Related papers: Linear equations in Piatetski-Shapiro primes

We consider the $d=1$ nonlinear Fokker-Planck-like equation with fractional derivatives $\frac{\partial}{\partial t}P(x,t)=D \frac{\partial^{\gamma}}{\partial x^{\gamma}}[P(x,t) ]^{\nu}$. Exact time-dependent solutions are found for $ \nu =…

We prove the existence of solutions for a class of quasilinear problems involving variable exponents and with nonlinearity having critical growth. The main tool used is the variational method, more precisely, Ekeland's Variational Principle…

We discuss spherically symmetric perfect fluid solutions of Einstein's equations which have equation of state ($p=\alpha \mu$) and which are self-similar in the sense that all dimensionless variables depend only upon $z\equiv r/t$. For each…

We show that there exists a bounded pattern of m consecutive primes for any m>0, that means a tuple H_m of m distinct non-negative integers h_i (i=1,2,...m) such that its translations contain arbitrarily long (finite) arithmetic…

\noindent Using the techniques connected with the measure of noncompactness we investigate the neutral difference equation of the following form \begin{equation*} \Delta \left(r_{n}\left(\Delta \left(x_{n}+p_{n}x_{n-k}\right) \right)…

Let $\mathcal{L}_\epsilon$ be a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients. We obtain the uniform $W^{1,p}$ estimate in a Lipschitz domain for solutions to the Dirichlet problem, where…

Let $[\, \cdot\,]$ be the floor function and $\|x\|$ denotes the distance from $x$ to the nearest integer. In this paper we show that whenever $\alpha$ is irrational and $\beta$ is real then for any fixed $1<c<12/11$ there exist infinitely…

We continue to study the distribution of prime numbers $p$, satisfying the condition $\{ p^{\alpha} \} \in I \subset [0; 1)$, in arithmetic progressions. In the paper, we prove an analogue of Bombieri-Vinogradov theorem for $0 < \alpha <…

Each real number $x\in[0,1]$ admits a unique power-2-decaying Gauss-like expansion (P2GLE for short) as $x=\sum_{i\in\mathbb{N}} 2^{-(d_1(x)+d_2(x)+\cdots+d_i(x))}$, where $d_i(x)\in\mathbb{N}$. For any $x\in(0,1]$, the Khintchine exponent…

In a previous paper of the authors, we showed that for any polynomials $P_1,\dots,P_k \in \Z[\mathbf{m}]$ with $P_1(0)=\dots=P_k(0)$ and any subset $A$ of the primes in $[N] = \{1,\dots,N\}$ of relative density at least $\delta>0$, one can…

For fixed positive integers $n$, we study the solution of the equation $n = k + p_k$, where $p_k$ denotes the $k$th prime number, by means of the iterative method \[ k_{j+1} = \pi(n-k_j), \qquad k_0 = \pi(n), \] which converges to the…

Inspired by the Erd\"os-Turan conjecture we consider subsets of the natural numbers that contains infinitely many aritmetic progressions (APs) of any given length - such sets will be called AP-sets and we know due to the Green-Tao Theorem…

We prove that $\mathop{\mathbb{E}}_{m \leq M} \mathop{\mathbb{E}}_{n \leq N} \Lambda(n) \Lambda\bigl(n + \lfloor m^c \rfloor\bigr) = 1 + \rm{O}(\log^{2 - Bc} N)$, where $c > 2$ is a non-integer, $B \geq 3/c$, and $M$ is of order $N^{1/c}…

We prove existence of a probability solution to the nonlinear stationary Fokker-Planck-Kolmogorov equation on an infinite dimensional space with a centered Gaussian measure $\gamma$ with a unit diffusion operator and a drift of the form…

Asymptotic expansions of series $\sum_{k=0}^\infty \epsilon^k(k+a)^\gamma e^{-(k+a)^\alpha x}$ and $\sum_{k=0}^\infty \epsilon^k(k+a)^\gamma / (x(k+a)^\alpha+1)^\mu}$ in powers of $x$ as $x\to+0$ are found, where $\epsilon=1$ or…

We study the stationary solutions of the Gross-Pitaevskii equation that reduce, in the limit of vanishing non-linearity, to the eigenfunctions of the associated Schr\"odinger equation. By providing analytical and numerical support, we…

New bounds on the number of similar or directly similar copies of a pattern within a finite subset of the line or the plane are proved. The number of equilateral triangles whose vertices all lie within an $n$-point subset of the plane is…

We consider discrete linear Chebyshev approximation problems in which the unknown parameters of linear function are fitted by minimizing the maximum absolute deviation of errors. Such problems find application in the solution of…

For $p \in (1, \infty),$ for an integer $N \geq 2$ and for a bounded Lipschitz domain $\Omega$, we consider the following nonlinear Steklov bifurcation problem \begin{equation*} \begin{aligned} -\Delta_p \phi & = 0 \; \text{in} \ \Omega, \\…

We prove the infinitude of shifted primes $p-1$ without prime factors above $p^{0.2844}$. This refines $p^{0.2961}$ from Baker and Harman in 1998. Consequently, we obtain an improved lower bound on the the distribution of Carmichael…