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c_gv73525r0wzh | In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of... | Fractal geometry |
c_zg045y4btc9h | One way that fractals are different from finite geometric figures is how they scale. Doubling the edge lengths of a filled polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). Likewise, if the radiu... | Fractal geometry |
c_t67gu1dezfk2 | However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer and is in general greater than its conventional dimension. This power is called the fractal dimension of the geometric object, to distinguish it from the conventional ... | Fractal geometry |
c_q7q8tttl474h | Starting in the 17th century with notions of recursion, fractals have moved through increasingly rigorous mathematical treatment to the study of continuous but not differentiable functions in the 19th century by the seminal work of Bernard Bolzano, Bernhard Riemann, and Karl Weierstrass, and on to the coining of the wo... | Fractal geometry |
c_mvgl9mr4p9p1 | That's fractals." More formally, in 1982 Mandelbrot defined fractal as follows: "A fractal is by definition a set for which the Hausdorff–Besicovitch dimension strictly exceeds the topological dimension." Later, seeing this as too restrictive, he simplified and expanded the definition to this: "A fractal is a rough or ... | Fractal geometry |
c_7x4mukhcia8c | Still later, Mandelbrot proposed "to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants".The consensus among mathematicians is that theoretical fractals are infinitely self-similar iterated and detailed mathematical constructs, of which many examples hav... | Fractal geometry |
c_jb5xqcn1n6es | In mathematics, a fractal is a geometrical shape that exhibits invariance under scaling. A piece of the whole, if enlarged, has the same geometrical features as the entire object itself. A fractal ambigram is a sort of space-filling ambigrams where the tiled word branches from itself and then shrinks in a self-similar ... | Ambigram |
c_mqvkgq5avbrb | In mathematics, a fractal sequence is one that contains itself as a proper subsequence. An example is 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, ...If the first occurrence of each n is deleted, the remaining sequence is identical to the original. The process can be repeated indefinitely, so that act... | Fractal sequence |
c_pjo6w1fii481 | In mathematics, a frame bundle is a principal fiber bundle F(E) associated to any vector bundle E. The fiber of F(E) over a point x is the set of all ordered bases, or frames, for Ex. The general linear group acts naturally on F(E) via a change of basis, giving the frame bundle the structure of a principal GL(k, R)-bun... | Unitary frame bundle |
c_odbulgoa4ive | In mathematics, a free Boolean algebra is a Boolean algebra with a distinguished set of elements, called generators, such that: Each element of the Boolean algebra can be expressed as a finite combination of generators, using the Boolean operations, and The generators are as independent as possible, in the sense that t... | Free Boolean algebra |
c_byuwh2g44ilk | In mathematics, a free Lie algebra over a field K is a Lie algebra generated by a set X, without any imposed relations other than the defining relations of alternating K-bilinearity and the Jacobi identity. | Free Lie algebra |
c_xafihn11xkpb | In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subset such that every element of the group can be uniquely expressed as an integ... | Free abelian group |
c_1tsufv9iby8y | Free abelian groups have properties which make them similar to vector spaces, and may equivalently be called free Z {\displaystyle \mathbb {Z} } -modules, the free modules over the integers. Lattice theory studies free abelian subgroups of real vector spaces. | Free abelian group |
c_zz4rq8g22d59 | In algebraic topology, free abelian groups are used to define chain groups, and in algebraic geometry they are used to define divisors. The elements of a free abelian group with basis B {\displaystyle B} may be described in several equivalent ways. These include formal sums over B {\displaystyle B} , which are expressi... | Free abelian group |
c_0kvdz7mu3m9b | Alternatively, the elements of a free abelian group may be thought of as signed multisets containing finitely many elements of B {\displaystyle B} , with the multiplicity of an element in the multiset equal to its coefficient in the formal sum. Another way to represent an element of a free abelian group is as a functio... | Free abelian group |
c_atox0m3vmnkh | This group is unique in the sense that every two free abelian groups with the same basis are isomorphic. Instead of constructing it by describing its individual elements, a free abelian group with basis B {\displaystyle B} may be constructed as a direct sum of copies of the additive group of the integers, with one copy... | Free abelian group |
c_43kqf6duse8h | The rank of a free abelian group is the cardinality of a basis; every two bases for the same group give the same rank, and every two free abelian groups with the same rank are isomorphic. Every subgroup of a free abelian group is itself free abelian; this fact allows a general abelian group to be understood as a quotie... | Free abelian group |
c_9bdgsdm8oon7 | In mathematics, a free boundary problem (FB problem) is a partial differential equation to be solved for both an unknown function u {\displaystyle u} and an unknown domain Ω {\displaystyle \Omega } . The segment Γ {\displaystyle \Gamma } of the boundary of Ω {\displaystyle \Omega } which is not known at the outset of t... | Free boundary problem |
c_6zkoo7hgzb7u | An interesting aspect of such a criticality is the so-called sandpile dynamic (or Internal DLA). The most classical example is the melting of ice: Given a block of ice, one can solve the heat equation given appropriate initial and boundary conditions to determine its temperature. But, if in any region the temperature i... | Free boundary problem |
c_1ruiuw85lu7f | In mathematics, a free module is a module that has a basis, that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules. Given any set S an... | Free module |
c_677rof0ovbw4 | In mathematics, a frieze or frieze pattern is a two-dimensional design that repeats in one direction. Such patterns occur frequently in architecture and decorative art. Frieze patterns can be classified into seven types according to their symmetries. | Frieze pattern |
c_v7ysyl1oxj7k | The set of symmetries of a frieze pattern is called a frieze group. Frieze groups are two-dimensional line groups, having repetition in only one direction. They are related to the more complex wallpaper groups, which classify patterns that are repetitive in two directions, and crystallographic groups, which classify pa... | Frieze pattern |
c_gw80p8oiphsm | In mathematics, a full subcategory A of a category B is said to be reflective in B when the inclusion functor from A to B has a left adjoint. : 91 This adjoint is sometimes called a reflector, or localization. Dually, A is said to be coreflective in B when the inclusion functor has a right adjoint. Informally, a reflec... | Reflective subcategory |
c_u5l41jgx1rvb | In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism. | Proper map |
c_bvemgzrtkr5u | In mathematics, a function defined on a region of the complex plane is said to be of bounded type if it is equal to the ratio of two analytic functions bounded in that region. But more generally, a function is of bounded type in a region Ω {\displaystyle \Omega } if and only if f {\displaystyle f} is analytic on Ω {\di... | Bounded type (mathematics) |
c_95pf7yit4ow8 | In mathematics, a function defined on an inner product space is said to have rotational invariance if its value does not change when arbitrary rotations are applied to its argument. | Rotational invariance |
c_s7x5exiaqgtv | In mathematics, a function f defined on some set X with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number M such that | f ( x ) | ≤ M {\displaystyle |f(x)|\leq M} for all x in X. A function that is not bounded is said to be unbounded.If f is real-va... | Bounded sequences |
c_9pmjq9tpb8dy | In mathematics, a function f is cofunction of a function g if f(A) = g(B) whenever A and B are complementary angles. This definition typically applies to trigonometric functions. The prefix "co-" can be found already in Edmund Gunter's Canon triangulorum (1620).For example, sine (Latin: sinus) and cosine (Latin: cosinu... | Cofunction |
c_g6yeh4edewkg | In mathematics, a function f is logarithmically convex or superconvex if log ∘ f {\displaystyle {\log }\circ f} , the composition of the logarithm with f, is itself a convex function. | Logarithmically convex function |
c_tt7cfkjvqqae | In mathematics, a function f of n variables x1, ..., xn leads to a Chisini mean M if, for every vector ⟨x1, ..., xn⟩, there exists a unique M such that f(M,M, ..., M) = f(x1,x2, ..., xn).The arithmetic, harmonic, geometric, generalised, Heronian and quadratic means are all Chisini means, as are their weighted variants.... | Chisini mean |
c_lxq7i88imcyr | In mathematics, a function f on the interval has the Luzin N property, named after Nikolai Luzin (also called Luzin property or N property) if for all N ⊂ {\displaystyle N\subset } such that λ ( N ) = 0 {\displaystyle \lambda (N)=0} , there holds: λ ( f ( N ) ) = 0 {\displaystyle \lambda (f(N))=0} , where λ {\display... | Luzin N property |
c_ssza3hf4n8c2 | In mathematics, a function f {\displaystyle f} is superadditive if for all x {\displaystyle x} and y {\displaystyle y} in the domain of f . {\displaystyle f.} Similarly, a sequence a 1 , a 2 , … {\displaystyle a_{1},a_{2},\ldots } is called superadditive if it satisfies the inequality for all m {\displaystyle m} and n ... | Superadditivity |
c_17h0g83at6fn | In mathematics, a function f {\displaystyle f} is weakly harmonic in a domain D {\displaystyle D} if ∫ D f Δ g = 0 {\displaystyle \int _{D}f\,\Delta g=0} for all g {\displaystyle g} with compact support in D {\displaystyle D} and continuous second derivatives, where Δ is the Laplacian. This is the same notion as a weak... | Weakly harmonic |
c_k1sskj4zaz78 | In mathematics, a function f: R k → R {\displaystyle f\colon \mathbb {R} ^{k}\to \mathbb {R} } is supermodular if f ( x ↑ y ) + f ( x ↓ y ) ≥ f ( x ) + f ( y ) {\displaystyle f(x\uparrow y)+f(x\downarrow y)\geq f(x)+f(y)} for all x {\displaystyle x} , y ∈ R k {\displaystyle y\in \mathbb {R} ^{k}} , where x ↑ y {\displa... | Supermodular function |
c_djii5pomv5nf | In mathematics, a function f: R n → R {\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} } is said to be closed if for each α ∈ R {\displaystyle \alpha \in \mathbb {R} } , the sublevel set { x ∈ dom f | f ( x ) ≤ α } {\displaystyle \{x\in {\mbox{dom}}f\vert f(x)\leq \alpha \}} is a closed set. Equivalently, if th... | Closed convex function |
c_0upnoujmsdru | In mathematics, a function f: R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } is symmetrically continuous at a point x if lim h → 0 f ( x + h ) − f ( x − h ) = 0. {\displaystyle \lim _{h\to 0}f(x+h)-f(x-h)=0.} The usual definition of continuity implies symmetric continuity, but the converse is not true. | Symmetrically continuous function |
c_wwh9z7e0xnpe | For example, the function x − 2 {\displaystyle x^{-2}} is symmetrically continuous at x = 0 {\displaystyle x=0} , but not continuous. Also, symmetric differentiability implies symmetric continuity, but the converse is not true just like usual continuity does not imply differentiability. The set of the symmetrically con... | Symmetrically continuous function |
c_oo060ul8hnq2 | In mathematics, a function f: V → W {\displaystyle f:V\to W} between two complex vector spaces is said to be antilinear or conjugate-linear if hold for all vectors x , y ∈ V {\displaystyle x,y\in V} and every complex number s , {\displaystyle s,} where s ¯ {\displaystyle {\overline {s}}} denotes the complex conjugate o... | Antidual space |
c_n7l1xs3fc88p | If the vector spaces are real then antilinearity is the same as linearity. Antilinear maps occur in quantum mechanics in the study of time reversal and in spinor calculus, where it is customary to replace the bars over the basis vectors and the components of geometric objects by dots put above the indices. Scalar-value... | Antidual space |
c_n4549umpt6vc | In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function.Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the... | Function evaluation |
c_9cx0vlu9llzf | A function is most often denoted by letters such as f, g and h, and the value of a function f at an element x of its domain is denoted by f(x); the numerical value resulting from the function evaluation at a particular input value is denoted by replacing x with this value; for example, the value of f at x = 4 is denote... | Function evaluation |
c_vmyx2xou0ffj | (x+1)^{2}\right\vert _{x=4}} (which results in 25).A function is uniquely represented by the set of all pairs (x, f (x)), called the graph of the function, a popular means of illustrating the function. When the domain and the codomain are sets of real numbers, each such pair may be thought of as the Cartesian coordinat... | Function evaluation |
c_o0lv89hi1vke | In mathematics, a function is a rule for taking an input (in the simplest case, a number or set of numbers) and providing an output (which may also be a number). A symbol that stands for an arbitrary input is called an independent variable, while a symbol that stands for an arbitrary output is called a dependent variab... | Response variable |
c_3o24g1piztqz | In mathematics, a function is locally bounded if it is bounded around every point. A family of functions is locally bounded if for any point in their domain all the functions are bounded around that point and by the same number. | Locally bounded |
c_5o37zhgv2l16 | In mathematics, a function is said to vanish at infinity if its values approach 0 as the input grows without bounds. There are two different ways to define this with one definition applying to functions defined on normed vector spaces and the other applying to functions defined on locally compact spaces. Aside from thi... | Rapidly decreasing function |
c_0qtwe8k12ges | In mathematics, a function of a motor variable is a function with arguments and values in the split-complex number plane, much as functions of a complex variable involve ordinary complex numbers. William Kingdon Clifford coined the term motor for a kinematic operator in his "Preliminary Sketch of Biquaternions" (1873).... | Motor variable |
c_amabypc7g1zu | For example, f ( z ) = u ( z ) + j v ( z ) , z = x + j y , x , y ∈ R , j 2 = + 1 , u ( z ) , v ( z ) ∈ R . {\displaystyle f(z)=u(z)+j\ v(z),\ z=x+jy,\ x,y\in R,\quad j^{2}=+1,\quad u(z),v(z)\in R.} | Motor variable |
c_j9cq94zsm18o | Functions of a motor variable provide a context to extend real analysis and provide compact representation of mappings of the plane. However, the theory falls well short of function theory on the ordinary complex plane. Nevertheless, some of the aspects of conventional complex analysis have an interpretation given with... | Motor variable |
c_4z2mhlpz3bq9 | In mathematics, a function of bounded deformation is a function whose distributional derivatives are not quite well-behaved-enough to qualify as functions of bounded variation, although the symmetric part of the derivative matrix does meet that condition. Thought of as deformations of elasto-plastic bodies, functions o... | Bounded deformation |
c_pjc3b8bzpmup | Suquet in 1978. BD is a strictly larger space than the space BV of functions of bounded variation. | Bounded deformation |
c_jti9f9vxqc26 | One can show that if u is of bounded deformation then the measure ε(u) can be decomposed into three parts: one absolutely continuous with respect to Lebesgue measure, denoted e(u) dx; a jump part, supported on a rectifiable (n − 1)-dimensional set Ju of points where u has two different approximate limits u+ and u−, tog... | Bounded deformation |
c_aklvhr98bgq0 | In mathematics, a function of n {\displaystyle n} variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f ( x 1 , x 2 ) {\displaystyle f\left(x_{1},x_{2}\right)} of two arguments is a symmetric function if and only if f ( x 1 , x 2 ) = f ( x 2 , x 1 ) {\displaysty... | Complete symmetric function |
c_g4h91ve1wim7 | A related notion is alternating polynomials, which change sign under an interchange of variables. Aside from polynomial functions, tensors that act as functions of several vectors can be symmetric, and in fact the space of symmetric k {\displaystyle k} -tensors on a vector space V {\displaystyle V} is isomorphic to the... | Complete symmetric function |
c_lz73aqcqzcbw | In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant function having only finitely many pieces. | Step function |
c_ko6oo8mntzec | In mathematics, a function or sequence is said to exhibit quadratic growth when its values are proportional to the square of the function argument or sequence position. "Quadratic growth" often means more generally "quadratic growth in the limit", as the argument or sequence position goes to infinity – in big Theta not... | Quadratic growth |
c_sgobzlc2yhuc | In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set X into a vector space has a natural vector space structure given by pointwise addition ... | Function space |
c_oxa7htlmjj1x | In mathematics, a functional (as a noun) is a certain type of function. The exact definition of the term varies depending on the subfield (and sometimes even the author). In linear algebra, it is synonymous with linear forms, which are linear mappings from a vector space V {\displaystyle V} into its field of scalars (t... | Functional (mathematics) |
c_9dmuaiitp2uf | Depending on the author, such mappings may or may not be assumed to be linear, or to be defined on the whole space X . {\displaystyle X.} In computer science, it is synonymous with higher-order functions, that is, functions that take functions as arguments or return them.This article is mainly concerned with the second... | Functional (mathematics) |
c_xplelnh90xab | The first concept, which is more modern and abstract, is discussed in detail in a separate article, under the name linear form. The third concept is detailed in the computer science article on higher-order functions. In the case where the space X {\displaystyle X} is a space of functions, the functional is a "function ... | Functional (mathematics) |
c_on52uqcz3hbv | In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch (more accurately, several related areas) of the field of functional analysis, connected with spectral theory. (Historically, the term was also used synonymously with calculus of v... | Functional calculus |
c_zzc3989puh1o | If f {\displaystyle f} is a function, say a numerical function of a real number, and M {\displaystyle M} is an operator, there is no particular reason why the expression f ( M ) {\displaystyle f(M)} should make sense. If it does, then we are no longer using f {\displaystyle f} on its original function domain. In the tr... | Functional calculus |
c_6ltzjo82xlx1 | This passes nearly unnoticed if we talk about 'squaring a matrix', though, which is the case of f ( x ) = x 2 {\displaystyle f(x)=x^{2}} and M {\displaystyle M} an n × n {\displaystyle n\times n} matrix. The idea of a functional calculus is to create a principled approach to this kind of overloading of the notation. Th... | Functional calculus |
c_3zgdj1tj1i86 | In the finite-dimensional case, the polynomial functional calculus yields quite a bit of information about the operator. For example, consider the family of polynomials which annihilates an operator T {\displaystyle T} . This family is an ideal in the ring of polynomials. | Functional calculus |
c_eyolc8q2e5d7 | Furthermore, it is a nontrivial ideal: let n {\displaystyle n} be the finite dimension of the algebra of matrices, then { I , T , T 2 , … , T n } {\displaystyle \{I,T,T^{2},\ldots ,T^{n}\}} is linearly dependent. So ∑ i = 0 n α i T i = 0 {\displaystyle \sum _{i=0}^{n}\alpha _{i}T^{i}=0} for some scalars α i {\displayst... | Functional calculus |
c_gyce0xtuvrh7 | Since the ring of polynomials is a principal ideal domain, this ideal is generated by some polynomial m {\displaystyle m} . Multiplying by a unit if necessary, we can choose m {\displaystyle m} to be monic. When this is done, the polynomial m {\displaystyle m} is precisely the minimal polynomial of T {\displaystyle T} ... | Functional calculus |
c_miwfcmjzcfau | This polynomial gives deep information about T {\displaystyle T} . For instance, a scalar α {\displaystyle \alpha } is an eigenvalue of T {\displaystyle T} if and only if α {\displaystyle \alpha } is a root of m {\displaystyle m} . | Functional calculus |
c_vwy2w98tpuln | Also, sometimes m {\displaystyle m} can be used to calculate the exponential of T {\displaystyle T} efficiently. The polynomial calculus is not as informative in the infinite-dimensional case. | Functional calculus |
c_b1tooz5em1kd | Consider the unilateral shift with the polynomials calculus; the ideal defined above is now trivial. Thus one is interested in functional calculi more general than polynomials. The subject is closely linked to spectral theory, since for a diagonal matrix or multiplication operator, it is rather clear what the definitio... | Functional calculus |
c_18e0t3mxx1wu | In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning is often used, where a functional equation is an equation that relates sever... | Abel's functional equation |
c_kgdtxx4cwhmb | {\displaystyle \log(xy)=\log(x)+\log(y).} If the domain of the unknown function is supposed to be the natural numbers, the function is generally viewed as a sequence, and, in this case, a functional equation (in the narrower meaning) is called a recurrence relation. | Abel's functional equation |
c_fpxvc1sq84pq | Thus the term functional equation is used mainly for real functions and complex functions. Moreover a smoothness condition is often assumed for the solutions, since without such a condition, most functional equations have very irregular solutions. For example, the gamma function is a function that satisfies the functio... | Abel's functional equation |
c_rcswh8q0wum8 | In mathematics, a functional square root (sometimes called a half iterate) is a square root of a function with respect to the operation of function composition. In other words, a functional square root of a function g is a function f satisfying f(f(x)) = g(x) for all x. | Functional square root |
c_40kqzlr19dge | In mathematics, a fundamental discriminant D is an integer invariant in the theory of integral binary quadratic forms. If Q(x, y) = ax2 + bxy + cy2 is a quadratic form with integer coefficients, then D = b2 − 4ac is the discriminant of Q(x, y). Conversely, every integer D with D ≡ 0, 1 (mod 4) is the discriminant of so... | Fundamental discriminant |
c_icfduin1sksg | Thus, all such integers are referred to as discriminants in this theory. There are explicit congruence conditions that give the set of fundamental discriminants. Specifically, D is a fundamental discriminant if and only if one of the following statements holds D ≡ 1 (mod 4) and is square-free, D = 4m, where m ≡ 2 or 3 ... | Fundamental discriminant |
c_wbvnownqogqx | In mathematics, a fundamental matrix of a system of n homogeneous linear ordinary differential equations is a matrix-valued function Ψ ( t ) {\displaystyle \Psi (t)} whose columns are linearly independent solutions of the system. Then every solution to the system can be written as x ( t ) = Ψ ( t ) c {\displaystyle \ma... | Fundamental matrix (linear differential equation) |
c_myt5334zg8hi | In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that defines a lattice in the complex plane. This type of lattice is the underlying object with which elliptic functions and modular forms are defined. | Lattice basis |
c_ovvi4860pyhm | In mathematics, a fundamental polygon can be defined for every compact Riemann surface of genus greater than 0. It encodes not only the topology of the surface through its fundamental group but also determines the Riemann surface up to conformal equivalence. By the uniformization theorem, every compact Riemann surface ... | Fundamental polygon |
c_nxirl8f5b2qz | In the last case of genus g > 1, the Riemann surface is conformally equivalent to H/Γ where Γ is a Fuchsian group of Möbius transformations. A fundamental domain for Γ is given by a convex polygon for the hyperbolic metric on H. These can be defined by Dirichlet polygons and have an even number of sides. The structure ... | Fundamental polygon |
c_vpque8osckn5 | Using the theory of quasiconformal mappings and the Beltrami equation, it can be shown there is a canonical convex Dirichlet polygon with 4g sides, first defined by Fricke, which corresponds to the standard presentation of Γ as the group with 2g generators a1, b1, a2, b2, ..., ag, bg and the single relation ⋅⋅⋅ = 1, ... | Fundamental polygon |
c_5kf6l60q7lpo | Moreover, it also follows from the theory of quasiconformal mappings that two compact Riemann surfaces are diffeomorphic if and only if they are homeomorphic. Consequently, two closed oriented 2-manifolds are homeomorphic if and only if they are diffeomorphic. Such a result can also be proved using the methods of diffe... | Fundamental polygon |
c_ehh353gilb0g | In mathematics, a fundamental solution for a linear partial differential operator L is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not address boundary conditions). In terms of the Dirac delta "function" δ(x), ... | Fundamental solution |
c_e8u5669dfnvf | It was investigated for all dimensions for the Laplacian by Marcel Riesz. The existence of a fundamental solution for any operator with constant coefficients — the most important case, directly linked to the possibility of using convolution to solve an arbitrary right hand side — was shown by Bernard Malgrange and Leon... | Fundamental solution |
c_xjlv7c2rhw3t | In mathematics, a fundamental theorem is a theorem which is considered to be central and conceptually important for some topic. For example, the fundamental theorem of calculus gives the relationship between differential calculus and integral calculus. The names are mostly traditional, so that for example the fundament... | Fundamental lemma |
c_zkl2rpdrn1gr | For instance, the fundamental theorem of curves describe classification of regular curves in space up to translation and rotation. Likewise, the mathematical literature sometimes refers to the fundamental lemma of a field. The term lemma is conventionally used to denote a proven proposition which is used as a stepping ... | Fundamental lemma |
c_exrivh8xfnyw | In mathematics, a fusion category is a category that is rigid, semisimple, k {\displaystyle k} -linear, monoidal and has only finitely many isomorphism classes of simple objects, such that the monoidal unit is simple. If the ground field k {\displaystyle k} is algebraically closed, then the latter is equivalent to H o ... | Fusion category |
c_s3x09j79f5lc | In mathematics, a fusion frame of a vector space is a natural extension of a frame. It is an additive construct of several, potentially "overlapping" frames. The motivation for this concept comes from the event that a signal can not be acquired by a single sensor alone (a constraint found by limitations of hardware or ... | Fusion frame |
c_bbsmsv5kbvhy | In mathematics, a general hypergeometric function or Aomoto–Gelfand hypergeometric function is a generalization of the hypergeometric function that was introduced by Gelfand (1986). The general hypergeometric function is a function that is (more or less) defined on a Grassmannian, and depends on a choice of some comple... | General hypergeometric function |
c_es88bo68surv | In mathematics, a generalized Kac–Moody algebra is a Lie algebra that is similar to a Kac–Moody algebra, except that it is allowed to have imaginary simple roots. Generalized Kac–Moody algebras are also sometimes called GKM algebras, Borcherds–Kac–Moody algebras, BKM algebras, or Borcherds algebras. The best known exam... | Borcherds algebra |
c_mx9arpwvy3h8 | In mathematics, a generalized Korteweg–De Vries equation (Masayoshi Tsutsumi, Toshio Mukasa & Riichi Iino 1970) is the nonlinear partial differential equation ∂ t u + ∂ x 3 u + ∂ x f ( u ) = 0. {\displaystyle \partial _{t}u+\partial _{x}^{3}u+\partial _{x}f(u)=0.\,} The function f is sometimes taken to be f(u) = uk+1/(... | Generalized Korteweg–De Vries equation |
c_o41es1dhz2ga | In mathematics, a generalized arithmetic progression (or multiple arithmetic progression) is a generalization of an arithmetic progression equipped with multiple common differences – whereas an arithmetic progression is generated by a single common difference, a generalized arithmetic progression can be generated by mu... | Multi-dimensional arithmetic progression |
c_vftjzb2dtgwg | In mathematics, a generalized conic is a geometrical object defined by a property which is a generalization of some defining property of the classical conic. For example, in elementary geometry, an ellipse can be defined as the locus of a point which moves in a plane such that the sum of its distances from two fixed po... | Generalized conic |
c_6k44a0pmt086 | In rectangular Cartesian coordinates, the equation y = x2 represents a parabola. The generalized equation y = x r, for r ≠ 0 and r ≠ 1, can be treated as defining a generalized parabola. The idea of generalized conic has found applications in approximation theory and optimization theory.Among the several possible ways ... | Generalized conic |
c_zq5g4r5chn7b | The starting point for this approach is to look upon an ellipse as a curve satisfying the 'two-focus property': an ellipse is a curve that is the locus of points the sum of whose distances from two given points is constant. The two points are the foci of the ellipse. The curve obtained by replacing the set of two fixed... | Generalized conic |
c_dqxujxnrgdqb | Generalized conics with three foci are called trifocal ellipses. This can be further generalized to curves which are obtained as the loci of points such that some weighted sum of the distances from a finite set of points is a constant. A still further generalization is possible by assuming that the weights attached to ... | Generalized conic |
c_pnpy4oz2ysav | Finally, the restriction that the set of fixed points, called the set of foci of the generalized conic, be finite may also be removed. The set may be assumed to be finite or infinite. | Generalized conic |
c_iz0ba8yztkrt | In the infinite case, the weighted arithmetic mean has to be replaced by an appropriate integral. Generalized conics in this sense are also called polyellipses, egglipses, or, generalized ellipses. Since such curves were considered by the German mathematician Ehrenfried Walther von Tschirnhaus (1651 – 1708) they are al... | Generalized conic |
c_g3vez6llzx0s | In mathematics, a generalized cwatset (GC-set) is an algebraic structure generalizing the notion of closure with a twist, the defining characteristic of the cwatset. | Closure with a twist |
c_5k77nbw4g304 | In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space V over a field F. When F is the real or complex numbers, a generalized flag variety is a smooth or complex manifold, called a real or complex flag manifold. Flag varieti... | Generalized flag manifold |
c_6q4djhlyt5my | For partial flags, one needs to specify the sequence of dimensions of the flags under consideration. For subgroups of the linear group, additional conditions must be imposed on the flags. In the most general sense, a generalized flag variety is defined to mean a projective homogeneous variety, that is, a smooth project... | Generalized flag manifold |
c_46tuvcm84b12 | If X has an F-rational point, then it is isomorphic to G/P for some parabolic subgroup P of G. A projective homogeneous variety may also be realised as the orbit of a highest weight vector in a projectivized representation of G. The complex projective homogeneous varieties are the compact flat model spaces for Cartan g... | Generalized flag manifold |
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