We introduce function spaces $B^{p,\lambda }$ with Morrey-Campanato norms, which unify $B^{p,\lambda }$, $CM{O}^{p,\lambda }$ and Morrey-Campanato spaces, and prove the boundedness of the fractional integral operator $I_{\alpha }$ on these spaces.

We consider a double analytic family of fractional integrals $S_z^{\gamma ,\alpha }$ along the curve ${t\mapsto \left|t\right|}^{\alpha }$, introduced for α = 2 by L. Grafakos in 1993 and defined by $\left(S_z^{\gamma ,\alpha }f\right)(x₁,x₂):=1/\Gamma (z+1/2){\int \int |u-1|}^z\psi (u-1){f(x₁-t,x₂-u|t|}^{\alpha }{\left)du\right|t|}^{\gamma }dt/t$, where ψ is a bump function on ℝ supported near the origin, $f{\in }_c^{\infty }\left(ℝ²\right)$, z,γ ∈ ℂ, Re γ ≥ 0, α ∈ ℝ, α ≥ 2. We determine the set of all (1/p,1/q,Re z) such that $S_z^{\gamma ,\alpha }$ maps $L^p\left(ℝ²\right)$ to $L^q\left(ℝ²\right)$ boundedly. Our proof is based on product-type kernel arguments. More precisely, we prove that the kernel $K_{-1+i\theta }^{i\varrho ,\alpha }$ is a product kernel on ℝ², adapted to the curve ${t\mapsto \left|t\right|}^{\alpha }$; as a consequence, we show...

Given a two-dimensional fractional multiplicative process ${\left(F_t\right)}_{t\in [0,1]}$ determined by two Hurst exponents $H_1$ and $H_2$, we show that there is an associated uniform Hausdorff dimension result for the images of subsets of $[0,1]$ by $F$ if and only if $H_1=H_2$.

Given a compact manifold $N^n\subset {ℝ}^{\nu }$ and real numbers $s\ge 1$ and $1\le p\<\infty$, we prove that the class $C^{\infty }({\overline{Q}}^m;N^n)$ of smooth maps on the cube with values into $N^n$ is strongly dense in the fractional Sobolev space $W^{s,p}(Q^m;N^n)$ when $N^n$ is $\lfloor sp\rfloor$ simply connected. For $sp$ integer, we prove weak sequential density of $C^{\infty }({\overline{Q}}^m;N^n)$ when $N^n$ is $sp-1$ simply connected. The proofs are based on the existence of a retraction of ${ℝ}^{\nu }$ onto $N^n$ except for a small subset of $N^n$ and on a pointwise estimate of fractional derivatives of composition of maps in $W^{s,p}\cap W^{1,sp}$.

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 $A_{\alpha }f\left(x\right)={\left|B\right(0,\left|x\right|\left)\right|}^{-\alpha /n}{\int }_{B(0,|x\left|\right)}f\left(t\right)dt$ be the n-dimensional fractional Hardy operator, where 0 < α ≤ n. It is well-known that $A_{\alpha }$ is bounded from $L^p$ to $L^{p_{\alpha }}$ with $p_{\alpha }=np/(\alpha p-np+n)$ when n(1-1/p) < α ≤ n. We improve this result within the framework of Banach function spaces, for instance, weighted Lebesgue spaces and Lorentz spaces. We in fact find a ’source’ space $S_{\alpha ,Y}$, which is strictly larger than X, and a ’target’ space $T_Y$, which is strictly smaller than Y, under the assumption that $A_{\alpha }$ is bounded from X into Y and the Hardy-Littlewood...

The following iterated commutators $T_{\ast ,\Pi b}$ of the maximal operator for multilinear singular integral operators and $I_{\alpha ,\Pi b}$ of the multilinear fractional integral operator are introduced and studied: $T_{\ast ,\Pi b}\left(f⃗\right)\left(x\right)=su{p}_{\delta >0}\left|[b₁,[b₂,\dots {[b_{m-1},[bₘ,T_{\delta }]ₘ]}_{m-1}\cdots ]₂]₁\left(f⃗\right)\left(x\right)\right|$, $I_{\alpha ,\Pi b}\left(f⃗\right)\left(x\right)=[b₁,[b₂,\dots {[b_{m-1},[bₘ,I_{\alpha }]ₘ]}_{m-1}\cdots ]₂]₁\left(f⃗\right)\left(x\right)$, where $T_{\delta }$ are the smooth truncations of the multilinear singular integral operators and $I_{\alpha }$ is the multilinear fractional integral operator, $b_i\in BMO$ for i = 1,…,m and f⃗ = (f1,…,fm). Weighted strong and L(logL) type end-point estimates for the above iterated commutators associated with two classes of multiple...

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 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 $G$ 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 $D_{\alpha }S\left(G\right)={\lambda }_1\left(D_{\alpha }\left(G\right)\right)-{\lambda }_n\left(D_{\alpha }\left(G\right)\right)$....

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 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 set D of vertices in a graph G = (V,E) is a locating-dominating set (LDS) if for every two vertices u,v of V-D the sets N(u)∩ D and N(v)∩ D are non-empty and different. The locating-domination number ${\gamma }_L\left(G\right)$ is the minimum cardinality of a LDS of G, and the upper locating-domination number, ${\Gamma }_L\left(G\right)$ is the maximum cardinality of a minimal LDS of G. We present different bounds on ${\Gamma }_L\left(G\right)$ and ${\gamma }_L\left(G\right)$.

We introduce the generalized fractional integrals $I{̃}_{\alpha ,d}$ and prove the strong and weak boundedness of $I{̃}_{\alpha ,d}$ on the central Morrey spaces $B^{p,\lambda }\left(ℝⁿ\right)$. In order to show the boundedness, the generalized λ-central mean oscillation spaces ${\Lambda }_{p,\lambda }^{\left(d\right)}\left(ℝⁿ\right)$ and the generalized weak λ-central mean oscillation spaces $W{\Lambda }_{p,\lambda }^{\left(d\right)}\left(ℝⁿ\right)$ play an important role.

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.

The generalized $k$-connectivity ${\kappa }_k\left(G\right)$ of a graph $G$ was introduced by Chartrand et al. in 1984. As a natural counterpart of this concept, Li et al. in 2011 introduced the concept of generalized $k$-edge-connectivity which is defined as ${\lambda }_k\left(G\right)=min\{\lambda \left(S\right):S\subseteq V\left(G\right)$ and $\left|S\right|=k\}$, where $\lambda \left(S\right)$ denotes the maximum number $\ell$ of pairwise edge-disjoint trees $T_1,T_2,...,T_{\ell }$ in $G$ such that $S\subseteq V\left(T_i\right)$ for $1\le i\le \ell$. In this paper we prove that for any two connected graphs $G$ and $H$ we have ${\lambda }_3(G\square H)\ge {\lambda }_3\left(G\right)+{\lambda }_3\left(H\right)$, where $G\square H$ is the Cartesian product of $G$ and $H$. Moreover, the bound is sharp. We also...

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...

Suppose that $A$ is a real symmetric matrix of order $n$. Denote by $m_A\left(0\right)$ the nullity of $A$. For a nonempty subset $\alpha$ of $\{1,2,...,n\}$, let $A\left(\alpha \right)$ be the principal submatrix of $A$ obtained from $A$ by deleting the rows and columns indexed by $\alpha$. When $m_{A\left(\alpha \right)}\left(0\right)=m_A\left(0\right)+\left|\alpha \right|$, we call $\alpha$ a P-set of $A$. It is known that every P-set of $A$ contains at most $\lfloor n/2\rfloor$ elements. The graphs of even order for which one can find a matrix attaining this bound are now completely characterized. However, the odd case turned out to be more difficult to tackle. As...