A Generalization of a Levitin and Parnovski Universal Inequality for Eigenvalues

In this paper, we derive “universal” inequalities for the sums of eigenvalues of the Hodge de Rham Laplacian on Euclidean closed submanifolds and of eigenvalues of the Kohn Laplacian on the Heisenberg group. These inequalities generalize the Levitin–Parnovski inequality obtained for the sums of eigenvalues of the Dirichlet Laplacian of a bounded Euclidean domain.     

The elliptic Apostol–Dedekind sums generate odd Dedekind symbols with Laurent polynomial reciprocity laws

Dedekind symbols are generalizations of the classical Dedekind sums (symbols). There is a natural isomorphism between the space of Dedekind symbols with Laurent polynomial reciprocity laws and the space of modular forms. We will define a new elliptic analogue of the Apostol–Dedekind sums. Then we will show that the newly defined sums generate all odd Dedekind symbols with Laurent polynomial reciprocity laws. Our construction is based on Machide’s result (J Number Theory 128:1060–1073, 2008) on his elliptic Dedekind–Rademacher sums. As an application of our results, we discover Eisenstein series identities which generalize certain formulas by Ramanujan (Collected Papers of Srinivasa Ramanujan, pp. 136–162. AMS Chelsea Publishing, Providence, 2000), van der Pol (Indag Math 13:261–271, 272–284, 1951), Rankin (Proc R Soc Edinburgh Sect A 76:107–117, 1976) and Skoruppa (J Number Theory 43:68–73, 1993).     

Bounds for exponential sums modulo <Emphasis Type="Italic">p</Emphasis><Superscript>2</Superscript>

In this paper we consider exponential sums over subgroups G ⊂ ℤ _q ^* . Using Stepanov’s method, we obtain nontrivial bounds for exponential sums in the case where q is a square of a prime number. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 6, pp. 81–94, 2005.     

Inequalities of Maximum of Partial Sums and Weak Convergence for a Class of Weak Dependent Random Variables

In this paper, we establish a Rosenthal-type inequality of the maximum of partial sums for ρ^- -mixing random fields. As its applications we get the Hájeck -Rènyi inequality and weak convergence of sums of ρ^- -mixing sequence. These results extend related results for NA sequence and p^＊ -mixing random fields,     

On the deviation between the distributions of sums and maximum sums of IID random vectors

Summary For a sequence of independent and identically distributed random vectors, with finite moment of order less than or equal to the second, the rate at which the deviation between the distribution functions of the vectors of partial sums and maximums of partial sums is obtained both when the sample size is fixed and when it is random, satisfying certain regularity conditions. When the second moments exist the rate is of ordern ^−1/4 (in the fixed sample size case). Two applications are given, first, we compliment some recent work of Ahmad (1979,J. Multivariate Anal.,9, 214–222) on rates of convergence for the vector of maximum sums and second, we obtain rates of convergence of the concentration functions of maximum sums for both the fixed and random sample size cases.     

Multiple sums and integrals as neutral BKP tau functions

We consider multiple sums and multiple integrals as tau functions of the so-called neutral Kadomtsev-Petviashvili hierarchy on a root lattice of type B; neutral fermions, as the simplest tool, are used to derive them. The sums are taken over projective Schur functions Q_α for strict partitions α. We consider two types of such sums: weighted sums of Q_α over strict partitions α and sums over products Q_αQ_γ. We thus obtain discrete analogues of the beta ensembles (β = 1, 2, 4). Continuous versions are represented as multiple integrals, which are interesting in several problems in mathematics and physics.     

Infinite paths with bounded or recurrent partial sums

We consider problems of the following type. Assign independently to each vertex of the square lattice the value +1, with probability p, or −1, with probability 1 −p. We ask whether an infinite path π exists, with the property that the partial sums of the ±1s along π are uniformly bounded, and whether there exists an infinite path π' with the property that the partial sums along π' are equal to zero infinitely often. The answers to these question depend on the type of path one allows, the value of p and the uniform bound specified. We show that phase transitions occur for these phenomena. Moreover, we make a surprising connection between the problem of finding a path to infinity (not necessarily self-avoiding, but visiting each vertex at most finitely many times) with a given bound on the partial sums, and the classical Boolean model with squares around the points of a Poisson process in the plane. For the recurrence problem, we also show that the probability of finding such a path is monotone in p, for p≥?. Received: 10 January 2000 / Revised version: 14 August 2000 / Published online: 9 March 2001     

Strong Limit Theorems for Weighted Sums of Negatively Associated Random Variables

In this paper, we establish strong laws for weighted sums of identically distributed negatively associated random variables. Marcinkiewicz-Zygmund’s strong law of large numbers is extended to weighted sums of negatively associated random variables. Furthermore, we investigate various limit properties of Cesàro’s and Riesz’s sums of negatively associated random variables. Some of the results in the i.i.d. setting, such as those in Jajte (Ann. Probab. 31(1), 409–412, 2003), Bai and Cheng (Stat. Probab. Lett. 46, 105–112, 2000), Li et al. (J. Theor. Probab. 8, 49–76, 1995) and Gut (Probab. Theory Relat. Fields 97, 169–178, 1993) are also improved and extended to the negatively associated setting.      

Rosenthal type inequalities for asymptotically almost negatively associated random variables and applications

In this paper, we establish some Rosenthal type inequalities for maximum partial sums of asymptotically almost negatively associated random variables, which extend the corresponding results for negatively associated random variables. As applications of these inequalities, by employing the notions of residual Cesàro α-integrability and strong residual Cesàro α-integrability, we derive some results on L _p convergence where 1 < p < 2 and complete convergence. In addition, we estimate the rate of convergence in Marcinkiewicz-Zygmund strong law for partial sums of identically distributed random variables.     

Parameterized Telescoping Proves Algebraic Independence of Sums

Usually creative telescoping is used to derive recurrences for sums. In this article we show that the non-existence of a creative telescoping solution, and more generally, of a parameterized telescoping solution, proves algebraic independence of certain types of sums. Combining this fact with summation-theory shows transcendence of whole classes of sums. Moreover, this result throws new light on the question why, for example, Zeilberger’s algorithm fails to find a recurrence with minimal order.     

Precise Large Deviations for Sums of Negatively Associated Random Variables with Common Dominatedly Varying Tails

In this paper, we obtain results on precise large deviations for non-random and random sums of negatively associated nonnegative random variables with common dominatedly varying tail distribution function. We discover that, under certain conditions, three precise large-deviation prob- abilities with different centering numbers are equivalent to each other. Furthermore, we investigate precise large deviations for sums of negatively associated nonnegative random variables with certain negatively dependent occurrences. The obtained results extend and improve the corresponding results of Ng, Tang, Yan and Yang （J. Appl. Prob., 41, 93-107, 2004）.     

All but Finitely Many Non-Trivial Zeros of the Approximations of the Epstein Zeta Function are Simple and on the Critical Line

The Chowla–Selberg formula is applied in approximatinga given Epstein zeta function. Partial sums of the series derivefrom the Chowla–Selberg formula, and although these partialsums satisfy a functional equation, as does an Epstein zetafunction, they do not possess an Euler product. What we callpartial sums throughout this paper may be considered as specialcases concerning a more general function satisfying a functionalequation only. In this article we study the distribution ofzeros of the function. We show that in any strip containingthe critical line, all but finitely many zeros of the functionare simple and on the critical line. For many Epstein zeta functionswe show that all but finitely many non-trivial zeros of partialsums in the Chowla–Selberg formula are simple and on thecritical line. 2000 Mathematics Subject Classification 11M26.     

On Exponential Sums Involving the Ideal Counting Function in Quadratic Number Fields

 Let χ be a Dirichlet character modulo k > 1, and F _χ(n) the arithmetical function which is generated by the product of the Riemann zeta-function and the Dirichlet L-function corresponding to χ in . In this paper we study the asymptotic behaviour of the exponential sums involving the arithmetical function F _χ(n). In particular, we study summation formulas for these exponential sums and mean square formulas for the error term. Received April 17, 2001; in revised form April 2, 2002     

Explicit and Efficient Formulas for the Lattice Point Count in Rational Polygons Using Dedekind—Rademacher Sums

Abstract. We give explicit, polynomial-time computable formulas for the number of integer points in any two-dimensional rational polygon. A rational polygon is one whose vertices have rational coordinates. We find that the basic building blocks of our formulas are Dedekind—Rademacher sums , which are polynomial-time computable finite Fourier series. As a by-product we rederive a reciprocity law for these sums due to Gessel, which generalizes the reciprocity law for the classical Dedekind sums. In addition, our approach shows that Gessel's reciprocity law is a special case of the one for Dedekind—Rademacher sums, due to Rademacher.     

Degree matrices and divisibility of exponential sums over finite fields

By using the degree matrix, we provide an elementary and algorithmic approach to estimating the divisibility of exponential sums over prime fields, which improves the Adolphson–Sperber theorem obtained by using the Newton polyhedron. Our result also improves the Ax–Katz theorem on estimating the number of rational points on hypersurfaces over prime fields.     

Ergodic transformations and sequences of integers

Using an ergodic transformation defined on an infinite measure space, we discuss complements in ℤ of the setA consisting of finite sums of odd powers of 2.     

Variations on the Theorem of Birkhoff – von Neumann and Extensions

 The theorem of Birkhoff – von Neumann concerns bistochastic matrices (i.e., matrices with nonnegative real entries such that all row sums and all column sums are equal to one). We consider here real matrices with entries unrestricted in sign and we extend the notion of permutation matrices (integral bistochastic matrices); some generalizations of the theorem are derived by using elementary properties of graph theory. Received: October 10, 2000 Final version received: April 11, 2002     

Semi-selfdecomposable distributions and a new class of limit theorems

Self-decomposable distributions are given as limits of normalized sums of independent random variables. We define semi-selfdecomposable distributions as limits of subsequences of normalized sums. More generally, we introduce a way of making a new class of limiting distributions derived from a class of distributions by taking the limits through subsequences of normalized sums, and define the class of semi-selfdecomposable distributions and a decreasing sequence of subclasses of it. We give two kinds of necessary and sufficient conditions for distributions belonging to those classes, one is in terms of the decomposability of random variables and another is in terms of Lévy measures. Received: 1 May 1997 / Revised version: 5 February 1998     

A-Codes from Rational Functions over Galois Rings

In this paper, we describe authentication codes via (generalized) Gray images of suitable codes over Galois rings. Exponential sums over these rings help determine—or bound—the parameters of such codes.