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In mathematics, more specifically in algebraic geometry, Parshin's conjecture (also referred to as the Beilinson–Parshin conjecture) states that for any smooth projective variety X defined over a finite field, the higher algebraic K-groups vanish up to torsion: K i ( X ) ⊗ Q = 0 , i > 0. {\displaystyle K_{i}(X)\otimes ... | https://en.wikipedia.org/wiki/Parshin's_conjecture |
In mathematics, more specifically in algebraic geometry, the Griffiths group of a projective complex manifold X measures the difference between homological equivalence and algebraic equivalence, which are two important equivalence relations of algebraic cycles. More precisely, it is defined as Griff k ( X ) := Z k ( ... | https://en.wikipedia.org/wiki/Griffiths_group |
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently from the method chosen for constructing them. For example, the definitions of ... | https://en.wikipedia.org/wiki/Universal_morphism |
In particular, the concept of universal property allows a simple proof that all constructions of real numbers are equivalent: it suffices to prove that they satisfy the same universal property. Technically, a universal property is defined in terms of categories and functors by means of a universal morphism (see § Forma... | https://en.wikipedia.org/wiki/Universal_morphism |
Universal properties occur almost everywhere in mathematics, and the use of the concept allows the use of general properties of universal properties for easily proving some properties that would need boring verifications otherwise. For example, given a commutative ring R, the field of fractions of the quotient ring of ... | https://en.wikipedia.org/wiki/Universal_morphism |
In mathematics, more specifically in category theory, internal categories are a generalisation of the notion of small category, and are defined with respect to a fixed ambient category. If the ambient category is taken to be the category of sets then one recovers the theory of small categories. In general, internal cat... | https://en.wikipedia.org/wiki/Internal_category |
In mathematics, more specifically in chromatic homotopy theory, the redshift conjecture states, roughly, that algebraic K-theory K ( R ) {\displaystyle K(R)} has chromatic level one higher than that of a complex-oriented ring spectrum R. It was formulated by John Rognes in a lecture at Schloss Ringberg, Germany, in Jan... | https://en.wikipedia.org/wiki/Redshift_conjecture |
In mathematics, more specifically in computational algebra, a straight-line program (SLP) for a finite group G = ⟨S⟩ is a finite sequence L of elements of G such that every element of L either belongs to S, is the inverse of a preceding element, or the product of two preceding elements. An SLP L is said to compute a gr... | https://en.wikipedia.org/wiki/Straight-line_program |
Babai and Szemerédi prove that every element of a finite group G has an SLP of length O(log2|G|) in every generating set. An efficient solution to the constructive membership problem is crucial to many group-theoretic algorithms. It can be stated in terms of SLPs as follows. | https://en.wikipedia.org/wiki/Straight-line_program |
Given a finite group G = ⟨S⟩ and g ∈ G, find a straight-line program computing g over S. The constructive membership problem is often studied in the setting of black box groups. The elements are encoded by bit strings of a fixed length. Three oracles are provided for the group-theoretic functions of multiplication, inv... | https://en.wikipedia.org/wiki/Straight-line_program |
A black box algorithm is one which uses only these oracles. Hence, straight-line programs for black box groups are black box algorithms. Explicit straight-line programs are given for a wealth of finite simple groups in the online ATLAS of Finite Groups. | https://en.wikipedia.org/wiki/Straight-line_program |
In mathematics, more specifically in differential geometry and geometric topology, the Milnor–Wood inequality is an obstruction to endow circle bundles over surfaces with a flat structure. It is named after John Milnor and John W. Wood. | https://en.wikipedia.org/wiki/Milnor–Wood_inequality |
In mathematics, more specifically in differential geometry and topology, various types of functions between manifolds are studied, both as objects in their own right and for the light they shed | https://en.wikipedia.org/wiki/Maps_of_manifolds |
In mathematics, more specifically in dynamical systems, the method of averaging (also called averaging theory) exploits systems containing time-scales separation: a fast oscillation versus a slow drift. It suggests that we perform an averaging over a given amount of time in order to iron out the fast oscillations and o... | https://en.wikipedia.org/wiki/Method_of_averaging |
It turns out to be a customary problem where there exists the trade off between how good is the approximated solution balanced by how much time it holds to be close to the original solution. More precisely, the system has the following form of a phase space variable x . {\displaystyle x.} | https://en.wikipedia.org/wiki/Method_of_averaging |
The fast oscillation is given by f {\displaystyle f} versus a slow drift of x ˙ {\displaystyle {\dot {x}}} . The averaging method yields an autonomous dynamical system which approximates the solution curves of x ˙ {\displaystyle {\dot {x}}} inside a connected and compact region of the phase space and over time of 1 / ε... | https://en.wikipedia.org/wiki/Method_of_averaging |
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always c... | https://en.wikipedia.org/wiki/Banach_Spaces |
In mathematics, more specifically in functional analysis, a K-space is an F-space V {\displaystyle V} such that every extension of F-spaces (or twisted sum) of the form is equivalent to the trivial one where R {\displaystyle \mathbb {R} } is the real line. | https://en.wikipedia.org/wiki/K-space_(functional_analysis) |
In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space ( V , ≤ ) {\displaystyle (V,\leq )} is a linear functional f {\displaystyle f} on V {\displaystyle V} so that for all positive elements v ∈ V , {\displaystyle v\in V,} that is v ≥ 0 , {\displaystyle v\geq 0... | https://en.wikipedia.org/wiki/Positive_linear_functional |
As in the case when V {\displaystyle V} is a C*-algebra with its partially ordered subspace of self-adjoint elements, sometimes a partial order is placed on only a subspace W ⊆ V , {\displaystyle W\subseteq V,} and the partial order does not extend to all of V , {\displaystyle V,} in which case the positive elements of... | https://en.wikipedia.org/wiki/Positive_linear_functional |
In mathematics, more specifically in functional analysis, a positive linear operator from an preordered vector space ( X , ≤ ) {\displaystyle (X,\leq )} into a preordered vector space ( Y , ≤ ) {\displaystyle (Y,\leq )} is a linear operator f {\displaystyle f} on X {\displaystyle X} into Y {\displaystyle Y} such that f... | https://en.wikipedia.org/wiki/Positive_linear_operator |
In mathematics, more specifically in functional analysis, a subset T {\displaystyle T} of a topological vector space X {\displaystyle X} is said to be a total subset of X {\displaystyle X} if the linear span of T {\displaystyle T} is a dense subset of X . {\displaystyle X.} This condition arises frequently in many theo... | https://en.wikipedia.org/wiki/Total_subset |
In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function whose domain is the natural numbers. The codomain of this function is usually some topological space. The motivation for general... | https://en.wikipedia.org/wiki/Moore–Smith_limit |
In particular, the following two conditions are, in general, not equivalent for a map f {\displaystyle f} between topological spaces X {\displaystyle X} and Y {\displaystyle Y}: The map f {\displaystyle f} is continuous in the topological sense; Given any point x {\displaystyle x} in X , {\displaystyle X,} and any sequ... | https://en.wikipedia.org/wiki/Moore–Smith_limit |
The concept of a net, first introduced by E. H. Moore and Herman L. Smith in 1922, is to generalize the notion of a sequence so that the above conditions (with "sequence" being replaced by "net" in condition 2) are in fact equivalent for all maps of topological spaces. In particular, rather than being defined on a coun... | https://en.wikipedia.org/wiki/Moore–Smith_limit |
Therefore, while sequences do not encode sufficient information about functions between topological spaces, nets do, because collections of open sets in topological spaces are much like directed sets in behavior. The term "net" was coined by John L. Kelley.Nets are one of the many tools used in topology to generalize c... | https://en.wikipedia.org/wiki/Moore–Smith_limit |
In mathematics, more specifically in general topology, the Tychonoff cube is the generalization of the unit cube from the product of a finite number of unit intervals to the product of an infinite, even uncountable number of unit intervals. The Tychonoff cube is named after Andrey Tychonoff, who first considered the ar... | https://en.wikipedia.org/wiki/Tychonoff_cube |
In mathematics, more specifically in geometric topology, the Kirby–Siebenmann class is an obstruction for topological manifolds to allow a PL-structure. | https://en.wikipedia.org/wiki/Kirby–Siebenmann_class |
In mathematics, more specifically in group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no non-trivial abelian quotients (equivalently, its abelianization, which is the universal abelian quotient, is trivial). In symbols, a perfect group is one such t... | https://en.wikipedia.org/wiki/Grün's_lemma |
In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information about the representation in a more condensed form. Georg Frobenius initially ... | https://en.wikipedia.org/wiki/Orthogonality_relations |
This is possible because a complex representation of a finite group is determined (up to isomorphism) by its character. The situation with representations over a field of positive characteristic, so-called "modular representations", is more delicate, but Richard Brauer developed a powerful theory of characters in this ... | https://en.wikipedia.org/wiki/Orthogonality_relations |
In mathematics, more specifically in harmonic analysis, Walsh functions form a complete orthogonal set of functions that can be used to represent any discrete function—just like trigonometric functions can be used to represent any continuous function in Fourier analysis. They can thus be viewed as a discrete, digital c... | https://en.wikipedia.org/wiki/Walsh_function |
The system of Walsh functions is known as the Walsh system. It is an extension of the Rademacher system of orthogonal functions.Walsh functions, the Walsh system, the Walsh series, and the fast Walsh–Hadamard transform are all named after the American mathematician Joseph L. Walsh. They find various applications in phy... | https://en.wikipedia.org/wiki/Walsh_function |
In mathematics, more specifically in homotopy theory, a simplicial presheaf is a presheaf on a site (e.g., the category of topological spaces) taking values in simplicial sets (i.e., a contravariant functor from the site to the category of simplicial sets). Equivalently, a simplicial presheaf is a simplicial object in ... | https://en.wikipedia.org/wiki/Homotopy_sheaf |
Thus, a simplicial scheme, a simplicial object in the site, represents a simplicial presheaf (in fact, often a simplicial sheaf). Example: Let G be a presheaf of groupoids. Then taking nerves section-wise, one obtains a simplicial presheaf B G {\displaystyle BG} . | https://en.wikipedia.org/wiki/Homotopy_sheaf |
For example, one might set B GL = lim → B G L n {\displaystyle B\operatorname {GL} =\varinjlim B\operatorname {GL_{n}} } . These types of examples appear in K-theory. If f: X → Y {\displaystyle f:X\to Y} is a local weak equivalence of simplicial presheaves, then the induced map Z f: Z X → Z Y {\displaystyle \mathbb {... | https://en.wikipedia.org/wiki/Homotopy_sheaf |
In mathematics, more specifically in linear algebra, the spark of a m × n {\displaystyle m\times n} matrix A {\displaystyle A} is the smallest integer k {\displaystyle k} such that there exists a set of k {\displaystyle k} columns in A {\displaystyle A} which are linearly dependent. If all the columns are linearly inde... | https://en.wikipedia.org/wiki/Spark_(mathematics) |
In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series. It is named after the French mathematician Augustin-Louis Cauchy. | https://en.wikipedia.org/wiki/Cesaro's_theorem |
In mathematics, more specifically in measure theory, the Baire sets form a σ-algebra of a topological space that avoids some of the pathological properties of Borel sets. There are several inequivalent definitions of Baire sets, but in the most widely used, the Baire sets of a locally compact Hausdorff space form the s... | https://en.wikipedia.org/wiki/Baire_set |
Every Baire set is a Borel set. The converse holds in many, but not all, topological spaces. Baire sets avoid some pathological properties of Borel sets on spaces without a countable base for the topology. In practice, the use of Baire measures on Baire sets can often be replaced by the use of regular Borel measures on... | https://en.wikipedia.org/wiki/Baire_set |
In mathematics, more specifically in multilinear algebra, an alternating multilinear map is a multilinear map with all arguments belonging to the same vector space (for example, a bilinear form or a multilinear form) that is zero whenever any pair of arguments is equal. More generally, the vector space may be a module ... | https://en.wikipedia.org/wiki/Alternating_multilinear_map |
In mathematics, more specifically in numerical linear algebra, the biconjugate gradient method is an algorithm to solve systems of linear equations A x = b . {\displaystyle Ax=b.\,} Unlike the conjugate gradient method, this algorithm does not require the matrix A {\displaystyle A} to be self-adjoint, but instead one n... | https://en.wikipedia.org/wiki/Biconjugate_gradient_method |
In mathematics, more specifically in point-set topology, the derived set of a subset S {\displaystyle S} of a topological space is the set of all limit points of S . {\displaystyle S.} It is usually denoted by S ′ . {\displaystyle S'.} The concept was first introduced by Georg Cantor in 1872 and he developed set theory... | https://en.wikipedia.org/wiki/Bendixson_derivative |
In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of integers. This generalized Euclidean algorithm can be put to many of the same uses a... | https://en.wikipedia.org/wiki/Euclidean_ring |
It is important to compare the class of Euclidean domains with the larger class of principal ideal domains (PIDs). An arbitrary PID has much the same "structural properties" of a Euclidean domain (or, indeed, even of the ring of integers), but when an explicit algorithm for Euclidean division is known, one may use the ... | https://en.wikipedia.org/wiki/Euclidean_ring |
So, given an integral domain R, it is often very useful to know that R has a Euclidean function: in particular, this implies that R is a PID. However, if there is no "obvious" Euclidean function, then determining whether R is a PID is generally a much easier problem than determining whether it is a Euclidean domain. Eu... | https://en.wikipedia.org/wiki/Euclidean_ring |
In mathematics, more specifically in ring theory, a cyclic module or monogenous module is a module over a ring that is generated by one element. The concept is a generalization of the notion of a cyclic group, that is, an Abelian group (i.e. Z-module) that is generated by one element. | https://en.wikipedia.org/wiki/Cyclic_module |
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all proper ideals. In other words, I is a maximal ideal of a ring R if there are no other ideals contained between I and R. Maximal ideals are important because the quotients of rings by... | https://en.wikipedia.org/wiki/Maximal_submodule |
In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. Local algebra is the... | https://en.wikipedia.org/wiki/Krull_intersection_theorem |
In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element. Equivalently, a ring is Noetherian if it satisfies the ascending chain condition on ideals; that is, given any chain:... | https://en.wikipedia.org/wiki/Commutative_ring_theory |
The notion of a Noetherian ring is of fundamental importance in both commutative and noncommutative ring theory, due to the role it plays in simplifying the ideal structure of a ring. For instance, the ring of integers and the polynomial ring over a field are both Noetherian rings, and consequently, such theorems as th... | https://en.wikipedia.org/wiki/Commutative_ring_theory |
In mathematics, more specifically in the context of geometric quantization, quantization commutes with reduction states that the space of global sections of a line bundle L satisfying the quantization condition on the symplectic quotient of a compact symplectic manifold is the space of invariant sections of L. This was... | https://en.wikipedia.org/wiki/Quantization_commutes_with_reduction |
In mathematics, more specifically in the field of analytic number theory, a Landau–Siegel zero or simply Siegel zero (also known as exceptional zero), named after Edmund Landau and Carl Ludwig Siegel, is a type of potential counterexample to the generalized Riemann hypothesis, on the zeros of Dirichlet L-functions asso... | https://en.wikipedia.org/wiki/Siegel_zero |
In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup. | https://en.wikipedia.org/wiki/Solvable_groups |
In mathematics, more specifically in the field of ring theory, a ring has the invariant basis number (IBN) property if all finitely generated free left modules over R have a well-defined rank. In the case of fields, the IBN property becomes the statement that finite-dimensional vector spaces have a unique dimension. | https://en.wikipedia.org/wiki/Invariant_basis_number |
In mathematics, more specifically in the study of dynamical systems and differential equations, a Liénard equation is a second order differential equation, named after the French physicist Alfred-Marie Liénard. During the development of radio and vacuum tube technology, Liénard equations were intensely studied as they ... | https://en.wikipedia.org/wiki/Lienard_equation |
In mathematics, more specifically in the theory of Lie algebras, the Poincaré–Birkhoff–Witt theorem (or PBW theorem) is a result giving an explicit description of the universal enveloping algebra of a Lie algebra. It is named after Henri Poincaré, Garrett Birkhoff, and Ernst Witt. The terms PBW type theorem and PBW the... | https://en.wikipedia.org/wiki/PBW_theorem |
In mathematics, more specifically in the theory of Monte Carlo methods, variance reduction is a procedure used to increase the precision of the estimates obtained for a given simulation or computational effort. Every output random variable from the simulation is associated with a variance which limits the precision of ... | https://en.wikipedia.org/wiki/Variance_reduction |
The main ones are common random numbers, antithetic variates, control variates, importance sampling, stratified sampling, moment matching, conditional Monte Carlo and quasi random variables. For simulation with black-box models subset simulation and line sampling can also be used. Under these headings are a variety of ... | https://en.wikipedia.org/wiki/Variance_reduction |
In mathematics, more specifically in topological groups, an extension of topological groups, or a topological extension, is a short exact sequence 0 → H → ı X → π G → 0 {\displaystyle 0\to H{\stackrel {\imath }{\to }}X{\stackrel {\pi }{\to }}G\to 0} where H , X {\displaystyle H,X} and G {\displaystyle G} are topologica... | https://en.wikipedia.org/wiki/Extension_of_a_topological_group |
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f: X → Y {\displaystyle f:X\to Y} is open if for any open set U {\displaystyle U} in X , {\displaystyle X,} the image f ( U ) {\displaystyle f(U)} is open in Y . ... | https://en.wikipedia.org/wiki/Closed_map |
A map may be open, closed, both, or neither; in particular, an open map need not be closed and vice versa.Open and closed maps are not necessarily continuous. Further, continuity is independent of openness and closedness in the general case and a continuous function may have one, both, or neither property; this fact re... | https://en.wikipedia.org/wiki/Closed_map |
Recall that, by definition, a function f: X → Y {\displaystyle f:X\to Y} is continuous if the preimage of every open set of Y {\displaystyle Y} is open in X . {\displaystyle X.} (Equivalently, if the preimage of every closed set of Y {\displaystyle Y} is closed in X {\displaystyle X} ). Early study of open maps was pio... | https://en.wikipedia.org/wiki/Closed_map |
In mathematics, more specifically in topology, the Volodin space X {\displaystyle X} of a ring R is a subspace of the classifying space B G L ( R ) {\displaystyle BGL(R)} given by X = ⋃ n , σ B ( U n ( R ) σ ) {\displaystyle X=\bigcup _{n,\sigma }B(U_{n}(R)^{\sigma })} where U n ( R ) ⊂ G L n ( R ) {\displaystyle U_{n}... | https://en.wikipedia.org/wiki/Volodin_space |
In mathematics, more specifically in topology, the equivariant stable homotopy theory is a subfield of equivariant topology that studies a spectrum with group action instead of a space with group action, as in stable homotopy theory. The field has become more active recently because of its connection to algebraic K-the... | https://en.wikipedia.org/wiki/Equivariant_stable_homotopy_theory |
In mathematics, more specifically measure theory, there are various notions of the convergence of measures. For an intuitive general sense of what is meant by convergence of measures, consider a sequence of measures μn on a space, sharing a common collection of measurable sets. Such a sequence might represent an attemp... | https://en.wikipedia.org/wiki/Portmanteau_theorem |
In mathematics, more specifically non-commutative ring theory, modern algebra, and module theory, the Jacobson density theorem is a theorem concerning simple modules over a ring R.The theorem can be applied to show that any primitive ring can be viewed as a "dense" subring of the ring of linear transformations of a vec... | https://en.wikipedia.org/wiki/Jacobson_density_theorem |
In mathematics, more specifically point-set topology, a Moore space is a developable regular Hausdorff space. That is, a topological space X is a Moore space if the following conditions hold: Any two distinct points can be separated by neighbourhoods, and any closed set and any point in its complement can be separated ... | https://en.wikipedia.org/wiki/Moore_space_(topology) |
(X is a developable space. )Moore spaces are generally interesting in mathematics because they may be applied to prove interesting metrization theorems. The concept of a Moore space was formulated by R. L. Moore in the earlier part of the 20th century. | https://en.wikipedia.org/wiki/Moore_space_(topology) |
In mathematics, more specifically ring theory and the theory of nil ideals, Levitzky's theorem, named after Jacob Levitzki, states that in a right Noetherian ring, every nil one-sided ideal is necessarily nilpotent. Levitzky's theorem is one of the many results suggesting the veracity of the Köthe conjecture, and indee... | https://en.wikipedia.org/wiki/Levitzky's_theorem |
In mathematics, more specifically ring theory, an atomic domain or factorization domain is an integral domain in which every non-zero non-unit can be written in at least one way as a finite product of irreducible elements. Atomic domains are different from unique factorization domains in that this decomposition of an e... | https://en.wikipedia.org/wiki/Atomic_domain |
In mathematics, more specifically ring theory, an ideal I of a ring R is said to be a nilpotent ideal if there exists a natural number k such that I k = 0. By I k, it is meant the additive subgroup generated by the set of all products of k elements in I. Therefore, I is nilpotent if and only if there is a natural numbe... | https://en.wikipedia.org/wiki/Nilpotent_ideal |
In mathematics, more specifically ring theory, the Jacobson radical of a ring R {\displaystyle R} is the ideal consisting of those elements in R {\displaystyle R} that annihilate all simple right R {\displaystyle R} -modules. It happens that substituting "left" in place of "right" in the definition yields the same idea... | https://en.wikipedia.org/wiki/Jacobson_radical |
The Jacobson radical of a ring has numerous internal characterizations, including a few definitions that successfully extend the notion to rings without unity. The radical of a module extends the definition of the Jacobson radical to include modules. The Jacobson radical plays a prominent role in many ring and module t... | https://en.wikipedia.org/wiki/Jacobson_radical |
In mathematics, more specifically sheaf theory, a branch of topology and algebraic geometry, the exceptional inverse image functor is the fourth and most sophisticated in a series of image functors for sheaves. It is needed to express Verdier duality in its most general form. | https://en.wikipedia.org/wiki/Exceptional_inverse_image_functor |
In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure. If f: X → Y {\displaystyle f:X\to Y} is a local homeomorphism, X {\displaystyle X} is said to be an étale space over Y . {\displaystyle... | https://en.wikipedia.org/wiki/Local_homeomorphism |
Typical examples of local homeomorphisms are covering maps. A topological space X {\displaystyle X} is locally homeomorphic to Y {\displaystyle Y} if every point of X {\displaystyle X} has a neighborhood that is homeomorphic to an open subset of Y . {\displaystyle Y.} | https://en.wikipedia.org/wiki/Local_homeomorphism |
For example, a manifold of dimension n {\displaystyle n} is locally homeomorphic to R n . {\displaystyle \mathbb {R} ^{n}.} If there is a local homeomorphism from X {\displaystyle X} to Y , {\displaystyle Y,} then X {\displaystyle X} is locally homeomorphic to Y , {\displaystyle Y,} but the converse is not always true.... | https://en.wikipedia.org/wiki/Local_homeomorphism |
In mathematics, more specifically, in convex geometry, the mixed volume is a way to associate a non-negative number to a tuple of convex bodies in R n {\displaystyle \mathbb {R} ^{n}} . This number depends on the size and shape of the bodies, and their relative orientation to each other. | https://en.wikipedia.org/wiki/Mixed_volume |
In mathematics, motivic L-functions are a generalization of Hasse–Weil L-functions to general motives over global fields. The local L-factor at a finite place v is similarly given by the characteristic polynomial of a Frobenius element at v acting on the v-inertial invariants of the v-adic realization of the motive. Fo... | https://en.wikipedia.org/wiki/Motivic_L-function |
In mathematics, multipliers and centralizers are algebraic objects in the study of Banach spaces. They are used, for example, in generalizations of the Banach–Stone theorem. | https://en.wikipedia.org/wiki/Multipliers_and_centralizers_(Banach_spaces) |
In mathematics, near sets are either spatially close or descriptively close. Spatially close sets have nonempty intersection. In other words, spatially close sets are not disjoint sets, since they always have at least one element in common. Descriptively close sets contain elements that have matching descriptions. | https://en.wikipedia.org/wiki/Near_sets |
Such sets can be either disjoint or non-disjoint sets. Spatially near sets are also descriptively near sets. | https://en.wikipedia.org/wiki/Near_sets |
The underlying assumption with descriptively close sets is that such sets contain elements that have location and measurable features such as colour and frequency of occurrence. The description of the element of a set is defined by a feature vector. Comparison of feature vectors provides a basis for measuring the close... | https://en.wikipedia.org/wiki/Near_sets |
Near set theory provides a formal basis for the observation, comparison, and classification of elements in sets based on their closeness, either spatially or descriptively. Near sets offer a framework for solving problems based on human perception that arise in areas such as image processing, computer vision as well as... | https://en.wikipedia.org/wiki/Near_sets |
From the beginning, descriptively near sets have proved to be useful in applications of topology, and visual pattern recognition , spanning a broad spectrum of applications that include camouflage detection, micropaleontology, handwriting forgery detection, biomedical image analysis, content-based image retrieval, popu... | https://en.wikipedia.org/wiki/Near_sets |
In mathematics, negacyclic convolution is a convolution between two vectors a and b. It is also called skew circular convolution or wrapped convolution. It results from multiplication of a skew circulant matrix, generated by vector a, with vector b. | https://en.wikipedia.org/wiki/Negacyclic_convolution |
In mathematics, negafibonacci coding is a universal code which encodes nonzero integers into binary code words. It is similar to Fibonacci coding, except that it allows both positive and negative integers to be represented. All codes end with "11" and have no "11" before the end. | https://en.wikipedia.org/wiki/Negafibonacci_coding |
In mathematics, negative definiteness is a property of any object to which a bilinear form may be naturally associated, which is negative-definite. See, in particular: Negative-definite bilinear form Negative-definite quadratic form Negative-definite matrix Negative-definite function | https://en.wikipedia.org/wiki/Negative_definite |
In mathematics, nilpotent orbits are generalizations of nilpotent matrices that play an important role in representation theory of real and complex semisimple Lie groups and semisimple Lie algebras. | https://en.wikipedia.org/wiki/Nilpotent_orbit |
In mathematics, non-Archimedean geometry is any of a number of forms of geometry in which the axiom of Archimedes is negated. An example of such a geometry is the Dehn plane. Non-Archimedean geometries may, as the example indicates, have properties significantly different from Euclidean geometry. There are two senses i... | https://en.wikipedia.org/wiki/Non-Archimedean_geometry |
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative,... | https://en.wikipedia.org/wiki/Non-Euclidean_space |
In mathematics, non-abelian class field theory is a catchphrase, meaning the extension of the results of class field theory, the relatively complete and classical set of results on abelian extensions of any number field K, to the general Galois extension L/K. While class field theory was essentially known by 1930, the ... | https://en.wikipedia.org/wiki/Non-abelian_class_field_theory |
In mathematics, nonabelian algebraic topology studies an aspect of algebraic topology that involves (inevitably noncommutative) higher-dimensional algebras. Many of the higher-dimensional algebraic structures are noncommutative and, therefore, their study is a very significant part of nonabelian category theory, and al... | https://en.wikipedia.org/wiki/Nonabelian_algebraic_topology |
An important part of nonabelian algebraic topology is concerned with the properties and applications of homotopy groupoids and filtered spaces. Noncommutative double groupoids and double algebroids are only the first examples of such higher-dimensional structures that are nonabelian. The new methods of Nonabelian Algeb... | https://en.wikipedia.org/wiki/Nonabelian_algebraic_topology |
Cubical omega-groupoids, higher homotopy groupoids, crossed modules, crossed complexes and Galois groupoids are key concepts in developing applications related to homotopy of filtered spaces, higher-dimensional space structures, the construction of the fundamental groupoid of a topos E in the general theory of topoi, a... | https://en.wikipedia.org/wiki/Nonabelian_algebraic_topology |
A related example is that of van Kampen theorems for categories of covering morphisms in lextensive categories. Other reports of generalisations of the van Kampen theorem include statements for 2-categories and a topos of topoi . Important results in higher-dimensional algebra are also the extensions of the Galois theo... | https://en.wikipedia.org/wiki/Nonabelian_algebraic_topology |
In mathematics, noncommutative harmonic analysis is the field in which results from Fourier analysis are extended to topological groups that are not commutative. Since locally compact abelian groups have a well-understood theory, Pontryagin duality, which includes the basic structures of Fourier series and Fourier tran... | https://en.wikipedia.org/wiki/Noncommutative_harmonic_analysis |
The interesting examples include many Lie groups, and also algebraic groups over p-adic fields. These examples are of interest and frequently applied in mathematical physics, and contemporary number theory, particularly automorphic representations. What to expect is known as the result of basic work of John von Neumann... | https://en.wikipedia.org/wiki/Noncommutative_harmonic_analysis |
He showed that if the von Neumann group algebra of G is of type I, then L2(G) as a unitary representation of G is a direct integral of irreducible representations. It is parametrized therefore by the unitary dual, the set of isomorphism classes of such representations, which is given the hull-kernel topology. The analo... | https://en.wikipedia.org/wiki/Noncommutative_harmonic_analysis |
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