Related papers: A Solution to Schroeder's Equation in Several Vari…

Let $B_1$ be the unit ball in $\mathbb{R}^N$ with $N \geq 2$. Let $f\in C^1([0, \infty), \mathbb{R})$, $f(0)=0$, $f(\beta) = \beta, \ f(s)<s\ \text{for}\ s\in (0,\beta), \ f(s)>s\ \text{for}\ s\in (\beta, \infty)$ and…

The Bloch equation is the fundamental dynamical model applicable to arbitrary two-level systems. Analytical solutions to date are incomplete for a number of reasons that motivate further investigation. The solution obtained here for the…

Kronecker observed that either all roots or only one root of a solvable irreducible equation of odd prime degree with integer coefficients are real. This gives a possibility to construct specific examples of equations not solvable by…

Let k be an algebraically closed field of characteristic zero. An element F from k(x_1,...,x_n) is called a closed rational function if the subfield k(F) is algebraically closed in the field k(x_1,...,x_n). We prove that a rational function…

We solve the stationary Schr\"odinger equation for a particle confined to a 3D spherical wedge -- the region $\{(r,\theta,\phi): 0 \leq r \leq R,\, 0 \leq \theta \leq \pi,\, 0 \leq \phi \leq \Phi\}$ with Dirichlet BCs on all surfaces. This…

We consider sparse polynomials in $N$ variables over a finite field, and ask whether they vanish on a set $S^N$, where $S$ is a set of nonzero elements of the field. We see that if for a polynomial $f$, there is $\mathbf{c}\in S^N$ with $f…

We consider a double power nonlinear Schr\"odinger equation which possesses the algebraically decaying stationary solution $\phi_0$ as well as exponentially decaying standing waves $e^{i\omega t}\phi_\omega(x)$ with $\omega>0$. It is…

We consider the one-dimensional Schr\"odinger equation with a potential satisfying the standard assumptions of the inverse scattering theory and supported on the half-line $x\ge 0$. For this equation at fixed positive energy we give…

Let $ \mathbb{Q}\mathcal{E}_{\mathbb{Z}} $ be the set of power sums whose characteristic roots belong to $ \mathbb{Z} $ and whose coefficients belong to $ \mathbb{Q} $, i.e. $ G : \mathbb{N} \rightarrow \mathbb{Q} $ satisfies…

For $c \in \mathbb{Q}$, consider the quadratic polynomial map $\varphi_c(x)=x^2-c$. Flynn, Poonen and Schaefer conjectured in 1997 that no rational cycle of $\varphi_c$ under iteration has length more than $3$. Here we discuss this…

In this paper we study the existence of solutions of a one-dimensional eigenvalue problem $-\left(|\phi_x|^{p-2}\phi_x\right)_x=\lambda \left(|\phi|^{q-2}\phi-f(\phi)\right)$ such that $\phi(0)=\phi(1)=0$, where $p,q>1$, $\lambda$ is a…

We prove that if $(\varphi_n)_{n=0}^\infty, \; \varphi_0 \equiv 1, $ is a basis in the space of entire functions of $d$ complex variables, $d\geq 1,$ then for every compact $K\subset \mathbb{C}^d$ there is a compact $K_1 \supset K$ such…

Let $f,g_1,\dots,g_m$ be polynomials with real coefficients in a vector of variables $x=(x_1,\dots,x_n)$. Denote by $\text{diag}(g)$ the diagonal matrix with coefficients $g=(g_1,\dots,g_m)$ and denote by $\nabla g$ the Jacobian of $g$. Let…

In this paper, by studying a class of 1-D Sturm-Liouville problems with periodic coefficients, we show and classify the solutions of periodic Schrodinger equations in a multidimensional case, which tells that not all the solutions are Bloch…

We prove, under certain conditions on $(\alpha,\beta)$, that each Schwartz function $f$ such that $f(\pm n^{\alpha}) = \hat{f}(\pm n^{\beta}) = 0, \forall n \ge 0$ must vanish identically, complementing a series of recent results involving…

In this work, we consider the 3D defocusing energy-critical nonlinear Schr\"odinger equation $i\partial_t u+\Delta u =|u|^4 u,\quad (t,x)\in \mathbb{R}\times \mathbb{R}^3$. Applying the outgoing and incoming decomposition presented in the…

The \emph{Chow parameters} of a Boolean function $f: \{-1,1\}^n \to \{-1,1\}$ are its $n+1$ degree-0 and degree-1 Fourier coefficients. It has been known since 1961 (Chow, Tannenbaum) that the (exact values of the) Chow parameters of any…

We consider the stationary semilinear Schr\"odinger equation $-\Delta u + a(x) u = f(x,u)$, $u\in H^1(\R^N)$, where $a$ and $f$ are continuous functions converging to some limits $a_\infty>0$ and $f_\infty=f_\infty(u)$ as $|x|\to\infty$. In…

In 1924 S.Bernstein asked for conditions on a uniformly bounded on $\mathbb{R}$ Borel function (weight) $w: \mathbb{R} \to [0, +\infty )$ which imply the denseness of algebraic polynomials ${\mathcal{P} }$ in the seminormed space $…

Behavior of solutions of $f''+Af=0$ is discussed under the assumption that $A$ is analytic in $\mathbb{D}$ and $\sup_{z\in\mathbb{D}}(1-|z|^2)^2|A(z)|<\infty$, where $\mathbb{D}$ is the unit disc of the complex plane. As a main result it is…