the village is set in a designated area of outstanding natural beauty (AONB). The Greensand Way runs fairly centrally through the parish, immediately south of the main village street which is part of the A25 road.

Operations on a set of network equations have a topological meaning which can aid visualisation of what is happening. Elimination of a node voltage from a set of network equations corresponds topologically to the elimination of that node from the graph. For a node connected to three other nodes, this corresponds to the well known Y-Δ transform. The transform can be extended to greater numbers of connected nodes and is then known as the star- mesh transform.

Solomon Lefschetz used his idea of a Lefschetz pencil to prove the theorem. Rather than considering the hyperplane section Y alone, he put it into a family of hyperplane sections Yt, where Y = Y0. Because a generic hyperplane section is smooth, all but a finite number of Yt are smooth varieties. After removing these points from the t-plane and making an additional finite number of slits, the resulting family of hyperplane sections is topologically trivial.

Most of the work on classification problems has focused on three dimensions, particularly on the classification of crystal nets, i.e., of periodic graphs that could serve as descriptions or designs for placement of atoms or molecular objects, with bonds indicated by edges, in a crystal. One of the more popular classification criteria is graph isomorphism, not to be confused with crystallographic isomorphism. Two periodic graphs are often called topologically equivalent if they are isomorphic, although not necessarily homotopic.

Path planning is an important issue as it allows a robot to get from point A to point B. Path planning algorithms are measured by their computational complexity. The feasibility of real-time motion planning is dependent on the accuracy of the map (or floorplan), on robot localization and on the number of obstacles. Topologically, the problem of path planning is related to the shortest path problem of finding a route between two nodes in a graph.

A sphere world is a mathematical concept used in robotic motion planning. Essentially, if the environment is represented as a sphere, and the robot and obstacles within the environment are represented as spheres, then it is possible to construct navigation functions which create paths from a start position to a goal position. Although the real world is rarely composed of spheres, sphere worlds are still useful for motion planning, as they are topologically equivalent to star worlds.

In mathematics, the tameness theorem states that every complete hyperbolic 3-manifold with finitely generated fundamental group is topologically tame, in other words homeomorphic to the interior of a compact 3-manifold. The tameness theorem was conjectured by . It was proved by and, independently, by Danny Calegari and David Gabai. It is one of the fundamental properties of geometrically infinite hyperbolic 3-manifolds, together with the density theorem for Kleinian groups and the ending lamination theorem.

The tameness theorem states that every complete hyperbolic 3-manifold with finitely generated fundamental group is topologically tame, in other words homeomorphic to the interior of a compact 3-manifold. The tameness theorem was conjectured by Marden. It was proved by Agol and, independently, by Danny Calegari and David Gabai. It is one of the fundamental properties of geometrically infinite hyperbolic 3-manifolds, together with the density theorem for Kleinian groups and the ending lamination theorem.

As the name suggests, the type of rod traditionally used with this hitch is a marlinespike. The advantages of this hitch over others which might serve the purpose are its quickness of tying and ease of releasing. Topologically it is a form of the noose, but in practice this hitch is not allowed to collapse into that shape. When it does capsize into a traditional noose, it can jam against the rod, making it much more difficult to release.

Since there is an algorithm to check if a 3-manifold is Haken (cf. Jaco–Oertel), the basic problem of recognition of 3-manifolds can be considered to be solved for Haken manifolds. proved that closed Haken manifolds are topologically rigid: roughly, any homotopy equivalence of Haken manifolds is homotopic to a homeomorphism (for the case of boundary, a condition on peripheral structure is needed). So these three-manifolds are completely determined by their fundamental group.

A stereographic projection of a Clifford torus performing a simple rotation Topologically a rectangle is the fundamental polygon of a torus, with opposite edges sewn together. In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the cartesian product of two circles S and S (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdon Clifford. It resides in R4, as opposed to in R3.

Bodies that are opaque to thermal radiation that falls on them are valuable in the study of heat radiation. Planck analyzed such bodies with the approximation that they be considered topologically to have an interior and to share an interface. They share the interface with their contiguous medium, which may be rarefied material such as air, or transparent material, through which observations can be made. The interface is not a material body and can neither emit nor absorb.

The Weyl tensor is algebraically special, in fact it has Petrov type D. The global structure is known. Topologically, the homotopy type of the Kerr spacetime can be simply characterized as a line with circles attached at each integer point. Note that the inner Kerr geometry is unstable with regard to perturbations in the interior region. This instability means that although the Kerr metric is axis-symmetric, a black hole created through gravitational collapse may not be so.

Peter Buser and Hermann Karcher. Gromov's almost flat manifolds. Astérisque, 81. Société Mathématique de France, Paris, 1981. 148 pp.Peter Buser and Hermann Karcher. The Bieberbach case in Gromov's almost flat manifold theorem. Global differential geometry and global analysis (Berlin, 1979), pp. 82–93, Lecture Notes in Math., 838, Springer, Berlin-New York, 1981. In 1979, Richard Schoen and Shing-Tung Yau showed that the class of smooth manifolds which admit Riemannian metrics of positive scalar curvature is topologically rich.

The FPP is also preserved by any retraction. According to Brouwer fixed point theorem every compact and convex subset of a Euclidean space has the FPP. More generally, according to the Schauder-Tychonoff fixed point theorem every compact and convex subset of a locally convex topological vector space has the FPP. Compactness alone does not imply the FPP and convexity is not even a topological property so it makes sense to ask how to topologically characterize the FPP.

Uniform synchrony, waves and spirals can readily be observed in two-dimensional Kuramoto networks with diffusive local coupling. The stability of waves in these models can be determined analytically using the methods of Turing stability analysis. Uniform synchrony tends to be stable when the local coupling is everywhere positive whereas waves arise when the long-range connections are negative (inhibitory surround coupling). Waves and synchrony are connected by a topologically distinct branch of solutions known as ripple.

He discovered the unstable sphaleron solution in the electroweak sector of the Standard Model of particle physics. The Higgs field is topologically twisted within a sphaleron. The sphaleron defines an energy scale for baryon and lepton number violation in the early universe — an energy scale within the range of the Large Hadron Collider. His other work includes the construction of a 10-dimensional theory containing supergravity and Yang–Mills theory, which is a low-energy limit of superstring theory.

Furthermore, a Higgs bundle is stable if and only if it admits an > irreducible Hermitian Yang–Mills connection, and therefore comes from an > irreducible representation of the fundamental group. Combined together, the correspondence can be phrased as follows: : > Nonabelian Hodge theorem: A Higgs bundle (which is topologically trivial) > arises from a semisimple representation of the fundamental group if and only > if it is polystable. Furthermore it arises from an irreducible > representation if and only if it is stable.

Intuitively, an open set provides a method to distinguish two points. For example, if about one point in a topological space there exists an open set not containing another (distinct) point, the two points are referred to as topologically distinguishable. In this manner, one may speak of whether two subsets of a topological space are "near" without concretely defining a metric on the topological space. Therefore, topological spaces may be seen as a generalization of metric spaces.

Spherical pentagonal icositetrahedron This polyhedron is topologically related as a part of sequence of polyhedra and tilings of pentagons with face configurations (V3.3.3.3.n). (The sequence progresses into tilings the hyperbolic plane to any n.) These face-transitive figures have (n32) rotational symmetry. The pentagonal icositetrahedron is second in a series of dual snub polyhedra and tilings with face configuration V3.3.4.3.n. The pentagonal icositetrahedron is one of a family of duals to the uniform polyhedra related to the cube and regular octahedron.

Stephen M. Omohundro, "Modelling Cellular Automata with Partial Differential Equations", Physica D, 10D (1984) 128-134. The asymptotic behavior of these PDEs is therefore logically undecidable. With John David Crawford he showed that the orbits of three- dimensional period doubling systems can form an infinite number of topologically distinct torus knots and described the structure of their stable and unstable manifolds.John David Crawford and Stephen M. Omohundro, "On the Global Structure of Period Doubling Flows", Physica D, 12D (1984), pp. 161-180.

15th monohedral convex pentagonal type, discovered in 2015 In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon. A regular pentagonal tiling on the Euclidean plane is impossible because the internal angle of a regular pentagon, 108°, is not a divisor of 360°, the angle measure of a whole turn. However, regular pentagons can tile the hyperbolic plane and the sphere; the latter produces a tiling topologically equivalent to the dodecahedron.

Keeping the research problem in mind, the behavior of elements in the system and their interactions are studied. Elements of a system can interact positively or negatively, that is, the level of an element may activate or reduce the rate of production of other elements or of itself. These interactions are represented as positive (activation) or negative (inhibition). When elements are connected in a topologically circular way, they exert an influence on their own rate of synthesis and they form a feedback loop.

If the control flow graph does not contain cycles (there were no explicit or implicit loops in the procedure) solving the equations is straightforward. The control flow graph can then be topologically sorted; running in the order of this sort, the entry states can be computed at the start of each block, since all predecessors of that block have already been processed, so their exit states are available. If the control flow graph does contain cycles, a more advanced algorithm is required.

A Weyl semimetal is a solid state crystal whose low energy excitations are Weyl fermions that carry electrical charge even at room temperatures. A Weyl semimetal enables realization of Weyl fermions in electronic systems. It is a topologically nontrivial phase of matter, together with Helium-3 A superfluid phase, that broadens the topological classification beyond topological insulators. The Weyl fermions at zero energy correspond to points of bulk band degeneracy, the Weyl nodes (or Fermi points), that are separated in momentum space.

In mathematics, a topological space (X, T) is called completely uniformizablee. g. Willard (or Dieudonné completeEncyclopedia of Mathematics) if there exists at least one complete uniformity that induces the topology T. Some authorse. g. Arkhangel'skii (in Encyclopedia of Mathematics), who uses the term Dieudonné complete additionally require X to be Hausdorff. Some authors have called these spaces topologically complete,Kelley although that term has also been used in other meanings like completely metrizable, which is a stronger property than completely uniformizable.

Ice XVI. White edges mark the unit cell (~17 Å). Ice XVI is the least dense (0.81 g/cm) experimentally obtained crystalline form of ice. It is topologically equivalent to the empty structure of sII clathrate hydrates. It was first obtained in 2014 by removing gas molecules from a neon clathrate under vacuum at temperatures below 147 K. The resulting empty water frame, ice XVI, is thermodynamically unstable at the experimental conditions, yet it can be preserved at cryogenic temperatures.

As a result of this good property, much of algebraic geometry studies an arbitrary variety by analyzing its codimension-1 subvarieties and the corresponding line bundles. On singular varieties, this good property can fail, and so one has to distinguish between codimension-1 subvarieties and varieties which can locally be defined by one equation. The former are Weil divisors while the latter are Cartier divisors. Topologically, Weil divisors play the role of homology classes, while Cartier divisors represent cohomology classes.

The Frucht graph is one of the five smallest cubic graphs possessing only a single graph automorphism, the identity (that is, every vertex can be distinguished topologically from every other vertex). Such graphs are called asymmetric (or identity) graphs. Frucht's theorem states that any group can be realized as the group of symmetries of a graph,. and a strengthening of this theorem also due to Frucht states that any group can be realized as the symmetries of a 3-regular graph;.

In the mathematical field of Lie theory, there are two definitions of a compact Lie algebra. Extrinsically and topologically, a compact Lie algebra is the Lie algebra of a compact Lie group; this definition includes tori. Intrinsically and algebraically, a compact Lie algebra is a real Lie algebra whose Killing form is negative definite; this definition is more restrictive and excludes tori,. A compact Lie algebra can be seen as the smallest real form of a corresponding complex Lie algebra, namely the complexification.

The ends of a topological space are, roughly speaking, the connected components of the “ideal boundary” of the space. That is, each end represents a topologically distinct way to move to infinity within the space. Adding a point at each end yields a compactification of the original space, known as the end compactification. The ends of a finitely generated group are defined to be the ends of the corresponding Cayley graph; this definition is independent of the choice of a finite generating set.

In a quantum field theory, there can be field configurations of bosons which are topologically twisted. These are coherent states (or solitons) which behave like a particle, and they can be fermionic even if all the constituent particles are bosons. This was discovered by Tony Skyrme in the early 1960s, so fermions made of bosons are named skyrmions after him. Skyrme's original example involved fields which take values on a three- dimensional sphere, the original nonlinear sigma model which describes the large distance behavior of pions.

Michelson redesigned Fizeau's apparatus with larger diameter tubes and a large reservoir providing three minutes of steady water flow. His common-path interferometer design provided automatic compensation of path length, so that white light fringes were visible at once as soon as the optical elements were aligned. Topologically, the light path was that of a Sagnac interferometer with an even number of reflections in each light path. This offered extremely stable fringes that were, to first order, completely insensitive to any movement of its optical components.

The physical arrangement of their components and values are symmetric or antimetric as in the ladder example above. However, it is not a necessary condition for electrical antimetry. For example, if the example networks of figure 1 have an additional identical T-section added to the left-hand side as shown in figure 2, then the networks remain topologically symmetric and antimetric. However, the network resulting from the application of Bartlett's bisection theoremBartlett, AC, "An extension of a property of artificial lines", Phil. Mag.

Photonic topological insulators are artificial electromagnetic materials that support topologically non-trivial, unidirectional states of light. Photonic topological phases are classical electromagnetic wave analogues of electronic topological phases studied in condensed matter physics. Similar to their electronic counterparts, they, can provide robust unidirectional channels for light propagation. The field that studies these phases of light is referred to as topological photonics, even though the working frequency of these electromagnetic topological insulators may fall in other parts of the electromagnetic spectrum such as the microwave range.

The field of real numbers is not algebraically closed, the geometry of even a plane curve C in the real projective plane. Assuming no singular points, the real points of C form a number of ovals, in other words submanifolds that are topologically circles. The real projective plane has a fundamental group that is a cyclic group with two elements. Such an oval may represent either group element; in other words we may or may not be able to contract it down in the plane.

We can generalize the rule from the figures above. Torus interconnect is a switch-less topology that can be seen as a mesh interconnect with nodes arranged in a rectilinear array of N = 2, 3, or more dimensions, with processors connected to their nearest neighbors, and corresponding processors on opposite edges of the array connected.[1] In this lattice, each node has 2N connections. This topology got the name from the fact that the lattice formed in this way is topologically homogeneous to an N-dimensional torus.

Costa's minimal surface, cropped by a sphere STL model of the surface In mathematics, Costa's minimal surface, is an embedded minimal surface discovered in 1982 by the Brazilian mathematician Celso José da Costa. It is also a surface of finite topology, which means that it can be formed by puncturing a compact surface. Topologically, it is a thrice-punctured torus. Until its discovery, the plane, helicoid and the catenoid were believed to be the only embedded minimal surfaces that could be formed by puncturing a compact surface.

The topological censorship theorem states that general relativity does not allow an observer to probe the topology of spacetime: any topological structure collapses too quickly to allow light to traverse it. More precisely, in a globally hyperbolic, asymptotically flat spacetime satisfying the null energy condition, every causal curve from past null infinity to future null infinity is fixed-endpoint homotopic to a curve in a topologically trivial neighbourhood of infinity. The topological censorship theorem was not proven in the original article because of a lacuna.

In the eukaryotic chromatin, DNA is rarely present in the free supercoiled form because nucleosomes restrain almost all negative supercoiling through tight binding of DNA to histones. Similarly, in E. coli, nucleoprotein complexes formed by NAPs restrain half of the supercoiling density of the nucleoid. In other words, if a NAP dissociates from a nucleoprotein complex, the DNA would adopt the free, plectonemic form. DNA binding of HU, Fis, and H-NS has been experimentally shown to restrain negative supercoiling in a relaxed but topologically constrained DNA.

Through gene silencing, H-NS acts as a global repressor preferentially inhibiting transcription of horizontally transferred genes. In another example, specific binding of HU at the gal operon facilitates the formation of a DNA loop that keeps the gal operon repressed in the absence of the inducer. The topologically distinct DNA micro-loop created by coherent bending of DNA by Fis at stable RNA promoters activates transcription. DNA bending by IHF differentially controls transcription from the two tandem promoters of the ilvGMEDA operon in E. coli.

The end of the 20th century had seen geocoding become more user- oriented, especially via open-source GIS software. Mapping applications and geospatial data had become more accessible over the Internet. Because the mail-out/mail-back technique was so successful in the 1980 Census, the U.S. Bureau of Census was able to put together a large geospatial database, using interpolated street geocoding. This database – along with the Census' nationwide coverage of households – allowed for the birth of TIGER (Topologically Integrated Geographic Encoding and Referencing).

Magnetic topological insulators are three dimensional magnetic materials with a non-trivial topological index protected by a symmetry other than time- reversal. In contrast with a non-magnetic topological insulator, a magnetic topological insulator can have naturally gapped surface states as long as the quantizing symmetry is broken at the surface. These gapped surfaces exhibit a topologically protected half-quantized surface anomalous Hall conductivity (e^2/2h) perpendicular to the surface. The sign of the half-quantized surface anomalous Hall conductivity depends on the specific surface termination.

Dual Independent Map Encoding (DIME) is an encoding scheme developed by the US Bureau of the Census for efficiently storing geographical data. The committee behind the case study that eventually resulted in DIME was established in 1965, although the term DIME itself was first coined by George Farnsworth in August 1967. The file format developed for storing the DIME-encoded data was known as Geographic Base Files (GBF). The Census Bureau replaced the data format with Topologically Integrated Geographic Encoding and Referencing (TIGER) in 1990.

If the molecules are large enough, the linking may occur in multiple topologically distinct ways, constituting different isomers. Cage compounds, such as helium enclosed in dodecahedrane (He@) and carbon peapods, are a similar type of topological isomerism involving molecules with large internal voids with restricted or no openings.Takahiro Iwamoto, Yoshiki Watanabe, Tatsuya Sadahiro, Takeharu Haino, and Shigeru Yamago (2011): "Size‐selective encapsulation of C60 by [10]cycloparaphenylene: Formation of the shortest fullerene‐peapod". Angewandte Chemie International Edition, volume 50, issue 36, pages 8342–8344.

This produces a geometric object called the Clifford torus, a surface in 4-space. In the field of topology, a torus is any topological space that is topologically equivalent to a torus. A coffee cup and a doughnut are both topological tori. An example of a torus can be constructed by taking a rectangular strip of flexible material, for example, a rubber sheet, and joining the top edge to the bottom edge, and the left edge to the right edge, without any half-twists (compare Möbius strip).

To achieve a high percentage of geometrically and topologically correct 3D building models, digital terrain surfaces and 2D footprint polygons are required by automated building reconstruction tools such as BREC. One key challenge is to find building parts with their corresponding roof geometry. "Since fully automatic image understanding is very hard to solve, semi- automatic components are usually required to at least support the recognition of very complex buildings by a human operator."N. Haala, M. Kada: An update on automatic 3D building reconstruction.

A node at which a graphlet is "touched" is topologically relevant, since it allows us to distinguish between nodes "touching", for example, a three node path at an end node or at the middle node. This is summarized by automorphism orbits (or just orbits, for brevity): by taking into account the "symmetries" between nodes of a graphlet, there are 73 different orbits across all 2-5-node graphlets (see [Pržulj, 2007] for details). For each orbit j, one needs to measure the jth GDD, dGj(k), i.e.

They showed its ground state has a rich structure of broken symmetries including one exhibiting canted anti- ferromagnetism. Rajaraman and PhD student Sankalpa Ghosh studied topologically non trivial "meron" and bi-meron excitations in layer-spin for bilayer Hall systems taking into account differences in interlayer and intra-layer coulomb energy. They also analyzed CP_{3} solitons arising in a four-component description of electrons carrying both spin and layer-spin. These solitons carry nontrivial intertwined windings of real spin and layer degrees of freedom.

T-square. The H tree is an example of a fractal canopy, in which the angle between neighboring line segments is always 180 degrees. In its property of coming arbitrarily close to every point of its bounding rectangle, it also resembles a space-filling curve, although it is not itself a curve. Topologically, an H tree has properties similar to those of a dendroid. However, they are not dendroids: dendroids must be closed sets, and H trees are not closed (their closure is the whole rectangle).

Mosaic of a dragon, third century BC, discovered in 1969 The first archaeological excavations were conducted between 1911 and 1913 by Paolo Orsi. The excavation area is named "Saggio SAS II" and topologically "San Marco nord-est". It is bordered by the Ionian Sea on the east, the Taranto-Reggio Calabria railway on the west, the Assi river on the north and the "casemate" area on the south. In 1969 a mosaic depicting a dragon was discovered in what is now called the "House of the Dragon".

In geometry, the barycentric subdivision is a standard way of dividing an arbitrary convex polygon into triangles, a convex polyhedron into tetrahedra, or, in general, a convex polytope into simplices with the same dimension, by connecting the barycenters of their faces in a specific way. The name is also used in topology for a similar operation on cell complexes. The result is topologically equivalent to that of the geometric operation, but the parts have arbitrary shape and size. This is an example of a finite subdivision rule.

Branch point in a polymer An important microstructural feature of a polymer is its architecture and shape, which relates to the way branch points lead to a deviation from a simple linear chain. A branched polymer molecule is composed of a main chain with one or more substituent side chains or branches. Types of branched polymers include star polymers, comb polymers, polymer brushes, dendronized polymers, ladder polymers, and dendrimers. There exist also two- dimensional polymers (2DP) which are composed of topologically planar repeat units.

The prototype here is the Jordan curve theorem, which topologically concerns the complement of a circle in the Riemann sphere. It also tells the same story. We have the honest Betti numbers :1, 1, 0 of the circle, and therefore :0, 1, 1 by flipping over and :1, 1, 0 by shifting to the left. This gives back something different from what the Jordan theorem states, which is that there are two components, each contractible (Schoenflies theorem, to be accurate about what is used here).

Changing the transcription pattern of genes changes the location of cohesin indicating that localization of cohesin may depend on transcription. # In another model, chromatin loop extrusion is pushed by transcription generated supercoiling ensuring also that cohesin relocalizes quickly and loops grow with reasonable speed and in a good direction. In addition, the supercoiling-driven loop extrusion mechanism is consistent with earlier explanations proposing why topologically associating domains (TADs) flanked by convergent CTCF binding sites form more stable chromatin loops than TADs flanked by divergent CTCF binding sites.

Linear tetraplex model proposal (1969)Even once the DNA duplex structure was solved, it was initially an open question whether additional DNA structures were needed to explain its overall topology. there were initially questions about how it might affect DNA replication. In 1963, autoradiographs of the E. coli chromosome demonstrated that it was a single circular molecule that is replicated at a pair of replication forks at which both new DNA strands are being synthesized. The two daughter chromosomes after replication would therefore be topologically linked.

The theorem often appears in filter theory where the lattice network is sometimes known as a filter X-section following the common filter theory practice of naming sections after alphabetic letters to which they bear a resemblance. The theorem as originally stated by Bartlett required the two halves of the network to be topologically symmetrical. The theorem was later extended by Wilhelm Cauer to apply to all networks which were electrically symmetrical. That is, the physical implementation of the network is not of any relevance.

At zero temperature, instantons are the name given to solutions of the classical equations of motion of the Euclidean version of the theory under consideration, and which are furthermore localized in Euclidean spacetime. They describe tunneling between different topological vacuum states of the Minkowski theory. One important example of an instanton is the BPST instanton, discovered in 1975 by Belavin, Polyakov, Schwartz and Tyupkin. This is a topologically stable solution to the four-dimensional SU(2) Yang–Mills field equations in Euclidean spacetime (i.e.

A surface is a space that looks topologically (i.e., up to homeomorphism) like ℝ². Consider, however, the space obtained by taking the quotient of two copies A,B of ℝ² under the identification of a closed half-space of each with a closed half-space of the other. This will be a surface except along a single line. Now, pick another copy C of ℝ and glue it and A together along halfspaces so that the singular line of this gluing is transverse in A to the previous singular line.

It is related by mutation to the Kinoshita–Terasaka knot, with which it shares the same Jones polynomial. Both knots also have the curious property of having the same Alexander polynomial and Conway polynomial as the unknot. The issue of the sliceness of the Conway knot was resolved in 2020 by Lisa Piccirillo, 50 years after John Horton Conway first proposed the knot. Her proof made use of Rasmussen's s-invariant, and showed that the knot is not a smoothly slice knot, though it is topologically slice (the Kinoshita–Terasaka knot is both).

Pure ice XII can be created from ice Ih at by rapid compression (0.81-1.00 GPa/min) or by warming high density amorphous ice at pressures between . The proton-ordered form of ice XII is ice XIV. While it is similar in density (1.29 g/cm3 at ) to ice IV (also found in the ice V space) it exists as a tetragonal crystal. Topologically it is a mix of seven- and eight- membered rings, a 4-connected net (4-coordinate sphere packing)—the densest possible arrangement without hydrogen bond interpenetration.

The area has Mesolithic, Neolithic, Bronze Age, Iron Age, Romano-British and Medieval history that spans a period of 7000 years or more. The oldest known ancient monuments at Coate are the undated stone circle and the Bronze Age burial mounds along Day House Lane. Further Middle Bronze Age cremations, a possible pond barrow, and two large ring ditches have been found on the opposite side of the small Day Brook valley. A large, regionally significant Mesolithic flint scatter, with some topologically late artifacts, is also present c.

The graph structure theorem provides such a "good reason" in the form of a rough description of the structure of G. In essence, every H-free graph G suffers from one of two structural deficiencies: either G is "too thin" to have H as a minor, or G can be (almost) topologically embedded on a surface that is too simple to embed H upon. The first reason applies if H is a planar graph, and both reasons apply if H is not planar. We first make precise these notions.

A Kepler–Poinsot polyhedron covers its circumscribed sphere more than once, with the centers of faces acting as winding points in the figures which have pentagrammic faces, and the vertices in the others. Because of this, they are not necessarily topologically equivalent to the sphere as Platonic solids are, and in particular the Euler relation :\chi=V-E+F=2\ does not always hold. Schläfli held that all polyhedra must have χ = 2, and he rejected the small stellated dodecahedron and great dodecahedron as proper polyhedra. This view was never widely held.

A manifold with boundary or open manifold is topologically distinct from a closed manifold and can be created by making a cut in any suitable closed manifold. For example, the disk or 1-ball B^1 is bounded by a circle S^1. It may be created by cutting a trivial cycle in any 2-manifold and keeping the piece removed, by piercing the sphere and stretching the puncture wide, or by cutting the projective plane. It can also be seen as filling-in the circle in the plane.

The KAM theorem established that periodic orbits can follow complex trajectories; in particular, they may form braids that can be investigated using Floer homology. In one class of finite element methods, boundary-value problems for differential equations involving the Hodge-Laplace operator may need to be solved on topologically nontrivial domains, for example, in electromagnetic simulations. In these simulations, solution is aided by fixing the cohomology class of the solution based on the chosen boundary conditions and the homology of the domain. FEM domains can be triangulated, from which the simplicial homology can be calculated.

Borges in "The Library of Babel" states that "The Library is a sphere whose exact center is any hexagon and whose circumference is unattainable". The library can then be visualized as being a 3-manifold, and if the only restriction is that of being locally euclidean, it can equally well be visualized as a topologically non-trivial manifold such as a torus or a Klein bottle. In his 1951 essay "Pascal's sphere" (La esfera de Pascal),La esfera de Pascal, en La Nación, Buenos Aires, 14 enero 1951, 2.ª sec.

The activated intermediate, LC3-Atg3 (E2), is recruited to the site where the lipidation takes place. Atg3 catalyses the conjugation of Atg8 and phosphatidylethanolamine (PE). Atg3 has an alpha/beta-fold, and its core region is topologically similar to canonical E2 enzymes. Atg3 has two regions inserted in the core region and another with a long alpha-helical structure that protrudes from the core region as far as 30 A. It interacts with atg8 through an intermediate thioester bond between Cys-288 and the C-terminal Gly of atg8.

Instead of a medieval city, the game took place on a contemporary oil platform with a red-headed protagonist nicknamed Red O'Hare collecting oil drums and putting out fires (a reference to Red Adair). This plotline was, however, quickly transplanted to a science-fictional setting, following the Piper Alpha disaster in 1988. The name Pipeline arose from the gameplay element of fast transport to distant parts of the map through topologically interwoven pipes (novel to the BBC Micro at the time, but see Super Mario Bros. or Sonic the Hedgehog for well-known examples).

This tricursal form can be seen on one of the Stora Hammars stones, as well as upon the Nene River Ring, and on the Oseberg ship bed post. Although other forms are topologically possible, these are the only attested forms found so far. In Norwegian Bokmål, the term valknute is used for a polygon with a loop on each of its corners.Municipal arms for Lødingen, blazoned in the Norwegian Royal Decree of 11 May 1984, quoted in Hans Cappelen og Knut Johannessen: Norske kommunevåpen, Oslo 1987, page 197.

The model is similar to the closed Friedmann–Lemaître–Robertson–Walker universe, in that spatial slices are positively curved and are topologically three-spheres S^3. However, in the FRW universe, the S^3 can only expand or contract: the only dynamical parameter is overall size of the S^3, parameterized by the scale factor a(t). In the Mixmaster universe, the S^3 can expand or contract, but also distort anisotropically. Its evolution is described by a scale factor a(t) as well as by two shape parameters \beta_\pm(t).

These numbers are always divisible by (because a cyclic permutation of a foldable stamp sequence is always itself foldable),. As cited by and the quotients of this division are :1, 1, 2, 4, 10, 24, 66, 174, 504, 1406, 4210, 12198, 37378, 111278, 346846, 1053874, ... , the number of topologically distinct ways that a half-infinite curve can make crossings with a line, called "semimeanders". In the 1960s, John E. Koehler and W. F. Lunnon implemented algorithms that, at that time, could calculate these numbers for up to 28 stamps.

Guillemin, pp. 127–132 The inverse of this transform is the Δ-Y transform which analytically corresponds to the elimination of a mesh current and topologically corresponds to the elimination of a mesh. However, elimination of a mesh current whose mesh has branches in common with an arbitrary number of other meshes will not, in general, result in a realisable graph. This is because the graph of the transform of the general star is a graph which will not map on to a sphere (it contains star polygons and hence multiple crossovers).

If there is a given Calabi-Yau manifold (basically a space with 6 or more dimensions curled up in a special way) then a sphere in the center can shrink down to an infinitesimal point that resembles a singularity. After reaching the singularity-like point, the sphere tears and then a new sphere "blows up" to replace the torn one. The sphere in the mirror image (from Mirror symmetry) merely undergoes topologically smooth transition. The mathematical results from the separate manifolds result in the same physics, so no laws of physics or mathematics are violated.

This curve has total curvature 6, and turning number 3, though it only has winding number 2 about p. Immersed plane curves have a well-defined turning number, which can be defined as the total curvature divided by 2\. This is invariant under regular homotopy, by the Whitney–Graustein theorem – topologically, it is the degree of the Gauss map, or equivalently the winding number of the unit tangent (which does not vanish) about the origin. Further, this is a complete set of invariants – any two plane curves with the same turning number are regular homotopic.

According to the Brouwer fixed-point theorem, every compact and convex subset of a Euclidean space has the FPP. Compactness alone does not imply the FPP and convexity is not even a topological property so it makes sense to ask how to topologically characterize the FPP. In 1932 Borsuk asked whether compactness together with contractibility could be a necessary and sufficient condition for the FPP to hold. The problem was open for 20 years until the conjecture was disproved by Kinoshita who found an example of a compact contractible space without the FPP.

In the common case of a real algebraic curve, where is the field of real numbers, an algebraic curve is a finite union of topological curves. When complex zeros are considered, one has a complex algebraic curve, which, from the topologically point of view, is not a curve, but a surface, and is often called a Riemann surface. Although not being curves in the common sense, algebraic curves defined over other fields have been widely studied. In particular, algebraic curves over a finite field are widely used in modern cryptography.

In singularity theory, there is a different meaning, of a decomposition of a topological space X into disjoint subsets each of which is a topological manifold (so that in particular a stratification defines a partition of the topological space). This is not a useful notion when unrestricted; but when the various strata are defined by some recognisable set of conditions (for example being locally closed), and fit together manageably, this idea is often applied in geometry. Hassler Whitney and René Thom first defined formal conditions for stratification. See Whitney stratification and topologically stratified space.

Weyl fermions may be realized as emergent quasiparticles in a low-energy condensed matter system. This prediction was first proposed by Conyers Herring in the context of electronic band structures of solid state systems such as electronic crystals. Topological materials in the vicinity of band inversion transition became a primary target in search of topologically protected bulk electronic band crossings. The first (non-electronic) liquid state which is suggested, has similarly emergent but neutral excitation and theoretically interpreted superfluid's chiral anomaly as observation of Fermi points is in Helium-3 A superfluid phase.

An annulus Illustration of Mamikon's visual calculus method showing that the areas of two annuli with the same chord length are the same regardless of inner and outer radii. In mathematics, an annulus (the Latin word for "little ring" is anulus / annulus, with plural anuli / annuli) is a ring-shaped object, a region bounded by two concentric circles; equivalently, it is the set difference between two concentric disks. The adjectival form is annular (as in annular eclipse). The open annulus is topologically equivalent to both the open cylinder and the punctured plane.

TIGER logo Topologically Integrated Geographic Encoding and Referencing, or TIGER, or TIGER/Line is a format used by the United States Census Bureau to describe land attributes such as roads, buildings, rivers, and lakes, as well as areas such as census tracts. TIGER was developed to support and improve the Bureau's process of taking the Decennial Census. The TIGER files do not contain the census demographic data, but merely the geospatial/map data. GIS can be used to merge census demographics or other data sources with the TIGER files to create maps and conduct analysis.

Let ƒ: S1 → S1 be an orientation-preserving diffeomorphism of the circle whose rotation number θ = ρ(ƒ) is irrational. Assume that it has positive derivative ƒ ′(x) > 0 that is a continuous function with bounded variation on the interval [0,1). Then ƒ is topologically conjugate to the irrational rotation by θ. Moreover, every orbit is dense and every nontrivial interval I of the circle intersects its forward image ƒ°q(I), for some q > 0 (this means that the non-wandering set of ƒ is the whole circle).

Atomic nuclei may exhibit solitonic behavior. Here the whole nuclear wave function is predicted to exist as a soliton under certain conditions of temperature and energy. Such conditions are suggested to exist in the cores of some stars in which the nuclei would not react but pass through each other unchanged, retaining their soliton waves through a collision between nuclei. The Skyrme Model is a model of nuclei in which each nucleus is considered to be a topologically stable soliton solution of a field theory with conserved baryon number.

Speaking informally, topological conjugation is a "change of coordinates" in the topological sense. However, the analogous definition for flows is somewhat restrictive. In fact, we are requiring the maps \varphi(\cdot, t) and \psi(\cdot, t) to be topologically conjugate for each t, which is requiring more than simply that orbits of \phi be mapped to orbits of \psi homeomorphically. This motivates the definition of topological equivalence, which also partitions the set of all flows in X into classes of flows sharing the same dynamics, again from the topological viewpoint.

South Newton is a village and civil parish about north-west of Salisbury in Wiltshire, England. Topologically it lies between chalk downs to the north- east, and downland with Grovely Wood to the southwest. The village straddles the A36 road and is on the left bank of the River Wylye, which defines much of the western boundary of the parish; the eastern boundary follows the A360 Salisbury-Devizes road. The parish includes the village of Stoford (not to be confused with Stoford, Somerset) and the hamlets of Little Wishford and Chilhampton.

Self-interacting (or self-associating) domains are found in many organisms – in bacteria, they are referred to as Chromosomal Interacting Domains (CIDs), whereas in mammalian cells, they are called Topologically Associating Domains (TADs). Self-interacting domains can range from the 1–2 mb scale in larger organisms to 10s of kb in single celled organisms. What characterizes a self-interacting domain is a set of common features. The first is that self-interacting domains have a higher of ratio of chromosomal contacts within the domain than outside it.

If the pentagrammic faces are broken into triangles, it is topologically related to the triakis icosahedron, with the same face connectivity, but much taller isosceles triangle faces. If the triangles are instead made to invert themselves and excavate the central icosahedron, the result is a great dodecahedron. The great stellated dodecahedron can be constructed analogously to the pentagram, its two-dimensional analogue, by attempting to stellate the n-dimensional pentagonal polytope which has pentagonal polytope faces and simplex vertex figures until it can no longer be stellated; that is, it is its final stellation.

Kauffman, supra have suggested the relevance of autocatalysis models for life processes. In this construct, a group of elements catalyse reactions in a cyclical, or topologically circular, fashion. Several investigators have used these insights to suggest essential elements of a thermodynamic definition of the life process, which might briefly be summarized as stable, patterned (correlated) processes which intake (and dissipate) energy, and reproduce themselves.See Brooks and Wylie, Smolin, Kauffman, supra, and Pearce Ulanowicz, a theoretical ecologist, has extended the relational analysis of life processes to ecosystems, using information theory tools.

This has led some to speculate that superconductivity arises dominantly from the α and β bands instead. Such a two-band superconductor, although having k-dependence phase winding in its order parameters on the two relevant bands, is topologically trivial with the two bands featuring opposite Chern numbers. Therefore, it could possibly give a much reduced if not completely cancelled supercurrent at the edge. However, this naive reasoning was later found not to be entirely correct: the magnitude of edge current is not directly related to the topological property of the chiral state.

Each gate receives two input wires and it has a single output wire which might be fan-out (i.e. be passed to multiple gates at the next level). Plain evaluation of the circuit is done by evaluating each gate in turn; assuming the gates have been topologically ordered. The gate is represented as a truth table such that for each possible pair of bits (those coming from the input wires' gate) the table assigns a unique output bit; which is the value of the output wire of the gate.

They are distinct from other families of polymers because 2D polymers can be isolated as multilayer crystals or as individual sheets. The term 2D polymer has also been used more broadly to include linear polymerizations performed at interfaces, layered non-covalent assemblies, or to irregularly cross-linked polymers confined to surfaces or layered films. 2D polymers can be organized based on these methods of linking (monomer interaction): covalently linked monomers, coordination polymers and supramolecular polymers. Topologically, 2DPs may thus be understood as structures made up from regularly tessellated regular polygons (the repeat units).

In the case when the N- and C-terminal repeats lie in close physical contact in a tandem repeat domain, the result is a topologically compact, closed structure. Such domains typically display a high rotational symmetry (unlike open repeats that only have translational symmetries), and assume a wheel-like shape. Because of the limitations of this structure, the number of individual repeats is not arbitrary. In the case of WD40 repeats (perhaps the largest family of closed solenoids) the number of repeats can range from 4 to 10 (more usually between 5 and 7).

In mammalian biology, insulated neighborhoods are chromosomal loop structures formed by the physical interaction of two DNA loci bound by the transcription factor CTCF and co-occupied by cohesin. Insulated neighborhoods are thought to be structural and functional units of gene control because their integrity is important for normal gene regulation. Current evidence suggests that these structures form the mechanistic underpinnings of higher-order chromosome structures, including topologically associating domains (TADs). Insulated neighborhoods are functionally important in understanding gene regulation in normal cells and dysregulated gene expression in disease.

A mechanistic way of modelling an HGT event on the reference tree is to first cut an internal branch—i.e., prune the tree—and then regraft it onto another edge, an operation referred to as subtree pruning and regrafting (SPR). If the gene tree was topologically consistent with the original reference tree, the editing results in an inconsistency. Similarly, when the original gene tree is inconsistent with the reference tree, it is possible to obtain a consistent topology by a series of one or more prune and regraft operations applied to the reference tree.

Senescent cells undergo chromatin landscape modifications as constitutive heterochromatin migrates to the center of the nucleus and displaces euchromatin and facultative heterochromatin to regions at the edge of the nucleus. This disrupts chromatin-lamin interactions and inverts of the pattern typically seen in a mitotically active cell. Individual Lamin-Associated Domains (LADs) and Topologically Associating Domains (TADs) are disrupted by this migration which can affect cis interactions across the genome. Additionally, there is a general pattern of canonical histone loss, particularly in terms of the nucleosome histones H3 and H4 and the linker histone H1.

The resulting shape is topologically equivalent to a Euclidean 2-torus (a doughnut shape). However, unlike the latter, all parts of its surface are identically deformed. On the doughnut, the surface around the 'doughnut hole' is deformed with negative curvature while the surface outside is deformed with positive curvature. The ridge of the duocylinder may be thought of as the actual global shape of the screens of video games such as Asteroids, where going off the edge of one side of the screen leads to the other side.

It is related to two star- tilings by the same vertex arrangement: the order-7 heptagrammic tiling, {7/2,7}, and heptagrammic-order heptagonal tiling, {7,7/2}. This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbol {3,p}. From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.

Gimbal lock occurs because any map is not a covering map. In particular, the relevant map carries any element of T3, that is, an ordered triple (a,b,c) of angles (real numbers mod 2), to the composition of the three coordinate axis rotations Rx(a)∘Ry(b)∘Rz(c) by those angles, respectively. Each of these rotations, and their composition, is an element of the rotation group SO(3), which is topologically RP3. This animation shows a set of three gimbals mounted together to allow three degrees of freedom.

To see why, let L(θ) denote the line through the origin at an angle θ to the positive x-axis. For each L(θ) there is the family P(θ) of all lines in the plane that are perpendicular to L(θ). Topologically, the family P(θ) is just a line (because each line in P(θ) intersects the line L(θ) in just one point). In this way, as θ increases in the range , the line L(θ) represents a line's worth of distinct lines in the plane.

Topologically this arranged terminals in a ring, with redundant sets of wires which allowed for a break in the wire to be tolerated simply by "turning back" the data on each side of the break. The 8100 also supported an "intelligent" terminal called the 8775 (which shared the same case as the 3279 colour display terminal for IBM's mainframes and, like the 3279 was designed at IBM's UK Development Lab at Hursley Park, England) which was the first to ship with the ability to download its functionality from the host computer to which it was attached.

In addition to entrapping DNA to ensure proper chromosome segregation during the cell cycle, SMC1A, as a component of cohesin, contributes to facilitating inter-chromatid contacts mediating distant-element interactions and to creating chromosome domains called topologically associating domains (TADs). It has been proposed that cohesin promotes the interaction between enhancers and promoters for regulating gene transcription regulation. The removal of cohesin triggers abnormal TAD topology because loops spanning multiple compartment intervals lead to mixing among loci in different compartments As a consequence, loop loss causes gene expression dysregulation. SMC1A also plays a role in spindle pole formation.

Circular (closed) solenoid domain In the case when the N- and C-terminal repeats lie in close physical contact in a solenoid domain, the result is a topologically compact, closed structure. Such domains typically display a high rotational symmetry (unlike open solenoids that only have translational symmetries), and assume a wheel-like shape. Because of the limitations of this structure, the number of individual repeats is not arbitrary. In the case of WD40 repeats (perhaps the largest family of closed solenoids) the number of repeats can range from 4 to 10 (more usually between 5 and 7).

Let G be an open domain in Rn with compact closure and smooth (n−1)-dimensional boundary. Consider the space X1(G) consisting of restrictions to G of C1 vector fields on Rn that are transversal to the boundary of G and are inward oriented. This space is endowed with the C1 metric in the usual fashion. A vector field F ∈ X1(G) is weakly structurally stable if for any sufficiently small perturbation F1, the corresponding flows are topologically equivalent on G: there exists a homeomorphism h: G → G which transforms the oriented trajectories of F into the oriented trajectories of F1.

In geometry, the crossed pentagrammic cupola is one of the nonconvex Johnson solid isomorphs, being topologically identical to the convex pentagonal cupola. It can be obtained as a slice of the great rhombicosidodecahedron or quasirhombicosidodecahedron. As in all cupolae, the base polygon has twice as many edges and vertices as the top; in this case the base polygon is a decagram. It may be seen as a cupola with a retrograde pentagrammic base, so that the squares and triangles connect across the bases in the opposite way to the pentagrammic cuploid, hence intersecting each other more deeply.

In geometry, the crossed square cupola is one of the nonconvex Johnson solid isomorphs, being topologically identical to the convex square cupola. It can be obtained as a slice of the nonconvex great rhombicuboctahedron or quasirhombicuboctahedron. As in all cupolae, the base polygon has twice as many edges and vertices as the top; in this case the base polygon is an octagram. It may be seen as a cupola with a retrograde square base, so that the squares and triangles connect across the bases in the opposite way to the square cupola, hence intersecting each other.

Alpha-prolamins are the major seed storage proteins of species of the grass tribe Andropogoneae. They are unusually rich in glutamine, proline, alanine, and leucine residues and their sequences show a series of tandem repeats presumed to be the result of multiple intragenic duplication. In Zea mays (Maize), the 22 kDa and 19 kDa zeins are encoded by a large multigene family and are the major seed storage proteins accounting for 70% of the total zein fraction. Structurally the 22 kDa and 19 kDa zeins are composed of nine adjacent, topologically antiparallel helices clustered within a distorted cylinder.

SED describes electromagnetic energy at absolute zero as a stochastic, fluctuating zero-point field. In SED the motion of a particle immersed in the stochastic zero-point radiation field generally results in highly nonlinear behaviour. Quantum effects emerge as a result of permanent matter-field interactions not possible to describe in QED The typical mathematical models used in classical electromagnetism, quantum electrodynamics (QED) and the standard model view electromagnetism as a U(1) gauge theory, which topologically restricts any complex nonlinear interaction. The electromagnetic vacuum in these theories is generally viewed as a linear system with no overall observable consequence.

The amalgamation problem has, historically, been pursued according to local topology. That is, rather than restricting K and L to be particular polytopes, they are allowed to be any polytope with a given topology, that is, any polytope tessellating a given manifold. If K and L are spherical (that is, tessellations of a topological sphere), then P is called locally spherical and corresponds itself to a tessellation of some manifold. For example, if K and L are both squares (and so are topologically the same as circles), P will be a tessellation of the plane, torus or Klein bottle by squares.

A tessellation of an n-dimensional manifold is actually a rank n + 1 polytope. This is in keeping with the common intuition that the Platonic solids are three dimensional, even though they can be regarded as tessellations of the two-dimensional surface of a ball. In general, an abstract polytope is called locally X if its facets and vertex figures are, topologically, either spheres or X, but not both spheres. The 11-cell and 57-cell are examples of rank 4 (that is, four-dimensional) locally projective polytopes, since their facets and vertex figures are tessellations of real projective planes.

As a vector space, the dimension of the cycle space of a graph with n vertices, m edges, and c connected components is m-n+c.. This number can be interpreted topologically as the first Betti number of the graph. In graph theory, it is known as the circuit rank, cyclomatic number, or nullity of the graph. Combining this formula for the rank with the fact that the cycle space is a vector space over the two- element field shows that the total number of elements in the cycle space is exactly 2^{m-n+c}.

Spurred by Weyl's brilliant results on compact groups, he developed new methods for the study of global properties of Lie groups; in particular he showed that topologically a connected Lie group is a product of a Euclidean space and a compact group, and for compact Lie groups he discovered that the possible fundamental groups of the underlying manifold can be read from the structure of the Lie algebra of the group. Finally, he outlined a method of determining the Betti numbers of compact Lie groups, again reducing the problem to an algebraic question on their Lie algebras, which has since been completely solved.

The mathematical models used in classical electromagnetism, quantum electrodynamics (QED) and the standard model all view the electromagnetic vacuum as a linear system with no overall observable consequence (e.g. in the case of the Casimir effect, Lamb shift, and so on) these phenomena can be explained by alternative mechanisms other than action of the vacuum by arbitrary changes to the normal ordering of field operators. See alternative theories section). This is a consequence of viewing electromagnetism as a U(1) gauge theory, which topologically does not allow the complex interaction of a field with and on itself.

The rhombille tiling is the dual of the trihexagonal tiling. It is one of many different ways of tiling the plane by congruent rhombi. Others include a diagonally flattened variation of the square tiling (with translational symmetry on all four sides of the rhombi), the tiling used by the Miura-ori folding pattern (alternating between translational and reflectional symmetry), and the Penrose tiling which uses two kinds of rhombi with 36° and 72° acute angles aperiodically. When more than one type of rhombus is allowed, additional tilings are possible, including some that are topologically equivalent to the rhombille tiling but with lower symmetry.

In Schwarz-type TQFTs, the correlation functions or partition functions of the system are computed by the path integral of metric-independent action functionals. For instance, in the BF model, the spacetime is a two-dimensional manifold M, the observables are constructed from a two-form F, an auxiliary scalar B, and their derivatives. The action (which determines the path integral) is :S=\int_M B F The spacetime metric does not appear anywhere in the theory, so the theory is explicitly topologically invariant. The first example appeared in 1977 and is due to A. Schwarz; its action functional is: :\int_M A\wedge dA.

Large molecules may have isomers that differ by the topology of their overall arrangement in space, even if there is no specific geometric constraint that separate them. For example, long chains may be twisted to form topologically distinct knots, with interconversion prevented by bulky substituents or cycle closing (as in circular DNA and RNA plasmids). Some knots may come in mirror-image enantiomer pairs. Such forms are called topological isomers or topoisomers Also, two or more such molecules may be bound together in a catenane by such topological linkages, even if there is no chemical bond between them.

Antillatoxin is a sodium channel gating modifier with special efficacy in cells expressing rNav1.2, rNav1.4 and rNav1.5 α subunits. It is suggested that ATX preferentially binds to the voltage-gated sodium channel in the inactivated state. The specific site of interaction of this neurotoxin is not yet known, however there is an allosteric interaction between ATX and brevetoxin (PbTx) at site 5 of the α subunit, which indicates that the neurotoxin site for ATX is topologically close and/or conformationally coupled to neurotoxin site 5. Additionally, sites 1, 2, 3, 5 and 7 were ruled out as possible binding sites.

For example, in the presence of Neveu–Schwarz H-flux or non-spin cycles some RR fluxes dictate the presence of D-branes. In the former case this is a consequence of the supergravity equation of motion which states that the product of a RR flux with the NS 3-form is a D-brane charge density. Thus the set of topologically distinct RR field strengths that can exist in brane-free configurations is only a subset of the cohomology with integral coefficients. This subset is still too big, because some of these classes are related by large gauge transformations.

Inspired by neuroscience, informatics, and the occupation with electronic calculating machines, but also by Wittgenstein's concept of the language-game, Bense tried to put into perspective or to extend the traditional view of literature. In that, he was one of the first philosophers of culture who integrated the technical possibilities of the computer into their thoughts and investigated them across disciplinary boundaries. He statistically and topologically analysed linguistic phenomena, subjected them to questions of semiotic, information theory, and communication theory using structuralistic approaches. Thus Bense became the first theoretician of concrete poetry, which was started by Eugen Gomringer in 1953, and encouraged e.g.

From the definition of vector spaces, it follows that subspaces are nonempty, and are closed under sums and under scalar multiples. Equivalently, subspaces can be characterized by the property of being closed under linear combinations. That is, a nonempty set W is a subspace if and only if every linear combination of finitely many elements of W also belongs to W. The equivalent definition states that it is also equivalent to consider linear combinations of two elements at a time. In a topological vector space X, a subspace W need not be topologically closed, but a finite-dimensional subspace is always closed.

In principle the resulting tree need not have a preferred tip ("main" latest revision) – just various different revisions – but in practice one tip is generally identified as HEAD. When a new revision is based on HEAD, it is either identified as the new HEAD, or considered a new branch.Note that if a new branch is based on HEAD, then topologically HEAD is no longer a tip, since it has a child. The list of revisions from the start to HEAD (in graph theory terms, the unique path in the tree, which forms a linear graph as before) is the trunk or mainline.

A toroidal graph is a graph that can be embedded without crossings on the torus. More generally, the genus of a graph is the minimum genus of a two-dimensional surface into which the graph may be embedded; planar graphs have genus zero and nonplanar toroidal graphs have genus one. Any graph may be embedded into three-dimensional space without crossings. However, a three-dimensional analogue of the planar graphs is provided by the linklessly embeddable graphs, graphs that can be embedded into three-dimensional space in such a way that no two cycles are topologically linked with each other.

He became a professor at the University of Basel in 1991 and a professor at the Ludwig Maximilian University of Munich in 1998. Kotschick has been a member of the Institute for Advanced Study three times (1989/90, 2008/09 and 2012/13).Kotschick, Dieter in the A community of scholars list of the IAS In 2012 he was elected a Fellow of the American Mathematical Society. In 2009, he solved a 55-year-old open problem posed in 1954 by Friedrich Hirzebruch, which asks "which linear combinations of Chern numbers of smooth complex projective varieties are topologically invariant".

Many scientifically important problems can be represented and empirically studied using networks. For example, biological and social patterns, the World Wide Web, metabolic networks, food webs, neural networks and pathological networks are real world problems that can be mathematically represented and topologically studied to reveal some unexpected structural features. Most of these networks possess a certain community structure that has substantial importance in building an understanding regarding the dynamics of the network. For instance, a closely connected social community will imply a faster rate of transmission of information or rumor among them than a loosely connected community.

Pittencrieff Park (known locally as "The Glen") is a public park in Dunfermline, Fife, Scotland. It was purchased in 1902 by the town's most famous son, Andrew Carnegie, and given to the people of Dunfermline in a ceremony the following year. Its lands include the historically significant and topologically rugged glen which interrupts the centre of Dunfermline and, accordingly, part of the intention of the purchase was to carry out civic development of the area in a way which also respected its heritage. The project notably attracted the attention of the urban planner and educationalist, Patrick Geddes.

Any two distinct points in [-1,1] are topologically distinguishable under the overlapping interval topology as one can always find an open set containing one but not the other point. However, every non-empty open set contains the point 0 which can therefore not be separated from any other point in [-1,1], making [-1,1] with the overlapping interval topology an example of a T0 space that is not a T1 space. The overlapping interval topology is second countable, with a countable basis being given by the intervals [-1,s), (r,s) and (r,1] with r < 0 < s and r and s rational.

Befunge-extensions and variants are called Fungeoids or just Funges. The Befunge-93 specification restricts each valid program to a grid of 80 instructions horizontally by 25 instructions vertically. Program execution which exceeds these limits "wraps around" to a corresponding point on the other side of the grid; a Befunge program is in this manner topologically equivalent to a torus. Since a Befunge-93 program can only have a single stack and its storage array is bounded, the Befunge-93 language is not Turing-complete (however, it has been shown that Befunge-93 is Turing Complete with unbounded stack word size).

In countries where one drives on the right, left turns are handled by semi- directional flyovers or under ramps. Vehicles first exit the main road to the right, then complete the turn via a ramp that crosses both highways, eventually merging with the traffic turning right from the opposite side of the interchange. A stack interchange therefore has two pairs of left-turning ramps, which may be "stacked" in various configurations above or below the two interchanging highways. In countries where one drives on the left, the appearance of the junction is topologically identical, but traffic flows are reversed.

Atomic force microscopy used in in vitro studies have shown that circular eukaryotic polysomes can be formed by free polyadenylated mRNA in the presence of initiation factor eIF4E bound to the 5’ cap and PABP bound to the 3’-poly(A) tail. However, this interaction between cap and poly(A)-tail mediated by a protein complex is not a unique way of circularizing polysomal mRNA. It has been found that topologically circular polyribosomes can be successfully formed in the translational system with mRNA with no cap and no poly(A) tail as well as a capped mRNA without a 3’-poly(A) tail.

Rhenium can cause superalloys to become microstructurally unstable, forming undesirable TCP (topologically close packed) phases. In 4th- and 5th- generation superalloys, ruthenium is used to avoid this effect. Among others the new superalloys are EPM-102 (with 3% Ru) and TMS-162 (with 6% Ru), as well as TMS-138 and TMS-174. CFM International CFM56 jet engine still with blades made with 3% rhenium For 2006, the consumption is given as 28% for General Electric, 28% Rolls-Royce plc and 12% Pratt & Whitney, all for superalloys, whereas the use for catalysts only accounts for 14% and the remaining applications use 18%.

Jeffrey Brock's research focuses on low- dimensional topology and geometry, particularly on spaces with hyperbolic geometry or negative curvature. His joint work with Richard Canary and Yair Minsky resulted in a solution to the "Ending Lamination Conjecture" of William Thurston, culminating in the geometric classification theorem for (topologically-finite) hyperbolic 3-manifolds in terms of their fundamental group and the structure of their ends. More recently, he has worked to understand applications of geometry and topology to the structure of massive and complex data sets and the risks and implications of the increasing use of 'black box' algorithms in science and society.

The existence of a topological defect can be demonstrated whenever the boundary conditions entail the existence of homotopically distinct solutions. Typically, this occurs because the boundary on which the conditions are specified has a non-trivial homotopy group which is preserved in differential equations; the solutions to the differential equations are then topologically distinct, and are classified by their homotopy class. Topological defects are not only stable against small perturbations, but cannot decay or be undone or be de-tangled, precisely because there is no continuous transformation that will map them (homotopically) to a uniform or "trivial" solution.

Evidence supporting such a concept of cell assemblies was later observed, both at the macroscopic level with the cortical columns in the somato-sensory areas, and at the microscopic level with the NMDA coding of coordinated activity in synapses. However, the mesoscopic level has remained elusive. Some authors, including Vernon Mountcastle, argued that the mesoscopic level of sensory brain areas might be topologically organized similarly to the macroscopic and microscopic level, in cortical minicolumns, specifically what has been termed the columnar functional organization. However, any exact mechanism of information encoding and decoding in these sensory cortical columns has remained elusive.

There does exist a higher-dimensional generalization due to and independently with , which is also called the generalized Schoenflies theorem. It states that, if an (n − 1)-dimensional sphere S is embedded into the n-dimensional sphere Sn in a locally flat way (that is, the embedding extends to that of a thickened sphere), then the pair (Sn, S) is homeomorphic to the pair (Sn, Sn−1), where Sn−1 is the equator of the n-sphere. Brown and Mazur received the Veblen Prize for their contributions. The Schoenflies problem can be posed in categories other than the topologically locally flat category, i.e.

A 3-sphere can be constructed topologically by "gluing" together the boundaries of a pair of 3-balls. The boundary of a 3-ball is a 2-sphere, and these two 2-spheres are to be identified. That is, imagine a pair of 3-balls of the same size, then superpose them so that their 2-spherical boundaries match, and let matching pairs of points on the pair of 2-spheres be identically equivalent to each other. In analogy with the case of the 2-sphere (see below), the gluing surface is called an equatorial sphere.

The 24-cell honeycomb is similar, but in addition to the vertices at integers (i,j,k,l), it has vertices at half integers (i+1/2,j+1/2,k+1/2,l+1/2) of odd integers only. It is a half-filled body centered cubic (a checkerboard in which the red 4-cubes have a central vertex but the black 4-cubes do not). The tesseract can make a regular tessellation of the 4-sphere, with three tesseracts per face, with Schläfli symbol {4,3,3,3}, called an order-3 tesseractic honeycomb. It is topologically equivalent to the regular polytope penteract in 5-space.

The surface of an ideal polyhedron (not including its vertices) forms a manifold, topologically equivalent to a punctured sphere, with a uniform two-dimensional hyperbolic geometry; the folds of the surface in its embedding into hyperbolic space are not detectable as folds in the intrinsic geometry of the surface. Because this surface can be partitioned into ideal triangles, its total area is finite. Conversely, and analogously to Alexandrov's uniqueness theorem, every two- dimensional manifold with uniform hyperbolic geometry and finite area, combinatorially equivalent to a finitely-punctured sphere, can be realized as the surface of an ideal polyhedron. (As with Alexandrov's theorem, such surfaces must be allowed to include ideal dihedra.); .

RD processors are spatially invariant, topologically invariant, analog, parallel processors characterized by reactions, where two agents can combine to create a third agent, and diffusion, the spreading of agents. RD processors are typically implemented through chemicals in a Petri dish (processor), light (input), and a camera (output) however RD processors can also be implemented through a multi-layer CNN processor. D processors can be used to create Voronoi diagrams and perform skeletonisation. The main difference between the chemical implementation and the CNN implementation is that CNN implementations are considerably faster than their chemical counterparts and chemical processors are spatially continuous whereas the CNN processors are spatially discrete.

Although no universally accepted mathematical definition of chaos exists, a commonly used definition, originally formulated by Robert L. Devaney, says that to classify a dynamical system as chaotic, it must have these properties: # it must be sensitive to initial conditions, # it must be topologically transitive, # it must have dense periodic orbits. In some cases, the last two properties above have been shown to actually imply sensitivity to initial conditions. In the discrete-time case, this is true for all continuous maps on metric spaces. In these cases, while it is often the most practically significant property, "sensitivity to initial conditions" need not be stated in the definition.

All the morphisms that can serve as composition of two given morphisms are related to each other by higher order invertible morphisms (2-simplices thought of as "homotopies"). These higher order morphisms can also be composed, but again the composition is well-defined only up to still higher order invertible morphisms, etc. The idea of higher category theory (at least, higher category theory when higher morphisms are invertible) is that, as opposed to the standard notion of a category, there should be a mapping space (rather than a mapping set) between two objects. This suggests that a higher category should simply be a topologically enriched category.

The classification of complex manifolds is much more subtle than that of differentiable manifolds. For example, while in dimensions other than four, a given topological manifold has at most finitely many smooth structures, a topological manifold supporting a complex structure can and often does support uncountably many complex structures. Riemann surfaces, two dimensional manifolds equipped with a complex structure, which are topologically classified by the genus, are an important example of this phenomenon. The set of complex structures on a given orientable surface, modulo biholomorphic equivalence, itself forms a complex algebraic variety called a moduli space, the structure of which remains an area of active research.

Cross and circle is a board game design used for race games played throughout the world. The design of most cross and circle games involves a circle divided into four equal portions by a cross inscribed inside it; the classic example of this design is the Korean game Yut. However, the term "cross and circle" is typically widened to include boards that replace the circle with a square, and cruciform boards that collapse the circle onto the cross; all three types are topologically equivalent. The Indian game Pachisi and its many descendants are perhaps the most well-known of all cross and circle games.

It was observed that in fact several edges may be drawn in the same "page"; the book thickness of the graph is the minimum number of halfplanes needed for such a drawing. Alternatively, any finite graph can be drawn with straight-line edges in three dimensions without crossings by placing its vertices in general position so that no four are coplanar. For instance, this may be achieved by placing the ith vertex at the point (i,i2,i3) of the moment curve. An embedding of a graph into three- dimensional space in which no two of the cycles are topologically linked is called a linkless embedding.

Even though the graph isomorphism problem is polynomial time reducible to crystal net topological equivalence (making topological equivalence a candidate for being "computationally intractable" in the sense of not being polynomial time computable), a crystal net is generally regarded as novel if and only if no topologically equivalent net is known. This has focused attention on topological invariants. One invariant is the array of minimal cycles (often called rings in the chemistry literature) arrayed about generic vertices and represented in a Schlafli symbol. The cycles of a crystal net are related to another invariant, that of the coordination sequence (or shell map in topology), which is defined as follows.

Given a knot or link, draw a diagram of the link using the convention that a gap in the line denotes an undercrossing. Call an arc in this diagram a bridge if it includes at least one overcrossing. Then the bridge number of a knot can be found as the minimum number of bridges required for any diagram of the knot.. Bridge number was first studied in the 1950s by Horst Schubert.. The bridge number can equivalently be defined geometrically instead of topologically. In bridge representation, a knot lies entirely in the plane apart for a finite number of bridges whose projections onto the plane are straight lines.

Cancellous bone is formed from groupings of trabeculated bone tissue. In cross sections, trabeculae of a cancellous bone can look like septa, but in three dimensions they are topologically distinct, with trabeculae being roughly rod or pillar-shaped and septa being sheet-like. When crossing fluid-filled spaces, trabeculae may have the function of resisting tension (as in the penis, see for example trabeculae of corpora cavernosa and trabeculae of corpus spongiosum) or providing a cell filter (as in the trabecular meshwork of the eye). Multiple perforations in a septum may reduce it to a collection of trabeculae, as happens to the walls of some of the pulmonary alveoli in emphysema.

Highly accessed. Alternatively, a parallel implementation is provided in PGD, a software library for computing graphlet-based network properties in large and massive networks. Graphlet degree vectors (signatures) and signature similarities were applied to biological networks to identify groups (or clusters) of topologically similar nodes in a network and predict biological properties of yet uncharacterized nodes based on known biological properties of characterized nodes. Specifically, they were applied to protein function prediction, cancer gene identification,Tijana Milenković, Vesna Memisević, Anand K. Ganesan, and Nataša Pržulj, Systems-level Cancer Gene Identification from Protein Interaction Network Topology Applied to Melanogenesis-related Interaction Networks, Journal of the Royal Society Interface 2009, .

If X is the circle S^1, the mapping cone C_f can be considered as the quotient space of the disjoint union of Y with the disk D^2 formed by identifying each point x on the boundary of D^2 to the point f(x) in Y. Consider, for example, the case where Y is the disk D^2, and f\colon S^1 \to Y = D^2 is the standard inclusion of the circle S^1 as the boundary of D^2. Then the mapping cone C_f is homeomorphic to two disks joined on their boundary, which is topologically the sphere S^2.

They must be derivable without error from axioms and already-proven statements. HoTT adds the univalence axiom, which relates the equality of logical-mathematical propositions to homotopy theory. An equation such as “a=b” is a mathematical proposition in which two different symbols have the same value. In homotopy type theory, this is taken to mean that the two shapes which represent the values of the symbols are topologically equivalent. These topological equivalence relationships, ETH Zürich Institute for Theoretical Studies director Giovanni Felder argues, can be better formulated in homotopy theory because it is more comprehensive: Homotopy theory explains not only why “a equals b” but also how to derive this.

Even when a pair of polyhedra cannot be obtained by reciprocation from each other, they may be called duals of each other as long as the vertices of one correspond to the faces of the other, and the edges of one correspond to the edges of the other, in an incidence-preserving way. Such pairs of polyhedra are still topologically or abstractly dual. The vertices and edges of a convex polyhedron form a graph (the 1-skeleton of the polyhedron), embedded on a topological sphere, the surface of the polyhedron. The same graph can be projected to form a Schlegel diagram on a flat plane.

In mathematics, the Narasimhan–Seshadri theorem, proved by , says that a holomorphic vector bundle over a Riemann surface is stable if and only if it comes from an irreducible projective unitary representation of the fundamental group. The main case to understand is that of topologically trivial bundles, i.e. those of degree zero (and the other cases are a minor technical extension of this case). This case of the Narasimhan–Seshadri theorem says that a degree zero holomorphic vector bundle over a Riemann surface is stable if and only if it comes from an irreducible unitary representation of the fundamental group of the Riemann surface.

"Members of the core group are under what could be termed 'positive control'—long relationships and similar mindsets make 'control' not so much of an issue, but there are distinct roles, and position (structural, financial, spiritual) determines authority, thus making the core group a hierarchy topologically." In the illustration of the core shown here, each member knows how to reach two other members, and also knows the member(s) he or she considers her or his ideological superior. Solid lines show basic communication, dotted red arrows show , and dotted blue arrows show a second level of ideological respect. If Osama, the most respected, died, the core would reconstitute itself.

The double-helical configuration of DNA strands makes them difficult to separate, which is required by helicase enzymes if other enzymes are to transcribe the sequences that encode proteins, or if chromosomes are to be replicated. In circular DNA, in which double- helical DNA is bent around and joined in a circle, the two strands are topologically linked, or knotted. Otherwise identical loops of DNA, having different numbers of twists, are topoisomers, and cannot be interconverted without the breaking of DNA strands. Topoisomerases catalyze and guide the unknotting or unlinking of DNA by creating transient breaks in the DNA using a conserved tyrosine as the catalytic residue.

In some games, a player can only use warps to travel to locations they have visited before. Because of this, a player has to make the journey by normal route at least once, but are not required to travel the same paths again if they need to revisit earlier areas in the game. Finding warp zones might become a natural goal of a gaming session, being used as a checkpoint. Though it is unclear which video game first made use of teleportation areas or devices, the element has been traced back to MUDs, where it allowed connected rooms to not be "topologically correct" if necessary.

Mathematicians have discovered that it is possible in principle to have multiple bodies in non-elliptical orbits that repeat periodically, although most such orbits are not stable regarding small perturbations in mass, position, or velocity. However, some special stable cases have been identified, including a planar figure-eight orbit occupied by three moving bodies. Further studies have discovered that nonplanar orbits are also possible, including one involving 12 masses moving in 4 roughly circular, interlocking orbits topologically equivalent to the edges of a cuboctahedron. Finding such orbits naturally occurring in the universe is thought to be extremely unlikely, because of the improbability of the required conditions occurring by chance.

Symmetrically, the origin of the ζ-chart plays the role of ∞ in the ξ-chart. Topologically, the resulting space is the one-point compactification of a plane into the sphere. However, the Riemann sphere is not merely a topological sphere. It is a sphere with a well-defined complex structure, so that around every point on the sphere there is a neighborhood that can be biholomorphically identified with C. On the other hand, the uniformization theorem, a central result in the classification of Riemann surfaces, states that every simply-connected Riemann surface is biholomorphic to the complex plane, the hyperbolic plane, or the Riemann sphere.

Side- scrapers slightly decrease in popularity towards the top of level C. The faunal assemblage, although fragmentary, again shows a completely modem aspect, with bones from wild goat, red deer, gazelle, field mouse, mole rat, hare, bat and several birds of woodland and scrub habitat. This evidence, and that from the presence of snails of the species Helix salomonica, indicates a mixed environment of woodland, grassland and scrub, much as exists today. A smaIl sounding in the adjacent Water Cave also revealed evidence of Mousterian occupation. Garrod did not keep all the excavated material and she only kept those pieces that were topologically informative.

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.

Let X = {a,b,c} be a set with 3 elements. There are 29 distinct topologies on X but only 9 inequivalent topologies: #{∅, {a,b,c}} #{∅, {c}, {a,b,c}} #{∅, {a,b}, {a,b,c}} #{∅, {c}, {a,b}, {a,b,c}} #{∅, {c}, {b,c}, {a,b,c}} #{∅, {c}, {a,c}, {b,c}, {a,b,c}} #{∅, {a}, {b}, {a,b}, {a,b,c}} #{∅, {b}, {c}, {a,b}, {b,c}, {a,b,c}} #{∅, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}} The last 5 of these are all T0. The first one is trivial, while in 2, 3, and 4 the points a and b are topologically indistinguishable.

This holds for a geometry over any field, and more generally over any division ring. In the real case, a point at infinity completes a line into a topologically closed curve. In higher dimensions, all the points at infinity form a projective subspace of one dimension less than that of the whole projective space to which they belong. A point at infinity can also be added to the complex line (which may be thought of as the complex plane), thereby turning it into a closed surface known as the complex projective line, CP1, also called the Riemann sphere (when complex numbers are mapped to each point).

The cyclooxygenase active site, which catalyzes the formation of prostaglandin G2 (PGG2) from arachidonic acid, resides at the apex of a long hydrophobic channel, extending from the membrane-binding domain to the center of the molecule. The peroxidase active site, which catalyzes the reduction of PGG2 to PGH2, is located on the other side of the molecule, at the heme binding site. Both MPO and the catalytic domain of PGHS are mainly alpha-helical, 19 helices being identified as topologically and spatially equivalent; PGHS contains 5 additional N-terminal helices that have no equivalent in MPO. In both proteins, three Asn residues in each monomer are glycosylated.

3D rendering of a car in CAD software with boundary representation Also important to the development of CAD was the development in the late 1980s and early 1990s of B-rep solid modeling kernels (engines for manipulating geometrically and topologically consistent 3D objects), Parasolid (ShapeData), and ACIS (Spatial Technology Inc.). These developments were inspired by the work of Ian Braid. This subsequently led to the release of mid-range packages such as SolidWorks and TriSpective (later known as IRONCAD) in 1995, Solid Edge (then Intergraph) in 1996, and Autodesk Inventor in 1999. An independent geometric modeling kernel has been evolving in Russia since the 1990s.

The infinite general linear group or stable general linear group is the direct limit of the inclusions as the upper left block matrix. It is denoted by either GL(F) or , and can also be interpreted as invertible infinite matrices which differ from the identity matrix in only finitely many places. It is used in algebraic K-theory to define K1, and over the reals has a well-understood topology, thanks to Bott periodicity. It should not be confused with the space of (bounded) invertible operators on a Hilbert space, which is a larger group, and topologically much simpler, namely contractible - see Kuiper's theorem.

Schematic view of the different current systems which shape the Earth's magnetosphere In many MHD systems most of the electric current is compressed into thin nearly-two-dimensional ribbons termed current sheets. These can divide the fluid into magnetic domains, inside of which the currents are relatively weak. Current sheets in the solar corona are thought to be between a few meters and a few kilometers in thickness, which is quite thin compared to the magnetic domains (which are thousands to hundreds of thousands of kilometers across). Another example is in the Earth's magnetosphere, where current sheets separate topologically distinct domains, isolating most of the Earth's ionosphere from the solar wind.

Emperor of China is a submarine volcano in the western part of the Banda Sea, Indonesia. This volcano is part of a chain with Nieuwerkerk volcano, known scientifically as the Emperor of China-Nieuwkerk (NEC) ridge, the depth of which is ranging from 3,100-2,700 metres (10,170-8,858 ft). The NEC ridge is lying on Damar Basin, a sea basin of which has 1-2 km Pliocene–Quaternary sediment thickness and is topologically flat. Damar Basin is located in Banda Sea, which is bounded by East Sunda and Banda volcanic arcs within the area of three major plates, namely Eurasian, Pacific and Indo-Australian plates; all of the three plates have been actively converging since Mesozoic times.

The basic design comprises a circle divided into four equal portions by a cross inscribed inside it like four spokes in a wheel; the classic example of this design is Yut. However, the term "cross and circle game" is also applied to boards that replace the circle with a square, and cruciform boards that collapse the circle onto the cross; all three types are topologically equivalent. Ludo and Parcheesi (both descendants of Pachisi) are examples of frequently played cruciform games. The category may also be expanded to include circular or square boards without a cross which are nevertheless quartered (Zohn Ahl), and boards that have more than four spokes (Aggravation, Trivial Pursuit).

In algebraic geometry, a local ring A is said to be unibranch if the reduced ring Ared (obtained by quotienting A by its nilradical) is an integral domain, and the integral closure B of Ared is also a local ring. A unibranch local ring is said to be geometrically unibranch if the residue field of B is a purely inseparable extension of the residue field of Ared. A complex variety X is called topologically unibranch at a point x if for all complements Y of closed algebraic subsets of X there is a fundamental system of neighborhoods (in the classical topology) of x whose intersection with Y is connected. In particular, a normal ring is unibranch.

Suppose the vacuum is the vacuum manifold Σ. Then, for finite energies, as we move along each direction towards spatial infinity, the state along the path approaches a point on the vacuum manifold Σ. Otherwise, we would not have a finite energy. In topologically trivial 3 + 1 dimensions, this means spatial infinity is homotopically equivalent to the topological sphere S. So, the superselection sectors are classified by the second homotopy group of Σ, π2(Σ). In the special case of a Yang–Mills–Higgs theory, the vacuum manifold is isomorphic to the quotient space G/H and the relevant homotopy group is π2(G/H). Note that this doesn't actually require the existence of a scalar Higgs field.

The tetrahemihexahedron is a projective polyhedron, and the only uniform projective polyhedron that immerses in Euclidean 3-space. Note that the prefix "hemi-" is also used to refer to hemipolyhedra, which are uniform polyhedra having some faces that pass through the center of symmetry. As these do not define spherical polyhedra (because they pass through the center, which does not map to a defined point on the sphere), they do not define projective polyhedra by the quotient map from 3-space (minus the origin) to the projective plane. Of these uniform hemipolyhedra, only the tetrahemihexahedron is topologically a projective polyhedron, as can be verified by its Euler characteristic and visually obvious connection to the Roman surface.

In recent years, the advent of a molecular method called chromosome conformation capture (3C) has allowed studying a high-resolution spatial organization of chromosomes in both bacteria and eukaryotes. 3C and its version that is coupled with deep sequencing (Hi-C) determine physical proximity, if any, between any two genomic loci in 3D space. A high-resolution contact map of bacterial chromosomes including the E. coli chromosome has revealed that a bacterial chromosome is segmented into many highly self- interacting regions called chromosomal interaction domains (CIDs). CIDs are equivalent to topologically associating domains (TADs) observed in many eukaryotic chromosomes, suggesting that the formation of CIDs is a general phenomenon of genome organization.

This topology was assumed to extend to all other filamentous phages, but it is not the case for phage Pf4, for which the DNA in the phage is single-stranded but topologically linear, not circular. During fd phage assembly, the phage DNA is first packaged into a linear intracellular nucleoprotein complex with many copies of the phage gene 5 replication/assembly protein. This protein also binds with high affinity to G-quadruplex structures (although they are not present in the phage DNA) and to similar hairpin structures in phage DNA. The gene 5 protein is then displaced by the gene 8 coat protein as the nascent phage is extruded across the bacterial plasma membrane without killing the bacterial host.

The theorem of the three geodesics says that for surfaces homeomorphic to the sphere, there exist at least three non-self-crossing closed geodesics. There may be more than three, for instance, the sphere itself has infinitely many. This result stems from the mathematics of ocean navigation, where the surface of the earth can be modeled accurately by an ellipsoid, and from the study of the geodesics on an ellipsoid, the shortest paths for ships to travel. In particular, a nearly-spherical triaxial ellipsoid has only three simple closed geodesics, its equators.. In 1905, Henri Poincaré conjectured that every smooth surface topologically equivalent to a sphere likewise contains at least three simple closed geodesics,.

In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory and the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for mathematical work related to topological field theory. In condensed matter physics, topological quantum field theories are the low-energy effective theories of topologically ordered states, such as fractional quantum Hall states, string-net condensed states, and other strongly correlated quantum liquid states.

Most of the above work is contained in Cauer's first and second monographs and is largely a treatment of one-ports. In his habilitation thesis Cauer begins to extend this work by showing that a global canonical form cannot be found in the general case for three-element kind multiports (that is, networks containing all three R, L and C elements) for the generation of realisation solutions, as it can be for the two-element kind case.E. Cauer et al., p6 Cauer extended the work of Bartlett and Brune on geometrically symmetric 2-ports to all symmetric 2-ports, that is 2-ports which are electrically symmetrical but not necessarily topologically symmetrical, finding a number of canonical circuits.

Ashoke Sen has conjectured that, in the absence of a topologically nontrivial NS 3-form flux, all IIB brane configurations can be obtained from stacks of spacefilling D9 and anti D9 branes via tachyon condensation. The topology of the resulting branes is encoded in the topology of the gauge bundle on the stack of the spacefilling branes. The topology of the gauge bundle of a stack of D9s and anti D9s can be decomposed into a gauge bundle on the D9's and another bundle on the anti D9's. Tachyon condensation transforms such a pair of bundles to another pair in which the same bundle is direct summed with each component in the pair.

A large number of MPTs is often seen as an analytical failure, and is widely believed to be related to the number of missing entries ("?") in the dataset, characters showing too much homoplasy, or the presence of topologically labile "wildcard" taxa (which may have many missing entries). Numerous methods have been proposed to reduce the number of MPTs, including removing characters or taxa with large amounts of missing data before analysis, removing or downweighting highly homoplastic characters (successive weighting) or removing wildcard taxa (the phylogenetic trunk method) a posteriori and then reanalyzing the data. Numerous theoretical and simulation studies have demonstrated that highly homoplastic characters, characters and taxa with abundant missing data, and "wildcard" taxa contribute to the analysis.

3D model of a crossed square cupola The crossed square cupola is one of the nonconvex Johnson solid isomorphs, being topologically identical to the convex square cupola. It can be obtained as a slice of the nonconvex great rhombicuboctahedron or quasirhombicuboctahedron, analogously to how the square cupola may be obtained as a slice of the rhombicuboctahedron. As in all cupolae, the base polygon has twice as many edges and vertices as the top; in this case the base polygon is an octagram. It may be seen as a cupola with a retrograde square base, so that the squares and triangles connect across the bases in the opposite way to the square cupola, hence intersecting each other.

In mathematics, the Gromov invariant of Clifford Taubes counts embedded (possibly disconnected) pseudoholomorphic curves in a symplectic 4-manifold, where the curves are holomorphic with respect to an auxiliary compatible almost complex structure. (Multiple covers of 2-tori with self-intersection 0 are also counted.) Taubes proved the information contained in this invariant is equivalent to invariants derived from the Seiberg–Witten equations in a series of four long papers. Much of the analytical complexity connected to this invariant comes from properly counting multiply covered pseudoholomorphic curves so that the result is invariant of the choice of almost complex structure. The crux is a topologically defined index for pseudoholomorphic curves which controls embeddedness and bounds the Fredholm index.

When a complex manifold X is blown up along a submanifold Z, the blow up locus Z is replaced by an exceptional divisor E and the rest of the manifold is left undisturbed. Topologically, this operation may also be viewed as the removal of an \epsilon-neighborhood of the blow up locus, followed by the collapse of the boundary by the Hopf map. Blowing up a symplectic manifold is more subtle, since the symplectic form must be adjusted in a neighborhood of the blow up locus in order to continue smoothly across the exceptional divisor in the blow up. The symplectic cut is an elegant means of making the neighborhood- deletion/boundary-collapse process symplectically rigorous.

A blastocoel (), also spelled blastocoele and blastocele, and also called blastocyst cavity (or cleavage or segmentation cavity) is a fluid-filled cavity that forms in the blastula (blastocyst) of early amphibian and echinoderm embryos, or between the epiblast and hypoblast of avian, reptilian, and mammalian blastoderm-stage embryos. It results from cleavage of the oocyte (ovum) after fertilization. It forms during embryogenesis, as what has been termed a "Third Stage" after the single-celled fertilized oocyte (zygote, ovum) has divided into 16-32 cells, via the process of mitosis. It can be described as the first cell cavity formed as the embryo enlarges, the essential precursor for the differentiated, topologically distinct, gastrula.

Ideas about race, ethnicity and identity have also evolved in the United States, and such changes warrant examination of how these shifts have impacted the accuracy of census data over time. The United States Census Bureau began pursuing technological innovations to improve the precision of its census data collection in the 1980s. Robert W. Marx, the Chief of the Geography Division of the USCB teamed up with the U.S. Geological Survey and oversaw the creation of the Topologically Integrated Geographic Encoding and Referencing (TIGER) database system. Census officials were able to evaluate the more sophisticated and detailed results that the TIGER system produced; furthermore, TIGER data is also available to the public.

Residual topology is a descriptive stereochemical term to classify a number of intertwined and interlocked molecules, which cannot be disentangled in an experiment without breaking of covalent bonds, while the strict rules of mathematical topology allow such a disentanglement. Examples of such molecules are rotaxanes, catenanes with covalently linked rings (so-called pretzelanes), and open knots (pseudoknots) which are abundant in proteins. The term "residual topology" was suggested on account of a striking similarity of these compounds to the well-established topologically nontrivial species, such as catenanes and knotanes (molecular knots). The idea of residual topological isomerism introduces a handy scheme of modifying the molecular graphs and generalizes former efforts of systemization of mechanically bound and bridged molecules.

Since topological indistinguishability is an equivalence relation on any topological space X, we can form the quotient space KX = X/≡. The space KX is called the Kolmogorov quotient or T0 identification of X. The space KX is, in fact, T0 (i.e. all points are topologically distinguishable). Moreover, by the characteristic property of the quotient map any continuous map f : X → Y from X to a T0 space factors through the quotient map q : X → KX. Although the quotient map q is generally not a homeomorphism (since it is not generally injective), it does induce a bijection between the topology on X and the topology on KX. Intuitively, the Kolmogorov quotient does not alter the topology of a space.

Vertex figure for the bialternatosnub 16-cell The bialternatosnub 16-cell or runcic snub rectified 16-cell, constructed by removing alternating long rectangles from the octagons, but is also not uniform. Like the omnisnub tesseract, it has a highest symmetry construction of order 192, with 8 rhombicuboctahedra (with Th symmetry), 16 icosahedra (with T symmetry), 24 rectangular trapezoprisms (topologically equivalent to a cube but with D2d symmetry), 32 triangular prisms, with 96 triangular prisms (as Cs-symmetry wedges) filling the gaps. A variant with regular icosahedra and uniform triangular prisms has two edge lengths in the ratio of 1 : 2, and occurs as a vertex-faceting of the scaliform runcic snub 24-cell.

In 2002, Jing Yang, Simon Liao and Mirek Pawlak found an explicit winning strategy for the first player on Hex boards of size 7×7 using a decomposition method with a set of reusable local patterns.On a decomposition method for finding winning strategy in Hex game , Jing Yang, Simon Liao and Mirek Pawlak, 2002 They extended the method to weakly solve the center pair of topologically congruent openings on 8×8 boards in 2002 and the center opening on 9×9 boards in 2003.Unpublished white papers, formerly @ www.ee.umanitoba.com/~jingyang/ In 2009, Philip Henderson, Broderick Arneson and Ryan B. Hayward completed the analysis of the 8×8 board with a computer search, solving all the possible openings.

Chiral magnetic effect (CME) is the generation of electric current along an external magnetic field induced by chirality imbalance. The CME is a macroscopic quantum phenomenon present in systems with charged chiral fermions, such as the quark-gluon plasma, or Dirac and Weyl semimetals; for review, see. The CME is a consequence of chiral anomaly in quantum field theory; unlike conventional superconductivity or superfluidity, it does not require a spontaneous symmetry breaking. The chiral magnetic current is non- dissipative, because it is topologically protected: the imbalance between the densities of left- and right-handed chiral fermions is linked to the topology of fields in gauge theory by the Atiyah-Singer index theorem.

Nazinga offered a topologically-varied environment, but endemic poaching was rapidly decimating the wildlife. He and Rob set up a conservation project with their families: using local labor they constructed dams, developed some 600 km of roads, built test paddocks and an administrative base, negotiated local subsistence farmers to abandon fields within the ranch boundaries, and banned all livestock from its land. They hired local poachers as game keepers in order to give them an incentive to protect it, established a fishing management program, and invited biologists from several countries to help them carry out wildlife and ecological studies. They also reintroduced native wildlife species which had been eradicated from the region.

The game board for the Aztec game Patolli consists of a collapsed circle without an interior cross and thus has the distinction of being a cross that is a circle (topologically), without being a cross plus circle. Tokens are moved around spaces drawn on the circle and on the cross, with the goal of being the first player to move all tokens all the way around the board. Generally the circle of the cross and circle forms the primary circuit followed by the players' pieces. The function of the cross is more variable; for example, in Yut the cross forms shortcuts to the finish, whereas in Pachisi the four spokes are used as player-specific exits and entrances to the pieces' home.

CNN processors have been used to research a variety of mathematical concepts, such as researching non-equilibrium systems, constructing non-linear systems of arbitrary complexity using a collection of simple, well-understood dynamic systems, studying emergent chaotic dynamics, generating chaotic signals, and in general discovering new dynamic behavior. They are often used in researching systemics, a trandisiplinary, scientific field that studies natural systems. The goal of systemics researchers is to develop a conceptual and mathematical framework necessary to analyze, model, and understand systems, including, but not limited to, atomic, mechanical, molecular, chemical, biological, ecological, social and economic systems. Topics explored are emergence, collective behavior, local activity and its impact on global behavior, and quantifying the complexity of an approximately spatial and topologically invariant system .

660x660px If we consider a rooted three-taxon tree, the simplest non-trivial phylogenetic tree, there are three different tree topologies but four possible gene trees. The existence of four distinct gene trees despite the smaller number of topologies reflects the fact that there are topologically identical gene tree that differ in their coalescent times. In the type 1 tree the alleles in species A and B coalesce after the speciation event that separated the A-B lineage from the C lineage. In the type 2 tree the alleles in species A and B coalesce before the speciation event that separated the A-B lineage from the C lineage (in other words, the type 2 tree is a deep coalescence tree).

Dynamic DNA nanotechnology: The complexes constructed in structural DNA nanotechnology use topologically branched nucleic acid structures containing junctions. (In contrast, most biological DNA exists as an unbranched double helix.) One of the simplest branched structures is a four-arm junction that consists of four individual DNA strands, portions of which are complementary in a specific pattern. Unlike in natural Holliday junctions, each arm in the artificial immobile four-arm junction has a different base sequence, causing the junction point to be fixed at a certain position. Multiple junctions can be combined in the same complex, such as in the widely used double-crossover (DX) structural motif, which contains two parallel double helical domains with individual strands crossing between the domains at two crossover points.

Each crossover point is, topologically, a four-arm junction, but is constrained to one orientation, in contrast to the flexible single four-arm junction, providing a rigidity that makes the DX motif suitable as a structural building block for larger DNA complexes. Dynamic DNA nanotechnology uses a mechanism called toehold-mediated strand displacement to allow the nucleic acid complexes to reconfigure in response to the addition of a new nucleic acid strand. In this reaction, the incoming strand binds to a single-stranded toehold region of a double-stranded complex, and then displaces one of the strands bound in the original complex through a branch migration process. The overall effect is that one of the strands in the complex is replaced with another one.

In guenon Brodmann area 5 is a subdivision of the parietal lobe defined on the basis of cytoarchitecture. It occupies primarily the superior parietal lobule. Brodmann-1909 considered it topologically and cytoarchitecturally homologous to the preparietal area 5 of the human. Distinctive features (Brodmann-1905): compared to area 4 of Brodmann-1909 area 5 has a thick self-contained internal granular layer (IV); lacks a distinct internal pyramidal layer (V); has a marked sublayer 3b of pyramidal cells in the external pyramidal layer (III); has a distinct boundary between the internal pyramidal layer (V) and the multiform layer (VI); and has ganglion cells in layer V beneath its boundary with layer IV that are separated from layer VI by a wide clear zone.

This is the Dirac quantization condition, and it is a topological condition that demands that the long distance U(1) gauge field configurations be consistent. # When the U(1) gauge group comes from breaking a compact Lie group, the path that winds around the U(1) group enough times is topologically trivial in the big group. In a non-U(1) compact Lie group, the covering space is a Lie group with the same Lie algebra, but where all closed loops are contractible. Lie groups are homogenous, so that any cycle in the group can be moved around so that it starts at the identity, then its lift to the covering group ends at , which is a lift of the identity.

Some authors restrict the phrase "toroidal polyhedra" to mean more specifically polyhedra topologically equivalent to the (genus 1) torus.. In this area, it is important to distinguish embedded toroidal polyhedra, whose faces are flat polygons in three-dimensional Euclidean space that do not cross themselves or each other, from abstract polyhedra, topological surfaces without any specified geometric realization.. Intermediate between these two extremes are polyhedra formed by geometric polygons or star polygons in Euclidean space that are allowed to cross each other. In all of these cases the toroidal nature of a polyhedron can be verified by its orientability and by its Euler characteristic being non-positive. The Euler characteristic generalizes to V − E + F = 2 − 2N, where N is the number of holes.

Newman has proposed a theory for the evolution of cell differentiation in animals. Based on a detailed consideration of gene regulatory components and processes that distinguish this group from all other forms of life, including their nearest holozoan relatives, he has suggested that the topologically associating domains found in the nuclei of metazoan cells had a unique propensity to amplify and exaggerate inherent physiological and structural functionalities of unicellular ancestors . With the evolutionary biologist Gerd B. Müller, Newman edited Origination of Organismal Form (MIT Press, 2003). This book on evolutionary developmental biology is a collection of papers by various researchers on generative mechanisms that were plausibly involved in the origination of disparate body forms during the Ediacaran and early Cambrian periods.

3D model of a crossed pentagrammic cupola In geometry, the crossed pentagrammic cupola is one of the nonconvex Johnson solid isomorphs, being topologically identical to the convex pentagonal cupola. It can be obtained as a slice of the nonconvex great rhombicosidodecahedron or quasirhombicosidodecahedron, analogously to how the pentagonal cupola may be obtained as a slice of the rhombicosidodecahedron. As in all cupolae, the base polygon has twice as many edges and vertices as the top; in this case the base polygon is a decagram. It may be seen as a cupola with a retrograde pentagrammic base, so that the squares and triangles connect across the bases in the opposite way to the pentagrammic cuploid, hence intersecting each other more deeply.

One of the most notorious pathologies in topology is the Alexander horned sphere, a counterexample showing that topologically embedding the sphere S2 in R3 may fail to separate the space cleanly. As a counter-example, it motivated the extra condition of tameness, which suppresses the kind of wild behavior the horned sphere exhibits. Like many other pathologies, the horned sphere in a sense plays on infinitely fine, recursively generated structure, which in the limit violates ordinary intuition. In this case, the topology of an ever-descending chain of interlocking loops of continuous pieces of the sphere in the limit fully reflects that of the common sphere, and one would expect the outside of it, after an embedding, to work the same.

The product molecule does not need to match orientation or specific layout of the molecules as long as the molecule is topologically equivalent with respect to atoms, bonds, and bond types; however, in larger puzzles, these factors will influence the inputs to downstream reactors. While the two waldos can cross over each other without harm, collision of atoms with one another or with the walls of the reactor is not allowed; such collisions stop the program and force the player to re-evaluate their solution. Similarly, if a waldo delivers the wrong product, the player will need to check their program. The player successfully completes each puzzle by constructing a program capable of repeatedly generating the required output, meeting a certain quota.

Topologically ordered states have some interesting properties, such as (1) topological degeneracy and fractional statistics or non-abelian statistics that can be used to realize topological quantum computer; (2) perfect conducting edge states that may have important device applications; (3) emergent gauge field and Fermi statistics that suggest a quantum information origin of elementary particles; See also (4) topological entanglement entropy that reveals the entanglement origin of topological order, etc. Topological order is important in the study of several physical systems such as spin liquids and the quantum Hall effect, along with potential applications to fault-tolerant quantum computation. Topological insulators and topological superconductors (beyond 1D) do not have topological order as defined above, their entanglements being only short-ranged.

In 1969, Branko Grünbaum conjectured that every 3-regular graph with a polyhedral embedding on any two-dimensional oriented manifold such as a torus must be of class one. In this context, a polyhedral embedding is a graph embedding such that every face of the embedding is topologically a disk and such that the dual graph of the embedding is simple, with no self-loops or multiple adjacencies. If true, this would be a generalization of the four color theorem, which was shown by Tait to be equivalent to the statement that 3-regular graphs with a polyhedral embedding on a sphere are of class one. However, showed the conjecture to be false by finding snarks that have polyhedral embeddings on high-genus orientable surfaces.

In that year, the University of Tübingen also successfully applied for Excellence Initiative support with its institutional strategy and research school. Ever since it moved in in early 2012 (official opening ceremony: 14 May 2012), the CIN has been at home in a building of its own on the Tübingen University Hospital's Schnarrenberg campus. It is situated next to institutions that are neighbours topologically as well as scientifically, nestled in between the Hertie Institute for Clinical Brain Research (HIH) on one side and the German Center for Neurodegenerative Diseases (DZNE) on the other. Before it moved into its current building, the CIN was installed in a building of the Tübingen- Reutlingen Technology Park, in close proximity to the Tübingen Max Planck campus.

Then, each crack spreads in two opposite directions along a line through the initiation point, with the slope of the line chosen uniformly at random. The cracks continue spreading at uniform speed until they reach another crack, at which point they stop, forming a T-junction. The result is a tessellation of the plane by irregular convex polygons. A variant of the model that has also been studied restricts the orientations of the cracks to be axis-parallel, resulting in a random tessellation of the plane by rectangles... write that, in comparison to alternative models in which cracks may cross each other or in which cracks are formed one at a time rather than simultaneously, "most mudcrack patterns in nature topologically resemble" the Gilbert model.

A few chloroplast genes found new homes in the mitochondrial genome—most became nonfunctional pseudogenes, though a few tRNA genes still work in the mitochondrion. Some transferred chloroplast DNA protein products get directed to the secretory pathway, though many secondary plastids are bounded by an outermost membrane derived from the host's cell membrane, and therefore topologically outside of the cell because to reach the chloroplast from the cytosol, the cell membrane must be crossed, which signifies entrance into the extracellular space. In those cases, chloroplast- targeted proteins do initially travel along the secretory pathway. Because the cell acquiring a chloroplast already had mitochondria (and peroxisomes, and a cell membrane for secretion), the new chloroplast host had to develop a unique protein targeting system to avoid having chloroplast proteins being sent to the wrong organelle.

Y is typically played on a triangular board with hexagonal spaces; the "official" Y board has three points with five-connectivity instead of six-connectivity, but it is just as playable on a regular triangle. Schensted and Titus' book Mudcrack Y & Poly-Y has a large number of boards for play of Y, all hand-drawn; most of them seem irregular but turn out to be topologically identical to a regular Y board. A simple board, 8 spaces per side As in most games of this type, one player takes the part of Black and one takes the part of White; they place stones on the board one at a time, neither removing nor moving any previously placed stones. The pie rule can be used to mitigate any first-move advantage.

The generalized Poincaré conjecture is true topologically, but false smoothly in some dimensions. This results in constructions of manifolds that are homeomorphic, but not diffeomorphic, to the standard sphere, which are known as the exotic spheres: you can interpret these as non-standard smooth structures on the standard (topological) sphere. Thus the homotopy spheres that John Milnor produced are homeomorphic (Top-isomorphic, and indeed piecewise linear homeomorphic) to the standard sphere S^n, but are not diffeomorphic (Diff-isomorphic) to it, and thus are exotic spheres: they can be interpreted as non-standard differentiable structures on the standard sphere. Michel Kervaire and Milnor showed that the oriented 7-sphere has 28 different smooth structures (or 15 ignoring orientations), and in higher dimensions there are usually many different smooth structures on a sphere.

Class I, including strains fd, f1, M13, If1 and IKe, has a rotation axis relating the gene 8 coat proteins, whereas Class II, including strains Pf1, Pf3, Pf4 and PH75, this rotation axis is replaced by a helix axis. This technical difference has little noticeable effect on the overall phage structure, but the extent of independent diffraction data is greater for symmetry Class II than for Class I. This assisted the determination of the Class II phage Pf1 structure, and by extension the Class I structure. The DNA isolated from fd phage is single-stranded, and topologically a circle. That is, the DNA single strand extends from one end of the phage particle to the other and then back again to close the circle, although the two strands are not base-paired.

A black string is a higher dimensional (D>4) generalization of a black hole in which the event horizon is topologically equivalent to S2 × S1 and spacetime is asymptotically Md−1 × S1. Perturbations of black string solutions were found to be unstable for L (the length around S1) greater than some threshold L′. The full non-linear evolution of a black string beyond this threshold might result in a black string breaking up into separate black holes which would coalesce into a single black hole. This scenario seems unlikely because it was realized a black string could not pinch off in finite time, shrinking S2 to a point and then evolving to some Kaluza–Klein black hole. When perturbed, the black string would settle into a stable, static non-uniform black string state.

One of the first examples of an exotic sphere found by was the following: Take two copies of B4×S3, each with boundary S3×S3, and glue them together by identifying (a,b) in the boundary with (a, a2ba−1), (where we identify each S3 with the group of unit quaternions). The resulting manifold has a natural smooth structure and is homeomorphic to S7, but is not diffeomorphic to S7. Milnor showed that it is not the boundary of any smooth 8-manifold with vanishing 4th Betti number, and has no orientation-reversing diffeomorphism to itself; either of these properties implies that it is not a standard 7-sphere. Milnor showed that this manifold has a Morse function with just two critical points, both non-degenerate, which implies that it is topologically a sphere.

The 2-dimensional worldsheet theory is an N = (2,2) supersymmetric sigma model, the (2,2) supersymmetry means that the fermionic generators of the supersymmetry algebra, called supercharges, may be assembled into a single Dirac spinor, which consists of two Majorana–Weyl spinors of each chirality. This sigma model is topologically twisted, which means that the Lorentz symmetry generators that appear in the supersymmetry algebra simultaneously rotate the physical spacetime and also rotate the fermionic directions via the action of one of the R-symmetries. The R-symmetry group of a 2-dimensional N = (2,2) field theory is U(1) × U(1), twists by the two different factors lead to the A and B models respectively. The topological twisted construction of topological string theories was introduced by Edward Witten in his 1988 paper.

All four Mur ligases are topologically similar to one another, even though they display low sequence identity. They are each composed of three domains: an N-terminal Rossmann-fold domain responsible for binding the UDPMurNAc substrate; a central domain (similar to ATP-binding domains of several ATPases and GTPases); and a C-terminal domain (similar to dihydrofolate reductase fold) that appears to be associated with binding the incoming amino acid. The conserved sequence motifs found in the four Mur enzymes also map to other members of the Mur ligase family, including folylpolyglutamate synthetase, cyanophycin synthetase and the capB enzyme from Bacillales. This family includes UDP-N-acetylmuramate-L-alanine ligase (MurC), UDP-N-acetylmuramoylalanyl-D-glutamate-2,6-diaminopimelate ligase (MurE), and UDP-N-acetylmuramoyl-tripeptide-D-alanyl-D-alanine ligase (MurF).

A lattice in the complex plane and its fundamental domain, with quotient a torus. Given an action of a group G on a topological space X by homeomorphisms, a fundamental domain for this action is a set D of representatives for the orbits. It is usually required to be a reasonably nice set topologically, in one of several precisely defined ways. One typical condition is that D is almost an open set, in the sense that D is the symmetric difference of an open set in G with a set of measure zero, for a certain (quasi)invariant measure on X. A fundamental domain always contains a free regular set U, an open set moved around by G into disjoint copies, and nearly as good as D in representing the orbits.

The Weierstrass functions are doubly periodic; that is, they are periodic with respect to a lattice Λ; in essence, the Weierstrass functions are naturally defined on a torus T = C/Λ. This torus may be embedded in the complex projective plane by means of the map :z \mapsto [1 :\wp(z) : \wp'(z)/2] This map is a group isomorphism of the torus (considered with its natural group structure) with the chord-and-tangent group law on the cubic curve which is the image of this map. It is also an isomorphism of Riemann surfaces from the torus to the cubic curve, so topologically, an elliptic curve is a torus. If the lattice Λ is related by multiplication by a non-zero complex number c to a lattice cΛ, then the corresponding curves are isomorphic.

In mathematics, the Schwarz alternating method or alternating process is an iterative method introduced in 1869-1870 by Hermann Schwarz in the theory of conformal mapping. Given two overlapping regions in the complex plane in each of which the Dirichlet problem could be solved, Schwarz described an iterative method for solving the Dirichlet problem in their union, provided their intersection was suitably well behaved. This was one of several constructive techniques of conformal mapping developed by Schwarz as a contribution to the problem of uniformization, posed by Riemann in the 1850s and first resolved rigorously by Koebe and Poincaré in 1907. It furnished a scheme for uniformizing the union of two regions knowing how to uniformize each of them separately, provided their intersection was topologically a disk or an annulus.

To turn a rectangle into a Möbius strip, join the edges labelled A so that the directions of the arrows match. Topologically, the Möbius strip can be defined as the square [0, 1] \times [0, 1] with its top and bottom sides identified by the relation (x, 0) \sim (1 - x, 1) for 0 \le x \le 1, as in the diagram. A less used presentation of the Möbius strip is as the topological quotient of a torus.Tony Phillips, Tony Phillips' Take on Math in the Media, American Mathematical Society, October 2006 A torus can be constructed as the square [0, 1] \times [0, 1] with the edges identified as (0, y) \sim (1, y) (glue left to right) and (x, 0) \sim (x, 1) (glue bottom to top).

Comparative studies of proteins classified as jelly roll and Greek key structures suggest that the Greek key proteins evolved significantly earlier than their more topologically complex jelly roll counterparts. Structural bioinformatics studies comparing virus capsid jelly-roll proteins to other proteins of known structure indicates that the capsid proteins form a well-separated cluster, suggesting that they are subject to a distinctive set of evolutionary constraints. One of the most notable features of viral capsid jelly roll proteins is their ability to form oligomers in a repeated tiling pattern to produce a closed protein shell; the cellular proteins that are most similar in fold and topology are mostly also oligomers. It has been proposed that viral jelly-roll capsid proteins have evolved from cellular jelly-roll proteins, potentially on several independent occasions, at the earliest stages of cellular evolution.

The first part introduces the subject visually, encouraging the reader to think about packings not just as static objects but as dynamic systems of circles that change in predictable ways when the conditions under which they are formed (their patterns of adjacency) change. The second part concerns the proof of the circle packing theorem itself, and of the associated rigidity theorem: every maximal planar graph can be associated with a circle packing that is unique up to Möbius transformations of the plane. More generally the same result holds for any triangulated manifold, with a circle packing on a topologically equivalent Riemann surface that is unique up to conformal equivalence. The third part of the book concerns the degrees of freedom that arise when the pattern of adjacencies is not fully triangulated (it is a planar graph, but not a maximal planar graph).

Escher would not have been familiar with Brückner's work and H. S. M. Coxeter writes that "It is remarkable that Escher, without any knowledge of algebra or analytic geometry, was able to rediscover this highly symmetrical figure." Earlier in 1948, Escher had made a preliminary woodcut with a similar theme, Study for Stars, but instead of using the compound of three regular octahedra in the study he used a different but related shape, a stellated rhombic dodecahedron (sometimes called Escher's solid), which can be formed as a compound of three flattened octahedra.The compound of three octahedra and a remarkable compound of three square dipyramids, the Escher's solid, Livio Zefiro, University of Genova. This form as a polyhedron is topologically identical to the disdyakis dodecahedron, which can be seen as rhombic dodecahedron with shorter pyramids on the rhombic faces.

A structure theorem for minimal distal flows proved by Hillel Furstenberg in the early 1960s inspired much work on classification of minimal flows. A lot of research in the 1970s and 1980s was devoted to topological dynamics of one-dimensional maps, in particular, piecewise linear self-maps of the interval and the circle. Unlike the theory of smooth dynamical systems, where the main object of study is a smooth manifold with a diffeomorphism or a smooth flow, phase spaces considered in topological dynamics are general metric spaces (usually, compact). This necessitates development of entirely different techniques but allows extra degree of flexibility even in the smooth setting, because invariant subsets of a manifold are frequently very complicated topologically (cf limit cycle, strange attractor); additionally, shift spaces arising via symbolic representations can be considered on an equal footing with more geometric actions.

The mathematical term perverse sheaves refers to a certain abelian category associated to a topological space X, which may be a real or complex manifold, or a more general topologically stratified space, usually singular. This concept was introduced in the thesis of Zoghman Mebkhout, gaining more popularity after the (independent) work of Joseph Bernstein, Alexander Beilinson, and Pierre Deligne (1982) as a formalisation of the Riemann-Hilbert correspondence, which related the topology of singular spaces (intersection homology of Mark Goresky and Robert MacPherson) and the algebraic theory of differential equations (microlocal calculus and holonomic D-modules of Joseph Bernstein, Masaki Kashiwara and Takahiro Kawai). It was clear from the outset that perverse sheaves are fundamental mathematical objects at the crossroads of algebraic geometry, topology, analysis and differential equations. They also play an important role in number theory, algebra, and representation theory.

For n = 4, the h-cobordism theorem is true topologically (proved by Michael Freedman using a 4-dimensional Whitney trick) but is false PL and smoothly (as shown by Simon Donaldson). For n = 3, the h-cobordism theorem for smooth manifolds has not been proved and, due to the 3-dimensional Poincaré conjecture, is equivalent to the hard open question of whether the 4-sphere has non-standard smooth structures. For n = 2, the h-cobordism theorem is equivalent to the Poincaré conjecture stated by Poincaré in 1904 (one of the Millennium Problems) and was proved by Grigori Perelman in a series of three papers in 2002 and 2003, where he follows Richard S. Hamilton's program using Ricci flow. For n = 1, the h-cobordism theorem is vacuously true, since there is no closed simply- connected 1-dimensional manifold.

A few chloroplast genes found new homes in the mitochondrial genome—most became nonfunctional pseudogenes, though a few tRNA genes still work in the mitochondrion. Some transferred chloroplast DNA protein products get directed to the secretory pathway (though many secondary plastids are bounded by an outermost membrane derived from the host's cell membrane, and therefore topologically outside of the cell, because to reach the chloroplast from the cytosol, you have to cross the cell membrane, just like if you were headed for the extracellular space. In those cases, chloroplast-targeted proteins do initially travel along the secretory pathway). Because the cell acquiring a chloroplast already had mitochondria (and peroxisomes, and a cell membrane for secretion), the new chloroplast host had to develop a unique protein targeting system to avoid having chloroplast proteins being sent to the wrong organelle.

In such a case, the value that is used must be recalculated earlier than the expression that uses it. Topologically ordering the dependency graph, and using this topological order to schedule the cell updates, allows the whole spreadsheet to be updated with only a single evaluation per cell.. Similar problems of task ordering arise in makefiles for program compilation and instruction scheduling for low-level computer program optimization.. PERT chart for a project with five milestones (labeled 10–50) and six tasks (labeled A–F). There are two critical paths, ADF and BC. A somewhat different DAG-based formulation of scheduling constraints is used by the program evaluation and review technique (PERT), a method for management of large human projects that was one of the first applications of DAGs. In this method, the vertices of a DAG represent milestones of a project rather than specific tasks to be performed.

Other topics covered through the book include the rigid geometric chirality of tree-like molecular structures such as tartaric acid, and the stronger topological chirality of molecules that cannot be deformed into their mirror image without breaking and re-forming some of their molecular bonds. It discusses results of Flapan and Jonathan Simon on molecules with the molecular structure of Möbius ladders, according to which every embedding of a Möbius ladder with an odd number of rungs is chiral while Möbius ladders with an even number of rungs have achiral embeddings. It uses the symmetries of graphs, in a result that the symmetries of certain graphs can always be extended to topological symmetries of three-dimensional space, from which it follows that non-planar graphs with no self-inverse symmetry are always chiral. It discusses graphs for which every embedding is topologically knotted or linked.

In this way, Beltrami attempted to demonstrate that two-dimensional non-Euclidean geometry is as valid as the Euclidean geometry of the space, and in particular, that Euclid's parallel postulate could not be derived from the other axioms of Euclidean geometry. It is often stated that this proof was incomplete due to the singularities of the pseudosphere, which means that geodesics could not be extended indefinitely. However, John Stillwell remarks that Beltrami must have been well aware of this difficulty, which is also manifested by the fact that the pseudosphere is topologically a cylinder, and not a plane, and he spent a part of his memoir designing a way around it. By a suitable choice of coordinates, Beltrami showed how the metric on the pseudosphere can be transferred to the unit disk and that the singularity of the pseudosphere corresponds to a horocycle on the non-Euclidean plane.

Here there is one exception: the Hopf circle containing the projection point maps to a straight line in R3 -- a "circle through infinity". The fibers over a circle of latitude on S2 form a torus in S3 (topologically, a torus is the product of two circles) and these project to nested toruses in R3 which also fill space. The individual fibers map to linking Villarceau circles on these tori, with the exception of the circle through the projection point and the one through its opposite point: the former maps to a straight line, the latter to a unit circle perpendicular to, and centered on, this line, which may be viewed as a degenerate torus whose minor radius has shrunken to zero. Every other fiber image encircles the line as well, and so, by symmetry, each circle is linked through every circle, both in R3 and in S3.

Situated ventral to the pallium in the basic vertebrate forebrain plan (though representing a topologically rostral field in neural plate fate maps) is another region of telencephalic gray matter known as the subpallium, which is the progenitor area for the basal ganglia, a set of structures that play a crucial role in the executive control of behavior. The subpallium region has distinct striatal, pallidal, diagonal and preoptic subregions, which are stretched obliquely between the septal midline and the amygdala at the posterior pole of the telencephalon. At least the striatum, pallidum and diagonal domains extend into the amygdala, representing there the subpallial amygdala, forming its central and medial nucleis, as well as the amygdaloid end of the bed nucleus stria terminalis complex. The amygdala thus encompasses an heterogeneous group of subpallial nuclei and hypopallial olfactory and amygdalohippocampal corticonuclear cell masses which are on the whole heavily involved in emotion and motivation.

A T3 space or regular Hausdorff space is a topological space that is both regular and a Hausdorff space. (A Hausdorff space or T2 space is a topological space in which any two distinct points are separated by neighbourhoods.) It turns out that a space is T3 if and only if it is both regular and T0. (A T0 or Kolmogorov space is a topological space in which any two distinct points are topologically distinguishable, i.e., for every pair of distinct points, at least one of them has an open neighborhood not containing the other.) Indeed, if a space is Hausdorff then it is T0, and each T0 regular space is Hausdorff: given two distinct points, at least one of them misses the closure of the other one, so (by regularity) there exist disjoint neighborhoods separating one point from (the closure of) the other.

In supersymmetric and nonsupersymmetric theories, the nonrenormalization of a quantity subject to the Dirac quantization condition is often a consequence of the fact that possible renormalizations would be inconsistent with the quantization condition, for example the quantization of the level of a Chern–Simons theory implies that it may only be renormalized at one-loop. In the 1994 article Nonrenormalization Theorem for Gauge Coupling in 2+1D the authors find the renormalization of the level can only be a finite shift, independent of the energy scale, and extended this result to topologically massive theories in which one includes a kinetic term for the gluons. In Notes on Superconformal Chern-Simons-Matter Theories the authors then showed that this shift needs to occur at one loop, because any renormalization at higher loops would introduce inverse powers of the level, which are nonintegral and so would be in conflict with the quantization condition.

Then one orientation-reversing isometry g of is given by , where denotes the complex conjugate of z. These facts imply that the mapping given by is an orientation-reversing isometry of that generates an infinite cyclic group G of isometries. (It can be expressed as , and its square is the isometry , which can be expressed as .) The quotient of the action of this group can easily be seen to be topologically a Möbius band. But it is also easy to verify that it is complete and non-compact, with constant negative curvature equal to −1. The group of isometries of this Möbius band is 1-dimensional and is isomorphic to the special orthogonal group SO(2). (Constant) zero curvature: This may also be constructed as a complete surface, by starting with portion of the plane R2 defined by and identifying with for all x in R (the reals).

As a simple example from the physics of magnetically confined plasmas, consider an axisymmetric system with circular, concentric magnetic flux surfaces of radius r (a crude approximation to the magnetic field geometry in an early tokamak but topologically equivalent to any toroidal magnetic confinement system with nested flux surfaces) and denote the toroidal angle by \zeta and the poloidal angle by \theta. Then the Toroidal/Poloidal coordinate system relates to standard Cartesian Coordinates by these transformation rules: : x = (R_0 +r \cos \theta) \cos\zeta : y = s_\zeta (R_0 + r \cos \theta) \sin\zeta : z = s_\theta r \sin \theta. where s_\theta = \pm 1, s_\zeta = \pm 1. The natural choice geometrically is to take s_\theta = s_\zeta = +1, giving the toroidal and poloidal directions shown by the arrows in the figure above, but this makes r,\theta,\zeta a left-handed curvilinear coordinate system.

The origin of the Rashba–Edelstein effect relies on the presence of spin-split surface or interface states, which can arise for a structural inversion asymmetry or because the material exhibits a topologically protected surface, being a topological insulator. In both cases, the material surface displays the spin polarization locked to the momentum, meaning that these two quantities are univocally linked and orthogonal one to the other (this is clearly visible from the Fermi countours). It is worth noticing that also a bulk inversion asymmetry could be present, which would result in the Dresselhaus effect. In fact, if, in addition to the spatial inversion asymmetry or to the topological insulator band structure, also a bulk inversion asymmetry is present, the spin and momentum are still locked but their relative orientation is not straightforwardly determinable (since also the orientation of the charge current with respect to the crystallographic axes plays a relevant role).

The book is organized historically, and reviewer Robert Bradley divides the topics of the book into three parts. The first part discusses the earlier history of polyhedra, including the works of Pythagoras, Thales, Euclid, and Johannes Kepler, and the discovery by René Descartes of a polyhedral version of the Gauss–Bonnet theorem (later seen to be equivalent to Euler's formula). It surveys the life of Euler, his discovery in the early 1750s that the Euler characteristic V-E+F is equal to two for all convex polyhedra, and his flawed attempts at a proof, and concludes with the first rigorous proof of this identity in 1794 by Adrien-Marie Legendre, based on Girard's theorem relating the angular excess of triangles in spherical trigonometry to their area. Although polyhedra are geometric objects, Euler's Gem argues that Euler discovered his formula through being the first to view them topologically (as abstract incidence patterns of vertices, faces, and edges), rather than through their geometric distances and angles.

For, if a graph is embedded on a single page, and another half-plane is attached to the spine to extend its page to a complete plane, then the outer face of the embedding includes the entire added half-plane, and all vertices lie on this outer face. Every two-page book embedding is a special case of a planar embedding, because the union of two pages of a book is a space topologically equivalent to the whole plane. Therefore, every graph with book thickness two is automatically a planar graph. More precisely, the book thickness of a graph is at most two if and only if is a subgraph of a planar graph that has a Hamiltonian cycle.. If a graph is given a two-page embedding, it can be augmented to a planar Hamiltonian graph by adding (into any page) extra edges between any two consecutive vertices along the spine that are not already adjacent, and between the first and last spine vertices.

M24 can be constructed from symmetries of the Klein quartic, augmented by a (non-geometric) symmetry of its immersion as the small cubicuboctahedron. M24 can be constructed starting from the symmetries of the Klein quartic (the symmetries of a tessellation of the genus three surface), which is PSL(2,7), which can be augmented by an additional permutation. This permutation can be described by starting with the tiling of the Klein quartic by 56 triangles (with 24 vertices – the 24 points on which the group acts), then forming squares of out some of the 2 triangles, and octagons out of 6 triangles, with the added permutation being "interchange the two endpoints of those edges of the original triangular tiling which bisect the squares and octagons". This can be visualized by coloring the triangles – the corresponding tiling is topologically but not geometrically the t0,1{4, 3, 3} tiling, and can be (polyhedrally) immersed in Euclidean 3-space as the small cubicuboctahedron (which also has 24 vertices).

Moreover, Kosslyn's work showed that there are considerable similarities between the neural mappings for imagined stimuli and perceived stimuli. The authors of these studies concluded that, while the neural processes they studied rely on mathematical and computational underpinnings, the brain also seems optimized to handle the sort of mathematics that constantly computes a series of topologically-based images rather than calculating a mathematical model of an object. Recent studies in neurology and neuropsychology on mental imagery have further questioned the "mind as serial computer" theory, arguing instead that human mental imagery manifests both visually and kinesthetically. For example, several studies have provided evidence that people are slower at rotating line drawings of objects such as hands in directions incompatible with the joints of the human body,Parsons 1987; 2003 and that patients with painful, injured arms are slower at mentally rotating line drawings of the hand from the side of the injured arm.

The Pythagorean tiling is the unique tiling by squares of two different sizes that is both unilateral (no two squares have a common side) and equitransitive (each two squares of the same size can be mapped into each other by a symmetry of the tiling).. Topologically, the Pythagorean tiling has the same structure as the truncated square tiling by squares and regular octagons.. The smaller squares in the Pythagorean tiling are adjacent to four larger tiles, as are the squares in the truncated square tiling, while the larger squares in the Pythagorean tiling are adjacent to eight neighbors that alternate between large and small, just as the octagons in the truncated square tiling. However, the two tilings have different sets of symmetries, because the truncated square tiling is symmetric under mirror reflections whereas the Pythagorean tiling isn't. Mathematically, this can be explained by saying that the truncated square tiling has dihedral symmetry around the center of each tile, while the Pythagorean tiling has a smaller cyclic set of symmetries around the corresponding points, giving it p4 symmetry., p. 42.

One of these algorithms, first described by , works by choosing vertices in the same order as the eventual topological sort. First, find a list of "start nodes" which have no incoming edges and insert them into a set S; at least one such node must exist in a non-empty acyclic graph. Then: L ← Empty list that will contain the sorted elements S ← Set of all nodes with no incoming edge while S is not empty do remove a node n from S add n to L for each node m with an edge e from n to m do remove edge e from the graph if m has no other incoming edges then insert m into S if graph has edges then return error (graph has at least one cycle) else return L (a topologically sorted order) If the graph is a DAG, a solution will be contained in the list L (the solution is not necessarily unique). Otherwise, the graph must have at least one cycle and therefore a topological sort is impossible.

Devaney is known for formulating a simple and widely used definition of chaotic systems, one that does not need advanced concepts such as measure theory. In his 1989 book An Introduction to Chaotic Dynamical Systems, Devaney defined a system to be chaotic if it has sensitive dependence on initial conditions, it is topologically transitive (for any two open sets, some points from one set will eventually hit the other set), and its periodic orbits form a dense set.. Later, it was observed that this definition is redundant: sensitive dependence on initial conditions follows automatically as a mathematical consequence of the other two properties.. Devaney hairs, a fractal structure in certain Julia sets, are named after Devaney, who was the first to investigate them.. As well as research and teaching in mathematics, Devaney's mathematical activities have included organizing one-day immersion programs in mathematics for thousands of Boston- area high school students, and consulting on the mathematics behind media productions including the 2008 film 21 and the 1993 play Arcadia. He was president of the Mathematical Association of America from 2013 to 2015..

The problem of gimbal lock appears when one uses Euler angles in applied mathematics; developers of 3D computer programs, such as 3D modeling, embedded navigation systems, and video games must take care to avoid it. In formal language, gimbal lock occurs because the map from Euler angles to rotations (topologically, from the 3-torus T3 to the real projective space RP3 which is the same as the space of 3d rotations SO3) is not a local homeomorphism at every point, and thus at some points the rank (degrees of freedom) must drop below 3, at which point gimbal lock occurs. Euler angles provide a means for giving a numerical description of any rotation in three- dimensional space using three numbers, but not only is this description not unique, but there are some points where not every change in the target space (rotations) can be realized by a change in the source space (Euler angles). This is a topological constraint – there is no covering map from the 3-torus to the 3-dimensional real projective space; the only (non-trivial) covering map is from the 3-sphere, as in the use of quaternions.

Connectors in which non- hermaphroditic contacts are arranged in a hermaphroditic arrangement (such as bullet connectors used in low end solar equipment) can be electrically incompatible (reverse polarity damages equipment) and non-hermaphroditic connectors can be mechanically incompatible with each other (won't mate). Many pieces of amateur radio equipment run on 12-volt DC automotive voltage, which is also called 13.8-volt DC. The voltage delivered by a lead–acid battery with six-cells used as an automotive battery will vary depending on various electrical loads in a vehicle. Without loads the battery will float from 11.7 to 12.8 volts, and while charging from an alternator the voltage will increase to 13.8–14.4 volts DC. For use in amateur radio, the community has adopted a standard color code, polarity, and specific physical arrangement for assembling pairs of Powerpole connectors. One red and one black powerpole housing can be physically arranged in 4 topologically different mechanical orientations (red left, red right, red top, red bottom - when viewed from contact side with tongue up), 2 of which are mechanically incompatible (connectors won't mate with ARES) and 1 is electrically incompatible (will mate but reverse polarity) with the ARES standard; there are also additional unusual configurations in which one housing is rotated 90 degrees.