We extend the classical theorems of I. I. Privalov and A. Zygmund from single to multiple conjugate functions in terms of the multiplicative modulus of continuity. A remarkable corollary is that if a function f belongs to the multiplicative Lipschitz class $Lip(\alpha ₁,...,{\alpha }_N)$ for some $0<\alpha ₁,...,{\alpha }_N<1$ and its marginal functions satisfy $f(·,x₂,...,x_N)\in Lip\beta ₁,...,f(x₁,...,x_{N-1},·)\in Lip{\beta }_N$ for some $0<\beta ₁,...,{\beta }_N<1$ uniformly in the indicated variables $x_l$, 1 ≤ l ≤ N, then $f{̃}^{(\eta ₁,...,{\eta }_N)}\in Lip(\alpha ₁,...,{\alpha }_N)$ for each choice of $(\eta ₁,...,{\eta }_N)$ with ${\eta }_l=0$ or 1 for 1 ≤ l ≤ N.

We find conditions on a real function f:[a,b] → ℝ equivalent to being Lebesgue equivalent to an n-times differentiable function (n ≥ 2); a simple solution in the case n = 2 appeared in an earlier paper. For that purpose, we introduce the notions of $CBV{G}_{1/n}$ and $SBV{G}_{1/n}$ functions, which play analogous rôles for the nth order differentiability to the classical notion of a VBG⁎ function for the first order differentiability, and the classes $CB{V}_{1/n}$ and $SB{V}_{1/n}$ (introduced by Preiss and Laczkovich) for Cⁿ smoothness....

Si dimostra che ogni funzione multivoca lipschitziana con costante di Lipschitz $M$, definita su un sottoinsieme di uno spazio di Hilbert $H$ a valori compatti e convessi in ${ℝ}^n$, può essere estesa su tutto $H$ ad una funzione multivoca lipschitziana con costante minore di 7 nM. In generale, non esistono invece estensioni aventi la stessa costante di Lipschitz $M$.

We prove a result about the rectifiability of class $C^2$ of the set of regular values (in the sense of Clarke) of a Lipschitz map $\phi :{ℝ}^n\to {ℝ}^N$ with

It is proved (independently of the result of Holmes [Fund. Math. 140 (1992)]) that the dual space of the uniform closure $CFL{(}_r)$ of the linear span of the maps x ↦ d(x,a) - d(x,b), where d is the metric of the Urysohn space ${}_r$ of diameter r, is (isometrically if r = +∞) isomorphic to the space $LIP{(}_r)$ of equivalence classes of all real-valued Lipschitz maps on ${}_r$. The space of all signed (real-valued) Borel measures on ${}_r$ is isometrically embedded in the dual space of $CFL{(}_r)$ and it is shown that the image...

By $F_n\left(X\right)$, $n\ge 1$, we denote the $n$-th symmetric product of a metric space $(X,d)$ as the space of the non-empty finite subsets of $X$ with at most $n$ elements endowed with the Hausdorff metric $d_H$. In this paper we shall describe that every isometry from the $n$-th symmetric product $F_n\left(X\right)$ into itself is induced by some isometry from $X$ into itself, where $X$ is either the Euclidean space or the sphere with the usual metrics. Moreover, we study the $n$-th symmetric product of the Euclidean space up to bi-Lipschitz equivalence...

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 (*)...

In this note we shall prove that for a continuous function $\varphi :\Delta \to {ℝ}^n$, where $\Delta \subset ℝ$,  the paratingent of $\varphi$ at $a\in \Delta$ is a non-empty and compact set in ${ℝ}^n$ if and only if $\varphi$ satisfies Lipschitz condition in a neighbourhood of $a$. Moreover, in this case the paratingent is a connected set.

For 0 < γ ≤ 1, let $\Lambda {⁺}_{\gamma }$ be the big Lipschitz algebra of functions analytic on the open unit disc which satisfy a Lipschitz condition of order γ on ̅. For a closed set E on the unit circle and an inner function Q, let $J_{\gamma }(E,Q)$ be the closed ideal in $\Lambda {⁺}_{\gamma }$ consisting of those functions $f\in \Lambda {⁺}_{\gamma }$ for which (i) f = 0 on E, (ii) $|f\left(z\right)-f\left(w\right)|=o\left(\right|z-w{|}^{\gamma })$ as d(z,E),d(w,E) → 0, (iii) $f/Q\in \Lambda {⁺}_{\gamma }$. Also, for a closed ideal I in $\Lambda {⁺}_{\gamma }$, let $E_I$ = z ∈ : f(z) = 0 for every f ∈ I and let $Q_I$ be the greatest common divisor of the inner parts of non-zero functions...

Si dimostra che ogni funzione multivoca lipschitziana con costante di Lipschitz $M$, definita su un sottoinsieme di uno spazio di Hilbert $H$ a valori compatti e convessi in ${ℝ}^n$, può essere estesa su tutto $H$ ad una funzione multivoca lipschitziana con costante minore di 7 nM. In generale, non esistono invece estensioni aventi la stessa costante di Lipschitz $M$.

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 $ex(n,3P_4)$.

Let $T_1,\cdots ,T_d$ be homogeneous trees with degrees $q_1+1,\cdots ,q_d+1\ge 3$, respectively. For each tree, let $𝔥:T_j\to \mathbb{Z}$ be the Busemann function with respect to a fixed boundary point (end). Its level sets are the horocycles. The horocyclic product of $T_1,\cdots ,T_d$ is the graph $\mathrm{𝖣𝖫}(q_1,\cdots ,q_d)$ consisting of all $d$-tuples $x_1\cdots x_d\in T_1\times \cdots \times T_d$ with $𝔥\left(x_1\right)+\cdots +𝔥\left(x_d\right)=0$, equipped with a natural neighbourhood relation. In the present paper, we explore the geometric, algebraic, analytic and probabilistic properties of these graphs and their isometry groups. If $d=2$ and $q_1=q_2=q$ then we obtain a Cayley graph...

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.

A graph $X$, with a group $G$ of automorphisms of $X$, is said to be $(G,s)$-transitive, for some $s\ge 1$, if $G$ is transitive on $s$-arcs but not on $(s+1)$-arcs. Let $X$ be a connected $(G,s)$-transitive graph of prime valency $p\ge 5$, and $G_v$ the vertex stabilizer of a vertex $v\in V\left(X\right)$. Suppose that $G_v$ is solvable. Weiss (1974) proved that $|G_v{|\mid p(p-1)}^2$. In this paper, we prove that $G_v\cong ({\mathbb{Z}}_p⋊{\mathbb{Z}}_m)\times {\mathbb{Z}}_n$ for some positive integers $m$ and $n$ such that $ndivm$ and $m\mid p-1$.

Let α ∈ (0,1) and let $G=(V_G,E_G$) be a graph. According to Dunbar, Hoffman, Laskar and Markus [3] a set $D\subseteq V_G$ is called an α-dominating set of G, if $|N_G\left(u\right)\cap D|\ge \alpha d_G\left(u\right)$ for all $u\in V_G\setminus D$. We prove a series of upper bounds on the α-domination number of a graph G defined as the minimum cardinality of an α-dominating set of G.

Let $R$ be a commutative ring with nonzero identity and $J\left(R\right)$ the Jacobson radical of $R$. The Jacobson graph of $R$, denoted by ${𝔍}_R$, is defined as the graph with vertex set $R\setminus J\left(R\right)$ such that two distinct vertices $x$ and $y$ are adjacent if and only if $1-xy$ is not a unit of $R$. The genus of a simple graph $G$ is the smallest nonnegative integer $n$ such that $G$ can be embedded into an orientable surface $S_n$. In this paper, we investigate the genus number of the compact Riemann surface in which ${𝔍}_R$ can be embedded and...

Let $G$ be a simple connected graph with vertex set $V\left(G\right)=\{v_1,v_2,\cdots ,v_n\}$ and edge set $E\left(G\right)$, and let $d_{v_i}$ be the degree of the vertex $v_i$. Let $D\left(G\right)$ be the distance matrix and let $T_r\left(G\right)$ be the diagonal matrix of the vertex transmissions of $G$. The generalized distance matrix of $G$ is defined as $D_{\alpha }\left(G\right)=\alpha T_r\left(G\right)+(1-\alpha )D\left(G\right)$, where $0\le \alpha \le 1$. Let ${\lambda }_1\left(D_{\alpha }\left(G\right)\right)\ge {\lambda }_2\left(D_{\alpha }\left(G\right)\right)\ge ...\ge {\lambda }_n\left(D_{\alpha }\left(G\right)\right)$ be the generalized distance eigenvalues of $G$, and let $k$ be an integer with $1\le k\le n$. We denote by $S_k\left(D_{\alpha }\left(G\right)\right)={\lambda }_1\left(D_{\alpha }\left(G\right)\right)+{\lambda }_2\left(D_{\alpha }\left(G\right)\right)+...+{\lambda }_k\left(D_{\alpha }\left(G\right)\right)$ the sum of the $k$ largest generalized distance eigenvalues. The generalized distance spread of a graph $G$ is defined as $D_{\alpha }S\left(G\right)={\lambda }_1\left(D_{\alpha }\left(G\right)\right)-{\lambda }_n\left(D_{\alpha }\left(G\right)\right)$....

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 $G$.

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.

Let $G$ be a finite group. The intersection graph ${\Delta }_G$ of $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of $G$, and two distinct vertices $X$ and $Y$ are adjacent if $X\cap Y\ne 1$, where $1$ denotes the trivial subgroup of order $1$. A question was posed by Shen (2010) whether the diameters of intersection graphs of finite non-abelian simple groups have an upper bound. We answer the question and show that the diameters...

Let n be a nonnegative integer and let u ∈ (n,n+1]. We say that f is u-times Peano bounded in the approximate (resp. $L^p$, 1 ≤ p ≤ ∞) sense at $x\in {ℝ}^m$ if there are numbers $f_{\alpha }\left(x\right)$, |α| ≤ n, such that $f(x+h)-{\sum }_{\left|\alpha \right|\le n}{f}_{\alpha }\left(x\right)h^{\alpha }/\alpha !$ is $O\left(h^u\right)$ in the approximate (resp. $L^p$) sense as h → 0. Suppose f is u-times Peano bounded in either the approximate or $L^p$ sense at each point of a bounded measurable set E. Then for every ε > 0 there is a perfect set Π ⊂ E and a smooth function g such that the Lebesgue measure of E∖Π is less than ε and...

Let E be a compact set in the complex plane, $g_E$ be the Green function of the unbounded component of ${ℂ}_{\infty }\setminus E$ with pole at infinity and $Mₙ\left(E\right)=sup\left(\right||P^{\text{'}}{\left|\right|}_E{)/(\left|\right|P\left|\right|}_E)$ where the supremum is taken over all polynomials ${P|}_E\not\equiv 0$ of degree at most n, and ${\left|\right|f\left|\right|}_E=sup\left|f\right(z\left)\right|:z\in E$. The paper deals with recent results concerning a connection between the smoothness of $g_E$ (existence, continuity, Hölder or Lipschitz continuity) and the growth of the sequence ${Mₙ\left(E\right)}_{n=1,2,...}$. Some additional conditions are given for special classes of sets.