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Hilbert's Nineteenth Problem
Hilbert's nineteenth problem is one of the 23 Hilbert problems, set out in a list compiled by David Hilbert in 1900. It asks whether the solutions of regular problems in the calculus of variations are always analytic. Informally, and perhaps less directly, since Hilbert's concept of a "''regular variational problem''" identifies this precisely as a variational problem whose Euler–Lagrange equation is an elliptic partial differential equation with analytic coefficients, Hilbert's nineteenth problem, despite its seemingly technical statement, simply asks whether, in this class of partial differential equations, any solution inherits the relatively simple and well understood property of being an analytic function from the equation it satisfies. Hilbert's nineteenth problem was solved independently in the late 1950s by Ennio De Giorgi and John Forbes Nash, Jr. History The origins of the problem David Hilbert presented what is now called his nineteenth problem in his speech at th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hilbert Problems
Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21, and 22) at the Paris conference of the International Congress of Mathematicians, speaking on August 8 at the Sorbonne. The complete list of 23 problems was published later, in English translation in 1902 by Mary Frances Winston Newson in the ''Bulletin of the American Mathematical Society''. Earlier publications (in the original German) appeared in ''Archiv der Mathematik und Physik''. and Of the cleanly formulated Hilbert problems, numbers 3, 7, 10, 14, 17, 18, 19, 21, and 20 have resolutions that are accepted by consensus of the mathematical community. Problems 1, 2, 5, 6, 9, 11, 12, 15, and 22 have solutions that have partial acceptance, but there exists some controversy as to whether ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Gaussian Curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a smooth Surface (topology), surface in three-dimensional space at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. For example, a sphere of radius has Gaussian curvature everywhere, and a flat plane and a cylinder have Gaussian curvature zero everywhere. The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus. Gaussian curvature is an ''intrinsic'' measure of curvature, meaning that it could in principle be measured by a 2-dimensional being living entirely within the surface, because it depends only on distances that are measured “within” or along the surface, not on the way it is isometrically embedding, embedded in Euclidean space. This is the content of the ''Theorema Egregium''. Gaussian curvature is named after Carl Friedrich Gauss, who published the ''Theorema Egregium'' in 1827. Informal definit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Regularity Problem
Regularity is a topic of the mathematical study of partial differential equations (PDE) such as Laplace's equation, about the integrability and differentiability of weak solutions. Hilbert's nineteenth problem was concerned with this concept. The motivation for this study is as follows. It is often difficult to construct a classical solution satisfying the PDE in regular sense, so we search for a weak solution at first, and then find out whether the weak solution is smooth enough to be qualified as a classical solution. Several theorems have been proposed for different types of PDEs. Elliptic regularity theory Let U be an open, bounded subset of \mathbb^n, denote its boundary as \partial U and the variables as x=(x_1,...,x_n). Representing the PDE as a partial differential operator L acting on an unknown function u=u(x) of x\in U results in a BVP of the form \left\{ \begin{align} L u &= f & &\text{in } U\\ u &=0 & &\text{on } \partial U, \end{align}\right. where f: U \r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Potential Theory
In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that the two fundamental forces of nature known at the time, namely gravity and the electrostatic force, could be modeled using functions called the gravitational potential and electrostatic potential, both of which satisfy Poisson's equation—or in the vacuum, Laplace's equation. There is considerable overlap between potential theory and the theory of Poisson's equation to the extent that it is impossible to draw a distinction between these two fields. The difference is more one of emphasis than subject matter and rests on the following distinction: potential theory focuses on the properties of the functions as opposed to the properties of the equation. For example, a result about the Mathematical singularity, singularities of harmonic functions would be said to belong to potential theory whilst a result ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Italic Type
In typography, italic type is a cursive font based on a stylised form of calligraphic handwriting. Along with blackletter and roman type, it served as one of the major typefaces in the history of Western typography. Owing to the influence from calligraphy, italics normally slant slightly to the right, ''like so''. Different glyph shapes from roman type are usually usedanother influence from calligraphyand upper-case letters may have Swash (typography), swashes, flourishes inspired by ornate calligraphy. Historically, italics were a distinct style of type used entirely separately from roman type, but they have come to be used in conjunction—most fonts now come with a roman type and an oblique type, oblique version (generally called "italic" though often not true italics). In this usage, italics are a way to emphasise key points in a printed text, to identify many types of creative works, to cite foreign words or phrases, or, when quoting a speaker, a way to show which w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hessian Determinant
In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named after him. Hesse originally used the term "functional determinants". The Hessian is sometimes denoted by H or \nabla\nabla or \nabla^2 or \nabla\otimes\nabla or D^2. Definitions and properties Suppose f : \R^n \to \R is a function taking as input a vector \mathbf \in \R^n and outputting a scalar f(\mathbf) \in \R. If all second-order partial derivatives of f exist, then the Hessian matrix \mathbf of f is a square n \times n matrix, usually defined and arranged as \mathbf H_f= \begin \dfrac & \dfrac & \cdots & \dfrac \\ .2ex \dfrac & \dfrac & \cdots & \dfrac \\ .2ex \vdots & \vdots & \ddots & \vdots \\ .2ex \dfrac & \d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Strong Derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. The process of finding a derivative is called differentiation. There are multiple different notations for differentiation. ''Leibniz notation'', named after Gottfried Wilhelm Leibniz, is represented as the ratio of two differentials, whereas ''prime notation'' is written by adding a prime mark. Higher order notations represent repeated differentiation, and they are usually denoted in Leibniz n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Weak Derivative
In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (''strong derivative'') for functions not assumed differentiable, but only integrable, i.e., to lie in the L''p'' space L^1( ,b. The method of integration by parts holds that for smooth functions u and \varphi we have :\begin \int_a^b u(x) \varphi'(x) \, dx & = \Big (x) \varphi(x)\Biga^b - \int_a^b u'(x) \varphi(x) \, dx. \\ pt \end A function ''u''' being the weak derivative of ''u'' is essentially defined by the requirement that this equation must hold for all smooth functions \varphi vanishing at the boundary points (\varphi(a)=\varphi(b)=0). Definition Let u be a function in the Lebesgue space L^1( ,b. We say that v in L^1( ,b is a weak derivative of u if :\int_a^b u(t)\varphi'(t) \, dt=-\int_a^b v(t)\varphi(t) \, dt for ''all'' infinitely differentiable functions \varphi with \varphi(a)=\varphi(b)=0. Generalizing to n dimensions, if u and v are in the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. The process of finding a derivative is called differentiation. There are multiple different notations for differentiation. '' Leibniz notation'', named after Gottfried Wilhelm Leibniz, is represented as the ratio of two differentials, whereas ''prime notation'' is written by adding a prime mark. Higher order notations represent repeated differentiation, and they are usually denoted in Leib ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Functional (mathematics)
In mathematics, a functional is a certain type of function. The exact definition of the term varies depending on the subfield (and sometimes even the author). * In linear algebra, it is synonymous with a linear form, which is a linear mapping from a vector space V into its field of scalars (that is, it is an element of the dual space V^*) "Let ''E'' be a free module over a commutative ring ''A''. We view ''A'' as a free module of rank 1 over itself. By the dual module ''E''∨ of ''E'' we shall mean the module Hom(''E'', ''A''). Its elements will be called functionals. Thus a functional on ''E'' is an ''A''-linear map ''f'' : ''E'' → ''A''." * In functional analysis and related fields, it refers to a mapping from a space X into the field of real or complex numbers. "A numerical function ''f''(''x'') defined on a normed linear space ''R'' will be called a ''functional''. A functional ''f''(''x'') is said to be ''linear'' if ''f''(α''x'' + β''y'') = α''f''(''x'') + β ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Elliptic Partial Differential Equation
In mathematics, an elliptic partial differential equation is a type of partial differential equation (PDE). In mathematical modeling, elliptic PDEs are frequently used to model steady states, unlike parabolic PDE and hyperbolic PDE which generally model phenomena that change in time. The canonical examples of elliptic PDEs are Laplace's Equation and Poisson's Equation. Elliptic PDEs are also important in pure mathematics, where they are fundamental to various fields of research such as differential geometry and optimal transport. Definition Elliptic differential equations appear in many different contexts and levels of generality. First consider a second-order linear PDE for an unknown function of two variables u = u(x,y), written in the form Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0, where , , , , , , and are functions of (x,y), using subscript notation for the partial derivatives. The PDE is called elliptic if B^2-AC 0 are hyperbolic. For a general linear second-order ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
