Related papers: On some conjectures on Generalized quadratic Gauss…
We discuss a possible generalisation of a conjecture of Bley, Burns and Hahn concerning the relation between the second Adams-operator twisted Galois-Gauss sums of weakly ramified Artin characters and the square root of the inverse…
We show that various identities from [1] and [3] involving Gould-Hopper polynomials can be deduced from the real but also complex orthogonal invariance of multivariate Gaussian distributions. We also deduce from this principle a useful…
The present article is devoted to the generalized Salem functions, the generailed shift operator, and certain related problems. A description of further investigations of the author of this article is given.These investigations (in terms of…
We consider $q$-analytic derivations of the $q$-Gauss summation formula for a $\, _2\phi _1$ that respect the symmetry in its upper parameters.
We give more evidence for Patterson's conjecture on sums of exponential sums, by getting an asymptotic for a sum of quartic exponential sums over $\Q[i].$ Previously, the strongest evidence of Patterson's conjecture over a number field is…
This article proposes a unified analytical approach leading to a partial resolution of the Erdos-Straus, Sierpinski conjectures, and their generalization. We introduce an equivalent reformulation of these conjectures while constructing two…
The aim of this paper is twofold. The first is to give a quantitative version of Schmidt's subspace theorem for arbitrary families of higher degree polynomials. The second is to give a generalization of the subspace theorem for arbitrary…
An analysis of the characteristic function of Gaussian quadratic forms is presented in [1] to study the performance of multichannel communication systems. This technical report reviews this analysis, obtaining alternative expressions to…
The purpose of this note is to provide an alternative proof of two quadratic transformation formulas contiguous to that of Gauss using a differential equation approach.
Motivated by a dictionary between polynomials and finite Blaschke products, we study both Smale's mean value conjecture and its dual conjecture for finite Blaschke products in this paper. Our result on the dual conjecture for finite…
A combinatorial methods are used to investigate some properties of certain generalized Stirling numbers, including explicit formula and recurrence relations. Furthermore, an expression of these numbers with symmetric function is deduced.
One of the fundamental tasks of science is to find explainable relationships between observed phenomena. One approach to this task that has received attention in recent years is based on probabilistic graphical modelling with sparsity…
We present a unified formulation for higher gauge theory using generalized forms, encompassing higher connections, curvatures, and gauge transformations. We begin by developing the calculus of generalized forms valued in higher algebras and…
Conventional approximations to Bayesian inference rely on either approximations by statistics such as mean and covariance or by point particles. Recent advances such as the ensemble Gaussian mixture filter have generalized these notions to…
We obtain in closed form averages of polynomials, taken over hermitian matrices with the Gaussian measure involved in the Kontsevich integral, and prove a conjecture of Witten enabling one to express analogous averages with the full (cubic…
This paper proposes a generalized ABC conjecture and assuming its validity settles a generalized version of Fermats last theorem.
We prove strong estimates for averages of shifted convolution sums consisting of quadratic twists of $\mathrm{GL}_{2}$ $L$-functions. The key input involves the circle method together with standard tools such as Vorono\u{\i}, quadratic…
New results on comparison of distributions of Gaussian quadratic forms are presented
This article is part of an ongoing investigation of the two-dimensional Jacobian conjecture. In the first paper of this series, we proved the generalized Magnus' formula. In this paper, inspired by cluster algebras, we introduce a sequence…
In this paper, the formulas of some exponential sums over finite field, related to the Coulter's polynomial, are settled based on the Coulter's theorems on Weil sums, which may have potential application in the construction of linear codes…