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Hard Hexagon Model
In statistical mechanics, the hard hexagon model is a 2-dimensional lattice model of a gas, where particles are allowed to be on the vertices of a triangular lattice but no two particles may be adjacent. The model was solved by , who found that it was related to the Rogers–Ramanujan identities. The partition function of the hard hexagon model The hard hexagon model occurs within the framework of the grand canonical ensemble, where the total number of particles (the "hexagons") is allowed to vary naturally, and is fixed by a chemical potential. In the hard hexagon model, all valid states have zero energy, and so the only important thermodynamic control variable is the ratio of chemical potential to temperature ''μ''/(''kT''). The exponential of this ratio, ''z'' = exp(''μ''/(''kT'')) is called the activity and larger values correspond roughly to denser configurations. For a triangular lattice with ''N'' sites, the grand partition function is :\displaystyle \mathcal Z(z) = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Statistical Mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applications include many problems in a wide variety of fields such as biology, neuroscience, computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ..., information theory and sociology. Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion. Statistical mechanics arose out of the development of classical thermodynamics, a field for which it was successful in explaining macroscopic physical properties—such as temperature, pressure, and heat capacity—in terms of microscop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Eric Weisstein
Eric Wolfgang Weisstein (born March 18, 1969) is an American scientist, mathematician, and encyclopedist who created and maintains the encyclopedias ''MathWorld'' and ''ScienceWorld''. In addition, he is the author of the '' CRC Concise Encyclopedia of Mathematics''. He works for Wolfram Research. Education Weisstein holds a Ph.D. in planetary astronomy, which he obtained from the California Institute of Technology's Division of Geological and Planetary Sciences in 1996 as well as an M.S. in planetary astronomy in 1993 also from Caltech. Weisstein graduated cum laude from Cornell University with a B.A. in physics and a minor in astronomy in 1990. During his summers away from Cornell, Weisstein participated in research at the Arecibo Observatory, a radio telescope facility in Puerto Rico operated by Cornell. As a graduate student, Weisstein also participated in research at Goddard Space Flight Center in Greenbelt, Maryland. During his time at Goddard, Weisstein participated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Lattice Models
Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an ornamental pattern of crossing strips of pastry Companies * Lattice Engines, a technology company specializing in business applications for marketing and sales * Lattice Group, a former British gas transmission business * Lattice Semiconductor, a US-based integrated circuit manufacturer Science, technology, and mathematics Mathematics * Lattice (group), a repeating arrangement of points ** Lattice (discrete subgroup), a discrete subgroup of a topological group whose quotient carries an invariant finite Borel measure ** Lattice (module), a module over a ring that is embedded in a vector space over a field ** Lattice graph, a graph that can be drawn within a repeating arrangement of points ** Lattice-based cryptography, encryp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Statistical Mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applications include many problems in a wide variety of fields such as biology, neuroscience, computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ..., information theory and sociology. Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion. Statistical mechanics arose out of the development of classical thermodynamics, a field for which it was successful in explaining macroscopic physical properties—such as temperature, pressure, and heat capacity—in terms of microscop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Exactly Solvable Models
Exact may refer to: * Exaction, a concept in real property law * ''Ex'Act'', 2016 studio album by Exo * Schooner Exact, the ship which carried the founders of Seattle Companies * Exact (company), a Dutch software company * Exact Change, an American independent book publishing company * Exact Editions, a content management platform Mathematics * Exact differentials, in multivariate calculus * Exact algorithms, in computer science and operations research * Exact colorings, in graph theory * Exact couples, a general source of spectral sequences * Exact sequences, in homological algebra * Exact functor, a function which preserves exact sequences See also * *Exactor (other) *XACT (other) *EXACTO, a sniper rifle {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Annals Of Combinatorics
''Annals of Combinatorics'' is a quarterly peer-reviewed scientific journal covering research in combinatorics. It was established in 1997 by William Chen and is published by Birkhäuser. The journal publishes articles in combinatorics and related areas with a focus on algebraic combinatorics, analytic combinatorics, graph theory, and matroid theory. Until December 2019, the journal was edited by George Andrews, William Chen, and Peter Paule. The current editors-in-chief are Frédérique Bassino, Kolja Knauer, and Matjaž Konvalinka. Abstracting and indexing The journal is abstracted and indexed in *MathSciNet, *Science Citation Index Expanded, *Scopus, and *ZbMATH Open. According to the ''Journal Citation Reports'', the journal has a 2022 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their field. The Im ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Harold Exton
Harold Exton is a mathematician at University of Central Lancashire (called Preston Polytechnic while he was there) working on hypergeometric function In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is ...s, who introduced the Hahn–Exton q-Bessel function. References * * * Academics of the University of Central Lancashire English mathematicians Living people Year of birth missing (living people) {{UK-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Proceedings Of The National Academy Of Sciences
''Proceedings of the National Academy of Sciences of the United States of America'' (often abbreviated ''PNAS'' or ''PNAS USA'') is a peer-reviewed multidisciplinary scientific journal. It is the official journal of the National Academy of Sciences, published since 1915, and publishes original research, scientific reviews, commentaries, and letters. According to ''Journal Citation Reports'', the journal has a 2022 impact factor of 9.4. ''PNAS'' is the second most cited scientific journal, with more than 1.9 million cumulative citations from 2008 to 2018. In the past, ''PNAS'' has been described variously as "prestigious", "sedate", "renowned" and "high impact". ''PNAS'' is a delayed open-access journal, with an embargo period of six months that can be bypassed for an author fee ( hybrid open access). Since September 2017, open access articles are published under a Creative Commons license. Since January 2019, ''PNAS'' has been online-only, although print issues are available ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Algebraic Number
In mathematics, an algebraic number is a number that is a root of a function, root of a non-zero polynomial in one variable with integer (or, equivalently, Rational number, rational) coefficients. For example, the golden ratio (1 + \sqrt)/2 is an algebraic number, because it is a root of the polynomial X^2 - X - 1, i.e., a solution of the equation x^2 - x - 1 = 0, and the complex number 1 + i is algebraic as a root of X^4 + 4. Algebraic numbers include all integers, rational numbers, and nth root, ''n''-th roots of integers. Algebraic complex numbers are closed under addition, subtraction, multiplication and division, and hence form a field (mathematics), field, denoted \overline. The set of algebraic real numbers \overline \cap \R is also a field. Numbers which are not algebraic are called transcendental number, transcendental and include pi, and . There are countable set, countably many algebraic numbers, hence almost all real (or complex) numbers (in the sense of Lebesgue ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Modular Function
In mathematics, a modular form is a holomorphic function on the complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modular forms has origins in complex analysis, with important connections with number theory. Modular forms also appear in other areas, such as algebraic topology, sphere packing, and string theory. Modular form theory is a special case of the more general theory of automorphic forms, which are functions defined on Lie groups that transform nicely with respect to the action of certain discrete subgroups, generalizing the example of the modular group \mathrm_2(\mathbb Z) \subset \mathrm_2(\mathbb R). Every modular form is attached to a Galois representation. The term "modular form", as a systematic description, is usually attributed to Erich Hecke. The importance of modular forms across multiple field of mathematics has been humorously repre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Lattice Model (physics)
In mathematical physics, a lattice model is a mathematical model of a physical system that is defined on a lattice, as opposed to a continuum, such as the continuum of space or spacetime. Lattice models originally occurred in the context of condensed matter physics, where the atoms of a crystal automatically form a lattice. Currently, lattice models are quite popular in theoretical physics, for many reasons. Some models are exactly solvable, and thus offer insight into physics beyond what can be learned from perturbation theory. Lattice models are also ideal for study by the methods of computational physics, as the discretization of any continuum model automatically turns it into a lattice model. The exact solution to many of these models (when they are solvable) includes the presence of solitons. Techniques for solving these include the inverse scattering transform and the method of Lax pairs, the Yang–Baxter equation and quantum groups. The solution of these models has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Dedekind Eta Function
In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string theory. Definition For any complex number with , let ; then the eta function is defined by, :\eta(\tau) = e^\frac \prod_^\infty \left(1-e^\right) = q^\frac \prod_^\infty \left(1 - q^n\right) . Raising the eta equation to the 24th power and multiplying by gives :\Delta(\tau)=(2\pi)^\eta^(\tau) where is the modular discriminant. The presence of 24 can be understood by connection with other occurrences, such as in the 24-dimensional Leech lattice. The eta function is holomorphic on the upper half-plane but cannot be continued analytically beyond it. The eta function satisfies the functional equations :\begin \eta(\tau+1) &=e^\frac\eta(\tau),\\ \eta\left(-\frac\right) &= \sqrt\, \eta(\tau).\, \end In the second equation the b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
