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We show how to carry out a sieve of Eratosthenes up to N in space O(N^{1/3} (log N)^{2/3}) and time O(N log N). These bounds constitute an improvement over the usual versions of the sieve, which take space about O(sqrt{N}) and time at least…

Motivated by a question of V. Bergelson and F. K. Richter (2017), we obtain asymptotic formulas for the number of relatively prime tuples composed of positive integers $n\le N$ and integer parts of polynomials evaluated at $n$. The error…

We study a logarithmic fractional Schr\"odinger--Poisson system in \(\R^{3}\): \begin{equation*} \begin{cases} \varepsilon^{2\alpha}(-\Delta)^{\alpha}u+V(x)u+\phi u=u\log u^{2}+|u|^{p-2}u, & \text{in }\R^{3},\\…

The Generalised Baker-Schmidt Problem (1970) concerns the Hausdorff measure of the set of $\psi$-approximable points on a nondegenerate manifold. Beresnevich-Dickinson-Velani (in 2006, for the homogeneous setting) and…

We consider the average-case complexity of some otherwise undecidable or open Diophantine problems. More precisely, we show that the following two problems can be solved in the complexity class PSPACE: (I) Given polynomials f_1,...,f_m in…

We propose a duality scheme for solving constrained nonsmooth and nonconvex optimization problems in a reflexive Banach space. We establish strong duality for a very general type of augmented Lagrangian, in which we assume a less…

We study the initial value problem of the Einstein-Dirac system, and show the stability of the Minkowski solution in the massless case with the use of generalized wave coordinates. This requires the understanding of the Dirac equation in…

This article concerns the dressing method for solving of multidimensional nonlinear Partial Differential Equations. In particular, we join hierarchy of matrix Burgers type equation with hierarchies of equations integrable by the Inverse…

We study an infinite system of ordinary differential equations that models the evolution of coagulating and fragmenting clusters, which we assume to be composed of identical units. Under very mild assumptions on the coefficients we prove…

We consider radial solutions of the slightly subcritical problem $-\Delta u_\varepsilon = |u_\varepsilon|^{\frac{4}{n-2}-\varepsilon}u_\varepsilon$ either on $\mathbb R^n$ ($n\geq 3$) or in a ball $B$ satisfying Dirichlet or Neumann…

We prove $S$-arithmetic inhomogeneous Khintchine type theorems on analytic nondegenerate manifolds. The divergence case, which constitutes the main substance of this paper, is proved in the general context of Hausdorff measures using…

In this work, we develop and analyse a novel Hybrid High-Order discretisation of the Brinkman problem. The method hinges on hybrid discrete velocity unknowns at faces and elements and on discontinuous pressures. Based on the discrete…

We improve upon an Omega result due to Soundararajan with respect to general trigonometric polynomials having positive Fourier coefficients. Instead of Dirichlet's approximation theorem we employ the resonance method and this leads to…

Several new methods of numerical integration of Cauchy problems with blow-up solutions for nonlinear ordinary differential equations of the first- and second-order are described. Solutions of such problems have singularities whose positions…

We study solvability of the Diophantine equation \begin{equation*} \frac{n}{2^{n}}=\sum_{i=1}^{k}\frac{a_{i}}{2^{a_{i}}}, \end{equation*} in integers $n, k, a_{1},\ldots, a_{k}$ satisfying the conditions $k\geq 2$ and $a_{i}<a_{i+1}$ for…

We prove that the set of large values of the trigonometric polynomial over a subset of density of the primes has some additive structure, similarly to what happens for subsets of densities in $\mathbb{Z}/{N}\mathbb{Z}$ but in a weaker form.…

Complex bands $\vec{k}^{\perp}(E)$ in a semiconductor crystal, along a general direction $\vec{n}$, can be computed by casting Schr\"odinger's equation as a generalized polynomial eigenvalue problem. When working with primitive lattice…

Let $A\subset \N_{+}$ and by $P_{A}(n)$ denotes the number of partitions of an integer $n$ into parts from the set $A$. The aim of this paper is to prove several result concerning the existence of integer solutions of Diophantine equations…

Let $u_n$ be a fixed non-degenerate binary recurrence sequence with positive discriminant, $w$ a fixed non-zero integer and $p_1,p_2,\dots,p_s$ fixed, distinct prime numbers. In this paper we consider the Diophantine equation $u_n+u_m=w…

Let $\mathbf{f} = (f_1, \ldots, f_R)$ be a system of polynomials with integer coefficients in which the degrees need not all be the same. We provide sufficient conditions for which the system of equations $f_j (x_1, \ldots, x_n) = 0 \ (1…