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Universal Differential Equation
A universal differential equation (UDE) is a non-trivial differential algebraic equation with the property that its solutions can approximate any continuous function on any interval of the real line to any desired level of accuracy. Precisely, a (possibly implicit) differential equation P(y', y'', y, ..., y^) = 0 is a UDE if for any continuous real-valued function f and for any positive continuous function \varepsilon there exist a smooth solution y of P(y', y'', y, ..., y^) = 0 with , y(x) - f(x), 3. * Briggs proposed another family of UDEs whose construction is based on Jacobi elliptic functions: :y^ y^-3 y^ y^ y^+2\left(1-n^\right) y^=0, where ''n'' > 3. * Bournez and Pouly proved the existence of a fixed polynomial vector field ''p'' such that for any ''f'' and ''ε'' there exists some initial condition of the differential equation y' = p(y) that yields a unique and analytic solution satisfying , ''y''(''x'') − ''f''(''x''), < ''ε''(''x'') for all ''x'' in R. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Algebraic Equation
In mathematics, a differential-algebraic system of equations (DAE) is a system of equations that either contains differential equations and algebraic equations, or is equivalent to such a system. The set of the solutions of such a system is a ''differential algebraic variety'', and corresponds to an ideal in a differential algebra of differential polynomials. In the univariate case, a DAE in the variable ''t'' can be written as a single equation of the form :F(\dot x, x, t)=0, where x(t) is a vector of unknown functions and the overdot denotes the time derivative, i.e., \dot x = \frac. They are distinct from ordinary differential equation (ODE) in that a DAE is not completely solvable for the derivatives of all components of the function ''x'' because these may not all appear (i.e. some equations are algebraic); technically the distinction between an implicit ODE system hat may be rendered explicitand a DAE system is that the Jacobian matrix \frac is a singular matrix ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximation Theory
In mathematics, approximation theory is concerned with how function (mathematics), functions can best be approximation, approximated with simpler functions, and with quantitative property, quantitatively characterization (mathematics), characterizing the approximation error, errors introduced thereby. What is meant by ''best'' and ''simpler'' will depend on the application. A closely related topic is the approximation of functions by generalized Fourier series, that is, approximations based upon summation of a series of terms based upon orthogonal polynomials. One problem of particular interest is that of approximating a function in a computer mathematical library, using operations that can be performed on the computer or calculator (e.g. addition and multiplication), such that the result is as close to the actual function as possible. This is typically done with polynomial or Rational function, rational (ratio of polynomials) approximations. The objective is to make the approxi ... [...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] |
Smooth Function
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain. A function of class C^k is a function of smoothness at least ; that is, a function of class C^k is a function that has a th derivative that is continuous in its domain. A function of class C^\infty or C^\infty-function (pronounced C-infinity function) is an infinitely differentiable function, that is, a function that has derivatives of all orders (this implies that all these derivatives are continuous). Generally, the term smooth function refers to a C^-function. However, it may also mean "sufficiently differentiable" for the problem under consideration. Differentiability classes Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Turing Machine
In computer science, a universal Turing machine (UTM) is a Turing machine capable of computing any computable sequence, as described by Alan Turing in his seminal paper "On Computable Numbers, with an Application to the Entscheidungsproblem". Common sense might say that a universal machine is impossible, but Turing proves that it is possible. He suggested that we may compare a human in the process of computing a real number to a machine which is only capable of a finite number of conditions ; which will be called "-configurations". He then described the operation of such machine, as described below, and argued: Turing introduced the idea of such a machine in 1936–1937. Introduction Martin Davis makes a persuasive argument that Turing's conception of what is now known as "the stored-program computer", of placing the "action table"—the instructions for the machine—in the same "memory" as the input data, strongly influenced John von Neumann's conception of the first Amer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Purpose Analog Computer
The general purpose analog computer (GPAC) is a mathematical model of analog computers first introduced in 1941 by Claude Shannon. This model consists of circuits where several basic units are interconnected in order to compute some function. The GPAC can be implemented in practice through the use of mechanical devices or analog electronics. Although analog computers have fallen almost into oblivion due to emergence of the digital computer, the GPAC has recently been studied as a way to provide evidence for the physical Church–Turing thesis. This is because the GPAC is also known to model a large class of dynamical systems defined with ordinary differential equations, which appear frequently in the context of physics. In particular it was shown in 2007 that (a deterministic variant of) the GPAC is equivalent, in computability terms, to Turing machines, thereby proving the physical Church–Turing thesis for the class of systems modelled by the GPAC. This was recently strengthened ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Elliptic Functions
In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum, as well as in the design of electronic elliptic filters. While trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation \operatorname for \sin. The Jacobi elliptic functions are used more often in practical problems than the Weierstrass elliptic functions as they do not require notions of complex analysis to be defined and/or understood. They were introduced by . Carl Friedrich Gauss had already studied special Jacobi elliptic functions in 1797, the lemniscate elliptic functions in particular, but his work was published much later. Overview There are twelve Jacobi elliptic functions denoted by \operatorna ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zeta Function Universality
In mathematics, the universality of zeta functions is the remarkable ability of the Riemann zeta function and other similar functions (such as the Dirichlet L-functions) to approximate arbitrary non-vanishing holomorphic functions arbitrarily well. The universality of the Riemann zeta function was first proven by in 1975 and is sometimes known as Voronin's universality theorem. Formal statement A mathematically precise statement of universality for the Riemann zeta function follows. Let be a compact subset of the strip :\left\ such that the complement of is connected. Let be a continuous function on which is holomorphic on the interior of and does not have any zeros in . Then for any there exists a such that for all \ s \in U ~. Even more: The lower density of the set of values satisfying the above inequality is positive. Precisely \ 0 ~ < ~ \liminf_ ~ \frac \ \lambda\!\left( \left\ \right)\ , where is the |
Hölder's Theorem
In mathematics, Hölder's theorem states that the gamma function does not satisfy any algebraic differential equation whose coefficients are rational functions. This result was first proved by Otto Hölder in 1887; several alternative proofs have subsequently been found. The theorem also generalizes to the q -gamma function. Statement of the theorem For every n \in \N_0, there is no non-zero polynomial P \in \Complex ;Y_0,Y_1,\ldots,Y_n such that \forall z \in \Complex \setminus \Z _: \qquad P \left( z;\Gamma(z),\Gamma'(z),\ldots,(z) \right) = 0, where \Gamma is the gamma function. For example, define P \in \Complex ;Y_0,Y_1,Y_2 by P ~ \stackrel ~ X^2 Y_2 + X Y_1 + (X^2 - \nu^2) Y_0. Then the equation P \left (z;f(z),f'(z),f''(z) \right ) = z^2 f''(z) + z f'(z) + \left (z^2 - \nu^2 \right ) f(z) \equiv 0 is called an ''algebraic differential equation'', which, in this case, has the solutions f = J_ and f = Y_ — the Bessel functions of the first and second kind respec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
