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A classical problem in analytic number theory is to study the distribution of fractional part $\alpha p^k+\beta,\,k\ge 1$ modulo 1, where $\alpha$ is irrational and $p$ runs over the set of primes. For $k=2$ we consider the subsequence…
We estimate the frequency of singular matrices and of matrices of a given rank whose entries are parametrised by arbitrary polynomials over the integers and modulo a prime $p$. In particular, in the integer case, we improve a recent bound…
We consider the complexity of two questions on polynomials given by arithmetic circuits: testing whether a monomial is present and counting the number of monomials. We show that these problems are complete for subclasses of the counting…
Over 300 sequences and many unsolved problems and conjectures related to them are presented herein together with theorems corollaries, formulae, examples, mathematical criteria, etc. (about integer sequences, numbers, quotients, residues,…
Quantile aggregation with dependence uncertainty has a long history in probability theory with wide applications in finance, risk management, statistics, and operations research. Using a recent result on inf-convolution of quantile-based…
Although NP-Complete problems are the most difficult decisional problems, it is possible to discover in them polynomial (or easy) observables. We study the Graph Partitioning Problem showing that it is possible to recognize in it two…
These highly informal lecture notes aim at introducing and explaining several closely related problems on zeros of analytic functions defined by ordinary differential equations and systems of such equations. The main incentive for this…
In this survey article we discuss key open problems which could serve as a guidance for further research directions of multiplicative ideal theory and factorization theory.
In this article, we give an account of some recent irreducibility testing criteria for polynomials having integer coefficients over the field of rational numbers.
The research area of algorithms with predictions has seen recent success showing how to incorporate machine learning into algorithm design to improve performance when the predictions are correct, while retaining worst-case guarantees when…
We employ the notions of `sequential function' and `interrogation' (dialogue) in order to define new partial combinatory algebra structures on sets of functions. These structures are analyzed using J. Longley's preorder-enriched category of…
We prove a function field analogue of Maynard's result about primes with restricted digits. That is, for certain ranges of parameters n and q, we prove an asymptotic formula for the number of irreducible polynomials of degree n over a…
In [1, 2], we have explored the theoretical aspects of feature extraction optimization processes for solving largescale problems and overcoming machine learning limitations. Majority of optimization algorithms that have been introduced in…
The auto-correlations of arithmetic functions, such as the von Mangoldt function, the M\"obius function and the divisor function, are the subject of classical problems in analytic number theory. The function field analogues of these…
In this paper we obtain bounds for integer solutions of quadratic polynomials in two variables that represent a natural number. Also we get some results on twin prime numbers. In addition, we use linear functionals to prove some results of…
Evidences have suggested that counting representations are sometimes tractable even when the corresponding classification problem is almost impossible, or "wild" in a precise sense. Such counting problems are directly related to matrix…
Compared with constraint satisfaction problems, counting problems have received less attention. In this paper, we survey research works on the problems of counting the number of solutions to constraints. The constraints may take various…
Motivated by some algorithmic problems, we give lower bounds on the size of the multiplicative groups containing rational function images of low-dimensional affine subspaces of a finite field~$\mathbb{F}_{q^n}$ considered as a linear space…
We investigate the techniques and ideas used in the convergence analysis of two proximal ADMM algorithms for solving convex optimization problems involving compositions with linear operators. Besides this, we formulate a variant of the ADMM…
In this paper, we investigate the relative power of several conjectures that attracted recently lot of interest. We establish a connection between the Network Coding Conjecture (NCC) of Li and Li and several data structure like problems…