|
Volterra Lattice
In mathematics, the Volterra lattice, also known as the discrete KdV equation, the Kac–van Moerbeke lattice, and the Langmuir lattice, is a system of ordinary differential equations with variables indexed by some of the points of a 1-dimensional lattice. It was introduced by and and is named after Vito Volterra. The Volterra lattice is a special case of the generalized Lotka–Volterra equation describing predator–prey interactions, for a sequence of species with each species preying on the next in the sequence. The Volterra lattice also behaves like a discrete version of the KdV equation. The Volterra lattice is an integrable system, and is related to the Toda lattice. It is also used as a model for Langmuir wave Langmuir may refer to: * Langmuir (crater), an impact crater on the Moon's far side * ''Langmuir'' (journal), an academic journal on colloids, surfaces and interfaces, published by the American Chemical Society * Langmuir (unit), a unit of expos ...s in plasmas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice (group)
In geometry and group theory, a lattice in the real coordinate space \mathbb^n is an infinite set of points in this space with the properties that coordinate wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point. Closure under addition and subtraction means that a lattice must be a subgroup of the additive group of the points in the space, and the requirements of minimum and maximum distance can be summarized by saying that a lattice is a Delone set. More abstractly, a lattice can be described as a free abelian group of dimension n which spans the vector space \mathbb^n. For any basis of \mathbb^n, the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice, and every lattice can be formed from a basis in this way. A lattice may be viewed as a regula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vito Volterra
Vito Volterra (, ; 3 May 1860 – 11 October 1940) was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations, being one of the founders of functional analysis. Biography Born in Ancona, then part of the Papal States, into a very poor Jewish family: his father was Abramo Volterra and his mother, Angelica Almagià. Abramo Volterra died in 1862 when Vito was two years old. The family moved to Turin, and then to Florence, where he studied at the Dante Alighieri Technical School and the Galileo Galilei Technical Institute. Volterra showed early promise in mathematics before attending the University of Pisa, where he fell under the influence of Enrico Betti, and where he became professor of rational mechanics in 1883. He immediately started work developing his theory of functionals which led to his interest and later contributions in integral and integro-differential equations. His work is summarised in his book ''Theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Lotka–Volterra Equation
The generalized Lotka–Volterra equations are a set of equations which are more general than either the competitive or predator–prey examples of Lotka–Volterra types. They can be used to model direct competition and trophic relationships between an arbitrary number of species. Their dynamics can be analysed analytically to some extent. This makes them useful as a theoretical tool for modeling food webs. However, they lack features of other ecological models such as predator preference and nonlinear functional responses, and they cannot be used to model mutualism without allowing indefinite population growth. The generalised Lotka-Volterra equations model the dynamics of the populations x_1, x_2, \dots of n biological species. Together, these populations can be considered as a vector \mathbf. They are a set of ordinary differential equations given by : \frac = x_i f_i(\mathbf), where the vector \mathbf is given by : \mathbf = \mathbf + A\mathbf, where \mathbf is a v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integrable System
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals, such that its behaviour has far fewer degrees of freedom than the dimensionality of its phase space; that is, its evolution is restricted to a submanifold within its phase space. Three features are often referred to as characterizing integrable systems: * the existence of a ''maximal'' set of conserved quantities (the usual defining property of complete integrability) * the existence of algebraic invariants, having a basis in algebraic geometry (a property known sometimes as algebraic integrability) * the explicit determination of solutions in an explicit functional form (not an intrinsic property, but something often referred to as solvability) Integrable systems may be seen as very different in qualitative character from mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toda Lattice
The Toda lattice, introduced by , is a simple model for a one-dimensional crystal in solid state physics. It is famous because it is one of the earliest examples of a non-linear completely integrable system. It is given by a chain of particles with nearest neighbor interaction, described by the Hamiltonian :\begin H(p,q) &= \sum_ \left(\frac +V(q(n+1,t)-q(n,t))\right) \end and the equations of motion :\begin \frac p(n,t) &= -\frac = e^ - e^, \\ \frac q(n,t) &= \frac = p(n,t), \end where q(n,t) is the displacement of the n-th particle from its equilibrium position, and p(n,t) is its momentum (mass m=1), and the Toda potential V(r)=e^+r-1. Soliton solutions Soliton solutions are solitary waves spreading in time with no change to their shape and size and interacting with each other in a particle-like way. The general N-soliton solution of the equation is : \begin q_N(n,t)=q_+ + \log \frac , \end where :C_N(n,t)=\Bigg(\frac\Bigg)_, with :\gamma_j(n,t)=\gamma_j\,e^ where \kappa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Langmuir Wave
Langmuir may refer to: * Langmuir (crater), an impact crater on the Moon's far side * ''Langmuir'' (journal), an academic journal on colloids, surfaces and interfaces, published by the American Chemical Society * Langmuir (unit), a unit of exposure of an adsorbate/gas to a substrate used in surface science to study adsorption * Langmuir Cove, a cove in the north end of Arrowsmith Peninsula, Graham Land, Antarctica * Langmuir monolayer, a one-molecule thick layer of an insoluble organic material spread onto an aqueous subphase in a Langmuir-Blodgett trough People * Alexander Langmuir Alexander Duncan Langmuir (12 September 1910 – 22 November 1993) was an American epidemiologist. He is renowned for creating the Epidemic Intelligence Service. Biography Alexander D. Langmuir was born in Santa Monica, California. He received h ... (1910–1993), American epidemiologist * Gavin I. Langmuir (1924–2005), Canadian veteran of World War II, historian of anti-Semitism and medieva ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed at publishing articles addressed to a broader "mathematical community", and not only to mathematicians in the author's field. Herbert Busemann writes, in the preface of the first issue, "The need for expository articles addressing either all mathematicians or only those in somewhat related fields has long been felt, but little has been done outside of the USSR. The serial publication ''Advances in Mathematics'' was created in response to this demand." Abstracting and indexing The journal is abstracted and indexed in: * |
