Effective Domain
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Effective Domain
In convex analysis Convex analysis is the branch of mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These ..., a branch of mathematics, the effective domain is an extension of the domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f) or \operatornamef, where is the function. More precisely, given a fu ... defined for functions that take values in the extended real number line In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra on ... \infty, \infty= \mathbb \cup \. In convex analysis ...
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Convex Analysis
Convex analysis is the branch of mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ... devoted to the study of properties of convex function In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of a function, graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigra ...s and convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the Real number, reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set o ...s, often with applications in convex minimization, a subdomain of optimization theory Mathematical optimization ( ...
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Domain Of A Function
In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f) or \operatornamef, where is the function. More precisely, given a function f\colon X\to Y, the domain of is . Note that in modern mathematical language, the domain is part of the definition of a function rather than a property of it. In the special case that and are both subsets of \R, the function can be graphed in the Cartesian coordinate system. In this case, the domain is represented on the -axis of the graph, as the projection of the graph of the function onto the -axis. For a function f\colon X\to Y, the set is called the codomain, and the set of values attained by the function (which is a subset of ) is called its Range of a function, range or Image (mathematics), image. Any function can be restricted to a subset of its domain. The Restriction (mathematics), restriction of f \colon X \to Y to ...
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Extended Real Number Line
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra on infinities and the various Limit of a function, limiting behaviors in calculus and mathematical analysis, especially in the theory of Measure (mathematics), measure and Integral, integration. The affinely extended real number system is denoted \overline or [-\infty, +\infty] or It is the Dedekind–MacNeille completion of the real numbers. When the meaning is clear from context, the symbol +\infty is often written simply as Motivation Limits It is often useful to describe the behavior of a function f, as either the argument x or the function value f gets "infinitely large" in some sense. For example, consider the function f defined by :f(x) = \frac. The graph of this function has a horizontal asymptote at y = 0. Geometrically, when m ...
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Variational Analysis
In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ..., the term variational analysis usually denotes the combination and extension of methods from convex optimization Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization proble ... and the classical calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of f ... to a more general theory. This includes the more general problems of ...
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Extended Real Number Line
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra on infinities and the various Limit of a function, limiting behaviors in calculus and mathematical analysis, especially in the theory of Measure (mathematics), measure and Integral, integration. The affinely extended real number system is denoted \overline or [-\infty, +\infty] or It is the Dedekind–MacNeille completion of the real numbers. When the meaning is clear from context, the symbol +\infty is often written simply as Motivation Limits It is often useful to describe the behavior of a function f, as either the argument x or the function value f gets "infinitely large" in some sense. For example, consider the function f defined by :f(x) = \frac. The graph of this function has a horizontal asymptote at y = 0. Geometrically, when m ...
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Global Minimum Point
In mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ..., the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range Range may refer to: Geography * Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra) ** Mountain range, a group of mountains bordered by lowlands * Range, a term used to i ... (the ''local'' or ''relative'' extrema), or on the entire domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Na ...
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Concave Function
In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ..., a concave function is the negative of a convex function. A concave function is also synonym A synonym is a word, morpheme, or phrase In syntax and grammar, a phrase is a group of words or singular word acting as a grammatical unit. For instance, the English language, English expression "the very happy squirrel" is a noun phrase whi ...ously called concave downwards, concave down, convex upwards, convex cap, or upper convex. Definition A real-valued function f on an interval (or, more generally, a convex set in vector space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, ...
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Canonical Projection
In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ..., when the elements of some set S have a notion of equivalence (formalized as an equivalence relation In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the same equivalence class if, and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either bot ...
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Image (function)
In mathematics, the image of a Function (mathematics), function is the set of all output values it may produce. More generally, evaluating a given function f at each Element (mathematics), element of a given subset A of its Domain of a function, domain produces a set, called the "image of A under (or through) f". Similarly, the inverse image (or preimage) of a given subset B of the codomain of f, is the set of all elements of the domain that map to the members of B. Image and inverse image may also be defined for general Binary relation#Operations on binary relations, binary relations, not just functions. Definition The word "image" is used in three related ways. In these definitions, f : X \to Y is a Function (mathematics), function from the Set (mathematics), set X to the set Y. Image of an element If x is a member of X, then the image of x under f, denoted f(x), is the Value (mathematics), value of f when applied to x. f(x) is alternatively known as the output of f for a ...
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Epigraph (mathematics)
In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ..., the epigraph or supergraph of a function f : X \to \infty, \infty/math> valued in the extended real numbers \infty, \infty= \R \cup \ is the set, denoted by \operatorname f, of all points in the Cartesian product In mathematics, specifically set theory, the Cartesian product of two Set (mathematics), sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notatio ... X \times \R lying on or above its graph. The strict epigraph \operatorname_S f is the set of points in X \times \R lying strictly above its graph. Importantly, although both the graph and epigraph of f consists of points in X \times \infty, \infty th ...
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Hypograph (mathematics)
In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ..., the hypograph or subgraph of a function f:\R^\rightarrow \R is the set of points lying on or below its graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre .... A related definition is that of such a function's epigraph, which is the set of points on or above the function's graph. The domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomo ...
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Real Number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limit (mathematics), limits, continuous function, continuity and derivatives. The set of real numbers is mathematical notation, denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers subset, include t ...
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