On the Representations of a Number as the Sum of Three Cubes and a Fourth or Fifth Power

Let R _k (n) denote the number of representations of a natural number n as the sum of three cubes and a kth power. In this paper, we show that R ₃ (n) n ^5/9+, and that R ₄ (n) n ^47/90+, where > 0 is arbitrary. This extends work of Hooley concerning sums of four cubes, to the case of sums of mixed powers. To achieve these bounds, we use a variant of the Selberg sieve method introduced by Hooley to study sums of two kth powers, and we also use various exponential sum estimates.     

Matrix Continued Fraction for the Resolvent Function of the Band Operator

The aim of this paper is the expansion of a matrix function in terms of a matrix-continued fraction as defined by Sorokin and Van Iseghem. The function under study is the Weyl function or resolvent function of an operator, given in the standard basis by a bi-infinite band matrix, with p subdiagonals and q superdiagonals, where the p + q – 1 intermediate diagonals are zero. The main goal of this paper is to find, for the moments, an explicit form in terms of the coefficients of the continued fraction, called genetic sums, which lead to a generalization of the notion of a Stieltjes continued fraction. These results are extension of some results already found for the vector case (p = 1) and are a step in the direction towards the solution of the direct and inverse spectral problem. The actual computation of the approximants of the given function as the convergents of the continued fraction is shown to be effective. Some possible extensions are considered.     

NOTES ON GLAISHER＇S CONGRUENCES

Let p be an odd prime and let n≥1,k≥0 and r be integers,denote by Bk the kth Bernoulli number,It is proved that(i) If r≥1 is odd and suppose 1≥r+4,then ∑j=1^p-1 1/(np+j)^r=-(2n+1)r(r+1)/2(r+2)Bp-r-2p^2(mod p^3).(ii)If r≥2 is even and suppose p≥r+3, then p-1∑j=1 1/(np+j)^r=r/r+1Bv-r-1p(mod P^2).(iii) p-1∑j=1 1/(np+j)p-2=-(2n+1)p(mod P^2).This result generalizes the Glaisher‘s congruence. As a corollary, a generalization of the Wolsten-holme‘s theorem is obtained.     

Strong Nearness Frames Having Enough Cauchy Filters

We prove results establishing sufficient conditions for the sum of two nearness frames to have enough Cauchy filters. From these results and the fact that, in the category of strong nearness frames having enough Cauchy filters and uniform frame maps, complete spatial frames form a coreflective subcategory, follow a variety of results where the open-sets contravariant functor from topological spaces to frames transforms products into sums and inverse limits into direct limits.     

Perturbed Dolbeault operators and the homology Todd class

This paper discusses the role played by perturbed Dolbeault operators in relating the coherent sheaf and elliptic operator perspectives on the homology of projective varieties. Among the consequences are index formulas for perturbed Dolbeault operators.

     

Finite groups of matrices over group rings

We investigate certain finite subgroups of , where is a finite nilpotent group. Such a group gives rise to a -module; we study the characters of these modules to limit the structure of . We also exhibit some exotic subgroups .

     

氧化亚氮3000—0200和1001—0110跃迁带在高温下的线强度

采用乘积近似法计算了氧化亚氮分子的总配分函数，其中转动配分函数考虑了离心扭曲修正，振动配分函数采用谐振子近似. 利用计算所得的配分函数和实验振动跃迁矩平方及Herman-Wallis因子系数，计算了氧化亚氮3000—0200和1001—0110跃迁带在常温和高温下的线强度. 结果显示，当温度高达3000K时，计算所得线强度与实验值及HITRAN数据库提供的结果仍符合较好. 这表明高温下的分子配分函数和线强度的计算是可靠的. 还进一步计算了氧化亚氮3000—0200和1001—0110跃迁带在更高温度(40 关键词： 氧化亚氮 配分函数 线强度 高温     

A Generalization on the Difference Between an Integer and Its Inverse Modulo Q

The main purpose of this paper is to use the properties of the Gauss sums, primitive characters and the mean value of Dirichlet L-functions to study the hybrid mean value of the error term E（n, l, c, q） and the hyper-Kloosterman sums K（h,n＋1,q）, the asymptotic property of the mean square value ∑^p c=1 E^2（n, 1, c, p）, and give two interesting mean value formulae.     

Central limit theorem and almost sure central limit theorem for the product of some partial sums

In this paper, we give the central limit theorem and almost sure central limit theorem for products of some partial sums of independent identically distributed random variables.     

Ramanujan-Slater type identities related to the moduli 18 and 24

We present several new families of Rogers-Ramanujan type identities related to the moduli 18 and 24. A few of the identities were found by either Ramanujan, Slater, or Dyson, but most are believed to be new. For one of these families, we discuss possible connections with Lie algebras. We also present two families of related false theta function identities.