Related papers: The Poincar\'e Inequality does not improve with bl…
We study the stability of recently constructed self-similar blow-up solutions to the incompressible Euler equation. A consequence of our work is the existence of finite-energy $C^{1,\alpha}$ solutions that become singular in finite time in…
A simple classical, deterministic, local situation violating the Bell inequality is described. The detectors used in the experiment are ideal and the observers who decide which pair of measuring devices to choose for a given pair of…
We introduce a family of information leakage measures called maximal $\alpha,\beta$-leakage, parameterized by real numbers $\alpha$ and $\beta$. The measure is formalized via an operational definition involving an adversary guessing an…
An exponent $\beta$ which characterises non-equilibrium coarsening processes is calculated in a deterministic solvable model of coarsening for a 1D q-state Potts system. We study how the fraction of sites P which have never changed their…
The blow-up for semilinear wave equations with the scale invariant damping has been well-studied for sub-Fujita exponent. However, for super-Fujita exponent, there is only one blow-up result which is obtained in 2014 by Wakasugi in the case…
In this work, we study inequalities and enumerative formulas for flags of Pfaff systems on $\mathbb{P}^n_{\mathbb{C}}$. More specifically, we find the number of independent Pfaff systems that leave invariant a one-dimensional holomorphic…
We show that the Poincar\'{e} inequality holds on an open set $D\subset\mathbb{R}^n$ if and only if $D$ admits a smooth, bounded function whose Laplacian has a positive lower bound on $D$. Moreover, we prove that the existence of such a…
We study positive blowing-up solutions of systems of the form: $$u_t=\delta_1 \Delta u+e^{pv},\quad v_t= \delta_2\Delta v+e^{qu},$$ with $\delta_1,\delta_2>0$ and $p, q>0$. We prove single-point blow-up for large classes of radially…
The paper investigates a class of a semilinear wave equation with time-dependent damping term ($-\frac{1}{{(1+t)}^{\beta}}\Delta u_t$) and a nonlinearity $|u|^p$. We will show the influence of the the parameter $\beta$ in the blow-up…
Let $(\mathcal{X}, \rho, \mu)$ be a metric measure space of homogeneous type which supports a certain Poincar\'e inequality. Denote by the symbol $\mathcal{C}_{\mathrm{c}}^\ast(\mathcal{X})$ the space of all continuous functions $f$ with…
Let $\alpha, \beta \geq 0$ and $\alpha + \beta < 1$. In this short note, we show that $\liminf_{n \to \infty} p_n^\beta(p_{n+1}^\alpha - p_n^\alpha) = 0$, where $p_n$ is the $n$th prime. This notes an improvement over results of S\'{a}ndor…
Let $\Omega \subset \mathbb{R}^n$ be a convex. If $u: \Omega \rightarrow \mathbb{R}$ has mean 0, then we have the classical Poincar\'{e} inequality $$ \|u \|_{L^p} \leq c_p \mbox{diam}(\Omega) \| \nabla u \|_{L^p}$$ with sharp constants…
We construct a class of Fourier multipliers whose associated operators are weak (1,1) bounded but fail to be weak (p, p) bounded for any 1 < p \leq \infty. Moreover, we show that this result is sharp.
The correspondence principle suggests that quantum systems grow classical when large. Classical systems cannot violate Bell inequalities. Yet agents given substantial control can violate Bell inequalities proven for large-scale systems. We…
We introduce a straightforward method to analyze the blow-up of systems of ordinary differential inequalities, and apply it to study the blow- up of solutions to a weakly coupled system of semilinear heat equations. We prove that the…
This article studies the finite time blow-up of weak solutions to a structural acoustics model consisting of a semilinear wave equation defined on a bounded domain $\Omega\subset\mathbb{R}^3$ which is strongly coupled with a Berger plate…
We establish Sobolev-Poincar\'e inequalities for piecewise $W^{1,p}$ functions over families of fairly general polytopic (thence also shape-regular simplicial and Cartesian) meshes in any dimension; amongst others, they cover the case of…
We are concerned with the existence of blowing-up solutions to the following boundary value problem $$-\Delta u= \la a(x) e^u-4\pi N \delta_0\;\hbox{ in } \Omega,\quad u=0 \;\hbox{ on }\partial \Omega,$$ where $\Omega$ is a smooth and…
This is a continuation of our previous work [13]. Let $(\Sigma,g)$ be a closed Riemann surface, where the metric $g$ has conical singularities at finite points. Suppose $\mathbf{G}$ is a group whose elements are isometries acting on…
We establish a general inequality comparing the $F$-thresholds of a local ring and its associated graded ring. As an application, we deduce that the $F$-rationality of the graded ring descends to the local ring.