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Cauchy's Functional Equation
Cauchy's functional equation is the functional equation: f(x+y) = f(x) + f(y).\ A function f that solves this equation is called an additive function. Over the rational numbers, it can be shown using elementary algebra that there is a single family of solutions, namely f \colon x\mapsto cx for any rational constant c. Over the real numbers, the family of linear maps f : x \mapsto cx, now with c an arbitrary real constant, is likewise a family of solutions; however there can exist other solutions not of this form that are extremely complicated. However, any of a number of regularity conditions, some of them quite weak, will preclude the existence of these pathological solutions. For example, an additive function f \colon \R\to\R is linear if: * f is continuous ( Cauchy, 1821). In fact, it suffices for f to be continuous at one point ( Darboux, 1875). * f(x)\ge0 or f(x)\le0 for all x\ge0. * f is monotonic on any interval. * f is bounded above or below on any interval. * f is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Equation
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 several values of the same function. For example, the logarithm functions are essentially characterized by the ''logarithmic functional equation'' \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. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamel Bases
In mathematics, a set of elements of a vector space is called a basis (: bases) if every element of can be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to . The elements of a basis are called . Equivalently, a set is a basis if its elements are linearly independent and every element of is a linear combination of elements of . In other words, a basis is a linearly independent spanning set. A vector space can have several bases; however all the bases have the same number of elements, called the dimension of the vector space. This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces. Basis vectors find applications in the study of crystal structures and frames of reference. Definition A basis of a vector space over a field (such as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zorn's Lemma
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least one maximal element. The lemma was proved (assuming the axiom of choice) by Kazimierz Kuratowski in 1922 and independently by Max Zorn in 1935. It occurs in the proofs of several theorems of crucial importance, for instance the Hahn–Banach theorem in functional analysis, the theorem that every vector space has a basis, Tychonoff's theorem in topology stating that every product of compact spaces is compact, and the theorems in abstract algebra that in a ring with identity every proper ideal is contained in a maximal ideal and that every field has an algebraic closure. Zorn's lemma is equivalent to the well-ordering theorem and also to the axiom of choice, in the sense that within ZF ( Zermelo–Fraenkel set theory without th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basis (linear Algebra)
In mathematics, a Set (mathematics), set of elements of a vector space is called a basis (: bases) if every element of can be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to . The elements of a basis are called . Equivalently, a set is a basis if its elements are linearly independent and every element of is a linear combination of elements of . In other words, a basis is a linearly independent spanning set. A vector space can have several bases; however all the bases have the same number of elements, called the dimension (vector space), dimension of the vector space. This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces. Basis vectors find applications in the study of crystal structures and frame of reference, frames of reference. De ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set (mathematics), set on which addition, subtraction, multiplication, and division (mathematics), division are defined and behave as the corresponding operations on rational number, rational and real numbers. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as field of rational functions, fields of rational functions, algebraic function fields, algebraic number fields, and p-adic number, ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many element (set), elements. The theory of fields proves that angle trisection and squaring the circle cannot be done with a compass and straighte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called scalar (mathematics), ''scalars''. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. Real vector spaces and complex vector spaces are kinds of vector spaces based on different kinds of scalars: real numbers and complex numbers. Scalars can also be, more generally, elements of any field (mathematics), field. Vector spaces generalize Euclidean vectors, which allow modeling of Physical quantity, physical quantities (such as forces and velocity) that have not only a Magnitude (mathematics), magnitude, but also a Orientation (geometry), direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix (mathematics), matrices, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Domain Of A Function
In mathematics, the domain of a function is the Set (mathematics), set of inputs accepted by the Function (mathematics), function. It is sometimes denoted by \operatorname(f) or \operatornamef, where is the function. In layman's terms, the domain of a function can generally be thought of as "what x can be". More precisely, given a function f\colon X\to Y, the domain of is . 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 sets of real numbers, 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'': the set to which all outputs must belong. The set of specific outputs the function assigns to elements of is called its ''Range of a function, range'' or ''Image (mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Of A Function
In mathematics, the graph of a function f is the set of ordered pairs (x, y), where f(x) = y. In the common case where x and f(x) are real numbers, these pairs are Cartesian coordinates of points in a plane (geometry), plane and often form a Plane curve, curve. The graphical representation of the graph of a Function (mathematics), function is also known as a ''Plot (graphics), plot''. In the case of Bivariate function, functions of two variables – that is, functions whose Domain of a function, domain consists of pairs (x, y) –, the graph usually refers to the set of ordered triples (x, y, z) where f(x,y) = z. This is a subset of three-dimensional space; for a continuous real-valued function of two real variables, its graph forms a Surface (mathematics), surface, which can be visualized as a ''surface plot (graphics), surface plot''. In science, engineering, technology, finance, and other areas, graphs are tools used for many purposes. In the simplest case one variable is p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pathological (mathematics)
In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition, it is sometimes called well-behaved or nice. These terms are sometimes useful in mathematical research and teaching, but there is no strict mathematical definition of pathological or well-behaved. In analysis A classic example of a pathology is the Weierstrass function, a function that is continuous everywhere but differentiable nowhere. The sum of a differentiable function and the Weierstrass function is again continuous but nowhere differentiable; so there are at least as many such functions as differentiable functions. In fact, using the Baire category theorem, one can show that continuous functions are generically nowhere differentiable. Such examples were deemed pathological when they were first discovered. To quote Henri Poincaré: Since Poincaré, nowhere differe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplicative Functional Equation
Multiplicative may refer to: *Multiplication *Multiplicative function *Multiplicative group *Multiplicative identity *Multiplicative inverse *Multiplicative order *Multiplicative partition *Multiplicative case * For the multiplicative numerals once, twice, and thrice, see English numerals English number words include numerals and various words derived from them, as well as a large number of words borrowed from other languages. Cardinal numbers Cardinal numbers refer to the size of a group. In English, these words are numerals. ... {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithmic Functional Equation
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Logarithmic can refer to: * Logarithm, a transcendental function in mathematics * Logarithmic scale, the use of the logarithmic function to describe measurements * Logarithmic spiral, * Logarithmic growth * Logarithmic distribution, a discrete probability distribution * Natural logarithm The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Functional Equation
Exponential may refer to any of several mathematical topics related to exponentiation, including: * Exponential function, also: **Matrix exponential, the matrix analogue to the above *Exponential decay, decrease at a rate proportional to value *Exponential discounting, a specific form of the discount function, used in the analysis of choice over time *Exponential growth, where the growth rate of a mathematical function is proportional to the function's current value *Exponential map (Riemannian geometry), in Riemannian geometry *Exponential map (Lie theory), in Lie theory * Exponential notation, also known as scientific notation, or standard form *Exponential object, in category theory *Exponential time, in complexity theory *in probability and statistics: **Exponential distribution, a family of continuous probability distributions **Exponentially modified Gaussian distribution, describes the sum of independent normal and exponential random variables **Exponential family, a parametr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
