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wiki_4700_chunk_0 | Complemented group | In mathematics, in the realm of group theory, the term complemented group is used in two distinct, but similar ways. In (Hall 1937), a complemented group is one in which every subgroup has a group-theoretic complement. Such groups are called completely factorizable groups in the Russian literature, following (Baeva 195... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4701_chunk_0 | Complemented group | In (Costantini & Zacher 2004) it is shown that every finite simple group is a complemented group. Note that in the classification of finite simple groups, K-group is more used to mean a group whose proper subgroups only have composition factors amongst the known finite simple groups. An example of a group that is not c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4702_chunk_0 | Observable subgroup | In mathematics, in the representation theory of algebraic groups, an observable subgroup is an algebraic subgroup of a linear algebraic group whose every finite-dimensional rational representation arises as the restriction to the subgroup of a finite-dimensional rational representation of the whole group. An equivalent... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4703_chunk_0 | Limit cycle | In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such behavior is exhibited in some no... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4704_chunk_0 | Hartman–Grobman theorem | In mathematics, in the study of dynamical systems, the Hartman–Grobman theorem or linearisation theorem is a theorem about the local behaviour of dynamical systems in the neighbourhood of a hyperbolic equilibrium point. It asserts that linearisation—a natural simplification of the system—is effective in predicting qual... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4705_chunk_0 | Hutchinson operator | In mathematics, in the study of fractals, a Hutchinson operator is the collective action of a set of contractions, called an iterated function system. The iteration of the operator converges to a unique attractor, which is the often self-similar fixed set of the operator. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4706_chunk_0 | Periodic mapping | In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4707_chunk_0 | Hopf algebroid | In mathematics, in the theory of Hopf algebras, a Hopf algebroid is a generalisation of weak Hopf algebras, certain skew Hopf algebras and commutative Hopf k-algebroids. If k is a field, a commutative k-algebroid is a cogroupoid object in the category of k-algebras; the category of such is hence dual to the category of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4708_chunk_0 | Hopf algebroid | The generalization of Hopf algebroids and its main part of the structure, associative bialgebroids, to the noncommutative base algebra was introduced by J.-H. Lu in 1996 as a result on work on groupoids in Poisson geometry (later shown equivalent in nontrivial way to a construction of Takeuchi from the 1970s and anothe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4709_chunk_0 | Linear stability | In mathematics, in the theory of differential equations and dynamical systems, a particular stationary or quasistationary solution to a nonlinear system is called linearly unstable if the linearization of the equation at this solution has the form d r / d t = A r {\displaystyle dr/dt=Ar} , where r is the perturbation t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4710_chunk_0 | Superrigidity theorem | In mathematics, in the theory of discrete groups, superrigidity is a concept designed to show how a linear representation ρ of a discrete group Γ inside an algebraic group G can, under some circumstances, be as good as a representation of G itself. That this phenomenon happens for certain broadly defined classes of lat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4711_chunk_0 | Superrigidity theorem | One simplified statement is this: take G to be a simply connected semisimple real algebraic group in GLn, such that the Lie group of its real points has real rank at least 2 and no compact factors. Suppose Γ is an irreducible lattice in G. For a local field F and ρ a linear representation of the lattice Γ of the Lie gr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4712_chunk_0 | Brauer tree | In mathematics, in the theory of finite groups, a Brauer tree is a tree that encodes the characters of a block with cyclic defect group of a finite group. In fact, the trees encode the group algebra up to Morita equivalence. Such algebras coming from Brauer trees are called Brauer tree algebras. Feit (1984) described t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4713_chunk_0 | Domain of holomorphy | In mathematics, in the theory of functions of several complex variables, a domain of holomorphy is a domain which is maximal in the sense that there exists a holomorphic function on this domain which cannot be extended to a bigger domain. Formally, an open set Ω {\displaystyle \Omega } in the n-dimensional complex spac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4714_chunk_0 | Lax form | In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that satisfy a corresponding differential equation, called the Lax equation. Lax pairs were introduced by Peter Lax to discuss solitons in continuous media. The inverse scattering transform makes use of the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4715_chunk_0 | Regular singular points | In mathematics, in the theory of ordinary differential equations in the complex plane C {\displaystyle \mathbb {C} } , the points of C {\displaystyle \mathbb {C} } are classified into ordinary points, at which the equation's coefficients are analytic functions, and singular points, at which some coefficient has a singu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4716_chunk_0 | Newman's lemma | In mathematics, in the theory of rewriting systems, Newman's lemma, also commonly called the diamond lemma, states that a terminating (or strongly normalizing) abstract rewriting system (ARS), that is, one in which there are no infinite reduction sequences, is confluent if it is locally confluent. In fact a terminating... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4717_chunk_0 | Levi problem | In mathematics, in the theory of several complex variables and complex manifolds, a Stein manifold is a complex submanifold of the vector space of n complex dimensions. They were introduced by and named after Karl Stein (1951). A Stein space is similar to a Stein manifold but is allowed to have singularities. Stein spa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4718_chunk_0 | Loop theorem | In mathematics, in the topology of 3-manifolds, the loop theorem is a generalization of Dehn's lemma. The loop theorem was first proven by Christos Papakyriakopoulos in 1956, along with Dehn's lemma and the Sphere theorem. A simple and useful version of the loop theorem states that if for some 3-dimensional manifold M ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4719_chunk_0 | Loop theorem | Let N ⊂ π 1 ( S ) {\displaystyle N\subset \pi _{1}(S)} be a normal subgroup such that k e r ( π 1 ( S ) → π 1 ( M ) ) − N ≠ ∅ {\displaystyle \mathop {\mathrm {ker} } (\pi _{1}(S)\to \pi _{1}(M))-N\neq \emptyset } . Let f: D 2 → M {\displaystyle f\colon D^{2}\to M} be a continuous map such that f ( ∂ D 2 ) ⊂ S {\displ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4720_chunk_0 | Loop theorem | Then there exists an embedding g: D 2 → M {\displaystyle g\colon D^{2}\to M} such that g ( ∂ D 2 ) ⊂ S {\displaystyle g(\partial D^{2})\subset S} and ∉ N . {\displaystyle \notin N.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4721_chunk_0 | Loop theorem | Furthermore if one starts with a map f in general position, then for any neighborhood U of the singularity set of f, we can find such a g with image lying inside the union of image of f and U. Stalling's proof utilizes an adaptation, due to Whitehead and Shapiro, of Papakyriakopoulos' "tower construction". The "tower" ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4722_chunk_0 | Loop theorem | There is also a version of Dehn's lemma for minimal discs due to Meeks and S.-T. Yau, which also crucially relies on the tower construction. A proof not utilizing the tower construction exists of the first version of the loop theorem. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4723_chunk_0 | Loop theorem | This was essentially done 30 years ago by Friedhelm Waldhausen as part of his solution to the word problem for Haken manifolds; although he recognized this gave a proof of the loop theorem, he did not write up a detailed proof. The essential ingredient of this proof is the concept of Haken hierarchy. Proofs were later ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4724_chunk_0 | Sphere theorem (3-manifolds) | In mathematics, in the topology of 3-manifolds, the sphere theorem of Christos Papakyriakopoulos (1957) gives conditions for elements of the second homotopy group of a 3-manifold to be represented by embedded spheres. One example is the following: Let M {\displaystyle M} be an orientable 3-manifold such that π 2 ( M ) ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4725_chunk_0 | Sphere theorem (3-manifolds) | Another more general version (also called the projective plane theorem, and due to David B. A. Epstein) is: Let M {\displaystyle M} be any 3-manifold and N {\displaystyle N} a π 1 ( M ) {\displaystyle \pi _{1}(M)} -invariant subgroup of π 2 ( M ) {\displaystyle \pi _{2}(M)} . If f: S 2 → M {\displaystyle f\colon S^{2}\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4726_chunk_0 | Incidence geometry | In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An incidence structure is what is obtained when all other concepts are removed and all... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4727_chunk_0 | Incidence geometry | Such fundamental results remain valid when additional concepts are added to form a richer geometry. It sometimes happens that authors blur the distinction between a study and the objects of that study, so it is not surprising to find that some authors refer to incidence structures as incidence geometries.Incidence stru... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4728_chunk_0 | Incidence geometry | In graph theory they are called hypergraphs, and in combinatorial design theory they are called block designs. Besides the difference in terminology, each area approaches the subject differently and is interested in questions about these objects relevant to that discipline. Using geometric language, as is done in incid... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4729_chunk_0 | Incidence geometry | It is, however, possible to translate the results from one discipline into the terminology of another, but this often leads to awkward and convoluted statements that do not appear to be natural outgrowths of the topics. In the examples selected for this article we use only those with a natural geometric flavor. A speci... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4730_chunk_0 | Arrow notation (Ramsey theory) | In mathematics, infinitary combinatorics, or combinatorial set theory, is an extension of ideas in combinatorics to infinite sets. Some of the things studied include continuous graphs and trees, extensions of Ramsey's theorem, and Martin's axiom. Recent developments concern combinatorics of the continuum and combinator... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4731_chunk_0 | Infinite compositions of analytic functions | In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the convergence/divergence of these expansions. Some functions can actually ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4732_chunk_0 | Infinite compositions of analytic functions | For infinite compositions of a single function see Iterated function. For compositions of a finite number of functions, useful in fractal theory, see Iterated function system. Although the title of this article specifies analytic functions, there are results for more general functions of a complex variable as well. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4733_chunk_0 | Infinite-dimensional holomorphy | In mathematics, infinite-dimensional holomorphy is a branch of functional analysis. It is concerned with generalizations of the concept of holomorphic function to functions defined and taking values in complex Banach spaces (or Fréchet spaces more generally), typically of infinite dimension. It is one aspect of nonline... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4734_chunk_0 | Lemma (mathematics) | In mathematics, informal logic and argument mapping, a lemma (PL: lemmas or lemmata) is a generally minor, proven proposition which is used as a stepping stone to a larger result. For that reason, it is also known as a "helping theorem" or an "auxiliary theorem". In many cases, a lemma derives its importance from the t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4735_chunk_0 | Euclid's orchard | In mathematics, informally speaking, Euclid's orchard is an array of one-dimensional "trees" of unit height planted at the lattice points in one quadrant of a square lattice. More formally, Euclid's orchard is the set of line segments from (x, y, 0) to (x, y, 1), where x and y are positive integers. The trees visible f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4736_chunk_0 | Euclid's orchard | If the orchard is projected relative to the origin onto the plane x + y = 1 (or, equivalently, drawn in perspective from a viewpoint at the origin) the tops of the trees form a graph of Thomae's function. The point (x, y, 1) projects to ( x x + y , y x + y , 1 x + y ) . {\displaystyle \left({\frac {x}{x+y}},{\frac {y}{... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4737_chunk_0 | Bijection, injection and surjection | In mathematics, injections, surjections, and bijections are classes of functions distinguished by the manner in which arguments (input expressions from the domain) and images (output expressions from the codomain) are related or mapped to each other. A function maps elements from its domain to elements in its codomain.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4738_chunk_0 | Bijection, injection and surjection | Notationally: ∀ x , x ′ ∈ X , f ( x ) = f ( x ′ ) ⟹ x = x ′ , {\displaystyle \forall x,x'\in X,f(x)=f(x')\implies x=x',} or, equivalently (using logical transposition), ∀ x , x ′ ∈ X , x ≠ x ′ ⟹ f ( x ) ≠ f ( x ′ ) . {\displaystyle \forall x,x'\in X,x\neq x'\implies f(x)\neq f(x').} The function is surjective, or onto,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4739_chunk_0 | Bijection, injection and surjection | That is, the image and the codomain of the function are equal. A surjective function is a surjection. Notationally: ∀ y ∈ Y , ∃ x ∈ X such that y = f ( x ) . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4740_chunk_0 | Bijection, injection and surjection | {\displaystyle \forall y\in Y,\exists x\in X{\text{ such that }}y=f(x).} The function is bijective (one-to-one and onto, one-to-one correspondence, or invertible) if each element of the codomain is mapped to by exactly one element of the domain. That is, the function is both injective and surjective. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4741_chunk_0 | Bijection, injection and surjection | A bijective function is also called a bijection. That is, combining the definitions of injective and surjective, ∀ y ∈ Y , ∃ ! x ∈ X such that y = f ( x ) , {\displaystyle \forall y\in Y,\exists !x\in X{\text{ such that }}y=f(x),} where ∃ ! | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4742_chunk_0 | Bijection, injection and surjection | x {\displaystyle \exists !x} means "there exists exactly one x".In any case (for any function), the following holds: ∀ x ∈ X , ∃ ! y ∈ Y such that y = f ( x ) . {\displaystyle \forall x\in X,\exists !y\in Y{\text{ such that }}y=f(x).} An injective function need not be surjective (not all elements of the codomain may be... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4743_chunk_0 | Completely integrable system | In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals that its motion is confined to a submanifold of much smaller dime... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4744_chunk_0 | Completely integrable system | Other elementary examples include the motion of a rigid body about its center of mass (the Euler top) and the motion of an axially symmetric rigid body about a point in its axis of symmetry (the Lagrange top). In the late 1960's, it was realized that there are completely integrable systems in physics having an infinite... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4745_chunk_0 | Completely integrable system | In the special case of Hamiltonian systems, if there are enough independent Poisson commuting first integrals for the flow parameters to be able to serve as a coordinate system on the invariant level sets (the leaves of the Lagrangian foliation), and if the flows are complete and the energy level set is compact, this i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4746_chunk_0 | Singular integral equations | In mathematics, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: where I i ( u ) {\displaystyle I^{i}(u)} is an integral operator acting on u. Hence, integral equations may be viewed as ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4747_chunk_0 | Singular integral equations | For example, one method of solving a boundary value problem is by converting the differential equation with its boundary conditions into an integral equation and solving the integral equation. In addition, because one can convert between the two, differential equations in physics such as Maxwell's equations often have ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4748_chunk_0 | Integral geometry | In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant (or equivariant) transformations from the space of functions on one geometrical space to the space of f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4749_chunk_0 | Integral of inverse functions | In mathematics, integrals of inverse functions can be computed by means of a formula that expresses the antiderivatives of the inverse f − 1 {\displaystyle f^{-1}} of a continuous and invertible function f {\displaystyle f} , in terms of f − 1 {\displaystyle f^{-1}} and an antiderivative of f {\displaystyle f} . This f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4750_chunk_0 | Unlikely intersections | In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem on curves and elimination theory. On the other hand, the topological theory m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4751_chunk_0 | Mathematical anti-realism | In mathematics, intuitionism is a program of methodological reform whose motto is that "there are no non-experienced mathematical truths" (L. E. J. Brouwer). From this springboard, intuitionists seek to reconstruct what they consider to be the corrigible portion of mathematics in accordance with Kantian concepts of bei... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4752_chunk_0 | Mathematical anti-realism | The axiom of choice is also rejected in most intuitionistic set theories, though in some versions it is accepted. In intuitionism, the term "explicit construction" is not cleanly defined, and that has led to criticisms. Attempts have been made to use the concepts of Turing machine or computable function to fill this ga... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4753_chunk_0 | Factorization system | In mathematics, it can be shown that every function can be written as the composite of a surjective function followed by an injective function. Factorization systems are a generalization of this situation in category theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4754_chunk_0 | Comparison operator | In mathematics, it is common practice to chain relational operators, such as in 3 < x < y < 20 (meaning 3 < x and x < y and y < 20). The syntax is clear since these relational operators in mathematics are transitive. However, many recent programming languages would see an expression like 3 < x < y as consisting of two ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4755_chunk_0 | Comparison operator | However, it does compile in C/C++ and some other languages, yielding surprising result (as true would be represented by the number 1 here). It is possible to give the expression x < y < z its familiar mathematical meaning, and some programming languages such as Python and Raku do that. Others, such as C# and Java, do n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4756_chunk_0 | Iterated forcing | In mathematics, iterated forcing is a method for constructing models of set theory by repeating Cohen's forcing method a transfinite number of times. Iterated forcing was introduced by Solovay and Tennenbaum (1971) in their construction of a model of set theory with no Suslin tree. They also showed that iterated forcin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4757_chunk_0 | Iterated forcing | If α+1 is a successor ordinal then Pα+1 is often constructed from Pα using a forcing notion in VPα, while if α is a limit ordinal then Pα is often constructed as some sort of limit (such as the direct limit) of the Pβ for β<α. A key consideration is that, typically, it is necessary that ω 1 {\displaystyle \omega _{1}} ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4758_chunk_0 | Iterated forcing | This is often accomplished by the use of a preservation theorem such as: Finite support iteration of c.c.c. forcings (see countable chain condition) are c.c.c. and thus preserve ω 1 {\displaystyle \omega _{1}} . Countable support iterations of proper forcings are proper (see Fundamental Theorem of Proper Forcing) and t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4759_chunk_0 | Iterated Function Systems | In mathematics, iterated function systems (IFSs) are a method of constructing fractals; the resulting fractals are often self-similar. IFS fractals are more related to set theory than fractal geometry. They were introduced in 1981. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4760_chunk_0 | Iterated Function Systems | IFS fractals, as they are normally called, can be of any number of dimensions, but are commonly computed and drawn in 2D. The fractal is made up of the union of several copies of itself, each copy being transformed by a function (hence "function system"). The canonical example is the Sierpiński triangle. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4761_chunk_0 | Iterated Function Systems | The functions are normally contractive, which means they bring points closer together and make shapes smaller. Hence, the shape of an IFS fractal is made up of several possibly-overlapping smaller copies of itself, each of which is also made up of copies of itself, ad infinitum. This is the source of its self-similar f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4762_chunk_0 | Iteration | In mathematics, iteration may refer to the process of iterating a function, i.e. applying a function repeatedly, using the output from one iteration as the input to the next. Iteration of apparently simple functions can produce complex behaviors and difficult problems – for examples, see the Collatz conjecture and jugg... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4763_chunk_0 | Hessian equation | In mathematics, k-Hessian equations (or Hessian equations for short) are partial differential equations (PDEs) based on the Hessian matrix. More specifically, a Hessian equation is the k-trace, or the kth elementary symmetric polynomial of eigenvalues of the Hessian matrix. When k ≥ 2, the k-Hessian equation is a fully... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4764_chunk_0 | Least-squares function approximation | In mathematics, least squares function approximation applies the principle of least squares to function approximation, by means of a weighted sum of other functions. The best approximation can be defined as that which minimizes the difference between the original function and the approximation; for a least-squares appr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4765_chunk_0 | Combining like terms | In mathematics, like terms are summands in a sum that differ only by a numerical factor. Like terms can be regrouped by adding their coefficients. Typically, in a polynomial expression, like terms are those that contain the same variables to the same powers, possibly with different coefficients. More generally, when so... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4766_chunk_0 | Combining like terms | For example, when considering a quadratic equation, one considers often the expression ( x − r ) ( x − s ) , {\displaystyle (x-r)(x-s),} where r {\displaystyle r} and s {\displaystyle s} are the roots of the equation and may be considered as parameters. Then, expanding the above product and regrouping the like terms gi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4767_chunk_0 | Weak limit cardinal | In mathematics, limit cardinals are certain cardinal numbers. A cardinal number λ is a weak limit cardinal if λ is neither a successor cardinal nor zero. This means that one cannot "reach" λ from another cardinal by repeated successor operations. These cardinals are sometimes called simply "limit cardinals" when the co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4768_chunk_0 | Discontinuous linear functional | In mathematics, linear maps form an important class of "simple" functions which preserve the algebraic structure of linear spaces and are often used as approximations to more general functions (see linear approximation). If the spaces involved are also topological spaces (that is, topological vector spaces), then it ma... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4769_chunk_0 | Linearization | In mathematics, linearization is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium poin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4770_chunk_0 | Local class field theory | In mathematics, local class field theory, introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite residue field: hence every local field is isomorphic (as a topological fie... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4771_chunk_0 | Log-polar coordinates | In mathematics, log-polar coordinates (or logarithmic polar coordinates) is a coordinate system in two dimensions, where a point is identified by two numbers, one for the logarithm of the distance to a certain point, and one for an angle. Log-polar coordinates are closely connected to polar coordinates, which are usual... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4772_chunk_0 | Logarithmic Sobolev inequalities | In mathematics, logarithmic Sobolev inequalities are a class of inequalities involving the norm of a function f, its logarithm, and its gradient ∇ f {\displaystyle \nabla f} . These inequalities were discovered and named by Leonard Gross, who established them in dimension-independent form, in the context of constructiv... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4773_chunk_0 | Logarithmic growth | In mathematics, logarithmic growth describes a phenomenon whose size or cost can be described as a logarithm function of some input. e.g. y = C log (x). Any logarithm base can be used, since one can be converted to another by multiplying by a fixed constant. Logarithmic growth is the inverse of exponential growth and i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4774_chunk_0 | Logarithmic growth | In more advanced mathematics, the partial sums of the harmonic series 1 + 1 2 + 1 3 + 1 4 + 1 5 + ⋯ {\displaystyle 1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+\cdots } grow logarithmically. In the design of computer algorithms, logarithmic growth, and related variants, such as log-linear, or linearith... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4775_chunk_0 | Logarithmic growth | Petersburg paradox.In microbiology, the rapidly growing exponential growth phase of a cell culture is sometimes called logarithmic growth. During this bacterial growth phase, the number of new cells appearing is proportional to the population. This terminological confusion between logarithmic growth and exponential gro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4776_chunk_0 | Recursive language | In mathematics, logic and computer science, a formal language (a set of finite sequences of symbols taken from a fixed alphabet) is called recursive if it is a recursive subset of the set of all possible finite sequences over the alphabet of the language. Equivalently, a formal language is recursive if there exists a T... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4777_chunk_0 | Recursive language | Recursive languages are also called decidable. The concept of decidability may be extended to other models of computation. For example, one may speak of languages decidable on a non-deterministic Turing machine. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4778_chunk_0 | Recursive language | Therefore, whenever an ambiguity is possible, the synonym used for "recursive language" is Turing-decidable language, rather than simply decidable. The class of all recursive languages is often called R, although this name is also used for the class RP. This type of language was not defined in the Chomsky hierarchy of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4779_chunk_0 | Recognizable language | In mathematics, logic and computer science, a formal language is called recursively enumerable (also recognizable, partially decidable, semidecidable, Turing-acceptable or Turing-recognizable) if it is a recursively enumerable subset in the set of all possible words over the alphabet of the language, i.e., if there exi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4780_chunk_0 | Impredicativity | The opposite of impredicativity is predicativity, which essentially entails building stratified (or ramified) theories where quantification over lower levels results in variables of some new type, distinguished from the lower types that the variable ranges over. A prototypical example is intuitionistic type theory, whi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4781_chunk_0 | System of types | In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general, type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a foundation of mathematics. Two influential type theories that were proposed as f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4782_chunk_0 | Undefined term | Formal theories cannot dispense with primitive notions, under pain of infinite regress (per the regress problem). For example, in contemporary geometry, point, line, and contains are some primitive notions. Instead of attempting to define them, their interplay is ruled (in Hilbert's axiom system) by axioms like "For ev... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4783_chunk_0 | Loop algebra | In mathematics, loop algebras are certain types of Lie algebras, of particular interest in theoretical physics. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4784_chunk_0 | Uniform spanning tree | In mathematics, loop-erased random walk is a model for a random simple path with important applications in combinatorics, physics and quantum field theory. It is intimately connected to the uniform spanning tree, a model for a random tree. See also random walk for more general treatment of this topic. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4785_chunk_0 | Low-rank approximation | In mathematics, low-rank approximation is a minimization problem, in which the cost function measures the fit between a given matrix (the data) and an approximating matrix (the optimization variable), subject to a constraint that the approximating matrix has reduced rank. The problem is used for mathematical modeling a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4786_chunk_0 | Majorization | The order of the entries of the vectors x {\displaystyle \mathbf {x} } or y {\displaystyle \mathbf {y} } does not affect the majorization, e.g., the statement ( 1 , 2 ) ≺ ( 0 , 3 ) {\displaystyle (1,2)\prec (0,3)} is simply equivalent to ( 2 , 1 ) ≺ ( 3 , 0 ) {\displaystyle (2,1)\prec (3,0)} . As a consequence, majoriz... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4787_chunk_0 | Majorization | For example, a wealth distribution is Lorenz-greater than another if its Lorenz curve lies below the other. As such, a Lorenz-greater wealth distribution has a higher Gini coefficient, and has more income disparity. Various other generalizations of majorization are discussed in chapters 14 and 15 of. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4788_chunk_0 | Zero-based numbering | In mathematics, many sequences of numbers or of polynomials are indexed by nonnegative integers, for example, the Bernoulli numbers and the Bell numbers. In both mechanics and statistics, the zeroth moment is defined, representing total mass in the case of physical density, or total probability, i.e. one, for a probabi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4789_chunk_0 | Mathematical economics | In mathematics, mathematical optimization (or optimization or mathematical programming) refers to the selection of a best element from some set of available alternatives. In the simplest case, an optimization problem involves maximizing or minimizing a real function by selecting input values of the function and computi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4790_chunk_0 | Mathematical economics | More generally, optimization includes finding the best available element of some function given a defined domain and may use a variety of different computational optimization techniques.Economics is closely enough linked to optimization by agents in an economy that an influential definition relatedly describes economic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4791_chunk_0 | Mathematical economics | Theory posits that consumers maximize their utility, subject to their budget constraints and that firms maximize their profits, subject to their production functions, input costs, and market demand.Economic equilibrium is studied in optimization theory as a key ingredient of economic theorems that in principle could be... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4792_chunk_0 | Mathematical economics | Many others may be sufficiently complex to require numerical methods of solution, aided by software. Still others are complex but tractable enough to allow computable methods of solution, in particular computable general equilibrium models for the entire economy.Linear and nonlinear programming have profoundly affected... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4793_chunk_0 | Harmonic function | In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U → R , {\displaystyle f:U\to \mathbb {R} ,} where U is an open subset of R n , {\displaystyle \mathbb {R} ^{n},} that satisfies Laplace's equation, that is, ∂ 2 f ∂ x 1 2 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4794_chunk_0 | Spin current | In mathematics, mathematical physics, and theoretical physics, the spin tensor is a quantity used to describe the rotational motion of particles in spacetime. The spin tensor has application in general relativity and special relativity, as well as quantum mechanics, relativistic quantum mechanics, and quantum field the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4795_chunk_0 | Named set theory | In mathematics, mathematical structures can have more than one definition. Therefore, there are several definitions of named sets, each representing a specific construction of named set theory. The informal definition is the most general. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4796_chunk_0 | Matrix addition | In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. For a vector, v → {\displaystyle {\vec {v}}\!} , adding two matrices would have the geometric effect of applying each matrix transformation separately onto v → {\displaystyle {\vec {v}}\!} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4797_chunk_0 | Matrix addition | , then adding the transformed vectors. A v → + B v → = ( A + B ) v → {\displaystyle \mathbf {A} {\vec {v}}+\mathbf {B} {\vec {v}}=(\mathbf {A} +\mathbf {B} ){\vec {v}}\!} However, there are other operations that could also be considered addition for matrices, such as the direct sum and the Kronecker sum. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4798_chunk_0 | Matrix differentiation | In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_4799_chunk_0 | Matrix differentiation | Two competing notational conventions split the field of matrix calculus into two separate groups. The two groups can be distinguished by whether they write the derivative of a scalar with respect to a vector as a column vector or a row vector. Both of these conventions are possible even when the common assumption is ma... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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