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Temperley–Lieb Algebra
In statistical mechanics, the Temperley–Lieb algebra is an algebra from which are built certain transfer matrices, invented by Neville Temperley and Elliott Lieb. It is also related to integrable models, knot theory and the braid group, quantum groups and subfactors of von Neumann algebras. Structure Generators and relations Let R be a commutative ring and fix \delta \in R. The Temperley–Lieb algebra TL_n(\delta) is the R-algebra generated by the elements e_1, e_2, \ldots, e_, subject to the Jones relations: *e_i^2 = \delta e_i for all 1 \leq i \leq n-1 *e_i e_ e_i = e_i for all 1 \leq i \leq n-2 *e_i e_ e_i = e_i for all 2 \leq i \leq n-1 *e_i e_j = e_j e_i for all 1 \leq i,j \leq n-1 such that , i-j, \neq 1 Using these relations, any product of generators e_i can be brought to Jones' normal form: : E= \big(e_e_\cdots e_\big)\big(e_e_\cdots e_\big)\cdots\big(e_e_\cdots e_\big) where (i_1,i_2,\dots,i_r) and (j_1,j_2,\dots,j_r) are two strictly increasing sequences in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic behavior of nature from the behavior of such ensembles. 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 microscopic parameters that fluctuate about average values and are characterized by probability distributions. This established the fields of statistical thermodynamics and statistical physics. The founding of the field of statistical mechanics is generally credited to three physicists: *Ludwig Boltzmann, who developed the fundamental interpretation of entropy in terms of a collection of microstates *James Clerk Maxwell, who developed models of probability distr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian (quantum Mechanics)
Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian with two-electron nature ** Molecular Hamiltonian, the Hamiltonian operator representing the energy of the electrons and nuclei in a molecule * Hamiltonian (control theory), a function used to solve a problem of optimal control for a dynamical system * Hamiltonian path, a path in a graph that visits each vertex exactly once * Hamiltonian group, a non-abelian group the subgroups of which are all normal * Hamiltonian economic program, the economic policies advocated by Alexander Hamilton, the first United States Secretary of the Treasury See also * Alexander Hamilton (1755 or 1757–1804), American statesman and one of the Founding Fathers of the US * Hamilton (other) Hamilton may refer to: People * Hamilton (name), a common ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 given i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hook Length Formula
In combinatorial mathematics, the hook length formula is a formula for the number of standard Young tableaux whose shape is a given Young diagram. It has applications in diverse areas such as representation theory, probability, and algorithm analysis; for example, the problem of longest increasing subsequences. A related formula gives the number of semi-standard Young tableaux, which is a specialization of a Schur polynomial. Definitions and statement Let \lambda=(\lambda_1\geq \cdots\geq \lambda_k) be a partition of n=\lambda_1+\cdots+\lambda_k. It is customary to interpret \lambda graphically as a Young diagram, namely a left-justified array of square cells with k rows of lengths \lambda_1,\ldots,\lambda_k. A (standard) Young tableau of shape \lambda is a filling of the n cells of the Young diagram with all the integers \, with no repetition, such that each row and each column form increasing sequences. For the cell in position (i,j), in the ith row and jth column, the hook H_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Branching Rule
In group theory, restriction forms a representation of a subgroup using a known representation of the whole group. Restriction is a fundamental construction in representation theory of groups. Often the restricted representation is simpler to understand. Rules for decomposing the restriction of an irreducible representation into irreducible representations of the subgroup are called branching rules, and have important applications in physics. For example, in case of explicit symmetry breaking, the symmetry group of the problem is reduced from the whole group to one of its subgroups. In quantum mechanics, this reduction in symmetry appears as a splitting of degenerate energy levels into multiplets, as in the Stark or Zeeman effect. The induced representation is a related operation that forms a representation of the whole group from a representation of a subgroup. The relation between restriction and induction is described by Frobenius reciprocity and the Mackey theorem. Restri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brauer Algebra
In mathematics, a Brauer algebra is an associative algebra introduced by Richard Brauer in the context of the representation theory of the orthogonal group. It plays the same role that the symmetric group does for the representation theory of the general linear group in Schur–Weyl duality. Structure The Brauer algebra \mathfrak_n(\delta) is a \mathbbdelta/math>-algebra depending on the choice of a positive integer n. Here \delta is an indeterminate, but in practice \delta is often specialised to the dimension of the fundamental representation of an orthogonal group O(\delta). The Brauer algebra has the dimension :\dim\mathfrak_n(\delta) = \frac = (2n-1)!! = (2n-1)(2n-3)\cdots 5\cdot 3\cdot 1 Diagrammatic definition A basis of \mathfrak_n(\delta) consists of all pairings on a set of 2n elements X_1, ..., X_n, Y_1, ..., Y_n (that is, all perfect matchings of a complete graph K_n: any two of the 2n elements may be matched to each other, regardless of their symbols). The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Coefficients
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the term in the polynomial expansion of the binomial power ; this coefficient can be computed by the multiplicative formula :\binom nk = \frac, which using factorial notation can be compactly expressed as :\binom = \frac. For example, the fourth power of is :\begin (1 + x)^4 &= \tbinom x^0 + \tbinom x^1 + \tbinom x^2 + \tbinom x^3 + \tbinom x^4 \\ &= 1 + 4x + 6 x^2 + 4x^3 + x^4, \end and the binomial coefficient \tbinom =\tfrac = \tfrac = 6 is the coefficient of the term. Arranging the numbers \tbinom, \tbinom, \ldots, \tbinom in successive rows for n=0,1,2,\ldots gives a triangular array called Pascal's triangle, satisfying the recurrence relation :\binom = \binom + \binom. The binomial coefficients occur in many areas of mathematics, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E 4 Temperley
E, or e, is the fifth letter and the second vowel letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''e'' (pronounced ); plural ''ees'', ''Es'' or ''E's''. It is the most commonly used letter in many languages, including Czech, Danish, Dutch, English, French, German, Hungarian, Latin, Latvian, Norwegian, Spanish, and Swedish. History The Latin letter 'E' differs little from its source, the Greek letter epsilon, 'Ε'. This in turn comes from the Semitic letter '' hê'', which has been suggested to have started as a praying or calling human figure ('' hillul'' 'jubilation'), and was most likely based on a similar Egyptian hieroglyph that indicated a different pronunciation. In Semitic, the letter represented (and in foreign words); in Greek, ''hê'' became the letter epsilon, used to represent . The various forms of the Old Italic script and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E 1 Temperley
E, or e, is the fifth letter and the second vowel letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''e'' (pronounced ); plural ''ees'', ''Es'' or ''E's''. It is the most commonly used letter in many languages, including Czech, Danish, Dutch, English, French, German, Hungarian, Latin, Latvian, Norwegian, Spanish, and Swedish. History The Latin letter 'E' differs little from its source, the Greek letter epsilon, 'Ε'. This in turn comes from the Semitic letter '' hê'', which has been suggested to have started as a praying or calling human figure ('' hillul'' 'jubilation'), and was most likely based on a similar Egyptian hieroglyph that indicated a different pronunciation. In Semitic, the letter represented (and in foreign words); in Greek, ''hê'' became the letter epsilon, used to represent . The various forms of the Old Italic script and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E 3 Temperley
E, or e, is the fifth Letter (alphabet), letter and the second vowel#Written vowels, vowel letter in the Latin alphabet, used in the English alphabet, modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is English alphabet#Letter names, ''e'' (pronounced ); plural ''ees'', ''Es'' or ''E's''. It is the most commonly used letter in many languages, including Czech language, Czech, Danish language, Danish, Dutch language, Dutch, English language, English, French language, French, German language, German, Hungarian language, Hungarian, Latin language, Latin, Latvian language, Latvian, Norwegian language, Norwegian, Spanish language, Spanish, and Swedish language, Swedish. History The Latin letter 'E' differs little from its source, the Greek alphabet, Greek letter epsilon, 'Ε'. This in turn comes from the Semitic alphabet, Semitic letter ''He (letter), hê'', which has been suggested to have started as a praying ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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