The lower semicontinuity of functionals of the type ${\int }_{\Omega }f(x,u,v,\nabla u)dx$ with respect to the $\left(W^{1,1}\times L^p\right)$-weak* topology is studied. Moreover, in absence of lower semicontinuity, an integral representation in $W^{1,1}\times L^p$ for the lower semicontinuous envelope is also provided.

In the paper we give an analogue of the Filippov Lemma for the fourth order differential inclusions y = y”” - (A² + B²)y” + A²B²y ∈ F(t,y), (*) with the initial conditions y(0) = y’(0) = y”(0) = y”’(0) = 0, (**) where the matrices $A,B\in {ℝ}^{d\times d}$ are commutative and the multifunction $F:[0,1]\times {ℝ}^d⇝cl\left({ℝ}^d\right)$ is Lipschitz continuous in y with a t-independent constant l < ||A||²||B||². Main theorem. Assume that $F:[0,1]\times {ℝ}^d⇝cl\left({ℝ}^d\right)ismeasurableintandintegrablybounded.Let$y₀ ∈ W4,1$beanarbitraryfunctionsatisfying(**)andsuchthat$ $d_H(y₀\left(t\right),F(t,y₀\left(t\right)))\le p₀\left(t\right)$ a.e. in [0,1], where p₀ ∈ L¹[0,1]. Then there exists a solution y ∈ W4,1 of (*)...

The paper deals with logarithmic capacities, an important tool in pluripotential theory. We show that a class of capacities, which contains the L-capacity, has the following product property: $C_{\nu }\left(E₁\times E₂\right)=min(C_{\nu ₁}\left(E₁\right),C_{\nu ₂}\left(E₂\right))$, where $E_j$ and ${\nu }_j$ are respectively a compact set and a norm in ${ℂ}^{N_j}$ (j = 1,2), and ν is a norm in ${ℂ}^{N₁+N₂}$, ν = ν₁⊕ₚ ν₂ with some 1 ≤ p ≤ ∞. For a convex subset E of ${ℂ}^N$, denote by C(E) the standard L-capacity and by ${\omega }_E$ the minimal width of E, that is, the minimal Euclidean distance between two supporting hyperplanes...

An investigation is launched into the fundamental characteristics of operations on and between sets, with a focus on compact convex sets and star sets (compact sets star-shaped with respect to the origin) in $n$-dimensional Euclidean space ${ℝ}^n$. It is proved that if $n\ge 2$, with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, $GL\left(n\right)$ covariant, and associative if and only if it is $L_p$ addition for some $1\le p\le \infty$. It is also demonstrated...

Let $H\subseteq Z\subseteq 2^{\omega }$. For n ≥ 2, we prove that if Selivanovski measurable functions from $2^{\omega }$ to Z give as preimages of H all Σₙ¹ subsets of $2^{\omega }$, then so do continuous injections.

Let X be a Polish space, and let C₀ and C₁ be disjoint coanalytic subsets of X. The pair (C₀,C₁) is said to be complete if for every pair (D₀,D₁) of disjoint coanalytic subsets of ${\omega }^{\omega }$ there exists a continuous function $f:{\omega }^{\omega }\to X$ such that $f^{-1}\left(C₀\right)=D₀$ and $f^{-1}\left(C₁\right)=D₁$. We give several explicit examples of complete pairs of coanalytic sets.

The aim of this paper is to establish an existence and uniqueness result for a class of the set functional differential equations of neutral type$$\left\{\begin{array}{c}{D}_{H}X\left(t\right)=F(t,X_t,D_H{X}_t),\hfill \\ {X|}_{[-r,0]}=\Psi ,\hfill \end{array}\right.$$ where $F:[0,b]\times {𝒞}_0\times {𝔏}_0^1\to K_c\left(E\right)$ is a given function, $K_c\left(E\right)$ is the family of all nonempty compact and convex subsets of a separable Banach space $E$, ${𝒞}_0$ denotes the space of all continuous set-valued functions $X$ from $[-r,0]$ into $K_c\left(E\right)$, ${𝔏}_0^1$ is the space of all integrally bounded set-valued functions $X:[-r,0]\to K_c\left(E\right)$, $\Psi \in {𝒞}_0$ and $D_H$ is the Hukuhara derivative. The continuous dependence of solutions on initial...

Let X be an arbitrary set, and γ: X × X → ℝ any function. Let Φ be a family of real-valued functions defined on X. Let $\Gamma :X\to 2^{\Phi }$ be a cyclic ${\Phi }^{\gamma (·,·)}$-monotone multifunction with non-empty values. It is shown that the following generalization of the Rockafellar theorem holds. There is a function f: X → ℝ such that Γ is contained in the ${\Phi }^{\gamma (·,·)}$-subdifferential of f, $\Gamma \left(x\right)\subset {\partial }_{\Phi }^{\gamma (·,·)}{f|}_x$.

We prove an interpolatory estimate linking the directional Haar projection $P^{\left(\epsilon \right)}$ to the Riesz transform in the context of Bochner-Lebesgue spaces $L^p(ℝⁿ;X)$, 1 < p < ∞, provided X is a UMD-space. If ${\epsilon }_{i₀}=1$, the result is the inequality $\left|\right|P^{\left(\epsilon \right)}{u\left|\right|}_{L^p(ℝⁿ;X)}{\le C\left|\right|u\left|\right|}_{L^p(ℝⁿ;X)}^{1/}\left|\right|R_{i₀}u{\left|\right|}_{L^p(ℝⁿ;X)}^{1-1/}$, (1) where the constant C depends only on n, p, the UMD-constant of X and the Rademacher type of $L^p(ℝⁿ;X)$. In order to obtain the interpolatory result (1) we analyze stripe operators $S_{\lambda }$, λ ≥ 0, which are used as basic building blocks to dominate the directional Haar projection....

We give a complete characterization of closed sets $F\subset {ℝ}^2$ whose distance function $d_F:=\mathrm{dist}(·,F)$ is DC (i.e., is the difference of two convex functions on ${ℝ}^2$). Using this characterization, a number of properties of such sets is proved.

Let ${\mu }_{n-1}\left(G\right)$ be the algebraic connectivity, and let ${\mu }_1\left(G\right)$ be the Laplacian spectral radius of a $k$-connected graph $G$ with $n$ vertices and $m$ edges. In this paper, we prove that $${\mu }_{n-1}\left(G\right)\ge \frac{2n{k}^2}{(n(n-1)-2m)(n+k-2)+2k^2},$$ with equality if and only if $G$ is the complete graph $K_n$ or $K_n-e$. Moreover, if $G$ is non-regular, then $${\mu }_1\left(G\right)<2\Delta -\frac{2(n\Delta -2m)k^2}{2(n\Delta -2m)(n^2-2n+2k)+n{k}^2},$$ where $\Delta$ stands for the maximum degree of $G$. Remark that in some cases, these two inequalities improve some previously known results.

A new class of functions called “$R_z$-supercontinuous functions” is introduced. Their basic properties are studied and their place in the hierarchy of strong variants of continuity that already exist in the literature is elaborated. The class of $R_z$-supercontinuous functions properly includes the class of $R_{\mathrm{cl}}$-supercontinuous functions, Tyagi, Kohli, Singh (2013), which in its turn contains the class of $\mathrm{cl}$-supercontinuous ($\equiv$ clopen continuous) functions, Singh (2007), Reilly, Vamanamurthy (1983),...

Let Ω be a measure space, and E, F be separable Banach spaces. Given a multifunction $f:\Omega \times E\to 2^F$, denote by $N_f\left(x\right)$ the set of all measurable selections of the multifunction $f(·,x\left(·\right)):\Omega \to 2^F$, s ↦ f(s,x(s)), for a function x: Ω → E. First, we obtain new theorems on H-upper/H-lower/lower semicontinuity (without assuming any conditions on the growth of the generating multifunction f(s,u) with respect to u) for the multivalued (Nemytskiĭ) superposition operator $N_f$ mapping some open domain G ⊂ X into $2^Y$, where X and Y are...

We define a new function space $B$, which contains in particular BMO, BV, and $W^{1/p,p}$, $1<p<\infty$. We investigate its embedding into Lebesgue and Marcinkiewicz spaces. We present several inequalities involving $L^p$ norms of integer-valued functions in $B$. We introduce a significant closed subspace, $B_0$, of $B$, containing in particular VMO and $W^{1/p,p}$, $1\le p<\infty$. The above mentioned estimates imply in particular that integer-valued functions belonging to $B_0$ are necessarily constant. This framework provides a “common roof”...

Let $ex(n,G)$ denote the maximum number of edges in a graph on $n$ vertices which does not contain $G$ as a subgraph. Let $P_i$ denote a path consisting of $i$ vertices and let $m{P}_i$ denote $m$ disjoint copies of $P_i$. In this paper we count $ex(n,3P_4)$.

In this paper, we study the size of the giant component $C_G$ in the random geometric graph $G=G(n,r_n,f)$ of $n$ nodes independently distributed each according to a certain density $f(·)$ in ${[0,1]}^2$ satisfying ${inf}_{x\in {[0,1]}^2}f\left(x\right)gt;$ $0$. If $\frac{c_1}{n}\le r_n^2\le c_2\frac{logn}{n}$ for some positive constants $c_1$, $c_2$ and $n{r}_n^2⟶\infty$ as $n\to \infty$, we show that the giant component of $G$ contains at least $n-\mathrm{o}\left(n\right)$ nodes with probability at least $1-{\mathrm{e}}^{-\beta n{r}_n^2}$ for all $n$ and for some positive constant $\beta$. We also obtain estimates on the diameter and number of the non-giant components of $G$.

We consider the integral functional $J\left(u\right)={\int }_{\Omega }\left[f\right(\left|Du\right|\left)-u\right]dx$, $u\in W_0^{1,1}\left(\Omega \right)$, where $\Omega \subset {ℝ}^n$, $n\ge 2$, is a nonempty bounded connected open subset of ${ℝ}^n$ with smooth boundary, and $ℝ\ni s\mapsto f\left(\right|s\left|\right)$ is a convex, differentiable function. We prove that if $J$ admits a minimizer in $W_0^{1,1}\left(\Omega \right)$ depending only on the distance from the boundary of $\Omega$, then $\Omega$ must be a ball.