Related papers: The binary Goldbach problem with arithmetic weight…

A classification of ordinary differential equations and finite-difference equations in one variable having polynomial solutions (the generalized Bochner problem) is given. The method used is based on the spectral problem for a polynomial…

We study relationship among versions of the Knapsack Problem where variables take values in Z and the number of them is fixed. In particular, we construct a finitely presented group where the problem of solvability of exponential equations…

Let $\mathbb{N}_0$ be a class of natural numbers whose binary expansions contain even numbers of ones. Waring problem in numbers of class $\mathbb{N}_0$ is solved.

We give a weighted sum formula for the double polylogarithm in two variables, from which we can recover the classical weighted sum formulas for double zeta values, double $T$-values, and some double $L$-values. Also presented is a…

This paper provides two parallel solutions on the mixed boundary value problem of a unit annulus subjected to a partially fixed outer periphery and an arbitrary traction acting along the inner periphery using the complex variable method.…

All sieve methods for the Goldbach problem sift out all the composite numbers; even though, strictly speaking, it is not necessary to do so and which is, in general, very difficult. Some new methods introduced in this paper show that the…

A complete characterization of near subnormality for bilateral weighted shifts is obtained. As an application of the main results, many new answers to the Hilbert space problem 160 are presented at the end of the paper.

We consider the two weight problem for the Hilbert transform, namely the question of finding real-variable characterization of those pair of weights for which the Hilbert transform acts boundedly on $ L ^2 $ of the weights. Such a…

We prove a sharp density theorem for quadratic Waring's problem over cyclic groups, when the number of variables is at least $5$. Also, we obtain some new improvements on the density version of the quadratic Waring--Goldbach problem over…

We derive combinatorial identities for variables satisfying specific systems of commutation relations, in particular elliptic commutation relations. The identities thus obtained extend corresponding ones for $q$-commuting variables $x$ and…

In the paper, there are new found methods to determine the range of every exceptional element in exceptional set, we can solve Twin primes problem and Goldbach Conjecture problem basically.

A commutative algebra $\mathbb{B}$ over the field of complex numbers with the bases $\{e_1,e_2\}$ satisfying the conditions $(e_1^2+e_2^2)^2=0$, $e_1^2+e_2^2\ne 0$, is considered. The algebra $\mathbb{B}$ is associated with the biharmonic…

Goldbach conjecture is one of the most famous open mathematical problems. It states that every even number, bigger than two, can be presented as a sum of 2 prime numbers. % In this work we present a deep learning based model that predicts…

In this work we cast the problem of binary classification in terms of estimating a partition on Bernoulli data. When the explanatory variables are all categorical, the problem can be modelled using the language of boolean functions. We…

We relate the graph isomorphism problem to the solvability of certain systems of linear equations with nonnegative variables. This version replaces the two previous versions of this paper.

We obtain an asymptotic formula for a weighted sum of the square of the tail of the singular series for the Goldbach and prime-pair problems.

We consider the exceptional set in the binary Goldbach problem for sums of two almost twin primes. Our main result is a power-saving bound for the exceptional set in the problem of representing $m=p_1+p_2$ where $p_1+2$ has at most $2$…

We develop an algebraic method of studying of Diophantine quadratic equations in three variables over the ring of Gaussian integers.

In this paper, we investigate exceptional sets in the Waring-Goldbach problem for unlike powers. For example, estimates are obtained for sufficiently large integers below a parameter subject to the necessary local conditions that do not…

In this note we devise and analyse well-posed variational formulations and operator theoretical methods for boundary value problems associated to the biharmonic operator. Of particular interest are Neumann type and over- and underdetermined…