Sum Of Quotients Research Articles

Rank deviations for overpartitions

We prove general fomulas for the deviations of two overpartition ranks from the average. These formulas are in terms of Appell–Lerch series and sums of quotients of theta functions and can be used, among other things, to recover any of the numerous overpartition rank difference identities in the literature. We give two illustrations.

Infinite Series and Logarithmic Integrals Associated to Differentiation with Respect to Parameters of the Whittaker Wκ,μx Function II

In the first part of this investigation, we considered the parameter differentiation of the Whittaker function Mκ,μx. In this second part, first derivatives with respect to the parameters of the Whittaker function Wκ,μx are calculated. Using the confluent hypergeometric function, these derivatives can be expressed as infinite sums of quotients of the digamma and gamma functions. Furthermore, it is possible to obtain these parameter derivatives in terms of infinite integrals, with integrands containing elementary functions (products of algebraic, exponential, and logarithmic functions), from the integral representation of Wκ,μx. These infinite sums and integrals can be expressed in closed form for particular values of the parameters. Finally, an integral representation of the integral Whittaker function wiκ,μx and its derivative with respect to κ, as well as some reduction formulas for the integral Whittaker functions Wiκ,μx and wiκ,μx, are calculated.

Infinite Series and Logarithmic Integrals Associated to Differentiation with Respect to Parameters of the Whittaker Mκ,μ(x) Function I

In this paper, first derivatives of the Whittaker function Mκ,μx are calculated with respect to the parameters. Using the confluent hypergeometric function, these derivarives can be expressed as infinite sums of quotients of the digamma and gamma functions. Moreover, from the integral representation of Mκ,μx it is possible to obtain these parameter derivatives in terms of finite and infinite integrals with integrands containing elementary functions (products of algebraic, exponential, and logarithmic functions). These infinite sums and integrals can be expressed in closed form for particular values of the parameters. For this purpose, we have obtained the parameter derivative of the incomplete gamma function in closed form. As an application, reduction formulas for parameter derivatives of the confluent hypergeometric function are derived, along with finite and infinite integrals containing products of algebraic, exponential, logarithmic, and Bessel functions. Finally, reduction formulas for the Whittaker functions Mκ,μx and integral Whittaker functions Miκ,μx and miκ,μx are calculated.

Differentiation of integral Mittag-Leffler and integral Wright functions with respect to parameters

Derivatives with respect to the parameters of the integral Mittag-Leffler function and the integral Wright function, recently introduced by us, are calculated. These derivatives can be expressed in the form of infinite sums of quotients of the digamma and gamma functions. In some particular cases, these infinite sums are calculated in closed-form with the help of MATHEMATICA. However, parameter differentiation reduction formulas are explicitly derived in order to check some of the results given by MATHEMATICA, as well as to provide many other new results. In addition, we present these infinite sums graphically for particular values of the parameters. Finally, new results for parameter derivatives of the Mittag-Leffler and Wright functions are reported in the Appendices.

Quadruples for Residue Number Systems with Sum of Quotients a Power of a Prime

We investigate Residue Number Systems (RNS) of special type which were recently shown to be useful for some type of computations in embedded systems. We develop an algorithm for derivation of all quadruples for RNS under investigation subject to certain restrictions.

Generalized Cesàro operators, fractional finite differences and Gamma functions

In this paper, we present a complete spectral research of generalized Cesàro operators on Sobolev–Lebesgue sequence spaces. The main idea is to subordinate such operators to suitable C0-semigroups on these sequence spaces. We introduce that family of sequence spaces using the fractional finite differences and we prove some structural properties similar to classical Lebesgue sequence spaces. In order to show the main results about fractional finite differences, we state equalities involving sums of quotients of Euler's Gamma functions. Finally, we display some graphical representations of the spectra of generalized Cesàro operators.

On sums involving products and quotients of $L$-functions over function fields

We estimate the sum of products or quotients of $L$-functions, where the sum is taken over all quadratic extensions of given genus over a fixed global function field. Our estimate for the sum of the quotient of two $L$-functions is analogous to a result of Schmidt where he estimates the sum of the quotient of two $L$-series, where the sum is over quadratic extensions of $\mathbb{Q}$ with absolute value of the discriminant less than a given bound.

Proximal operator of quotient functions with application to a feasibility problem in query optimization

In this paper we determine the proximity functions of the sum and the maximum of componentwise (reciprocal) quotients of positive vectors. For the sum of quotients, denoted by Q1, the proximity function is just a componentwise shrinkage function which we call q-shrinkage. This is similar to the proximity function of the ℓ1-norm which is given by componentwise soft shrinkage. For the maximum of quotients Q∞, the proximal function can be computed by first order primal–dual methods involving epigraphical projections.The proximity functions of Qν, ν=1,∞ are applied to solve convex problems of the form argminxQν(Axb) subject to x≥0, 1⊤x≤1. Such problems are of interest in selectivity estimation for cost-based query optimizers in database management systems.

Surjectivity of the global-to-local map defining a Selmer group

This article studies a map from a global Galois cohomology group to a direct sum of quotients of local Galois cohomology groups. The kernel of such a global-to-local map is often called a Selmer group. The objective of this article is to study the cokernel of such a map. We do so in a very general context. In particular, we find various sets of assumptions which imply that a global-to-local map is surjective.

Characteristic elements, pairings and functional equations over the false Tate curve extension

AbstractWe construct a pairing on the dual Selmer group over false Tate curve extensions of an elliptic curve with good ordinary reduction at a primep≥5. This gives a functional equation of the characteristic element which is compatible with the conjectural functional equation of thep-adicL-function. As an application we compute the characteristic elements of those modules – arising naturally in the Iwasawa-theory for elliptic curves over the false Tate curve extension – which have rank 1 over the subgroup of the Galois group fixing the cyclotomic extension of the ground field. We also show that the example of a non-principal reflexive left ideal of the Iwasawa algebra does not rule out the possibility that all torsion Iwasawa-modules are pseudo-isomorphic to the direct sum of quotients of the algebra by principal ideals.

Maskit combinations of Poincaré–Einstein metrics

We establish a boundary connected sum theorem for asymptotically hyperbolic Einstein metrics, and also show that if the two metrics have scalar positive conformal infinities, then the same is true for this boundary join. This construction is also extended to spaces with a finite number of interior conic singularities, and as a result we show that any 3-manifold which is a finite connected sum of quotients of S3 and S2×S1 bounds such a space (with conic singularities); putatively, any 3-manifold admitting a metric of positive scalar curvature is of this form.

Cohen–Macaulay quotients of normal semigroup rings via irreducible resolutions

Every quotient R/I of a semigroup ring R by a radical monomial ideal I has a unique minimal injective-like resolution by direct sums of quotients of R modulo prime monomial ideals. The quotient R/I is Cohen-Macaulay if and only if every summand in cohomological degree $i$ has dimension exactly dim(R/I) - i. This Cohen-Macaulay characterization reduces to the Eagon-Reiner theorem by Alexander duality when R is a polynomial ring. The proof exploits a graded ring-theoretic generalization of the Zeeman spectral sequence, thereby also providing a combinatorial topological version for polyhedral cell complexes, involving no commutative algebra.

Comments on "A tabulation of Hilbert transforms for electrical engineers"

A new table of Hilbert transforms was prepared and published by Alavi-Sereshki and Prabhakar (see ibid., vol.COM-20, no.6, p.1194, 1972). During preparation, it was found that some formulas in the above paper were in error. Therefore, the tables in that should be used with great care. Correct Hilbert transform pairs are presented by Hahn (1996). The Hilbert transform of an amplitude-modulated signal, given in the above paper, as the infinite sum of quotients of integrals, is derived in compact form.

Maximization of sums of quotients of quadratic forms and some generalizations

Monotonically convergent algorithms are described for maximizing six (constrained) functions of vectors x, or matrices X with columns x(1),..., x(r). These functions are h(1)(x) = Sigma(k) (x'A(k)x)(x'C(k)x)(-1), H-1(X) = Sigma(k) tr (X'A(k)X)(X'C(k)X)(-1), (h) over tilde(1)(X) = Sigma(k) Sigma(l)(x'(l)A(k)x(l))(x'(l)C(k)x(l))(-1) with X constrained to be columnwise orthonormal, h(2)(x) = Sigma(k) (x'A(k)x)(2)(x'C(k)X)(-1) subject to x'x = 1, H-2(X) = Sigma(k) tr (X'A(k)X)(X'A(k)X)'(X'C(k)X)(-1) subject to X'X = I, and (h) over tilde(2)(X) = Sigma(k) Sigma(l) (x'(l)A(k)x(l))(2)(x'(l)C(k)x(l))(-1) subject to X'X = I. In these functions the matrices C-k are assumed to be positive definite. The matrices A(k) can be arbitrary square matrices. The general formulation of the functions and the algorithms allows for application of the algorithms in various problems that arise in multivariate analysis. Several applications of the general algorithms are given. Specifically, algorithms are given for reciprocal principal components analysis, binormamin rotation, generalized discriminant analysis, variants of generalized principal components analysis, simple structure rotation for one of the latter variants, and set component analysis. For most of these methods the algorithms appear to be new, for the others the existing algorithms turn out to be special cases of the newly derived general algorithms.

Sums of reciprocals of certain additive functions

We obtain sharp estimates for sums of reciprocals and sums of quotients of certain non-negative integer-valued additive arithmetical functions.

Sums of Quotients of Additive Functions

Denote by $\omega (n)$ and $\Omega (n)$ the number of distinct prime factors of $n$ and the total number of prime factors of $n$, respectively. Given any positive integer $\alpha$, we prove that \[ \sum \limits _{2 \leqq n \leqq x} {\Omega (n)/\omega } (n) = x + x\sum \limits _{i = 1}^\alpha {{a_i}/{{(\log \log x)}^i} + O{{(x/\log \log x)}^{\alpha + 1}}),} \] where ${a_1} = \sum \nolimits _p {1/p(p - 1)}$ and all the other ${a_i}$’s are computable constants. This improves a previous result of R. L. Duncan.

Sums of quotients of additive functions

Denote by ω ( n ) \omega (n) and Ω ( n ) \Omega (n) the number of distinct prime factors of n n and the total number of prime factors of n n , respectively. Given any positive integer α \alpha , we prove that \[ ∑ 2 ≦ n ≦ x Ω ( n ) / ω ( n ) = x + x ∑ i = 1 α a i / ( log ⁡ log ⁡ x ) i + O ( x / log ⁡ log ⁡ x ) α + 1 ) , \sum \limits _{2 \leqq n \leqq x} {\Omega (n)/\omega } (n) = x + x\sum \limits _{i = 1}^\alpha {{a_i}/{{(\log \log x)}^i} + O{{(x/\log \log x)}^{\alpha + 1}}),} \] where a 1 = ∑ p 1 / p ( p − 1 ) {a_1} = \sum \nolimits _p {1/p(p - 1)} and all the other a i {a_i} ’s are computable constants. This improves a previous result of R. L. Duncan.