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Using elementary arguments based on the Fourier transform we prove that for $1 \leq q < p < \infty$ and $s \geq 0$ with $s > n(1/2-1/p)$, if $f \in L^{q,\infty}(\R^n) \cap \dot{H}^s(\R^n)$ then $f \in L^p(\R^n)$ and there exists a constant…

Let P be a linear, second order, elliptic operator satisfying a Hardy inequality with potential W (i.e. $P-W\geq0$) and best constant $\alpha$. We give conditions so that the spectrum of $W^{-1}P$ is $[\alpha,\infty)$. We apply this to…

Sparse, irregular graphs show up in various applications like linear algebra, machine learning, engineering simulations, robotic control, etc. These graphs have a high degree of parallelism, but their execution on parallel threads of modern…

We investigate the possibility of improving the $p$-Poincar\'e inequality $\|\nabla_{\mathbb{H}^N} u\|_p \ge \Lambda_p \|u\|_p$ on the hyperbolic space, where $p>2$ and $\Lambda_p:=[(N-1)/p]^{p}$ is the best constant for which such…

A mathematical analysis is established for the weak Galerkin finite element methods for the Poisson equation with Dirichlet boundary value when the curved elements are involved on the interior edges of the finite element partition or/and on…

We prove the existence of the weak solutions to the compressible Navier--Stokes system with barotropic pressure $p(\varrho)=\varrho^\gamma$ for $\gamma\geq 9/5$ in three space dimension. The novelty of the paper is the approximation scheme…

For $m \geq 2$, let $(\mathbb{Z}_{m+1}^N, |\cdot|)$ denote the group equipped with the so-called $l^0$ metric, \[ |y| = \left| \big( y(1), \dots, y(N) \big) \right| := | \{1 \leq i \leq N : y(i) \neq 0 \} |,\] and define the…

A new primal-dual weak Galerkin (PDWG) finite element method is introduced and analyzed for the ill-posed elliptic Cauchy problems with ultra-low regularity assumptions on the exact solution. The Euler-Lagrange formulation resulting from…

Let $G$ be a strongly connected directed graph. We consider the following three problems, where we wish to compute the smallest strongly connected spanning subgraph of $G$ that maintains respectively: the $2$-edge-connected blocks of $G$…

We define the optimal constant $Y ( p_1 , p_2 ; G )$ of Young's convolution inequality as \begin{align} Y ( p_1 , p_2 ; G ) := \sup \{ \| \phi_1 * ( \phi_2 \Delta^{1 / p_1'} ) \|_p \mid \phi_1 , \phi_2 \colon G \to \mathbb{C} , \; \| \phi_1…

To construct a parallel approach for solving optimization problems with orthogonality constraints is usually regarded as an extremely difficult mission, due to the low scalability of the orthonormalization procedure. However, such demand is…

We introduce a new stabilization for discontinuous Galerkin methods for the Poisson problem on polygonal meshes, which induces optimal convergence rates in the polynomial approximation degree $p$. In the setting of [S. Bertoluzza and D.…

This paper is concerned with Lagrange interpolation by total degree polynomials in moderate dimensions. In particular, we are interested in characterising the optimal choice of points for the interpolation problem, where we define the…

In this paper we study the best constant in a Hardy inequality for the p-Laplace operator on convex domains with Robin boundary conditions. We show, in particular, that the best constant equals $((p-1)/p)^p$ whenever Dirichlet boundary…

We study tightness properties of a Lagrangian dual (LD) bound for the nonconvex alternating current optimal power flow (ACOPF) problem. We show an LD bound that can be computed in a parallel, decentralized manner. Specifically, the proposed…

In this paper we establish an optimal Lorentz space estimate for the Riesz potential acting on curl-free vectors: There is a constant $C=C(\alpha,d)>0$ such that \[ \|I_\alpha F \|_{L^{d/(d-\alpha),1}(\mathbb{R}^d;\mathbb{R}^d)} \leq C…

Let \phi(G) be the minimum conductance of an undirected graph G, and let 0=\lambda_1 <= \lambda_2 <=... <= \lambda_n <= 2 be the eigenvalues of the normalized Laplacian matrix of G. We prove that for any graph G and any k >= 2, \phi(G) =…

Let $1<p<\infty$. Let $\{T_t\}_{t>0}$ be a noncommutative symmetric diffusion semigroup on a semifinite von Neumann algebra $\mathcal{M}$, and let $\{P_t\}_{t>0}$ be its associated subordinated Poisson semigroup. The celebrated…

The main goal of the paper is to establish time semidiscrete and space-time fully discrete maximal parabolic regularity for the lowest order time discontinuous Galerkin solution of linear parabolic equations with time-dependent…

We investigate strong and weak versions of maximum and comparison principles for a class of quasilinear parabolic equations with the $p$-Laplacian $$ \partial_t u - \Delta_p u = \lambda |u|^{p-2} u + f(x,t) $$ under zero boundary and…