On sums of degrees of the partial quotients in continued fraction expansions of Laurent series

For any formal Laurent series with coefficients cn lying in some given finite field, let x=[a₀(x);a₁(x),a₂(x),…] be its continued fraction expansion. It is known that, with respect to the Haar measure, almost surely, the sum of degrees of partial quotients grows linearly. In this note, we quantify the exceptional sets of points with faster growth orders than linear ones by their Hausdorff dimension, which covers an earlier result by J. Wu.     

交换半群环的分式环

设含幺交换环Ｒ对其乘法子集Ｔ的分式环为ＲＴ，交换幺半群Ｓ在其子半群∑处局部化为Ｓ∑本文证明了Ｒ［Ｓ］对于Ａ的分环式环Ｒ［Ｓ］ＡＭ 构于半群环ＲＴ［Ｓ∑］。     

Rook numbers and the normal ordering problem

For an element w in the Weyl algebra generated by D and U with relation DU=UD+1, the normally ordered form is w=∑c_i,jUⁱD^j. We demonstrate that the normal order coefficients c_i,j of a word w are rook numbers on a Ferrers board. We use this interpretation to give a new proof of the rook factorization theorem, which we use to provide an explicit formula for the coefficients c_i,j. We calculate the Weyl binomial coefficients: normal order coefficients of the element (D+U)ⁿ in the Weyl algebra. We extend these results to the q-analogue of the Weyl algebra. We discuss further generalizations using i-rook numbers.     

On the Approximation of Irrationals by Rationals

Let ξ be a real irrational number, and a ≧ 0, b ≧ 0, s > 1 be integers. A theorem of S. UCHIYAMA states that there are infinitely many pairs of integers u and v ≠ 0 such that OVBARξ?u/vOVBAR ≤ s²/4v² and u ? a, v ? b mod s, provided that it is not a ? 6 ? 0 mod s. It is shown that this result is best-possible for all integers s > 1.     

Continued fraction of e2 with confluent hypergeometric functions

The tanh-type, tan-type, and e-type Hurwitz continued fractions have been generalized by the author. In this paper, we study a generalized form of e²-type Hurwitz continued fractions by using confluent hypergeometric functions. We also obtain a similar type of Tasoev continued fractions. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 513–531, October–December, 2006.     

左分式半群的几个性质

讨论了若干半群类在取左分式半群下的封闭性，给出了关于左发式半群的一个同构定理。     

Changes in sol fraction during peroxide crosslinking of linear low-density polyethylenes with homogeneous distribution of short chain branching

FTIR measurements of sol fractions were carried out in order to assess side reactions during dicumyl peroxide (DCP) crosslinking of linear low density polyethylenes (LLDPEs) with homogeneous distribution of short chain branching. The changes of methyl and olefinic unsaturation concentration were measured as a function of degree of branching and DCP content. The changes of methyl and vinylidene concentration indicate that the scission probability has greater significance for the samples with initial branch content above 31 CH₃/1000 C and increases with increasing DCP concentration. The analysis of trans-vinylene concentration indicates that the probability of disproportionation also increases with increasing DCP content. The changes of methyl concentration in the sol fraction of the sample containing initially ca. 23 CH₃/1000 C with increasing DCP content could be interpreted using the dependence of crosslinking efficiency on branch content found previously for some range of branch concentration. © 1995 John Wiley & Sons, Inc.     

Difference equations in statistical mechanics. I. Cluster statistics models

A review and some new results are presented for several cluster statistics models, solutions of which can be reduced to difference equations. Mathematical techniques suitable for solving these equations are surveyed.     

On a new class of multiplicative pseudo-random number generators

In the computing literature, there are few detailed analytical studies of the global statistical characteristics of a class of multiplicative pseudo-random number generators.We comment briefly on normal numbers and study analytically the approximately uniform discrete distribution or (j,)-normality in the sense of Besicovitch for complete periods of fractional parts {x _{0 1} ⁱ /p} on [0, 1] fori=0, 1,..., (p–1)p^–1–1, i.e. in current terminology, generators given byx _{n+1 1} x _n mod p wheren=0, 1,..., (p–1)p ^–1–1,p is any odd prime, (x ₀,p)=1, ₁ is a primitive root modp ², and 1 is any positive integer.We derive the expectationsE(X, ),E(X ², ),E(X _nX_n+k); the varianceV(X, ), and the serial correlation coefficient k. By means of Dedekind sums and some results of H. Rademacher, we investigate the asymptotic properties of k for various lagsk and integers 1 and give numerical illustrations. For the frequently used case =1, we find comparable results to estimates of Coveyou and Jansson as well as a mathematical demonstration of a so-called rule of thumb related to the choice of ₁ for small k.Due to the number of parameters in this class of generators, it may be possible to obtain increased control over the statistical behavior of these pseudo-random sequences both analytically as well as computationally.     

Difference equations in statistical mechanics. II. Solid-on-solid models in two dimensions

A review and some new results are presented for the solid-on-solid models of wetting in two dimensions (i.e., line interfaces) with an emphasis on the difference equations arising in the transfer matrix calculations for these models. Methods for solving the appropriate difference equations exactly or approximately are surveyed, including specific results for short-range, long-range power-law, and applied field-like (binding) external potentials.