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matrix coefficient | They arise naturally from finite-dimensional representations of G as the matrix-entry functions of the corresponding matrix representations. The Peter–Weyl theorem says that the matrix coefficients on G are dense in the Hilbert space of square-integrable functions on G. Matrix coefficients of representations of Lie groups turned out to be intimately related with the theory of special functions, providing a unifying approach to large parts of this theory. Growth properties of matrix coefficients play a key role in the classification of irreducible representations of locally compact groups, in particular, reductive real and p-adic groups. The formalism of matrix coefficients leads to a generalization of the notion of a modular form. In a different direction, mixing properties of certain dynamical systems are controlled by the properties of suitable matrix coefficients. | wikipedia |
m-theory | In mathematics, a matrix is a rectangular array of numbers or other data. In physics, a matrix model is a particular kind of physical theory whose mathematical formulation involves the notion of a matrix in an important way. A matrix model describes the behavior of a set of matrices within the framework of quantum mechanics.One important example of a matrix model is the BFSS matrix model proposed by Tom Banks, Willy Fischler, Stephen Shenker, and Leonard Susskind in 1997. This theory describes the behavior of a set of nine large matrices. | wikipedia |
m-theory | In their original paper, these authors showed, among other things, that the low energy limit of this matrix model is described by eleven-dimensional supergravity. These calculations led them to propose that the BFSS matrix model is exactly equivalent to M-theory. The BFSS matrix model can therefore be used as a prototype for a correct formulation of M-theory and a tool for investigating the properties of M-theory in a relatively simple setting. | wikipedia |
gauge–gravity duality | In mathematics, a matrix is a rectangular array of numbers or other data. In physics, a matrix model is a particular kind of physical theory whose mathematical formulation involves the notion of a matrix in an important way. A matrix model describes the behavior of a set of matrices within the framework of quantum mechanics.One important example of a matrix model is the BFSS matrix model proposed by Tom Banks, Willy Fischler, Stephen Shenker, and Leonard Susskind in 1997. This theory describes the behavior of a set of nine large matrices. | wikipedia |
gauge–gravity duality | In their original paper, these authors showed, among other things, that the low energy limit of this matrix model is described by eleven-dimensional supergravity. These calculations led them to propose that the BFSS matrix model is exactly equivalent to M-theory. The BFSS matrix model can therefore be used as a prototype for a correct formulation of M-theory and a tool for investigating the properties of M-theory in a relatively simple setting.The development of the matrix model formulation of M-theory has led physicists to consider various connections between string theory and a branch of mathematics called noncommutative geometry. | wikipedia |
gauge–gravity duality | This subject is a generalization of ordinary geometry in which mathematicians define new geometric notions using tools from noncommutative algebra. In a paper from 1998, Alain Connes, Michael R. Douglas, and Albert Schwarz showed that some aspects of matrix models and M-theory are described by a noncommutative quantum field theory, a special kind of physical theory in which spacetime is described mathematically using noncommutative geometry. This established a link between matrix models and M-theory on the one hand, and noncommutative geometry on the other hand. It quickly led to the discovery of other important links between noncommutative geometry and various physical theories. | wikipedia |
meander (mathematics) | In mathematics, a meander or closed meander is a self-avoiding closed curve which crosses a given line a number of times, meaning that it intersects the line while passing from one side to the other. Intuitively, a meander can be viewed as a meandering river with a straight road crossing the river over a number of bridges. The points where the line and the curve cross are therefore referred to as "bridges". | wikipedia |
measure algebra | In mathematics, a measure algebra is a Boolean algebra with a countably additive positive measure. A probability measure on a measure space gives a measure algebra on the Boolean algebra of measurable sets modulo null sets. | wikipedia |
mock theta functions | In mathematics, a mock modular form is the holomorphic part of a harmonic weak Maass form, and a mock theta function is essentially a mock modular form of weight 1/2. The first examples of mock theta functions were described by Srinivasa Ramanujan in his last 1920 letter to G. H. Hardy and in his lost notebook. Sander Zwegers discovered that adding certain non-holomorphic functions to them turns them into harmonic weak Maass forms. | wikipedia |
modular lie algebra | In mathematics, a modular Lie algebra is a Lie algebra over a field of positive characteristic. The theory of modular Lie algebras is significantly different from the theory of real and complex Lie algebras. This difference can be traced to the properties of Frobenius automorphism and to the failure of the exponential map to establish a tight connection between properties of a modular Lie algebra and the corresponding algebraic group. Although serious study of modular Lie algebras was initiated by Nathan Jacobson in 1950s, their representation theory in the semisimple case was advanced only recently due to the influential Lusztig conjectures, which as of 2007 have been partially proved. | wikipedia |
elliptic modular form | In mathematics, a modular form is a (complex) analytic function on the upper half-plane that satisfies: a kind of functional equation with respect to the group action of the modular group, and a growth condition.The theory of modular forms therefore belongs to complex analysis. The main importance of the theory is its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory. | wikipedia |
elliptic modular form | Modular form theory is a special case of the more general theory of automorphic forms, which are functions defined on Lie groups that transform nicely with respect to the action of certain discrete subgroups, generalizing the example of the modular group S L 2 ( Z ) ⊂ S L 2 ( R ) {\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )\subset \mathrm {SL} _{2}(\mathbb {R} )} . The term "modular form", as a systematic description, is usually attributed to Hecke. Each modular form is attached to a Galois representation. | wikipedia |
monopole (mathematics) | In mathematics, a monopole is a connection over a principal bundle G with a section of the associated adjoint bundle. | wikipedia |
monotonically non-decreasing | In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. | wikipedia |
many-valued function | In mathematics, a multivalued function, also called multifunction and many-valued function, is a set-valued function with continuity properties that allow considering it locally as an ordinary function. Multivalued functions arise commonly in applications of the implicit function theorem, since this theorem can be viewed as asserting the existence of a multivalued function. In particular, the inverse function of a differentiable function is a multivalued function, and is single-valued only when the original function is monotonic. For example, the complex logarithm is a multivalued function, as the inverse of the exponential function. | wikipedia |
many-valued function | It cannot be considered as an ordinary function, since, when one follows one value of the logarithm along a circle centered at 0, one gets another value than the starting one after a complete turn. This phenomenon is called monodromy. Another common way for defining a multivalued function is analytic continuation, which generates commonly some monodromy: analytic continuation along a closed curve may generate a final value that differs from the starting value. Multivalued functions arise also as solutions of differential equations, where the different values are parametrized by initial conditions. | wikipedia |
isotope (jordan algebra) | In mathematics, a mutation, also called a homotope, of a unital Jordan algebra is a new Jordan algebra defined by a given element of the Jordan algebra. The mutation has a unit if and only if the given element is invertible, in which case the mutation is called a proper mutation or an isotope. Mutations were first introduced by Max Koecher in his Jordan algebraic approach to Hermitian symmetric spaces and bounded symmetric domains of tube type. | wikipedia |
isotope (jordan algebra) | Their functorial properties allow an explicit construction of the corresponding Hermitian symmetric space of compact type as a compactification of a finite-dimensional complex semisimple Jordan algebra. The automorphism group of the compactification becomes a complex subgroup, the complexification of its maximal compact subgroup. Both groups act transitively on the compactification. The theory has been extended to cover all Hermitian symmetric spaces using the theory of Jordan pairs or Jordan triple systems. Koecher obtained the results in the more general case directly from the Jordan algebra case using the fact that only Jordan pairs associated with period two automorphisms of Jordan algebras are required. | wikipedia |
near-field (mathematics) | In mathematics, a near-field is an algebraic structure similar to a division ring, except that it has only one of the two distributive laws. Alternatively, a near-field is a near-ring in which there is a multiplicative identity and every non-zero element has a multiplicative inverse. | wikipedia |
relativistic system (mathematics) | In mathematics, a non-autonomous system of ordinary differential equations is defined to be a dynamic equation on a smooth fiber bundle Q → R {\displaystyle Q\to \mathbb {R} } over R {\displaystyle \mathbb {R} } . For instance, this is the case of non-relativistic non-autonomous mechanics, but not relativistic mechanics. To describe relativistic mechanics, one should consider a system of ordinary differential equations on a smooth manifold Q {\displaystyle Q} whose fibration over R {\displaystyle \mathbb {R} } is not fixed. Such a system admits transformations of a coordinate t {\displaystyle t} on R {\displaystyle \mathbb {R} } depending on other coordinates on Q {\displaystyle Q} . | wikipedia |
relativistic system (mathematics) | Therefore, it is called the relativistic system. In particular, Special Relativity on the Minkowski space Q = R 4 {\displaystyle Q=\mathbb {R} ^{4}} is of this type. Since a configuration space Q {\displaystyle Q} of a relativistic system has no preferable fibration over R {\displaystyle \mathbb {R} } , a velocity space of relativistic system is a first order jet manifold J 1 1 Q {\displaystyle J_{1}^{1}Q} of one-dimensional submanifolds of Q {\displaystyle Q} . | wikipedia |
relativistic system (mathematics) | The notion of jets of submanifolds generalizes that of jets of sections of fiber bundles which are utilized in covariant classical field theory and non-autonomous mechanics. A first order jet bundle J 1 1 Q → Q {\displaystyle J_{1}^{1}Q\to Q} is projective and, following the terminology of Special Relativity, one can think of its fibers as being spaces of the absolute velocities of a relativistic system. | wikipedia |
relativistic system (mathematics) | Given coordinates ( q 0 , q i ) {\displaystyle (q^{0},q^{i})} on Q {\displaystyle Q} , a first order jet manifold J 1 1 Q {\displaystyle J_{1}^{1}Q} is provided with the adapted coordinates ( q 0 , q i , q 0 i ) {\displaystyle (q^{0},q^{i},q_{0}^{i})} possessing transition functions q ′ 0 = q ′ 0 ( q 0 , q k ) , q ′ i = q ′ i ( q 0 , q k ) , q ′ 0 i = ( ∂ q ′ i ∂ q j q 0 j + ∂ q ′ i ∂ q 0 ) ( ∂ q ′ 0 ∂ q j q 0 j + ∂ q ′ 0 ∂ q 0 ) − 1 . {\displaystyle q'^{0}=q'^{0}(q^{0},q^{k}),\quad q'^{i}=q'^{i}(q^{0},q^{k}),\quad {q'}_{0}^{i}=\left({\frac {\partial q'^{i}}{\partial q^{j}}}q_{0}^{j}+{\frac {\partial q'^{i}}{\partial q^{0}}}\right)\left({\frac {\partial q'^{0}}{\partial q^{j}}}q_{0}^{j}+{\frac {\partial q'^{0}}{\partial q^{0}}}\right)^{-1}.} The relativistic velocities of a relativistic system are represented by elements of a fibre bundle R × T Q {\displaystyle \mathbb {R} \times TQ} , coordinated by ( τ , q λ , a τ λ ) {\displaystyle (\tau ,q^{\lambda },a_{\tau }^{\lambda })} , where T Q {\displaystyle TQ} is the tangent bundle of Q {\displaystyle Q} . | wikipedia |
relativistic system (mathematics) | Then a generic equation of motion of a relativistic system in terms of relativistic velocities reads ( ∂ λ G μ α 2 … α 2 N 2 N − ∂ μ G λ α 2 … α 2 N ) q τ μ q τ α 2 ⋯ q τ α 2 N − ( 2 N − 1 ) G λ μ α 3 … α 2 N q τ τ μ q τ α 3 ⋯ q τ α 2 N + F λ μ q τ μ = 0 , {\displaystyle \left({\frac {\partial _{\lambda }G_{\mu \alpha _{2}\ldots \alpha _{2N}}}{2N}}-\partial _{\mu }G_{\lambda \alpha _{2}\ldots \alpha _{2N}}\right)q_{\tau }^{\mu }q_{\tau }^{\alpha _{2}}\cdots q_{\tau }^{\alpha _{2N}}-(2N-1)G_{\lambda \mu \alpha _{3}\ldots \alpha _{2N}}q_{\tau \tau }^{\mu }q_{\tau }^{\alpha _{3}}\cdots q_{\tau }^{\alpha _{2N}}+F_{\lambda \mu }q_{\tau }^{\mu }=0,} G α 1 … α 2 N q τ α 1 ⋯ q τ α 2 N = 1. {\displaystyle G_{\alpha _{1}\ldots \alpha _{2N}}q_{\tau }^{\alpha _{1}}\cdots q_{\tau }^{\alpha _{2N}}=1.} For instance, if Q {\displaystyle Q} is the Minkowski space with a Minkowski metric G μ ν {\displaystyle G_{\mu \nu }} , this is an equation of a relativistic charge in the presence of an electromagnetic field. | wikipedia |
normed algebra | In mathematics, a normed algebra A is an algebra over a field which has a sub-multiplicative norm: ∀ x , y ∈ A ‖ x y ‖ ≤ ‖ x ‖ ‖ y ‖ . {\displaystyle \forall x,y\in A\qquad \|xy\|\leq \|x\|\|y\|.} Some authors require it to have a multiplicative identity 1A such that ║1A║ = 1. | wikipedia |
nowhere continuous function | In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain. If f {\displaystyle f} is a function from real numbers to real numbers, then f {\displaystyle f} is nowhere continuous if for each point x {\displaystyle x} there is some ε > 0 {\displaystyle \varepsilon >0} such that for every δ > 0 , {\displaystyle \delta >0,} we can find a point y {\displaystyle y} such that | x − y | < δ {\displaystyle |x-y|<\delta } and | f ( x ) − f ( y ) | ≥ ε {\displaystyle |f(x)-f(y)|\geq \varepsilon } . Therefore, no matter how close we get to any fixed point, there are even closer points at which the function takes not-nearby values. More general definitions of this kind of function can be obtained, by replacing the absolute value by the distance function in a metric space, or by using the definition of continuity in a topological space. | wikipedia |
harmonic (mathematics) | In mathematics, a number of concepts employ the word harmonic. The similarity of this terminology to that of music is not accidental: the equations of motion of vibrating strings, drums and columns of air are given by formulas involving Laplacians; the solutions to which are given by eigenvalues corresponding to their modes of vibration. Thus, the term "harmonic" is applied when one is considering functions with sinusoidal variations, or solutions of Laplace's equation and related concepts.Mathematical terms whose names include "harmonic" include: Projective harmonic conjugate Cross-ratio Harmonic analysis Harmonic conjugate Harmonic form Harmonic function Harmonic mean Harmonic mode Harmonic number Harmonic series Alternating harmonic series Harmonic tremor Spherical harmonics | wikipedia |
data cube | In mathematics, a one-dimensional array corresponds to a vector, a two-dimensional array resembles a matrix; more generally, a tensor may be represented as an n-dimensional data cube. | wikipedia |
p-adic zeta function | In mathematics, a p-adic zeta function, or more generally a p-adic L-function, is a function analogous to the Riemann zeta function, or more general L-functions, but whose domain and target are p-adic (where p is a prime number). For example, the domain could be the p-adic integers Zp, a profinite p-group, or a p-adic family of Galois representations, and the image could be the p-adic numbers Qp or its algebraic closure. The source of a p-adic L-function tends to be one of two types. The first source—from which Tomio Kubota and Heinrich-Wolfgang Leopoldt gave the first construction of a p-adic L-function (Kubota & Leopoldt 1964)—is via the p-adic interpolation of special values of L-functions. | wikipedia |
p-adic zeta function | For example, Kubota–Leopoldt used Kummer's congruences for Bernoulli numbers to construct a p-adic L-function, the p-adic Riemann zeta function ζp(s), whose values at negative odd integers are those of the Riemann zeta function at negative odd integers (up to an explicit correction factor). p-adic L-functions arising in this fashion are typically referred to as analytic p-adic L-functions. The other major source of p-adic L-functions—first discovered by Kenkichi Iwasawa—is from the arithmetic of cyclotomic fields, or more generally, certain Galois modules over towers of cyclotomic fields or even more general towers. | wikipedia |
p-adic zeta function | A p-adic L-function arising in this way is typically called an arithmetic p-adic L-function as it encodes arithmetic data of the Galois module involved. The main conjecture of Iwasawa theory (now a theorem due to Barry Mazur and Andrew Wiles) is the statement that the Kubota–Leopoldt p-adic L-function and an arithmetic analogue constructed by Iwasawa theory are essentially the same. In more general situations where both analytic and arithmetic p-adic L-functions are constructed (or expected), the statement that they agree is called the main conjecture of Iwasawa theory for that situation. Such conjectures represent formal statements concerning the philosophy that special values of L-functions contain arithmetic information. | wikipedia |
pair of pants (mathematics) | In mathematics, a pair of pants is a surface which is homeomorphic to the three-holed sphere. The name comes from considering one of the removed disks as the waist and the two others as the cuffs of a pair of pants. Pairs of pants are used as building blocks for compact surfaces in various theories. Two important applications are to hyperbolic geometry, where decompositions of closed surfaces into pairs of pants are used to construct the Fenchel-Nielsen coordinates on Teichmüller space, and in topological quantum field theory where they are the simplest non-trivial cobordisms between 1-dimensional manifolds. | wikipedia |
cantor's pairing function | In mathematics, a pairing function is a process to uniquely encode two natural numbers into a single natural number.Any pairing function can be used in set theory to prove that integers and rational numbers have the same cardinality as natural numbers. | wikipedia |
parallelization (mathematics) | In mathematics, a parallelization of a manifold M {\displaystyle M\,} of dimension n is a set of n global smooth linearly independent vector fields. | wikipedia |
paratopological group | In mathematics, a paratopological group is a topological semigroup that is algebraically a group. In other words, it is a group G with a topology such that the group's product operation is a continuous function from G × G to G. This differs from the definition of a topological group in that the group inverse is not required to be continuous. As with topological groups, some authors require the topology to be Hausdorff.Compact paratopological groups are automatically topological groups. == References == | wikipedia |
parent function | In mathematics, a parent function is the core representation of a function type without manipulations such as translation and dilation. For example, for the family of quadratic functions having the general form y = a x 2 + b x + c , {\displaystyle y=ax^{2}+bx+c\,,} the simplest function is y = x 2 {\displaystyle y=x^{2}} .This is therefore the parent function of the family of quadratic equations. For linear and quadratic functions, the graph of any function can be obtained from the graph of the parent function by simple translations and stretches parallel to the axes. | wikipedia |
parent function | For example, the graph of y = x2 − 4x + 7 can be obtained from the graph of y = x2 by translating +2 units along the X axis and +3 units along Y axis. This is because the equation can also be written as y − 3 = (x − 2)2. For many trigonometric functions, the parent function is usually a basic sin(x), cos(x), or tan(x). | wikipedia |
parent function | For example, the graph of y = A sin(x) + B cos(x) can be obtained from the graph of y = sin(x) by translating it through an angle α along the positive X axis (where tan(α) = A⁄B), then stretching it parallel to the Y axis using a stretch factor R, where R2 = A2 + B2. This is because A sin(x) + B cos(x) can be written as R sin(x−α) (see List of trigonometric identities). The concept of parent function is less clear for polynomials of higher power because of the extra turning points, but for the family of n-degree polynomial functions for any given n, the parent function is sometimes taken as xn, or, to simplify further, x2 when n is even and x3 for odd n. Turning points may be established by differentiation to provide more detail of the graph. | wikipedia |
partial functions | In mathematics, a partial function f from a set X to a set Y is a function from a subset S of X (possibly the whole X itself) to Y. The subset S, that is, the domain of f viewed as a function, is called the domain of definition or natural domain of f. If S equals X, that is, if f is defined on every element in X, then f is said to be a total function. More technically, a partial function is a binary relation over two sets that associates every element of the first set to at most one element of the second set; it is thus a functional binary relation. It generalizes the concept of a (total) function by not requiring every element of the first set to be associated to exactly one element of the second set. A partial function is often used when its exact domain of definition is not known or difficult to specify. | wikipedia |
partial functions | This is the case in calculus, where, for example, the quotient of two functions is a partial function whose domain of definition cannot contain the zeros of the denominator. For this reason, in calculus, and more generally in mathematical analysis, a partial function is generally called simply a function. | wikipedia |
partial functions | In computability theory, a general recursive function is a partial function from the integers to the integers; no algorithm can exist for deciding whether an arbitrary such function is in fact total. When arrow notation is used for functions, a partial function f {\displaystyle f} from X {\displaystyle X} to Y {\displaystyle Y} is sometimes written as f: X ⇀ Y , {\displaystyle f:X\rightharpoonup Y,} f: X ↛ Y , {\displaystyle f:X\nrightarrow Y,} or f: X ↪ Y . {\displaystyle f:X\hookrightarrow Y.} However, there is no general convention, and the latter notation is more commonly used for inclusion maps or embeddings.Specifically, for a partial function f: X ⇀ Y , {\displaystyle f:X\rightharpoonup Y,} and any x ∈ X , {\displaystyle x\in X,} one has either: f ( x ) = y ∈ Y {\displaystyle f(x)=y\in Y} (it is a single element in Y), or f ( x ) {\displaystyle f(x)} is undefined.For example, if f {\displaystyle f} is the square root function restricted to the integers f: Z → N , {\displaystyle f:\mathbb {Z} \to \mathbb {N} ,} defined by: f ( n ) = m {\displaystyle f(n)=m} if, and only if, m 2 = n , {\displaystyle m^{2}=n,} m ∈ N , n ∈ Z , {\displaystyle m\in \mathbb {N} ,n\in \mathbb {Z} ,} then f ( n ) {\displaystyle f(n)} is only defined if n {\displaystyle n} is a perfect square (that is, 0 , 1 , 4 , 9 , 16 , … {\displaystyle 0,1,4,9,16,\ldots } ). So f ( 25 ) = 5 {\displaystyle f(25)=5} but f ( 26 ) {\displaystyle f(26)} is undefined. | wikipedia |
partial group algebra | In mathematics, a partial group algebra is an associative algebra related to the partial representations of a group. | wikipedia |
phase line (mathematics) | In mathematics, a phase line is a diagram that shows the qualitative behaviour of an autonomous ordinary differential equation in a single variable, d y d x = f ( y ) {\displaystyle {\tfrac {dy}{dx}}=f(y)} . The phase line is the 1-dimensional form of the general n {\displaystyle n} -dimensional phase space, and can be readily analyzed. | wikipedia |
piecewise algebraic space | In mathematics, a piecewise algebraic space is a generalization of a semialgebraic set, introduced by Maxim Kontsevich and Yan Soibelman. The motivation was for the proof of Deligne's conjecture on Hochschild cohomology. Robert Hardt, Pascal Lambrechts, Victor Turchin, and Ismar Volić later developed the theory. | wikipedia |
point process | In mathematics, a point process is a random element whose values are "point patterns" on a set S. While in the exact mathematical definition a point pattern is specified as a locally finite counting measure, it is sufficient for more applied purposes to think of a point pattern as a countable subset of S that has no limit points. | wikipedia |
point source | In mathematics, a point source is a singularity from which flux or flow is emanating. Although singularities such as this do not exist in the observable universe, mathematical point sources are often used as approximations to reality in physics and other fields. | wikipedia |
polylogarithmic function | In mathematics, a polylogarithmic function in n is a polynomial in the logarithm of n, a k ( log n ) k + a k − 1 ( log n ) k − 1 + ⋯ + a 1 ( log n ) + a 0 . {\displaystyle a_{k}(\log n)^{k}+a_{k-1}(\log n)^{k-1}+\cdots +a_{1}(\log n)+a_{0}.} The notation logkn is often used as a shorthand for (log n)k, analogous to sin2θ for (sin θ)2. | wikipedia |
polylogarithmic function | In computer science, polylogarithmic functions occur as the order of time or memory used by some algorithms (e.g., "it has polylogarithmic order"), such as in the definition of QPTAS (see PTAS). All polylogarithmic functions of n are o(nε) for every exponent ε > 0 (for the meaning of this symbol, see small o notation), that is, a polylogarithmic function grows more slowly than any positive exponent. This observation is the basis for the soft O notation Õ(n). | wikipedia |
polynomial multiplication | In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example with three indeterminates is x3 + 2xyz2 − yz + 1. | wikipedia |
polynomial multiplication | Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. | wikipedia |
positive-semidefinite function | In mathematics, a positive-definite function is, depending on the context, either of two types of function. | wikipedia |
pre-lie algebra | In mathematics, a pre-Lie algebra is an algebraic structure on a vector space that describes some properties of objects such as rooted trees and vector fields on affine space. The notion of pre-Lie algebra has been introduced by Murray Gerstenhaber in his work on deformations of algebras. Pre-Lie algebras have been considered under some other names, among which one can cite left-symmetric algebras, right-symmetric algebras or Vinberg algebras. | wikipedia |
principal subalgebra | In mathematics, a principal subalgebra of a complex simple Lie algebra is a 3-dimensional simple subalgebra whose non-zero elements are regular. A finite-dimensional complex simple Lie algebra has a unique conjugacy class of principal subalgebras, each of which is the span of an sl2-triple. | wikipedia |
mathematical product | In mathematics, a product is the result of multiplication, or an expression that identifies objects (numbers or variables) to be multiplied, called factors. For example, 21 is the product of 3 and 7 (the result of multiplication), and x ⋅ ( 2 + x ) {\displaystyle x\cdot (2+x)} is the product of x {\displaystyle x} and ( 2 + x ) {\displaystyle (2+x)} (indicating that the two factors should be multiplied together). When one factor is an integer, the product is called a multiple. The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication. | wikipedia |
mathematical product | When matrices or members of various other associative algebras are multiplied, the product usually depends on the order of the factors. Matrix multiplication, for example, is non-commutative, and so is multiplication in other algebras in general as well. There are many different kinds of products in mathematics: besides being able to multiply just numbers, polynomials or matrices, one can also define products on many different algebraic structures. | wikipedia |
progressive function | In mathematics, a progressive function ƒ ∈ L2(R) is a function whose Fourier transform is supported by positive frequencies only: s u p p f ^ ⊆ R + . {\displaystyle \mathop {\rm {supp}} {\hat {f}}\subseteq \mathbb {R} _{+}.} It is called super regressive if and only if the time reversed function f(−t) is progressive, or equivalently, if s u p p f ^ ⊆ R − . {\displaystyle \mathop {\rm {supp}} {\hat {f}}\subseteq \mathbb {R} _{-}.} | wikipedia |
progressive function | The complex conjugate of a progressive function is regressive, and vice versa. The space of progressive functions is sometimes denoted H + 2 ( R ) {\displaystyle H_{+}^{2}(R)} , which is known as the Hardy space of the upper half-plane. This is because a progressive function has the Fourier inversion formula f ( t ) = ∫ 0 ∞ e 2 π i s t f ^ ( s ) d s {\displaystyle f(t)=\int _{0}^{\infty }e^{2\pi ist}{\hat {f}}(s)\,ds} and hence extends to a holomorphic function on the upper half-plane { t + i u: t , u ∈ R , u ≥ 0 } {\displaystyle \{t+iu:t,u\in R,u\geq 0\}} by the formula f ( t + i u ) = ∫ 0 ∞ e 2 π i s ( t + i u ) f ^ ( s ) d s = ∫ 0 ∞ e 2 π i s t e − 2 π s u f ^ ( s ) d s . | wikipedia |
progressive function | {\displaystyle f(t+iu)=\int _{0}^{\infty }e^{2\pi is(t+iu)}{\hat {f}}(s)\,ds=\int _{0}^{\infty }e^{2\pi ist}e^{-2\pi su}{\hat {f}}(s)\,ds.} Conversely, every holomorphic function on the upper half-plane which is uniformly square-integrable on every horizontal line will arise in this manner. Regressive functions are similarly associated with the Hardy space on the lower half-plane { t + i u: t , u ∈ R , u ≤ 0 } {\displaystyle \{t+iu:t,u\in R,u\leq 0\}} . This article incorporates material from progressive function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. | wikipedia |
projection (mathematics) | In mathematics, a projection is an idempotent mapping of a set (or other mathematical structure) into a subset (or sub-structure). In this case, idempotent means that projecting twice is the same as projecting once. The restriction to a subspace of a projection is also called a projection, even if the idempotence property is lost. An everyday example of a projection is the casting of shadows onto a plane (sheet of paper): the projection of a point is its shadow on the sheet of paper, and the projection (shadow) of a point on the sheet of paper is that point itself (idempotency). | wikipedia |
projection (mathematics) | The shadow of a three-dimensional sphere is a closed disk. Originally, the notion of projection was introduced in Euclidean geometry to denote the projection of the three-dimensional Euclidean space onto a plane in it, like the shadow example. The two main projections of this kind are: The projection from a point onto a plane or central projection: If C is a point, called the center of projection, then the projection of a point P different from C onto a plane that does not contain C is the intersection of the line CP with the plane. | wikipedia |
projection (mathematics) | The points P such that the line CP is parallel to the plane does not have any image by the projection, but one often says that they project to a point at infinity of the plane (see Projective geometry for a formalization of this terminology). The projection of the point C itself is not defined. The projection parallel to a direction D, onto a plane or parallel projection: The image of a point P is the intersection with the plane of the line parallel to D passing through P. See Affine space § Projection for an accurate definition, generalized to any dimension.The concept of projection in mathematics is a very old one, and most likely has its roots in the phenomenon of the shadows cast by real-world objects on the ground. | wikipedia |
projection (mathematics) | This rudimentary idea was refined and abstracted, first in a geometric context and later in other branches of mathematics. Over time different versions of the concept developed, but today, in a sufficiently abstract setting, we can unify these variations.In cartography, a map projection is a map of a part of the surface of the Earth onto a plane, which, in some cases, but not always, is the restriction of a projection in the above meaning. The 3D projections are also at the basis of the theory of perspective.The need for unifying the two kinds of projections and of defining the image by a central projection of any point different of the center of projection are at the origin of projective geometry. However, a projective transformation is a bijection of a projective space, a property not shared with the projections of this article. | wikipedia |
projectionless c*-algebra | In mathematics, a projectionless C*-algebra is a C*-algebra with no nontrivial projections. For a unital C*-algebra, the projections 0 and 1 are trivial. While for a non-unital C*-algebra, only 0 is considered trivial. | wikipedia |
projectionless c*-algebra | The problem of whether simple infinite-dimensional C*-algebras with this property exist was posed in 1958 by Irving Kaplansky, and the first example of one was published in 1981 by Bruce Blackadar. For commutative C*-algebras, being projectionless is equivalent to its spectrum being connected. Due to this, being projectionless can be considered as a noncommutative analogue of a connected space. | wikipedia |
property (mathematics) | In mathematics, a property is any characteristic that applies to a given set. Rigorously, a property p defined for all elements of a set X is usually defined as a function p: X → {true, false}, that is true whenever the property holds; or equivalently, as the subset of X for which p holds; i.e. the set {x | p(x) = true}; p is its indicator function. However, it may be objected that the rigorous definition defines merely the extension of a property, and says nothing about what causes the property to hold for exactly those values. | wikipedia |
pseudo-reductive algebraic group | In mathematics, a pseudo-reductive group over a field k (sometimes called a k-reductive group) is a smooth connected affine algebraic group defined over k whose k-unipotent radical (i.e., largest smooth connected unipotent normal k-subgroup) is trivial. Over perfect fields these are the same as (connected) reductive groups, but over non-perfect fields Jacques Tits found some examples of pseudo-reductive groups that are not reductive. A pseudo-reductive k-group need not be reductive (since the formation of the k-unipotent radical does not generally commute with non-separable scalar extension on k, such as scalar extension to an algebraic closure of k). | wikipedia |
pseudo-reductive algebraic group | Pseudo-reductive groups arise naturally in the study of algebraic groups over function fields of positive-dimensional varieties in positive characteristic (even over a perfect field of constants). Springer (1998) gives an exposition of Tits' results on pseudo-reductive groups, while Conrad, Gabber & Prasad (2010) builds on Tits' work to develop a general structure theory, including more advanced topics such as construction techniques, root systems and root groups and open cells, classification theorems, and applications to rational conjugacy theorems for smooth connected affine groups over arbitrary fields. The general theory (with applications) as of 2010 is summarized in Rémy (2011), and later work in the second edition Conrad, Gabber & Prasad (2015) and in Conrad & Prasad (2016) provides further refinements. | wikipedia |
pseudogamma function | In mathematics, a pseudogamma function is a function that interpolates the factorial. The gamma function is the most famous solution to the problem of extending the notion of the factorial beyond the positive integers only. However, it is clearly not the only solution, as, for any set of points, an infinite number of curves can be drawn through those points. | wikipedia |
pseudogamma function | Such a curve, namely one which interpolates the factorial but is not equal to the gamma function, is known as a pseudogamma function. The two most famous pseudogamma functions are Hadamard's gamma function: H ( x ) = ψ ( 1 − x 2 ) − ψ ( 1 2 − x 2 ) 2 Γ ( 1 − x ) {\displaystyle H(x)={\frac {\psi \left(1-{\frac {x}{2}}\right)-\psi \left({\frac {1}{2}}-{\frac {x}{2}}\right)}{2\Gamma (1-x)}}} and the Luschny factorial: Γ ( x + 1 ) ( 1 − sin ( π x ) π x ( x 2 ( ψ ( x + 1 2 ) − ψ ( x 2 ) ) − 1 2 ) ) {\displaystyle \Gamma (x+1)\left(1-{\frac {\sin \left(\pi x\right)}{\pi x}}\left({\frac {x}{2}}\left(\psi \left({\frac {x+1}{2}}\right)-\psi \left({\frac {x}{2}}\right)\right)-{\frac {1}{2}}\right)\right)} where Γ(x) denotes the classical gamma function and ψ(x) denotes the digamma function. == References == | wikipedia |
quadratic algebra | In mathematics, a quadratic algebra is a filtered algebra generated by degree one elements, with defining relations of degree 2. It was pointed out by Yuri Manin that such algebras play an important role in the theory of quantum groups. The most important class of graded quadratic algebras is Koszul algebras. | wikipedia |
quadratic functions | In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before the 20th century, the distinction was unclear between a polynomial and its associated polynomial function; so "quadratic polynomial" and "quadratic function" were almost synonymous. This is still the case in many elementary courses, where both terms are often abbreviated as "quadratic". | wikipedia |
quadratic functions | For example, a univariate (single-variable) quadratic function has the form f ( x ) = a x 2 + b x + c , a ≠ 0 , {\displaystyle f(x)=ax^{2}+bx+c,\quad a\neq 0,} where x is its variable. The graph of a univariate quadratic function is a parabola, a curve that has an axis of symmetry parallel to the y-axis. If a quadratic function is equated with zero, then the result is a quadratic equation. | wikipedia |
quadratic functions | The solutions of a quadratic equation are the zeros of the corresponding quadratic function. The bivariate case in terms of variables x and y has the form f ( x , y ) = a x 2 + b x y + c y 2 + d x + e y + f , {\displaystyle f(x,y)=ax^{2}+bxy+cy^{2}+dx+ey+f,} with at least one of a, b, c not equal to zero. The zeros of this quadratic function is, in general (that is, if a certain expression of the coefficients is not equal to zero), a conic section (a circle or other ellipse, a parabola, or a hyperbola). | wikipedia |
quadratic functions | A quadratic function in three variables x, y, and z contains exclusively terms x2, y2, z2, xy, xz, yz, x, y, z, and a constant: f ( x , y , z ) = a x 2 + b y 2 + c z 2 + d x y + e x z + f y z + g x + h y + i z + j , {\displaystyle f(x,y,z)=ax^{2}+by^{2}+cz^{2}+dxy+exz+fyz+gx+hy+iz+j,} where at least one of the coefficients a, b, c, d, e, f of the second-degree terms is not zero. A quadratic function can have an arbitrarily large number of variables. The set of its zero form a quadric, which is a surface in the case of three variables and a hypersurface in general case. | wikipedia |
quadratic-linear algebra | In mathematics, a quadratic-linear algebra is an algebra over a field with a presentation such that all relations are sums of monomials of degrees 1 or 2 in the generators. They were introduced by Polishchuk and Positselski (2005, p.101). An example is the universal enveloping algebra of a Lie algebra, with generators a basis of the Lie algebra and relations of the form XY – YX – = 0. | wikipedia |
quadric (algebraic geometry) | In mathematics, a quadric or quadric hypersurface is the subspace of N-dimensional space defined by a polynomial equation of degree 2 over a field. Quadrics are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine space. | wikipedia |
quadric (algebraic geometry) | An example is the quadric surface x y = z w {\displaystyle xy=zw} in projective space P 3 {\displaystyle {\mathbf {P} }^{3}} over the complex numbers C. A quadric has a natural action of the orthogonal group, and so the study of quadrics can be considered as a descendant of Euclidean geometry. Many properties of quadrics hold more generally for projective homogeneous varieties. Another generalization of quadrics is provided by Fano varieties. | wikipedia |
quantum affine algebra | In mathematics, a quantum affine algebra (or affine quantum group) is a Hopf algebra that is a q-deformation of the universal enveloping algebra of an affine Lie algebra. They were introduced independently by Drinfeld (1985) and Jimbo (1985) as a special case of their general construction of a quantum group from a Cartan matrix. One of their principal applications has been to the theory of solvable lattice models in quantum statistical mechanics, where the Yang–Baxter equation occurs with a spectral parameter. Combinatorial aspects of the representation theory of quantum affine algebras can be described simply using crystal bases, which correspond to the degenerate case when the deformation parameter q vanishes and the Hamiltonian of the associated lattice model can be explicitly diagonalized. | wikipedia |
quantum groupoid | In mathematics, a quantum groupoid is any of a number of notions in noncommutative geometry analogous to the notion of groupoid. In usual geometry, the information of a groupoid can be contained in its monoidal category of representations (by a version of Tannaka–Krein duality), in its groupoid algebra or in the commutative Hopf algebroid of functions on the groupoid. Thus formalisms trying to capture quantum groupoids include certain classes of (autonomous) monoidal categories, Hopf algebroids etc. | wikipedia |
quantized enveloping algebra | In mathematics, a quantum or quantized enveloping algebra is a q-analog of a universal enveloping algebra. Given a Lie algebra g {\displaystyle {\mathfrak {g}}} , the quantum enveloping algebra is typically denoted as U q ( g ) {\displaystyle U_{q}({\mathfrak {g}})} . The notation was introduced by Drinfeld and independently by Jimbo.Among the applications, studying the q → 0 {\displaystyle q\to 0} limit led to the discovery of crystal bases. | wikipedia |
quasi-frobenius lie algebra | In mathematics, a quasi-Frobenius Lie algebra ( g , , β ) {\displaystyle ({\mathfrak {g}},,\beta )} over a field k {\displaystyle k} is a Lie algebra ( g , ) {\displaystyle ({\mathfrak {g}},)} equipped with a nondegenerate skew-symmetric bilinear form β: g × g → k {\displaystyle \beta :{\mathfrak {g}}\times {\mathfrak {g}}\to k} , which is a Lie algebra 2-cocycle of g {\displaystyle {\mathfrak {g}}} with values in k {\displaystyle k} . In other words, β ( , Z ) + β ( , Y ) + β ( , X ) = 0 {\displaystyle \beta \left(\left,Z\right)+\beta \left(\left,Y\right)+\beta \left(\left,X\right)=0} for all X {\displaystyle X} , Y {\displaystyle Y} , Z {\displaystyle Z} in g {\displaystyle {\mathfrak {g}}} . If β {\displaystyle \beta } is a coboundary, which means that there exists a linear form f: g → k {\displaystyle f:{\mathfrak {g}}\to k} such that β ( X , Y ) = f ( ) , {\displaystyle \beta (X,Y)=f(\left),} then ( g , , β ) {\displaystyle ({\mathfrak {g}},,\beta )} is called a Frobenius Lie algebra. | wikipedia |
quasi-lie algebra | In mathematics, a quasi-Lie algebra in abstract algebra is just like a Lie algebra, but with the usual axiom = 0 {\displaystyle =0} replaced by = − {\displaystyle =-} (anti-symmetry).In characteristic other than 2, these are equivalent (in the presence of bilinearity), so this distinction doesn't arise when considering real or complex Lie algebras. It can however become important, when considering Lie algebras over the integers. In a quasi-Lie algebra, 2 = 0. {\displaystyle 2=0.} Therefore, the bracket of any element with itself is 2-torsion, if it does not actually vanish. | wikipedia |
denjoy–carleman theorem | In mathematics, a quasi-analytic class of functions is a generalization of the class of real analytic functions based upon the following fact: If f is an analytic function on an interval ⊂ R, and at some point f and all of its derivatives are zero, then f is identically zero on all of . Quasi-analytic classes are broader classes of functions for which this statement still holds true. | wikipedia |
quasitopological space | In mathematics, a quasi-topology on a set X is a function that associates to every compact Hausdorff space C a collection of mappings from C to X satisfying certain natural conditions. A set with a quasi-topology is called a quasitopological space. They were introduced by Spanier, who showed that there is a natural quasi-topology on the space of continuous maps from one space to another. | wikipedia |
quasiconcave function | In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form ( − ∞ , a ) {\displaystyle (-\infty ,a)} is a convex set. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. The negative of a quasiconvex function is said to be quasiconcave. All convex functions are also quasiconvex, but not all quasiconvex functions are convex, so quasiconvexity is a generalization of convexity. Univariate unimodal functions are quasiconvex or quasiconcave, however this is not necessarily the case for functions with multiple arguments. For example, the 2-dimensional Rosenbrock function is unimodal but not quasiconvex and functions with star-convex sublevel sets can be unimodal without being quasiconvex. | wikipedia |
quasiperiodic function | In mathematics, a quasiperiodic function is a function that has a certain similarity to a periodic function. A function f {\displaystyle f} is quasiperiodic with quasiperiod ω {\displaystyle \omega } if f ( z + ω ) = g ( z , f ( z ) ) {\displaystyle f(z+\omega )=g(z,f(z))} , where g {\displaystyle g} is a "simpler" function than f {\displaystyle f} . What it means to be "simpler" is vague. A simple case (sometimes called arithmetic quasiperiodic) is if the function obeys the equation: f ( z + ω ) = f ( z ) + C {\displaystyle f(z+\omega )=f(z)+C} Another case (sometimes called geometric quasiperiodic) is if the function obeys the equation: f ( z + ω ) = C f ( z ) {\displaystyle f(z+\omega )=Cf(z)} An example of this is the Jacobi theta function, where ϑ ( z + τ ; τ ) = e − 2 π i z − π i τ ϑ ( z ; τ ) , {\displaystyle \vartheta (z+\tau ;\tau )=e^{-2\pi iz-\pi i\tau }\vartheta (z;\tau ),} shows that for fixed τ {\displaystyle \tau } it has quasiperiod τ {\displaystyle \tau } ; it also is periodic with period one. | wikipedia |
quasiperiodic function | Another example is provided by the Weierstrass sigma function, which is quasiperiodic in two independent quasiperiods, the periods of the corresponding Weierstrass ℘ function. Functions with an additive functional equation f ( z + ω ) = f ( z ) + a z + b {\displaystyle f(z+\omega )=f(z)+az+b\ } are also called quasiperiodic. An example of this is the Weierstrass zeta function, where ζ ( z + ω , Λ ) = ζ ( z , Λ ) + η ( ω , Λ ) {\displaystyle \zeta (z+\omega ,\Lambda )=\zeta (z,\Lambda )+\eta (\omega ,\Lambda )\ } for a z-independent η when ω is a period of the corresponding Weierstrass ℘ function. In the special case where f ( z + ω ) = f ( z ) {\displaystyle f(z+\omega )=f(z)\ } we say f is periodic with period ω in the period lattice Λ {\displaystyle \Lambda } . | wikipedia |
quaternion algebra | In mathematics, a quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes a matrix algebra by extending scalars (equivalently, tensoring with a field extension), i.e. for a suitable field extension K of F, A ⊗ F K {\displaystyle A\otimes _{F}K} is isomorphic to the 2 × 2 matrix algebra over K. The notion of a quaternion algebra can be seen as a generalization of Hamilton's quaternions to an arbitrary base field. The Hamilton quaternions are a quaternion algebra (in the above sense) over F = R {\displaystyle F=\mathbb {R} } , and indeed the only one over R {\displaystyle \mathbb {R} } apart from the 2 × 2 real matrix algebra, up to isomorphism. When F = C {\displaystyle F=\mathbb {C} } , then the biquaternions form the quaternion algebra over F. | wikipedia |
quintic equation | In mathematics, a quintic function is a function of the form g ( x ) = a x 5 + b x 4 + c x 3 + d x 2 + e x + f , {\displaystyle g(x)=ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+f,\,} where a, b, c, d, e and f are members of a field, typically the rational numbers, the real numbers or the complex numbers, and a is nonzero. In other words, a quintic function is defined by a polynomial of degree five. Because they have an odd degree, normal quintic functions appear similar to normal cubic functions when graphed, except they may possess one additional local maximum and one additional local minimum. | wikipedia |
quintic equation | The derivative of a quintic function is a quartic function. Setting g(x) = 0 and assuming a ≠ 0 produces a quintic equation of the form: a x 5 + b x 4 + c x 3 + d x 2 + e x + f = 0. {\displaystyle ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+f=0.\,} Solving quintic equations in terms of radicals (nth roots) was a major problem in algebra from the 16th century, when cubic and quartic equations were solved, until the first half of the 19th century, when the impossibility of such a general solution was proved with the Abel–Ruffini theorem. | wikipedia |
radial function | In mathematics, a radial function is a real-valued function defined on a Euclidean space Rn whose value at each point depends only on the distance between that point and the origin. The distance is usually the Euclidean distance. For example, a radial function Φ in two dimensions has the form Φ ( x , y ) = φ ( r ) , r = x 2 + y 2 {\displaystyle \Phi (x,y)=\varphi (r),\quad r={\sqrt {x^{2}+y^{2}}}} where φ is a function of a single non-negative real variable. Radial functions are contrasted with spherical functions, and any descent function (e.g., continuous and rapidly decreasing) on Euclidean space can be decomposed into a series consisting of radial and spherical parts: the solid spherical harmonic expansion. | wikipedia |
radial function | A function is radial if and only if it is invariant under all rotations leaving the origin fixed. That is, ƒ is radial if and only if f ∘ ρ = f {\displaystyle f\circ \rho =f\,} for all ρ ∈ SO(n), the special orthogonal group in n dimensions. This characterization of radial functions makes it possible also to define radial distributions. | wikipedia |
radial function | These are distributions S on Rn such that S = S {\displaystyle S=S} for every test function φ and rotation ρ. Given any (locally integrable) function ƒ, its radial part is given by averaging over spheres centered at the origin. To wit, ϕ ( x ) = 1 ω n − 1 ∫ S n − 1 f ( r x ′ ) d x ′ {\displaystyle \phi (x)={\frac {1}{\omega _{n-1}}}\int _{S^{n-1}}f(rx')\,dx'} where ωn−1 is the surface area of the (n−1)-sphere Sn−1, and r = |x|, x′ = x/r. It follows essentially by Fubini's theorem that a locally integrable function has a well-defined radial part at almost every r. The Fourier transform of a radial function is also radial, and so radial functions play a vital role in Fourier analysis. Furthermore, the Fourier transform of a radial function typically has stronger decay behavior at infinity than non-radial functions: for radial functions bounded in a neighborhood of the origin, the Fourier transform decays faster than R−(n−1)/2. The Bessel functions are a special class of radial function that arise naturally in Fourier analysis as the radial eigenfunctions of the Laplacian; as such they appear naturally as the radial portion of the Fourier transform. | wikipedia |
radially unbounded function | In mathematics, a radially unbounded function is a function f: R n → R {\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} } for which Or equivalently, Such functions are applied in control theory and required in optimization for determination of compact spaces. Notice that the norm used in the definition can be any norm defined on R n {\displaystyle \mathbb {R} ^{n}} , and that the behavior of the function along the axes does not necessarily reveal that it is radially unbounded or not; i.e. to be radially unbounded the condition must be verified along any path that results in: For example, the functions are not radially unbounded since along the line x 1 = x 2 {\displaystyle x_{1}=x_{2}} , the condition is not verified even though the second function is globally positive definite. == References == | wikipedia |
proper rational function | In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field K. In this case, one speaks of a rational function and a rational fraction over K. The values of the variables may be taken in any field L containing K. Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is L. The set of rational functions over a field K is a field, the field of fractions of the ring of the polynomial functions over K. | wikipedia |
read-once function | In mathematics, a read-once function is a special type of Boolean function that can be described by a Boolean expression in which each variable appears only once. More precisely, the expression is required to use only the operations of logical conjunction, logical disjunction, and negation. By applying De Morgan's laws, such an expression can be transformed into one in which negation is used only on individual variables (still with each variable appearing only once). By replacing each negated variable with a new positive variable representing its negation, such a function can be transformed into an equivalent positive read-once Boolean function, represented by a read-once expression without negations. | wikipedia |
uniform continuity | In mathematics, a real function f {\displaystyle f} of real numbers is said to be uniformly continuous if there is a positive real number δ {\displaystyle \delta } such that function values over any function domain interval of the size δ {\displaystyle \delta } are as close to each other as we want. In other words, for a uniformly continuous real function of real numbers, if we want function value differences to be less than any positive real number ϵ {\displaystyle \epsilon } , then there is a positive real number δ {\displaystyle \delta } such that | f ( x ) − f ( y ) | < ϵ {\displaystyle |f(x)-f(y)|<\epsilon } at any x {\displaystyle x} and y {\displaystyle y} in any function interval of the size δ {\displaystyle \delta } . The difference between uniform continuity and (ordinary) continuity is that, in uniform continuity there is a globally applicable δ {\displaystyle \delta } (the size of a function domain interval over which function value differences are less than ϵ {\displaystyle \epsilon } ) that depends on only ε {\displaystyle \varepsilon } , while in (ordinary) continuity there is a locally applicable δ {\displaystyle \delta } that depends on the both ε {\displaystyle \varepsilon } and x {\displaystyle x} . So uniform continuity is a stronger continuity condition than continuity; a function that is uniformly continuous is continuous but a function that is continuous is not necessarily uniformly continuous. | wikipedia |
uniform continuity | The concepts of uniform continuity and continuity can be expanded to functions defined between metric spaces. Continuous functions can fail to be uniformly continuous if they are unbounded on a bounded domain, such as f ( x ) = 1 x {\displaystyle f(x)={\tfrac {1}{x}}} on ( 0 , 1 ) {\displaystyle (0,1)} , or if their slopes become unbounded on an infinite domain, such as f ( x ) = x 2 {\displaystyle f(x)=x^{2}} on the real (number) line. | wikipedia |
uniform continuity | However, any Lipschitz map between metric spaces is uniformly continuous, in particular any isometry (distance-preserving map). Although continuity can be defined for functions between general topological spaces, defining uniform continuity requires more structure. The concept relies on comparing the sizes of neighbourhoods of distinct points, so it requires a metric space, or more generally a uniform space. | wikipedia |
singular function | In mathematics, a real-valued function f on the interval is said to be singular if it has the following properties: f is continuous on . (**) there exists a set N of measure 0 such that for all x outside of N the derivative f ′(x) exists and is zero, that is, the derivative of f vanishes almost everywhere. f is non-constant on .A standard example of a singular function is the Cantor function, which is sometimes called the devil's staircase (a term also used for singular functions in general). There are, however, other functions that have been given that name. | wikipedia |
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