Related papers: Unbounded $p$-convergence in Lattice-Normed Vector…

In this paper we study convexity properties for quasilinear Lane-Emden-Fowler equations of the type $$\begin{cases} -\Delta_p u = a(x) u^q & \quad \hbox{ in $\Omega$},\\ u >0 & \quad \hbox{ in $\Omega$}, \\ u =0 & \quad \hbox{ on $\partial…

Let $E$ be a sublattice of a vector lattice $F$. $\left( x_\alpha \right)\subseteq E$ is said to be $ F $-order convergent to a vector $ x $ (in symbols $ x_\alpha \xrightarrow{Fo} x $), whenever there exists another net $…

Let (X,d,p) be a pointed metric space. A pretangent space to X at p is a metric space consisting of some equivalence classes of convergent to p sequences (x_n), x_n \in X, whose degree of convergence is comparable with a given scaling…

Consider the d-dimensional lattice Z^d where each vertex is ``open'' or ``closed'' with probability p or 1-p, respectively. An open vertex v is connected by an edge to the closest open vertex w such that the dth co-ordinates of v and w…

A net $(x_\alpha)$ in an $f$-algebra $E$ is said to be multiplicative order convergent to $x\in E$ if $\x_\alpha-x\u\oc 0$ for all $u\in E_+$. In this paper, we introduce the notions $mo$-convergence, $mo$-Cauchy, $mo$-complete,…

We apply the linking method for cones in normed spaces to p-Laplace equations with various nonlinear boundary conditions. Some existence results are obtained.

We study the pointwise convergence and the $\Gamma$-convergence of a family of non-local, non-convex functionals $\Lambda_\delta$ in $L^p(\Omega)$ for $p>1$. We show that the limits are multiples of $\int_{\Omega} |\nabla u|^p$. This is a…

A linear operator $T$ between two vector lattices normed by locally solid Riesz spaces is said to be $p_\tau$-continuous if, for any $p_\tau$-null net $(x_\alpha)$, the net $(Tx_\alpha)$ is $p_\tau$-null, and $T$ is said to be…

Given a set C in R^d, let p(C) be the probability that a random d-dimensional unimodular lattice, chosen according to Haar measure on SL(d,Z)\SL(d,R), is disjoint from C\{0}. For special convex sets C we prove bounds on p(C) which are sharp…

For vector lattices $E$ and $F$, where $F$ is Dedekind complete and supplied with a locally solid topology, we introduce the corresponding locally solid absolute strong operator topology on the order bounded operators $\mathcal…

This paper extends the theory of rough convergence from normed linear spaces to the more abstract setting of Riesz spaces. We introduce and systematically develop the concept of rough $\mathbb{c}$-convergence ($rc$-convergence) for nets. A…

Let $X$ and $E$ be $f$-algebras and $p:X \to E_+$ be a monotone vector norm. Then the triple $(X,p,E)$ is called a lattice-normed $f$-algebraic space. In this paper, we show a generalization of the extension of the Hahn-Banach theorem for…

We consider vector lattices endowed with locally solid convergence structures, which are not necessarily topological. We show that such a convergence is defined by the convergence to $0$ on the positive cone. Some results on unbounded…

In the category \(\mathbf{V}\) of unital archimedean vector lattices, four notions of uniform completeness obtain. In all cases completeness requires the convergence of uniformly Cauchy sequences; the completions are distinguished by the…

In first-passage percolation on the integer lattice, the Shape Theorem provides precise conditions for convergence of the set of sites reachable within a given time from the origin, once rescaled, to a compact and convex limiting shape.…

In order to find an extension of Brezis -- Lieb's lemma to the case of nets, we replace the almost everywhere convergence by the unbounded order convergence and introduce the Brezis -- Lieb property in normed lattices. Then we identify a…

We study uO convergence on infinitely distributive lattices, extending key properties known from Riesz spaces. We show that order continuity of uO convergence characterizes infinite distributivity. We examine O-adherence and uO adherence of…

Let $\Omega$ be a bounded open set and $p,q,r>1$. The main observation of the present work is the following: $W_0^{1,p}(\Omega)$-solutions of the equation $-\Delta_p u = \mu |u|^{q-2}u + |u|^{r-2}u$ parameterized by $\mu$ are in bijection…

Unbounded order convergence has lately been systematically studied as a generalization of almost everywhere convergence to the abstract setting of vector and Banach lattices. This paper presents a duality theory for unbounded order…

Let $d\geq 3$ be a fixed integer, and a prime number $p$ such that $\gcd(p,d)=1$. Let $A$ be the adjacency matrix of a random $d$-regular directed graph on $n$ vertices. We show that as a random matrix in ${\mathbb F}_p$, \begin{equation}…