A graph is called supermagic if there is a labeling of edges where the edges are labeled with consecutive distinct positive integers such that the sum of the labels of all edges incident with any vertex is constant. A graph G is called degree-magic if there is a labeling of the edges by integers 1، 2، ...، |E(G)| such that the sum of the labels of the edges incident with any vertex v is equal to (1+|E(G)|)deg(v)/2. Degree-magic graphs extend supermagic regular graphs. In this paper we find the necessary and sufficient conditions for the existence of balanced degree-magic labelings of graphs obtained by taking the join، composition، Cartesian product، tensor product and strong product of complete bipartite graphs.

‎The sequences of the form ${E_{mb}g_{n}}_{m‎، ‎ninmathbb{Z}}$،‎ ‎where $E_{mb}$ is the modulation operator‎، ‎$b>0$ and $g_{n}$ is the‎ ‎window function in $L^{2}(mathbb{R})$‎، ‎construct Fourier-like‎ ‎systems‎. ‎We try to consider some sufficient conditions on the window‎ ‎functions of Fourier-like systems‎، ‎to make a frame and find a dual‎ ‎frame with the same structure‎. ‎We also extend the given two Bessel‎ ‎Fourier-like systems to make a pair of dual frames and prove that‎ ‎the window functions of Fourier-like Bessel sequences share the‎ ‎compactly supported property with their extensions‎. ‎But for‎ ‎polynomials windows‎، ‎a result of this type does not happen.

This paper studies some existence results for generalized epsilon-vector equilibrium problems and generalized epsilon-vector variational inequalities. The existence results for solutions are derived by using the celebrated KKM theorem. The results achieved in this paper generalize and improve the works of many authors in references.

Let $G$ be a simple connected graph. In this paper، Szeged dimension and PI$_v$ dimension of graph $G$ are introduced. It is proved that if $G$ is a graph of Szeged dimension $1$ then line graph of $G$ is 2-connected. The dimensions of five composite graphs: sum، corona، composition، disjunction and symmetric difference with strongly regular components is computed. Also explicit formulas of Szeged and PI$_v$ indices for these composite graphs is obtained.

Chen's biharmonic conjecture is well-known and stays open: The only biharmonic submanifolds of Euclidean spaces are the minimal ones. In this paper، we consider an advanced version of the conjecture، replacing $Delta$ by its extension، $L_1$-operator ($L_1$-conjecture). The $L_1$-conjecture states that any $L_1$-biharmonic Euclidean hypersurface is 1-minimal. We prove that the $L_1$-conjecture is true for $L_1$-biharmonic hypersurfaces with three distinct principal curvatures and constant mean curvature of a Euclidean space of arbitrary dimension.

Fej'{e}r Hadamard inequality is generalization of Hadamard inequality. In this paper we prove certain Fej'{e}r Hadamard inequalities for $k$-fractional integrals.
We deduce Fej'{e}r Hadamard-type inequalities for Riemann-Liouville fractional integrals. Also as special case Hadamard inequalities for $k$-fractional as well as fractional integrals are given.

The original aim of this paper is to construct a graph associated to a vector space. By inspiration of the classical definition for the Cayley graph related to a group we define Cayley graph of a vector space. The vector space Cayley graph ${rm Cay(mathcal{V}،S)}$ is a graph with the vertex set the whole vectors of the vector space $mathcal{V}$ and two vectors $v_1،v_2$ join by an edge whenever $v_1-v_2in S$ or $-S$، where $S$ is a basis of $mathcal{V}$. This fact causes a new connection between vector spaces and graphs. The vector space Cayley graph is made of copies of the cycles of length $t$، where $t$ is the cardinal number of the field that $mathcal{V}$ is constructed over it. The vector space Cayley graph is generalized to the graph $Gamma(mathcal{V}،S)$. It is a graph with vertex set whole vectors of $mathcal{V}$ and two vertices $v$ and $w$ are adjacent whenever $c_{1}upsilon+ c_{2}omega = sum^{n}_{i=1} alpha_{i}$، where $S={alpha_1،cdots،alpha_n}$ is an ordered basis for $mathcal{V}$ and $c_1،c_2$ belong to the field that the vector space $mathcal{V}$ is made of over. It is deduced that if $ S'$ is another basis for $mathcal{V}$ which is constructed by special invertible matrix $P$، then $Gamma(mathcal{V}،S)cong Gamma(mathcal{V}،S')$.

If M is a compact Riemannian manifold then we show that for typical continuous function defined on M، the upper box dimension of graph(f) is as big as possible and the lower box dimension of graph(f) is as small as possible.

In the present paper، we investigate the notion of I -statistical convergence and introduce I -st limit points and I -st cluster points of real number sequence and also studied some of its basic properties.

In this paper، a Bernoulli pseudo-spectral method for solving nonlinear fractional Volterra integro-differential equations is considered. First existence of a unique solution for the problem under study is proved. Then the Caputo fractional derivative and Riemman-Liouville fractional integral properties are employed to derive the new approximate formula for unknown function of the problem. The suggested technique transforms these types of equations to the solution of systems of algebraic equations. In the next step، the error analysis of the proposed method is investigated. Finally، the technique is applied to some problems to show its validity and applicability.

‎‎The equivalence relation isoclinism partitions the class of all pairs of groups‎ ‎into families‎. ‎In this paper‎، ‎a complete classification of the set of all pairs $(G،G')$ is established‏‎،‎ whenever‎ $G$ is a $p$-group of order at most $p^5‎‏$‎ and ‎$p$ is a prime number greater than 3. Moreover‎، ‎the classification of pairs $(H،H')$ for extra special $p$-groups $H$ is also given.

The aim of this paper is to introduce two new extensions of the Jacobi and Laguerre polynomials as the eigenfunctions of two non-classical Sturm-Liouville problems. We prove some important properties of these operators such as: These sets of functions are orthogonal with respect to a positive de nite inner product de ned over the compact intervals [-1، 1] and [0،1)، respectively and also these sequences form two new orthog-onal bases for the corresponding Hilbert spaces. Finally، the spectral and Rayleigh-Ritz methods are carry out using these basis functions to solve some examples. Our nu-merical results are compared with other existing results to con rm the eciency and accuracy of our method.

We define the notion of almost $n$-layered $QTAG$-modules and study their basic properties. One of the main result is that almost 1-layered modules are almost $(omega+1)$-projective exactly when they are almost direct sum of countably generated modules of length less than or equal to $(omega+1)$. Some other characterizations of this new class are also established.