Extended abstract. The paper deals with the generalized Ising model of ferromagnetism. Being generalized means that the model can be used for solving the 1D, 2D and 3D problems. The spins were divided into a mesh-lattice model. To calculate the temperature above transition temperature TC , we used series expansion approximation, for spin-to-spin interaction the mean field theory, enabeling us t...

The autocorrelation function and the run structure are two basic notions for binary sequences, and they have been used as two independent criterions to characterize the randomness of binary sequences for more than 30 years. In this paper, we establish a run series expansion formula for autocorrelation function of the binary sequence and show that the autocorrelation function is in fact complete...

We derive a set of new consistency conditions for the pion-pion scattering amplitude. These conditions hold for any s, t, u in the cube, 0 5 s, t, u 2 P2, with the four external mass variables off -mass -shell and restricted such that 4:. = 0, qt = s, 432 = t, and c$ = u. Using these consistency conditions, we determine the coefficients of the power series expansion of the pion-pion amplitude u...

We prove several identities relating three-variable Mahler measures to integrals of inverse trigonometric functions. After deriving closed forms for most of these integrals, we obtain ten explicit formulas for three-variable Mahler measures. Several of these results generalize formulas due to Condon and Laĺın. As a corollary, we also obtain three q-series expansions for the dilogarithm.

We establish q-analogues of Taylor series expansions in special polynomial bases for functions analytic in bounded domains and for entire functions whose maximum modulus Mðr; f Þ satisfies jln Mðr; f ÞjpA ln r: This solves the problem of constructing such entire functions from their values at 1⁄2aq þ q =a =2; for 0oqo1: Our technique is constructive and gives an explicit representation of the s...

An algorithm for the evaluation of the complex exponential function is proposed which is quasi-linear in time and linear in space. This algorithm is based on a modified binary splitting method for the hypergeometric series and a modified Karatsuba method for the fast evaluation of the exponential function. The time complexity of this algorithm is equal to that of the ordinary algorithm for the ...

We are concerned here with improving long range stereo by filtering image sequences. Traditionally, measurement errors from stereo camera systems have been approximated as 3-D Gaussians, where the mean is derived by triangulation and the covariance by linearized error propagation. However, there are two problems that arise when filtering such 3-D measurements. First, stereo triangulation suffer...

While the concept of Padé approximant is essentially several centuries old, its multivariate version dates only from the early seventies. In the last century many univariate convergence results were proven, describing the approximation power for several function classes. It is not our aim to give a general review of the univariate case, but to discuss only these theorems that have a multivariat...

To obtain insight in the quality of heavy-traffic approximations for queues with many servers, we consider the steady-state number of waiting customers in an M/D/s queue as s → ∞. In the Halfin-Whitt regime, it is well known that this random variable converges to the supremum of a Gaussian random walk. This paper develops methods that yield more accurate results in terms of series expansions an...

In this note we present a series expansion of inverse moments of a non-negative discrete random variate in terms of its factorial cumulants, based on the Poisson-Charlier expansion of a discrete distribution. We apply the general method to the positive binomial distribution and obtain a convergent series for its inverse moments with an error residual that is uniformly bounded on the entire inte...