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Gauss Sums of the Cubic Character over $GF(2^m)$: an elementary derivation
Gauss sum character binary finite fields
2011/2/25
An elementary approach is shown which derives the value of the Gauss sum of a cubic char-
acter over a finite field F2s without using Davenport-Hasse’s theorem (namely, if s is odd the
Gauss sum is ...
Some Congruences of Kloosterman Sums and their Minimal Polynomials
Congruences of Kloosterman Sums Minimal Polynomials
2011/1/18
We prove two results on Kloosterman sums over finite fields, using Stickelberger’s theorem and the Gross-Koblitz formula. The first result concerns the minimal polynomial over Q of a Kloosterman sum, ...
Note on harmonic sums and alternating Euler sums
Harmonic sums Alternating Euler sums Multiple zeta values
2011/2/25
The explicit formulas expressing harmonic sums via alternating Euler sums (colored multiple zeta values) are given, and some explicit evaluations are given as applications.
Best $\ell_1$-approximation of nonnegative polynomials by sums of squares
Polynomials sums of squares semidefinite programming
2011/2/21
Given a nonnegative polynomial f, we provide an explicit expres-sion for its best ℓ1-norm approximation by a sum of squares of given degree.
A strong invariance principle for nonconventional sums
strong approximations limit theorems martingale approximation
2011/1/21
In [12] we obtained a functional central limit theorem (known also as a weak invariance principle) for sums of the form P[Nt] n=1 FX(n),X(2n), ...,X(kn),X(qk+1(n)),X(qk+2(n)),
Random Gaussian sums on trees
Gaussian processes processes indexed by trees bounded processes
2011/1/21
Let T be a tree with induced partial order . We investigate centered Gaussian processes
X = (Xt)t∈T represented as Xt = σ(t) X vt α(v)ξv for given weight functions α and σ on T and with (ξv)v∈T i.i...
On unit root formulas for toric exponential sums
Exponential sum A-hypergeometric function
2011/1/19
Starting from a classical generating series for Bessel functions due to Schl¨omilch[4], we use Dwork’s relative dual theory to broadly generalize unit-root results of Dwork[3] on Kloosterman sums and ...
On the limit distributions of some sums of a random multiplicative function
limit distributions of some sums of random multiplicative function
2011/1/17
We study sums of a random multiplicative function; this is an example,of number-theoretic interest, of sums of products of independent random variables (chaoses).
(k+1)-sums versus k-sums
(k+1)-sums k-sums
2010/11/23
Given $k\in \mathbb{N}$ and a subset $A=\{a_{1},...,a_{n}\}$ of $n$ integers, we define $S_{k}=S_{k}(A)=\{\sum_{I}a_{i}:I\in [n]^{(k)}\}$, i.e. $S_{k}$ is the set of integers that may be expressed as...
Let $\{X_t, t \geq 1\}$ be a sequence of identically distributed and pairwise asymptotically independent random variables with regularly varying tails and $\{ \Theta_t, t\geq1 \}$ be a sequence of pos...
A Computer Algebra Toolbox for Harmonic Sums Related to Particle Physics
Computer Algebra Harmonic Sums Related Particle Physics
2010/11/11
In this work we present the computer algebra package HarmonicSums and its theoretical background for the manipulation of harmonic sums and some related quantities as for example Euler-Zagier sums and...
We present a hybrid approach to bounding exponential sums over kth powers via Vinogradov's mean value theorem, and derive estimates of utility for exponents k of intermediate size.
Logarithmic Integrals, Polylogarithmic Integrals and Euler sums
Logarithmic Integrals Polylogarithmic Integrals Euler sums
2010/11/9
Relations among integrals of logarithms, polylogarithms and Euler sums are presented. A unifying element being the introduction of Nielsen's generalized polylogarithms.
Sums of Laplace eigenvalues - rotationally symmetric maximizers in the plane
Isoperimetric membrane tight frame
2010/12/14
The sum of the first n 1 eigenvalues of the Laplacian is shown to be maximal among triangles for the equilateral triangle,maximal among parallelograms for the square, and maximal among ellipses for ...
Line arrangements and direct sums of free groups
Line arrangements direct sums of free groups
2010/12/13
We show that if the fundamental groups of the complements of two line arrangements in the complex projective plane are isomorphic to the same direct sum of free groups, then the complements of the arr...