Existence of equilibrium states of systems of hard spheres in the Boltzmann-Enskog limit within the frame work of the grand canonical ensemble

We study equilibrium states of systems of hard spheres in the Boltzmann-Enskog limit (d→0, 1/v→∞ (z→∞), and d ³ (1/v)=const (d ³ z=const)). For this purpose, we use the Kirkwood-Salsburg equations. We prove that, in the Boltzmann-Enskog limit, solutions of these equations exist and the limit distribution functions are constant. By using the cluster and compatibility conditions, we prove that all distribution functions are equal to the product of one-particle distribution functions, which can be represented as power series in z=d ³ z with certain coefficients. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 1, pp. 112–121, January, 1997.     

On an Identity of Mahler

We prove that certain multiple integrals depending on the complex parameter z can be expressed as polynomials in z and ln(1 ? z). Similar identities were first used by K. Mahler in connection with the proofs of certain results of the theory of transcendental numbers.     

Branch point area methods in conformal mapping

The classical estimate of Bieberbach that ?a ₂?≤2 for a given univalent function ?(z)=z+a ₂ z ²+… in the classS leads to the best possible pointwise estimates of the ratio ?"(z)/?'(z) for ?∈S, first obtained by K?be and Bieberbach. For the corresponding class Σ of univalent functions in the exterior disk, Goluzin found in 1943 by variational methods the corresponding best possible pointwise estimates of ?"(z)/?'(z) for ψ∈Σ. It was perhaps surprising that this time, the expressions involve elliptic integrals. Here, we obtain an area-type theorem which has Goluzin's pointwise estimate as a corollary. This shows that Goluzin's estimate, like the K?be-Bieberbach estimate, is firmly rooted in areabased methods. The appearance of elliptic integrals finds a natural explanation: they arise because a certain associated covering surface of the Riemann sphere is a torus.     

On certain generalized incomplete gamma functions

Recently, Chaudhry and Zubair have introduced a generalized incomplete gamma function Γ(v,x;z) which reduces to the incomplete gamma function Γ(v,x) when its variable z vanishes. We show that Γ(v,x;z) may be written essentially as a single Kampé de Fériet function which in turn may be expressed as a linear combination of two incomplete Weber integrals. Then by using properties of the latter integrals we deduce additional representations for Γ(v,x;z). In particular, we show that Γ(v,x;z) is essentially completely determined by a finite number of modified Bessel functions for all v ≠ 0 provided we know the values of the two incomplete Weber integrals when 0 < Re v ⩽ 1. When v = 0 we derive connections between the generalized incomplete gamma function and incomplete Lipschitz-Hankel integrals, and indicate that there exist connections with other special functions.     

Invariance principles for stochastic area and related stochastic integrals

Given an antisymmetric kernel K (K(z, z′) = ?K(z′, z)) and i.i.d. random variates Z_n, n?1, such that EK²(Z₁, Z₂)<∞, set A_n = ∑₁?i?j?nK(Z_i,Z_j), n?1. If the Z_n's are two-dimensional and K is the determinant function, A_n is a discrete analogue of Paul Lévy's so-called stochastic area. Using a general functional central limit theorem for stochastic integrals, we obtain limit theorems for the A_n's which mirror the corresponding results for the symmetric kernels that figure in theory of U-statistics.     

The evaluation of Tornheim double sums, Part 1

We provide an explicit formula for the Tornheim double series in terms of integrals involving the Hurwitz zeta function. We also study the limit when the parameters of the Tornheim sum become natural numbers, and show that in that case it can be expressed in terms of definite integrals of triple products of Bernoulli polynomials and the Bernoulli function A_k(q)?kζ^′(1-k,q).     

Ito formulas for Skorohod and Skorohod-Stratonovich integrals in the two-parameter case

We consider a two-parameter process X_z defined by the sum of multiple Skorohod integrals and ordinary Lebesgue integrals. A generalized Ito's formula is given. We also introduce a two-parameter analog of the SkorohodStratonovich integral and establish an Ito's formula in the Stratonovich form     

Orthogonality and recurrence for ordered Laurent polynomial sequences

We consider orderings of nested subspaces of the space of Laurent polynomials on the real line, more general than the balanced orderings associated with the ordered bases {1,z⁻¹,z,z⁻²,z²,…} and {1,z,z⁻¹,z²,z⁻²,…}. We show that with such orderings the sequence of orthonormal Laurent polynomials determined by a positive linear functional satisfies a three-term recurrence relation. Reciprocally, we show that with such orderings a sequence of Laurent polynomials which satisfies a recurrence relation of this form is orthonormal with respect to a certain positive functional.     

On the Uniform Approximation of a Class of Analytic Functions by Bruwier Series

For a class of analytic functions f(z) defined by Laplace–Stieltjes integrals the uniform convergence on compact subsets of the complex plane of the Bruwier series (B-series) ∑^∞_n=0 λ_n(f) , λ_n(f)=f⁽ⁿ⁾(nc)+cf⁽ⁿ⁺¹⁾(nc), generated by f(z) and the uniform approximation of the generating function f(z) by its B-series in cones |arg z|< is shown.     

Integral transformation of solutions for a Fuchsian-class equation corresponding to the Okamoto transformation of the Painlevé VI equation

We show that under the Euler integral transformation with the kernel (x−z)^−α, some solutions of the Fuchs equations (the original pair for the Painlevé VI equation) pass into solutions of a system of the same form with the parameters changed according to the Okamoto transformation. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 3, pp. 355–364, March, 2006.     

The riesz turndown collar theorem giving an asymptotic estimate of the powers of an operator under the ritt condition

For Banach space operatorsT satisfying the Tadmor-Ritt condition ‖(zI−T)⁻¹‖≤C|z−1|⁻¹, |z|>1, we show how to use the Riesz turndown collar theorem to estimate sup _n≥0‖T ⁿ‖. A similar estimate is shown for lim sup _n ‖T ⁿ‖ in terms of the Ritt constantM=lim sup _z→1‖(1−z)(zI−T)⁻¹‖. We also obtain an estimate of the functional calculus for these operators proving, in particular, that ‖f(T)‖≤C _q‖f‖ _Mult , where ‖·‖ _Mult stands for the multiplier norm of the Cauchy-Stieltjes integrals over a Lusin type cone domain depending onC and a parameterq, 0<q<1. Notation.D denotes the open unit disc of the complex plane,D={z∈ℂ:|z|<1}, andT={z∈ℂ:|z|=1} is the unit circle.H ^∞ is the Banach algebra of bounded analytic functions onD equipped with the supremum norm ‖.‖_∞.     

Low-Temperature Localisation for Continuous Spin-Systems

We study the free energy of continuous spin-systems on Z ^d , in the framework of Laplace integrals and transfer operators. Under a weak coupling condition, we show that the free energy in the low-temperature limit is determined, up to an exponentially small error, by the restriction to a neighbourhood of global minima of the energy. We have results for some single- and double-well problems.     

Minimum asymptotic error of algorithms for solving ODE

We deal with algorithms for solving systems z′(x) = f(x, z(x)), x ε [0, c], z(0) = η where f has r continuous bounded derivatives in [0, c) × ^s. We consider algorithms whose sole dependence on f is through the values of n linear continuous functionals at f. We show that if these functionals are defined by partial derivatives off then, roughly speaking, the error of an algorithm (for a fixed f) cannot converge to zero faster than n^−r as n → +∞. This minimal error is achieved by the Taylor algorithm. If arbitrary linear continuous functionals are allowed, then the error cannot converge to zero faster than n^−(r+1) as n → +∞. This minimal error is achieved by the Taylor-integral algorithm which uses integrals of f.     

Index transforms associated with generalized hypergeometric functions

We deal with a class of integral transformations whose kernels contain the Clausenian hypergeometric function ₃F₂(a₁,a₂,a₃;b₁,b₂;z). These transforms are defined in terms of integrals with respect to their parameters. It involves as particular cases the familiar Olevskii and generalized Mehler–Fock transforms which are key tools in the methods of the mathematical theory of elasticity. The main theorem of boundedness of these operators as a map of L₂(?₊)L₂(?₊;x^?1 dx) is proved. Some examples of the Olevskii and Mehler–Fock type integrals are given. Copyright © 2003 John Wiley & Sons, Ltd.     

Optimal tristance anticodes in certain graphs

For z₁,z₂,z₃∈Zⁿ, the tristance d₃(z₁,z₂,z₃) is a generalization of the L₁-distance on Zⁿ to a quantity that reflects the relative dispersion of three points rather than two. A tristance anticodeA_d of diameter d is a subset of Zⁿ with the property that d₃(z₁,z₂,z₃)?d for all z₁,z₂,z₃∈A_d. An anticode is optimal if it has the largest possible cardinality for its diameter d. We determine the cardinality and completely classify the optimal tristance anticodes in Z² for all diameters d?1. We then generalize this result to two related distance models: a different distance structure on Z² where d(z₁,z₂)=1 if z₁,z₂ are adjacent either horizontally, vertically, or diagonally, and the distance structure obtained when Z² is replaced by the hexagonal lattice A₂. We also investigate optimal tristance anticodes in Z³ and optimal quadristance anticodes in Z², and provide bounds on their cardinality. We conclude with a brief discussion of the applications of our results to multi-dimensional interleaving schemes and to connectivity loci in the game of Go.     

Differential Inequalities and Starlikeness

We study some differential inequalities in the unit disc which imply starlikeness: for example if ? (z) = z + Σ^∞ _n=2 a_n (?)zⁿ is analytic in D = {z | |z| < 1} and |z(?′′(z) ? 1)| ≤ 1 ? α, z?D for some α ] [0, 1), then ? is one-to-one on D and ? (D) is a starlike domain with respect to the origin.     

Differentiation formulas for stochastic integrals in the plane

For a one-parameter process of the form X_t=X₀+∫^t₀φ_sdW_s+∫^t₀ψ_sds, where W is a Wiener process and ∫φdW is a stochastic integral, a twice continuously differentiable function f(X_t) is again expressible as the sum of a stochastic integral and an ordinary integral via the Ito differentiation formula. In this paper we present a generalization for the stochastic integrals associated with a two-parameter Wiener process.Let {W₂, z∈R²₊} be a Wiener process with a two-dimensional parameter. Ertwhile, we have defined stochastic integrals ∫ φdWand ∫ψdWdW, as well as mixed integrals ∫h dz dW and ∫gdW dz. Now let X_z be a two-parameter process defined by the sum of these four integrals and an ordinary Lebesgue integral. The objective of this paper is to represent a suitably differentiable function f(X_z) as such a sum once again. In the process we will also derive the (basically one-dimensional) differentiation formulas of f(X_z) on increasing paths in $\text{R}{}^2{}_{\mathrm{+}}$.     

Uniform asymptotic expansion for a class of polynomials biorthogonal on the unit circle

An asymptotic expansion including error bounds is given for polynomials {P _n, Q_n} that are biorthogonal on the unit circle with respect to the weight function (1?e^iθ)^α+β(1?e^?iθ)^α?β. The asymptotic parameter isn; the expansion is uniform with respect toz in compact subsets ofC{0}. The pointz=1 is an interesting point, where the asymptotic behavior of the polynomials strongly changes. The approximants in the expansions are confluent hyper-geometric functions. The polynomials are special cases of the Gauss hyper-geometric functions. In fact, with the results of the paper it follows how (in a uniform way) the confluent hypergeometric function is obtained as the limit of the hypergeometric function₂ F ₁(a, b; c; z/b), asb→±∞,z≠b, withz=0 as “transition” point in the uniform expansion.     

The mean value of the zeta-function on <Emphasis Type="Italic">σ</Emphasis>=1

Let R(T): = ò₁^T|z(1+it)|² dt - z(2)T+plogTR(T):= \int_{1}^{T}|\zeta (1+it)|^{2}\,\mathrm{d}t - \zeta (2)T+\pi\log T. We derive a precise explicit expression for R(t) which is used to derive asymptotic formulas for ò₁^TR(t) dt\int_{1}^{T}R(t)\,\mathrm{d}t and ò₁^TR²(t) dt\int_{1}^{T}R^{2}(t)\,\mathrm{d}t. These results improve on earlier upper bounds of Balasubramanian, Ramachandra and the author for the integrals in question.     

Buffons Nadelproblem und verwandte probleme behandelt mit der Theorie der Gleichverteilung: Teil II

We denote with PC ^m the m-dimensional complex projective space, with U the unitary group acting on it with z _i(j=0, 1,..., m) the homogenous coordinates of a point [z] of PC ^m and assume that the z _i are normalized such that z ₀z₀ +...+z _mz_m=1. Furthermore we denote the U-invariant metric on PC ^m with d. We consider now a uniformly distributed sequence ([z] _k ; k=1,2,...) of points on PC ^m and study the sequence (d ^l([z] _k , [z]₀)), l0, [z]₀ a fixed point. We prove with the help of the theory of uniform distribution properties of this sequence. We consider furthermore a dual sequence suggested by the theory of H. Weyl and L. V. Ahlfors on meromorphic curves.