Related papers: Linear equations in Piatetski-Shapiro primes

We examine directed spanners through flow-based linear programming relaxations. We design an $\~O(n^{2/3})$-approximation algorithm for the directed $k$-spanner problem that works for all $k\geq 1$, which is the first sublinear…

For an irrational $\alpha\in \mathbb{R}$, we consider additive problems with the set of primes satisfying $\lVert\alpha p\rVert\leq \frac{1}{p^\tau}$ for some fixed $\tau>0$. In particular, we show that there exist infinitely many…

Let $[\, x\,]$ denote the integer part of a real number $x$. Assume that $\lambda_1,\lambda_2,\lambda_3$ are nonzero real numbers, not all of the same sign, that $\lambda_1/\lambda_2$ is irrational, and that $\eta$ is real. Let…

In this paper, it is proved that, for $\gamma\in(\frac{317}{320},1)$, every sufficiently large odd integer can be written as the sum of nine cubes of primes, each of which is of the form $[n^{1/\gamma}]$. This result constitutes an…

Almost sure convergence rates for linear algorithms $h_{k+1} = h_k +\frac{1}{k^\chi} (b_k-A_kh_k)$ are studied, where $\chi\in(0,1)$, $\{A_{k}\}_{k=1}^\infty$ are symmetric, positive semidefinite random matrices and $\{b_{k}\}_{k=1}^\infty$…

The purpose of this paper is to study subsequences of synchronizing $k$-automatic sequences $a(n)$ along Piatetski-Shapiro sequences $\lfloor n^c \rfloor$ with non-integer $c>1$. In particular, we show that $a(\lfloor n^c \rfloor)$…

In this paper, the problem of approximate symmetries of a class of non-linear reaction-diffusion equations called Kolmogorov-Petrovsky-Piskounov (KPP) equation is comprehensively analyzed. In order to compute the approximate symmetries, we…

The Sitnikov problem is a special case of the restricted three-body problem where the primaries moves in elliptic orbits of the two-body problem with eccentricity $e\in [0,1[$ and the massless body moves on a straight line perpendicular to…

We introduce a wide class of deterministic subsets of primes of zero relative density and we prove Roth's Theorem in these sets, namely, we show that any subset of them with positive relative upper density contains infinitely many…

We prove that the primes below $x$ are, on average, equidistributed in arithmetic progressions to smooth moduli of size up to $x^{1/2+1/40-\epsilon}$. The exponent of distribution $\tfrac{1}{2} + \tfrac{1}{40}$ improves on a result of…

We solve the modified Gursky-Streets equation, which is a fully nonlinear equation arising in conformal geometry with uniform $C^{1, 1}$ estimates when (i) $\gamma > 0$ and $1 \leq k \leq n$ or (ii) $r > 0$ and $2 s k \leq r n$. We also…

The Marchenko method is developed in the inverse scattering problem for a linear system of first-order differential equations containing potentials proportional to the spectral parameter. The corresponding Marchenko system of integral…

Let $[\,\cdot\,]$ denote the floor function. Assume that $\lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5$ are nonzero real numbers, not all of the same sign, that $\lambda_1/\lambda_2$ is irrational, and that $\eta$ is a real number.…

In this paper, we prove an extended version of the Minkowski Inequality, holding for any smooth bounded set $\Omega \subset \mathbb R^n$, $n\geq 3$. Our proof relies on the discovery of effective monotonicity formulas holding along the…

We study the infinite time dynamics of a class of nonlinear Schr\"odinger / Gross-Pitaevskii equations. In our previous paper, we prove the asymptotic stability of the nonlinear ground state in a general situation which admits degenerate…

Let $p_{k}$ denote the $k$-th prime and $d(p_{k}) = p_{k} - p_{k - 1}$, the difference between consecutive primes. We denote by $N_{\epsilon}(x)$ the number of primes $\leq x$ which satisfy the inequality $d(p_{k}) \leq (\log p_{k})^{2 +…

For $1<p<\infty$, we consider the following problem $$ -\Delta_p u=f(u),\quad u>0\text{ in }\Omega,\quad\partial_\nu u=0\text{ on }\partial\Omega, $$ where $\Omega\subset\mathbb R^N$ is either a ball or an annulus. The nonlinearity $f$ is…

We study arithmetic progressions in primes with common differences as small as possible. Tao and Ziegler showed that, for any $k \geq 3$ and $N$ large, there exist non-trivial $k$-term arithmetic progressions in (any positive density subset…

Recently, Grynkiewicz et al. [{\it Israel J. Math.} {\bf 193} (2013), 359--398], using tools from additive combinatorics and group theory, proved necessary and sufficient conditions under which the linear congruence $a_1x_1+\cdots…

A semiclassical approximation approach based on the Maslov complex germ method is considered in detail for the 1D nonlocal Fisher-Kolmogorov-Petrovskii-Piskunov equation under the supposition of weak diffusion. In terms of the semiclassical…