On Z4-Linear Goethals Codes and Kloosterman Sums

Studying the coset weight distributions of the Z₄-linear Goethals codes, e connect these codes with the Kloosterman sums. From one side, e obtain for some cases, of the cosets of weight four, the exact expressions for the number of code ords of weight four in terms of the Kloosterman sums. From the other side, e obtain some limitations for the possible values of the Kloosterman sums, hich improve the well known results due to Lachaud and Wolfmann kn:lac.     

关于广义3维Kloosterman和4次均值的注记

本文主要利用二次剩余理论和一类对称同余方程解的个数问题研究一类广义3维Kloosterman和4次均值的计算问题,并得到其精确的渐近公式.     

Kloosterman sum identities and low-weight codewords in a cyclic code with two zeros

We apply relations of n-dimensional Kloosterman sums to exponential sums over finite fields to count the number of low-weight codewords in a cyclic code with two zeros. As a corollary we obtain a new proof for a result of Carlitz which relates one- and two-dimensional Kloosterman sums. In addition, we count some power sums of Kloosterman sums over certain subfields.     

A hybrid mean value related to the Dedekind sums and Kloosterman sums

The main purpose of this paper is to use the properties of character sum and the analytic method to study a hybrid mean value problem related to the Dedekind sums and Kloosterman sums, and give some interesting mean value formulae and identities for it.     

Trivial factors for L-functions of symmetric products of Kloosterman sheaves

In this paper, we completely determine the trivial factors for L-functions of symmetric products of Kloosterman sheaves.     

3-Designs from the Z 4-Goethals Codes via a New Kloosterman Sum Identity

Recently, active research has been performed on constructing t-designs from linear codes over Z ₄. In this paper, we will construct a new simple 3 – (2 ^m , 7, 14/3 (2 ^m – 8)) design from codewords of Hamming weight 7 in the Z ₄-Goethals code for odd m 5. For 3 arbitrary positions, we will count the number of codewords of Hamming weight 7 whose support includes those 3 positions. This counting can be simplified by using the double-transitivity of the Goethals code and divided into small cases. It turns out interestingly that, in almost all cases, this count is related to the value of a Kloosterman sum. As a result, we can also prove a new Kloosterman sum identity while deriving the 3-design.     

A note on the Cochrane sum and its hybrid mean value formula

In this paper, we use the properties of Gauss sums, primitive characters and the mean value theorems of Dirichlet L-functions to study the hybrid mean value of Cochrane sums and general Kloosterman sums, and give two sharp asymptotic formulae.     

Trace Formula for p-Adics

We give a p-adic analogue of Selberg's trace formula relating the trace of a semigroup generated by a natural elliptic operator with a sum over contributions coming from closed geodesics. The construction uses probabilistic methods to define the generator.     

A New Proof of the Mullineux Conjecture

Let S _d denote the symmetric group on d letters. In 1979 Mullineux conjectured a combinatorial algorithm for calculating the effect of tensoring with an irreducible S _d-module with the one dimensional sign module when the ground field has positive characteristic. Kleshchev proved the Mullineux conjecture in 1996. In the present article we provide a new proof of the Mullineux conjecture which is entirely independent of Kleshchev's approach. Applying the representation theory of the supergroup GL(m | n) and the supergroup analogue of Schur-Weyl Duality it becomes straightforward to calculate the combinatorial effect of tensoring with the sign representation and, hence, to verify Mullineux's conjecture. Similar techniques also allow us to classify the irreducible polynomial representations of GL(m | n) of degree d for arbitrary m, n, and d.     

Hill-Type Formula and Krein-Type Trace Formula for Hamiltonian Systems

In this paper, we give a survey on the Hill-type formula and its applications.Moreover, we generalize the Hill-type formula for linear Hamiltonian systems and Sturm-Liouville systems with any self-adjoint boundary conditions, which include the standard Neumann, Dirichlet and periodic boundary conditions. The Hill-type formula connects the infinite determinant of the Hessian of the action functional with the determinant of matrices which depend on the monodromy matrix and boundary conditions. Further, based on the Hill-type formula, we derive the Krein-type trace formula. As applications, we give nontrivial estimations for the eigenvalue problem and the relative Morse index.     

Bayes Formula for Optimal Filter with n-ple Markov Gaussian Errors

We consider the nonlinear filtering problem where the observation noise process is n-ple Markov Gaussian. A Kallianpur–Striebel type Bayes formula for the optimal filter is obtained.     

On the mean value of

Let χ be the Dirichlet character modulo q3 and L(s,χ) denote the corresponding Dirichlet L-function. The mean value of is studied and a few asymptotic formulae are given. Hybrid mean value of , general Kloosterman sums and general quadratic Gauss sums are considered.     

A Determinantal Formula for Supersymmetric Schur Polynomials

We derive a new formula for the supersymmetric Schur polynomial s (x/y). The origin of this formula goes back to representation theory of the Lie superalgebra gl(m/n). In particular, we show how a character formula due to Kac and Wakimoto can be applied to covariant representations, leading to a new expression for s (x/y). This new expression gives rise to a determinantal formula for s (x/y). In particular, the denominator identity for gl(m/n) corresponds to a determinantal identity combining Cauchy's double alternant with Vandermonde's determinant. We provide a second and independent proof of the new determinantal formula by showing that it satisfies the four characteristic properties of supersymmetric Schur polynomials. A third and more direct proof ties up our formula with that of Sergeev-Pragacz.     

Moments of Generalized Quadratic Gauss Sums Weighted by L-Functions

The main purpose of this paper is using estimates for character sums and analytic methods to study the second, fourth, and sixth order moments of generalized quadratic Gauss sums weighted by L-functions. Three asymptotic formulae are obtained.     

On the LIL for Self-Normalized Sums of IID Random Variables

Let be i.i.d. random variables and let, for each and . It is shown that a.s. whenever the sequence of self-normalized sums S _n /V _n is stochastically bounded, and that this limsup is a.s. positive if, in addition, X is in the Feller class. It is also shown that, for X in the Feller class, the sequence of self-normalized sums is stochastically bounded if and only if      

On Dirichlet Series for Sums of Squares

Hardy and Wright (An Introduction to the Theory of Numbers, 5th edn., Oxford, 1979) recorded elegant closed forms for the generating functions of the divisor functions _k (n) and _k ²(n) in the terms of Riemann Zeta function (s) only. In this paper, we explore other arithmetical functions enjoying this remarkable property. In Theorem 2.1 below, we are able to generalize the above result and prove that if f _i and g _i are completely multiplicative, then we have where L _f(s) := _{n = 1} f(n)n ^–s is the Dirichlet series corresponding to f. Let r _N(n) be the number of solutions of x ₁ ² + ··· + x _N ² = n and r _2,P (n) be the number of solutions of x ² + Py ² = n. One of the applications of Theorem 2.1 is to obtain closed forms, in terms of (s) and Dirichlet L-functions, for the generating functions of r _N(n), r _N ²(n), r _2,P (n) and r _2,P (n)² for certain N and P. We also use these generating functions to obtain asymptotic estimates of the average values for each function for which we obtain a Dirichlet series.     

Estimates for Mixed Character Sums

We give sharp estimates for certain families of exponential sums in several variables over finite fields. Received: May 2007, Accepted: September 2007     

Projection representable relations on Menger (2, n)-semigroups

Abstract characterizations of relations of nonempty intersection, inclusion end equality of domains for partial n-place functions are presented. Representations of Menger (2, n)-semigroups by partial n-place functions closed with respect to these relations are investigated.     

Some Sets of Type (m,n) in Cubic Order Planes

A (4,9)-set of size 829 in (2,5³) is constructed, as is a (4,11)-set of size 3189 in (2,7³).