|
Liouville–Neumann Series
In mathematics, the Liouville–Neumann series is a function series that results from applying the resolvent formalism to solve Fredholm integral equations in Fredholm theory. Definition The Liouville–Neumann series is defined as :\phi\left(x\right) = \sum^\infty_ \lambda^n \phi_n \left(x\right) which, provided that \lambda is small enough so that the series converges, is the unique continuous solution of the Fredholm integral equation of the second kind, If the ''n''th iterated kernel is defined as ''n''−1 nested integrals of ''n'' operator kernels , :K_n\left(x,z\right) = \int\int\cdots\int K\left(x,y_1\right)K\left(y_1,y_2\right) \cdots K\left(y_, z\right) dy_1 dy_2 \cdots dy_ then :\phi_n\left(x\right) = \int K_n\left(x,z\right)f\left(z\right)dz with :\phi_0\left(x\right) = f\left(x\right)~, so ''K''0 may be taken to be , the kernel of the identity operator. The resolvent, also called the "solution kernel" for the integral operator, is then given by a generalization of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Resolvent Formalism
In mathematics, the resolvent formalism is a technique for applying concepts from complex analysis to the study of the spectrum of operators on Banach spaces and more general spaces. Formal justification for the manipulations can be found in the framework of holomorphic functional calculus. The resolvent captures the spectral properties of an operator in the analytic structure of the functional. Given an operator , the resolvent may be defined as : R(z;A)= (A-zI)^~. Among other uses, the resolvent may be used to solve the inhomogeneous Fredholm integral equations; a commonly used approach is a series solution, the Liouville–Neumann series. The resolvent of can be used to directly obtain information about the spectral decomposition of . For example, suppose is an isolated eigenvalue in the spectrum of . That is, suppose there exists a simple closed curve C_\lambda in the complex plane that separates from the rest of the spectrum of . Then the residue : -\frac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fredholm Integral Equation
In mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators. The integral equation was studied by Ivar Fredholm. A useful method to solve such equations, the Adomian decomposition method, is due to George Adomian. Equation of the first kind A Fredholm equation is an integral equation in which the term containing the kernel function (defined below) has constants as integration limits. A closely related form is the Volterra integral equation which has variable integral limits. An inhomogeneous Fredholm equation of the first kind is written as and the problem is, given the continuous kernel function K and the function g, to find the function f. An important case of these types of equation is the case when the kernel is a function only of the difference of its arguments, namely K(t,s)=K(t-s), and the limits of integration are ±∞, then the right hand side of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fredholm Theory
In mathematics, Fredholm theory is a theory of integral equations. In the narrowest sense, Fredholm theory concerns itself with the solution of the Fredholm integral equation. In a broader sense, the abstract structure of Fredholm's theory is given in terms of the spectral theory of Fredholm operators and Fredholm kernels on Hilbert space. It therefore forms a branch of operator theory and functional analysis. The theory is named in honour of Erik Ivar Fredholm. Fredholm equation of the first kind Much of Fredholm theory concerns itself with the following integral equation for ''f'' when ''g'' and ''K'' are given: :g(x)=\int_a^b K(x,y) f(y)\,dy. This equation arises naturally in many problems in physics and mathematics, as the inverse of a differential equation. That is, one is asked to solve the differential equation :Lg(x)=f(x) where the function is given and is unknown. Here, stands for a linear differential operator. For example, one might take to be an elliptic op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Until the 19th century, mathematicians largely relied on intuitive notions of continuity and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the most general continuous functions, and their d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kernel (integral Operator)
In mathematics, an integral transform is a type of transform that maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in the original function space. The transformed function can generally be mapped back to the original function space using the ''inverse transform''. General form An integral transform is any transform ''T'' of the following form: :(Tf)(u) = \int_^ f(t)\, K(t, u)\, dt The input of this transform is a function ''f'', and the output is another function ''Tf''. An integral transform is a particular kind of mathematical operator. There are numerous useful integral transforms. Each is specified by a choice of the function K of two variables, that is called the kernel or nucleus of the transform. Some kernels have an associated ''inverse kernel'' K^( u,t ) which (roughly speaking) yields an inverse transform: :f( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Operator
Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged. That is, when is the identity function, the equality is true for all values of to which can be applied. Definition Formally, if is a set, the identity function on is defined to be a function with as its domain and codomain, satisfying In other words, the function value in the codomain is always the same as the input element in the domain . The identity function on is clearly an injective function as well as a surjective function (its codomain is also its range), so it is bijective. The identity function on is often denoted by . In set theory, where a function is defined as a particular kind of binary relation, the identity function is given by the identity relation, or ''diagonal'' of . Algebraic properties If ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Resolvent Formalism
In mathematics, the resolvent formalism is a technique for applying concepts from complex analysis to the study of the spectrum of operators on Banach spaces and more general spaces. Formal justification for the manipulations can be found in the framework of holomorphic functional calculus. The resolvent captures the spectral properties of an operator in the analytic structure of the functional. Given an operator , the resolvent may be defined as : R(z;A)= (A-zI)^~. Among other uses, the resolvent may be used to solve the inhomogeneous Fredholm integral equations; a commonly used approach is a series solution, the Liouville–Neumann series. The resolvent of can be used to directly obtain information about the spectral decomposition of . For example, suppose is an isolated eigenvalue in the spectrum of . That is, suppose there exists a simple closed curve C_\lambda in the complex plane that separates from the rest of the spectrum of . Then the residue : -\frac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Series
In mathematics, a geometric series is a series (mathematics), series summing the terms of an infinite geometric sequence, in which the ratio of consecutive terms is constant. For example, 1/2 + 1/4 + 1/8 + 1/16 + ⋯, the series \tfrac12 + \tfrac14 + \tfrac18 + \cdots is a geometric series with common ratio , which converges to the sum of . Each term in a geometric series is the geometric mean of the term before it and the term after it, in the same way that each term of an arithmetic series is the arithmetic mean of its neighbors. While Ancient Greek philosophy, Greek philosopher Zeno's paradoxes about time and motion (5th century BCE) have been interpreted as involving geometric series, such series were formally studied and applied a century or two later by Greek mathematics, Greek mathematicians, for example used by Archimedes to Quadrature of the Parabola, calculate the area inside a parabola (3rd century BCE). Today, geometric series are used in mathematical finance, calculati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Volterra Integral Equation
In mathematics, the Volterra integral equations are a special type of integral equations. They are divided into two groups referred to as the first and the second kind. A linear Volterra equation of the first kind is : f(t) = \int_a^t K(t,s)\,x(s)\,ds where ''f'' is a given function and ''x'' is an unknown function to be solved for. A linear Volterra equation of the second kind is : x(t) = f(t) + \int_a^t K(t,s)x(s)\,ds. In operator theory, and in Fredholm theory, the corresponding operators are called Volterra operators. A useful method to solve such equations, the Adomian decomposition method, is due to George Adomian. A linear Volterra integral equation is a convolution equation if : x(t) = f(t) + \int_^t K(t-s)x(s)\,ds. The function K in the integral is called the kernel. Such equations can be analyzed and solved by means of Laplace transform techniques. For a weakly singular kernel of the form K(t,s) = (t^2-s^2)^ with 0Defining x_ = x(s_), f_ = f(t_), and K_ = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neumann Series
A Neumann series is a mathematical series that sums ''k''-times repeated applications of an operator T . This has the generator form : \sum_^\infty T^k where T^k is the ''k''-times repeated application of T ; T^0 is the identity operator I and T^k := T^\circ for k > 0 . This is a special case of the generalization of a geometric series of real or complex numbers to a geometric series of operators. The generalized initial term of the series is the identity operator T^0 = I and the generalized common ratio of the series is the operator T. The series is named after the mathematician Carl Neumann, who used it in 1877 in the context of potential theory. The Neumann series is used in functional analysis. It is closely connected to the resolvent formalism for studying the spectrum of bounded operators and, applied from the left to a function, it forms the Liouville-Neumann series that formally solves Fredholm integral equations. Properties Suppose that T is a bounded linear o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Series (mathematics)
In mathematics, a series is, roughly speaking, an addition of Infinity, infinitely many Addition#Terms, terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures in combinatorics through generating functions. The mathematical properties of infinite series make them widely applicable in other quantitative disciplines such as physics, computer science, statistics and finance. Among the Ancient Greece, Ancient Greeks, the idea that a potential infinity, potentially infinite summation could produce a finite result was considered paradoxical, most famously in Zeno's paradoxes. Nonetheless, infinite series were applied practically by Ancient Greek mathematicians including Archimedes, for instance in the Quadrature of the Parabola, quadrature of the parabola. The mathematical side of Zeno's paradoxes was resolved using the concept of a limit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
