Explicit formula

Xem 1-11 trên 11 kết quả Explicit formula
  • One of the important consequences of the mere existence of this formula is the following. Suppose that g is the Lie algebra of a Lie group G. Then the local structure of G near the identity, i.e. the rule for the product of two elements of G sufficiently closed to the identity is determined by its Lie algebra g. Indeed, the exponential map is locally a diffeomorphism from a neighborhood of the origin in g onto a neighborhood W of the identity, and if U ⊂ W is a (possibly smaller) neighborhood of the identity such that U · U ⊂ W, the the product of a...

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  • To the memory of Rodica Simion The goals of this paper are two-fold. First, we prove, for an arbitrary finite root system Φ, the periodicity conjecture of Al. B. Zamolodchikov [24] that concerns Y -systems, a particular class of functional relations playing an important role in the theory of thermodynamic Bethe ansatz. Algebraically, Y -systems can be viewed as families of rational functions defined by certain birational recurrences formulated in terms of the root system Φ.

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  • We study random surfaces which arise as height functions of random perfect matchings (a.k.a. dimer configurations) on a weighted, bipartite, doubly periodic graph G embedded in the plane. We derive explicit formulas for the surface tension and local Gibbs measure probabilities of these models. The answers involve a certain plane algebraic curve, which is the spectral curve of the Kasteleyn operator of the graph. For example, the surface tension is the Legendre dual of the Ronkin function of the spectral curve.

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  • Introduction In 1903 Voronoi [42] postulated the existence of explicit formulas for sums of the form (1.1) n≥1 an f (n) , for any “arithmetically interesting” sequence of coefficients (an )n≥1 and every f in a large class of test functions, including characteristic functions of bounded intervals. He actually established such a formula when an = d(n) is the number of positive divisors of n [43]. He also asserted a formula for (1.2) an = #{(a, b) ∈ Z2 | Q(a, b) = n} , where Q denotes a positive definite integral quadratic form [44]; ...

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  • BOUNDARY VALUE PROBLEMS FOR ANALYTIC FUNCTIONS IN THE CLASS OF CAUCHY-TYPE INTEGRALS WITH DENSITY IN L p(·) (Γ) V. KOKILASHVILI, V. PAATASHVILI, AND S. SAMKO Received 9 July 2004 We study the Riemann boundary value problem Φ+ (t) = G(t)Φ− (t) + g(t), for analytic functions in the class of analytic functions represented by the Cauchy-type integrals with density in the spaces L p(·) (Γ) with variable exponent. We consider both the case when the coefficient G is piecewise continuous and the case when it may be of a more general nature, admitting its oscillation.

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  • The Chebyshev polynomial of degree n is denoted Tn (x), and is given by the explicit formula This may look trigonometric at first glance (and there is in fact a close relation between the Chebyshev polynomials and the discrete Fourier transform); however (5.8.1) can

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  • We present explicit formulas representations of the real diamond Lie algebra obtained from the normal polarization on K-orbits. From this we have list irreducible unitary representations of the real diamond Lie group that is coincide with the representations via Fedosov deformation quantisation. Here the computations are more simple for use star-product.

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  • Selecting Cells by Reference Chọn các ô theo tham chiếu If a cell contains a formula, Excel defines the cell’s precedents as those cells that the formula refers to. For example, if cell A4 contains the formula = SUM(A1:A3), cells A1, A2, and A3 are the precedents of A4. A direct precedent is a cell referred to explicitly in the formula. In the preceding example, A1, A2, and A3 are direct precedents of A4. An indirect precedent is a cell referred to by a precedent. For example, if cell A1 contains the formula = B3*2, cell B3 is an indirect...

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  • This chapter gives an overview of some properties of the storage occupancy process in a buffer fed with ``fractional Brownian traf®c,'' a Gaussian self-similar process. This model, called here ``fractional Brownian storage,'' is the logically simplest long-range-dependent (LRD) storage system having strictly self-similar input variation. The impact of the self-similarity parameter H can be very clearly illustrated with this model.

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  • The purpose of this paper is to give an explicit local formula for the difference of two natural versions of equivariant analytic torsion in de Rham theory. This difference is the sum of the integral of a Chern-Simons current and of a new invariant, the V -invariant of an odd dimensional manifold equipped with an action of a compact Lie group.

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  • We study the Radon transform Rf of functions on Stiefel and Grassmann manifolds. We establish a connection between Rf and G˚ arding-Gindikin fractional integrals associated to the cone of positive definite matrices. By using this connection, we obtain Abel-type representations and explicit inversion formulae for Rf and the corresponding dual Radon transform. We work with the space of continuous functions and also with Lp spaces.

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