SoftWear's Sewbots rely on two things: high-speed, high-resolution cameras able to monitor the movement of individual threads in a piece of cloth, and software that takes those movements and generalises them to describe the distortion and orientation of the fabric which the threads in question are part of.
|
|
This generalises, but in some sense with loss of explicit information (as is typical of several complex variables).
|
|
This generalises the definition based on the scaling property of the volume with distance. The best definition depends on the application.
|
|
The theory of rough paths generalises the Young integral and Young differential equations and makes heavy use of the concept of p-variation.
|
|
This generalises the product rule for matrices. Further generalizations of the product rule have been demonstrated for appropriate products of hypermatrices of boundary format.
|
|
In mathematics, especially in higher-dimensional algebra and homotopy theory, a double groupoid generalises the notion of groupoid and of category to a higher dimension.
|
|
In applied probability, a dynamic contagion process is a point process with stochastic intensity that generalises the Hawkes process and Cox process with exponentially decaying shot noise intensity.
|
|
In mathematics, an ordered exponential field is an ordered field together with a function which generalises the idea of exponential functions on the ordered field of real numbers.
|
|
In mathematics, in particular the study of abstract algebra, a Dedekind–Hasse norm is a function on an integral domain that generalises the notion of a Euclidean function on Euclidean domains.
|
|
In mathematics, in the field of algebraic number theory, an S-unit generalises the idea of unit of the ring of integers of the field. Many of the results which hold for units are also valid for S-units.
|
|
19, No. 4, 1918, pp. 251–278.Vladimir P. Kostov, A mapping defined by the Schur–Szegő composition, Comptes Rendus Acad. Bulg. Sci. tome 63, No. 7, 2010, pp. 943–952. In the 1970s Askold Khovanskii developed the theory of fewnomials that generalises Descartes' rule.
|
|
In mathematics, the Tutte homotopy theorem, introduced by , generalises the concept of "path" from graphs to matroids, and states roughly that closed paths can be written as compositions of elementary closed paths, so that in some sense they are homotopic to the trivial closed path.
|
|
In an infinite-dimensional space V, as used in functional analysis, the flag idea generalises to a subspace nest, namely a collection of subspaces of V that is a total order for inclusion and which further is closed under arbitrary intersections and closed linear spans. See nest algebra.
|
|
Triangle-free planar graphs are word- representable . A K4-free near-triangulation is 3-colourable if and only if it is word-representable M. Glen. Colourability and word-representability of near-triangulations, Pure and Applied Mathematics, to appear, 2019.; this result generalises studies in P. Akrobotu, S. Kitaev, and Z. Masárová.
|
|
Grakn is a distributed knowledge graph for knowledge oriented system, i.e. a knowledge base. Under the hood, Grakn has built an expressive knowledge representation system with a transactional query interface. Grakn’s knowledge representation system is based on hypergraph theory, a subfield in mathematics that generalises an edge to be a set of vertices.
|
|
The multigrid reduction in time method (MGRIT) generalises the interpretation of Parareal as a multigrid-in-time algorithms to multiple levels using different smoothers. It is a more general approach but for a specific choice of parameters it is equivalent to Parareal. The XBraid library implementing MGRIT is being developed by Lawrence Livermore National Laboratory.
|
|
The General Polygon Clipper (GPC) is a software library providing for computing the results of clipping operations on sets of polygons. It generalises the computer graphics clipping problem of intersecting polygons with polygons. The first release of GPC was designed and implemented in 1997 by Alan Murta. the current GPC release was version 2.32.
|
|
Under the first operator (+) it forms an abelian group. Under the second operator (×) it is associative, but it does not need to have an identity, or inverse, so division is not required. The additive (+) identity element is written as 0 and the additive inverse of a is written as −a. Distributivity generalises the distributive law for numbers.
|
|
In mathematics, the Iwasawa decomposition (aka KAN from its expression) of a semisimple Lie group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix (a consequence of Gram–Schmidt orthogonalization). It is named after Kenkichi Iwasawa, the Japanese mathematician who developed this method.
|
|
MAX-3SAT is a problem in the computational complexity subfield of computer science. It generalises the Boolean satisfiability problem (SAT) which is a decision problem considered in complexity theory. It is defined as: Given a 3-CNF formula Φ (i.e. with at most 3 variables per clause), find an assignment that satisfies the largest number of clauses.
|
|
Gödel has a module system, and it supports arbitrary precision integers, arbitrary precision rationals, and also floating-point numbers. It can solve constraints over finite domains of integers and also linear rational constraints. It supports processing of finite sets. It also has a flexible computation rule and a pruning operator which generalises the commit of the concurrent logic programming languages.
|
|
On Rn or Cn, the closed sets of the Zariski topology are the solution sets of systems of polynomial equations. A linear graph has a natural topology that generalises many of the geometric aspects of graphs with vertices and edges. The Sierpiński space is the simplest non-discrete topological space. It has important relations to the theory of computation and semantics.
|
|
The concept of binary function generalises to ternary (or 3-ary) function, quaternary (or 4-ary) function, or more generally to n-ary function for any natural number n. A 0-ary function to Z is simply given by an element of Z. One can also define an A-ary function where A is any set; there is one input for each element of A.
|
|
This work simultaneously generalises and simplifies recent work of several authors, and it already has many applications. In one application, he develops a new theory of groups acting on dendrites. Building on previous contributions of Gilbert Levitt, G. Ananda Swarup and others, this led him to a solution of the 'cut-point conjecture'. This recent work also yields a characterisation of word- hyperbolic groups as convergence groups.
|
|
The dimension of a graph is written: \dim G. For example, the Petersen graph can be drawn with unit edges in E^2, but not in E^1: its dimension is therefore 2 (see the figure to the right). This concept was introduced in 1965 by Paul Erdős, Frank Harary and William Tutte. It generalises the concept of unit distance graph to more than 2 dimensions.
|
|
In fact, Danijela Kambasković-Sawers generalises, "The sonnet sequence genre constructs a double sense of immediacy: drawing on the lyricism of its constituent sonnets, it also often generates a perception of a personal narrative when the sequence is read from beginning to end".Kambasković-Sawers, Danijela. "Three themes in one, which wondrous scope affords: Ambiguous Speaker and Storytelling in Shakespeare's Sonnets." Criticism 49.3 (2007): 285-305.
|
|
In mathematics, an n-group, or n-dimensional higher group, is a special kind of n-category that generalises the concept of group to higher-dimensional algebra. Here, n may be any natural number or infinity. The thesis of Alexander Grothendieck's student Hoàng Xuân Sính was an in-depth study of 2-groups under the moniker 'gr-category'. The general definition of n-group is a matter of ongoing research.
|
|
The equilibria of Fig. 12 are not points at which curves are true tangents to each other. They do however have a property which generalises the definition in terms of tangents, which is that the two curves can be locally separated by a straight line. Arror and Debreu defined equilibrium in the same way as each other in their (independent) papers of 1951 without providing any source or rationale for their definition.
|
|
In abstract algebra, the idea of an inverse element generalises the concepts of negation (sign reversal) (in relation to addition) and reciprocation (in relation to multiplication). The intuition is of an element that can 'undo' the effect of combination with another given element. While the precise definition of an inverse element varies depending on the algebraic structure involved, these definitions coincide in a group. The word 'inverse' is derived from that means 'turned upside down', 'overturned'.
|
|
A magic hyperbeam (n-dimensional magic rectangle) is a variation on a magic hypercube where the orders along each direction may be different. As such a magic hyperbeam generalises the two dimensional magic rectangle and the three dimensional magic beam, a series that mimics the series magic square, magic cube and magic hypercube. This article will mimic the magic hypercubes article in close detail, and just as that article serves merely as an introduction to the topic.
|
|
Thomson went on to prove Stokes' theorem, which earned that name after Stokes asked students to prove in on a test in 1854. Stokes learned it from Thomson in a letter in 1850. Stokes' theorem generalises Green's theorem, which itself is a higher-dimensional version of the Fundamental Theorem of Calculus. Arthur Cayley is credited with the creation of the theory of matrices—rectangular arrays of numbers—as distinct objects from determinants, studied since the mid-eighteenth century.
|
|
Dinur has stated at Eurocrypt 2009 that Cube generalises and improves upon AIDA. However, Vielhaber contends that the cube attack is no more than his attack under another name. It is, however, acknowledged by all parties involved that Cube's use of an efficient linearity test such as the BLR test results in the new attack needing less time than AIDA, although how substantial this particular change is remains in dispute. It is not the only way in which Cube and AIDA differ.
|
|
The Lebesgue integral is deficient in one respect. The Riemann integral generalises to the improper Riemann integral to measure functions whose domain of definition is not a closed interval. The Lebesgue integral integrates many of these functions (always reproducing the same answer when it did), but not all of them. For functions on the real line, the Henstock integral is an even more general notion of integral (based on Riemann's theory rather than Lebesgue's) that subsumes both Lebesgue integration and improper Riemann integration.
|
|
There is a close relationship, assuming everything is smooth, with vector fields: given a vector field on that is never zero, its integral curves will give a 1-dimensional foliation. (i.e. a codimension foliation). This observation generalises to the Frobenius theorem, saying that the necessary and sufficient conditions for a distribution (i.e. an dimensional subbundle of the tangent bundle of a manifold) to be tangent to the leaves of a foliation, is that the set of vector fields tangent to the distribution are closed under Lie bracket.
|
|
Einstein's general theory of relativity places space and time on equal footing, so that one considers the geometry of a unified spacetime instead of considering space and time separately. The cases of spacetime of constant curvature are de Sitter space (positive), Minkowski space (zero), and anti-de Sitter space (negative). As such, they are exact solutions of Einstein's field equations for an empty universe with a positive, zero, or negative cosmological constant, respectively. Anti-de Sitter space generalises to any number of space dimensions.
|
|
If is a subset of a Euclidean space, then is an interior point of if there exists an open ball centered at which is completely contained in . (This is illustrated in the introductory section to this article.) This definition generalizes to any subset of a metric space with metric : is an interior point of if there exists , such that is in whenever the distance . This definition generalises to topological spaces by replacing "open ball" with "open set". Let be a subset of a topological space .
|
|
The complete row of Pascal's triangle for the hypercube in this construction runs 1 (single vertex), 4 (tetrahedron tetrad), 6 (hexany), 4 (another tetrad), 1. The idea generalises to other numbers of dimensions, for instance, the cross-sections of a five-dimensional cube give two versions of the dekany, a ten-note scale rich in tetrads, triads and dyads, which also contains many hexanies. In six dimensions the same construction gives the twenty-note eikosany, which is even richer in chords. It has pentads, tetrads, and triads as well as hexanies and dekanies.
|
|
" EDL members perceive the burqa and niqab as being intimidating towards non-Muslims, degrading to women, and facilitating the concealment of criminals and terrorists. The EDL believed Muslims wanted to impose sharia law on Britain and Western society. The group generalises sharia as a uniform set of rules, ignoring the fact that it represents a diverse and often contradictory range of approaches to Islamic jurisprudence. In opposing sharia, which it regards as inherently misogynistic, the EDL positions itself as the defenders of women, employing the slogan "EDL Angels [i.e.
|
|
Portable Arbitrary Map (PAM) is an extension of the older binary P4...P6 graphics formats. PAM generalises all features of PBM, PGM and PPM, and provides for extensions. PAM defines two new attributes; depth and tuple type: #The depth attribute defines the number of channels in the image, such as 1 for greyscale images and 3 for RGB images. #The tuple type attribute specifies what kind of image the PAM file represents, thus enabling it to stand for the older Netpbm formats, as well as to be extended to new uses, e.g.
|
|
Just as one can generalise a vector bundle to the notion of a Higgs bundle, it is possible to formulate a definition of a principal G-Higgs bundle. The above definition of stability for principal bundles generalises to these objects by requiring the reductions of structure group are compatible with the Higgs field of the principal Higgs bundle. It was shown by Anchouche and Biswas that the analogue of the nonabelian Hodge correspondence for Higgs vector bundles is true for principal G-Higgs bundles in the case where the base manifold (X,\omega) is a complex projective variety.
|
|
Two greedy colorings of the same crown graph using different vertex orders. The right example generalises to 2-colorable graphs with vertices, where the greedy algorithm expends colors. In the study of graph coloring problems in mathematics and computer science, a greedy coloring or sequential coloring is a coloring of the vertices of a graph formed by a greedy algorithm that considers the vertices of the graph in sequence and assigns each vertex its first available color. Greedy colorings can be found in linear time, but they do not in general use the minimum number of colors possible.
|
|
A figure illustrating the vehicle routing problem The vehicle routing problem (VRP) is a combinatorial optimization and integer programming problem which asks "What is the optimal set of routes for a fleet of vehicles to traverse in order to deliver to a given set of customers?". It generalises the well-known travelling salesman problem (TSP). It first appeared in a paper by George Dantzig and John Ramser in 1959, in which the first algorithmic approach was written and was applied to petrol deliveries. Often, the context is that of delivering goods located at a central depot to customers who have placed orders for such goods.
|
|
Few results are known for the general G/G/k model as it generalises the M/G/k queue for which few metrics are known. Bounds can be computed using mean value analysis techniques, adapting results from the M/M/c queue model, using heavy traffic approximations, empirical results or approximating distributions by phase type distributions and then using matrix analytic methods to solve the approximate systems. In a G/G/2 queue with heavy-tailed job sizes, the tail of the delay time distribution is known to behave like the tail of an exponential distribution squared under low loads and like the tail of an exponential distribution for high loads.
|
|
In mathematics, specifically in algebraic geometry, the Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is itself a generalisation of the classical Riemann–Roch theorem for line bundles on compact Riemann surfaces. Riemann–Roch type theorems relate Euler characteristics of the cohomology of a vector bundle with their topological degrees, or more generally their characteristic classes in (co)homology or algebraic analogues thereof. The classical Riemann–Roch theorem does this for curves and line bundles, whereas the Hirzebruch–Riemann–Roch theorem generalises this to vector bundles over manifolds.
|
|
The notion of ideal generalises to any Mal'cev algebra (as linear subspace in the case of vector spaces, normal subgroup in the case of groups, two-sided ideals in the case of rings, and submodule in the case of modules). It turns out that ker f is not a subalgebra of A, but it is an ideal. Then it makes sense to speak of the quotient algebra G/(ker f). The first isomorphism theorem for Mal'cev algebras states that this quotient algebra is naturally isomorphic to the image of f (which is a subalgebra of B). The connection between this and the congruence relation for more general types of algebras is as follows.
|
|
The first reason that can be cited for that is that it did not provide fresh information on the splitting of prime ideals in a Galois extension; a common way to explain the objective of a non-abelian class field theory is that it should provide a more explicit way to express such patterns of splitting.On the statistical level, the classical result on primes in arithmetic progressions of Dirichlet generalises to Chebotaryov's density theorem; what is asked for is a generalisation, of the same scope of quadratic reciprocity. The cohomological approach therefore was of limited use in even formulating non-abelian class field theory. Behind the history was the wish of Chevalley to write proofs for class field theory without using Dirichlet series: in other words to eliminate L-functions.
|
|
Lebesgue integration has the property that every function defined over a bounded interval with a Riemann integral also has a Lebesgue integral, and for those functions the two integrals agree. Furthermore, every bounded function on a closed bounded interval has a Lebesgue integral and there are many functions with a Lebesgue integral that have no Riemann integral. As part of the development of Lebesgue integration, Lebesgue invented the concept of measure, which extends the idea of length from intervals to a very large class of sets, called measurable sets (so, more precisely, simple functions are functions that take a finite number of values, and each value is taken on a measurable set). Lebesgue's technique for turning a measure into an integral generalises easily to many other situations, leading to the modern field of measure theory.
|
|
The computation of the homology groups of the product of two topological spaces involves the tensor product; but only in the simplest cases, such as a torus, is it directly calculated in that fashion (see Künneth theorem). The topological phenomena were subtle enough to need better foundational concepts; technically speaking, the Tor functors had to be defined. The material to organise was quite extensive, including also ideas going back to Hermann Grassmann, the ideas from the theory of differential forms that had led to de Rham cohomology, as well as more elementary ideas such as the wedge product that generalises the cross product. The resulting rather severe write-up of the topic (by Bourbaki) entirely rejected one approach in vector calculus (the quaternion route, that is, in the general case, the relation with Lie groups).
|
|
Access routes to La Pique d'Endron summit via the créneau d'Endron Le Pas de l'Échelle, route de Gavarnie (Hautes Pyrénées) Pyreneism, in this meaning, is distinct from alpinism only by the mountain range in which it is practised. Difficulty pyreneism was not born in the 20th century. Its father is certainly Henri Brulle who, as early as 1878, generalises the use of lifeline and short ice pick during his ascents. With Bazillac, de Monts, d'Astorg, led by guides Célestin Passet and François Bernat-Salles, he achieves many firsts, the north face of Monte Perdido, le corridor of Gaube at the Vignemale, ... Undeniably the pyreneist enterprise, the adventure, the attraction of the unknown and of the conquest of first order summits, the exploration of new massifs, shrunk as time passed.
|
|
While there is some debate regarding whether the "Standard Interpretation" is that described by Turing or, instead, based on a misreading of his paper, these three versions are not regarded as equivalent, and their strengths and weaknesses are distinct. Huma Shah points out that Turing himself was concerned with whether a machine could think and was providing a simple method to examine this: through human-machine question-answer sessions. Shah argues there is one imitation game which Turing described could be practicalised in two different ways: a) one-to-one interrogator-machine test, and b) simultaneous comparison of a machine with a human, both questioned in parallel by an interrogator. Since the Turing test is a test of indistinguishability in performance capacity, the verbal version generalises naturally to all of human performance capacity, verbal as well as nonverbal (robotic).
|
|
For one variable, the local form of an analytic function f(z) near 0 is zkh(z) where h(0) is not 0, and k is the order of the zero of f at 0. This is the result that the preparation theorem generalises. We pick out one variable z, which we may assume is first, and write our complex variables as (z, z2, ..., zn). A Weierstrass polynomial W(z) is :zk \+ gk−1zk−1 \+ ... + g0 where gi(z2, ..., zn) is analytic and gi(0, ..., 0) = 0. Then the theorem states that for analytic functions f, if :f(0, ...,0) = 0, and :f(z, z2, ..., zn) as a power series has some term only involving z, we can write (locally near (0, ..., 0)) :f(z, z2, ..., zn) = W(z)h(z, z2, ..., zn) with h analytic and h(0, ..., 0) not 0, and W a Weierstrass polynomial.
|
|
For nilpotent groups the theory simplifies much from the general case, and stays similar to the case of Abelian groups. All lattices in a nilpotent Lie group are uniform, and if N is a connected simply connected nilpotent Lie group (equivalently it does not contain a nontrivial compact subgroup) then a discrete subgroup is a lattice if and only if it is not contained in a proper connected subgroup (this generalises the fact that a discrete subgroup in a vector space is a lattice if and only if it spans the vector space). A nilpotent Lie group contains a lattice if and only if it can be defined over the rationals, that is if and only if its structure constants are rational numbers. More precisely, in a nilpotent group satisfying this condition lattices correspond via the exponential map to lattices (in the more elementary sense of Lattice (group)) in the Lie algebra.
|
|
Busemann functions can be used to determine special visual metrics on the class of CAT(-1) spaces. These are complete geodesic metric spaces in which the distances between points on the boundary of a geodesic triangle are less than or equal to the comparison triangle in the hyperbolic upper half plane or equivalently the unit disk with the Poincaré metric. In the case of the unit disk the chordal metric can be recovered directly using Busemann functions Bγ and the special theory for the disk generalises completely to any proper CAT(-1) space X. The hyperbolic upper half plane is a CAT(0) space, as lengths in a hyperbolic geodesic triangle are less than lengths in the Euclidean comparison triangle: in particular a CAT(-1) space is a CAT(0) space, so the theory of Busemann functions and the Gromov boundary applies. From the theory of the hyperbolic disk, it follows in particular that every geodesic ray in a CAT(-1) space extends to a geodesic line and given two points of the boundary there is a unique geodesic γ such that has these points as the limits γ(±∞).
|
|