|
Bateman Transform
In the mathematical study of partial differential equations, the Bateman transform is a method for solving the Laplace equation in four dimensions and wave equation in three by using a line integral of a holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ... in three complex variables. It is named after the mathematician Harry Bateman, who first published the result in . The formula asserts that if ''ƒ'' is a holomorphic function of three complex variables, then : \phi(w, x, y, z) = \oint_\gamma f\big((w + ix) + (iy + z)\zeta, (iy - z) + (w - ix)\zeta, \zeta\big) \,d\zeta is a solution of the Laplace equation, which follows by differentiation under the integral. Furthermore, Bateman asserted that the most general solution of the Laplace equation arises in this ... [...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] |
Partial Differential Equations
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how is thought of as an unknown number solving, e.g., an algebraic equation like . However, it is usually impossible to write down explicit formulae for solutions of partial differential equations. There is correspondingly a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity and stability. Among the many open questions are the existence an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace Equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties in 1786. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \nabla \cdot \nabla = \nabla^2 is the Laplace operator,The delta symbol, Δ, is also commonly used to represent a finite change in some quantity, for example, \Delta x = x_1 - x_2. Its use to represent the Laplacian should not be confused with this use. \nabla \cdot is the divergence operator (also symbolized "div"), \nabla is the gradient operator (also symbolized "grad"), and f (x, y, z) is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function. If the right-hand side is specified as a given function, h(x, y, z), we have \Delta f = h This is called Poisson's equation, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wave Equation
The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves). It arises in fields like acoustics, electromagnetism, and fluid dynamics. This article focuses on waves in classical physics. Quantum physics uses an operator-based wave equation often as a relativistic wave equation. Introduction The wave equation is a hyperbolic partial differential equation describing waves, including traveling and standing waves; the latter can be considered as linear superpositions of waves traveling in opposite directions. This article mostly focuses on the scalar wave equation describing waves in scalars by scalar functions of a time variable (a variable representing time) and one or more spatial variables (variables representing a position in a space under discussion). At the same time, there a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Integral
In mathematics, a line integral is an integral where the function (mathematics), function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, although that is typically reserved for #Complex line integral, line integrals in the complex plane. The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the Dot product, scalar product of the vector field with a Differential (infinitesimal), differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on interval (mathematics), intervals. Many simple formulae in physics, such as the definition of Work (physics), work as have natural continuous analogues in terms of l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Holomorphic Function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivative in a neighbourhood is a very strong condition: It implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (is '' analytic''). Holomorphic functions are the central objects of study in complex analysis. Though the term '' analytic function'' is often used interchangeably with "holomorphic function", the word "analytic" is defined in a broader sense to denote any function (real, complex, or of more general type) that can be written as a convergent power series in a neighbourhood of each point in its domain. That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis. Holomorphic functions are also sometimes referred to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harry Bateman
Harry Bateman FRS (29 May 1882 – 21 January 1946) was an English mathematician with a specialty in differential equations of mathematical physics. With Ebenezer Cunningham, he expanded the views of spacetime symmetry of Lorentz and Poincare to a more expansive conformal group of spacetime leaving Maxwell's equations invariant. Moving to the US, he obtained a Ph.D. in geometry with Frank Morley and became a professor of mathematics at California Institute of Technology. There he taught fluid dynamics to students going into aerodynamics with Theodore von Karman. Bateman made a broad survey of applied differential equations in his Gibbs Lecture in 1943 titled, "The control of an elastic fluid". Biography Bateman was born in Manchester, England, on 29 May 1882. He first gained an interest in mathematics during his time at Manchester Grammar School. In his final year, he won a scholarship to Trinity College, Cambridge. Bateman studied with coach Robert Alfred Herman to prepa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical Society and the Operational Research Society (ORS). History The Society was established on 16 January 1865, the first president being Augustus De Morgan. The earliest meetings were held in University College, but the Society soon moved into Burlington House, Piccadilly. The initial activities of the Society included talks and publication of a journal. The LMS was used as a model for the establishment of the American Mathematical Society in 1888. Mary Cartwright was the first woman to be President of the LMS (in 1961–62). The Society was granted a royal charter in 1965, a century after its foundation. In 1998 the Society moved from rooms in Burlington House into De Morgan House (named after the society's first president), at 57–5 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Sciences Research Institute
The Simons Laufer Mathematical Sciences Institute (SLMath), formerly the Mathematical Sciences Research Institute (MSRI), is an independent nonprofit mathematical research institution on the University of California campus in Berkeley, California. It is a mathematical center for collaborative research, drawing thousands of researchers each year. The institute was founded in 1982, and its funding sources include the National Science Foundation, private foundations, corporations, and more than 90 universities and institutions. The institute is located at 17 Gauss Way on the Berkeley campus, close to Grizzly Peak in the Berkeley Hills. Given its contribution to the nation's scientific potential, the institute is supported by the National Science Foundation and the National Security Agency. Private individuals, foundations, and nearly 100 Academic Sponsor Institutions, including mathematics departments in the United States, also provide support and flexibility. Jim Simons, f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Analysis
Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded domains such as the full real line or by Fourier series for functions on bounded domains, especially periodic functions on finite intervals. Generalizing these transforms to other domains is generally called Fourier analysis, although the term is sometimes used interchangeably with harmonic analysis. Harmonic analysis has become a vast subject with applications in areas as diverse as number theory, representation theory, signal processing, quantum mechanics, tidal analysis, spectral analysis, and neuroscience. The term "harmonics" originated from the Ancient Greek word ''harmonikos'', meaning "skilled in music". In physical eigenvalue problems, it began to mean waves whose frequencies are integer multiples of one another, as are the freq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Geometry
In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant (or equivariant) transformations from the space of functions on one geometrical space to the space of functions on another geometrical space. Such transformations often take the form of integral transforms such as the Radon transform and its generalizations. Classical context Integral geometry as such first emerged as an attempt to refine certain statements of geometric probability theory. The early work of Luis Santaló and Wilhelm Blaschke was in this connection. It follows from the classic theorem of Crofton expressing the length of a plane curve as an expectation of the number of intersections with a random line. Here the word 'random' must be interpreted as subject to correct symmetry considerations. There is a sample space of lines, one on which the a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
