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This short paper is motivated by the paper of Bu et al. [C. Bu, C. Feng, P. Dong, A note on computational formulas for the Drazin inverse of certain block matrices, J. Appl. Math. Comput.(38) (2012) 631-640], where the authors gave additive formula for Drazin inverse for matrices under new conditions, and two representations under some specific conditions. Here is shown that the additive formula is not valid for all matrices which satisfy given conditions. Also, here is proved that the representations which were given in mentioned paper do not extend the results given by Hartwig et al. [R. Hartwig, X. Li, Y. Wei, Representations for the Drazin inverse of a 2 _ 2 block matrix, SIAM J. Matrix. Anal. Appl. (27)(2006) 757-771 ], in fact they are equivalent.

In this short paper, we will show that the space of real valued uniformly continuous functions defined on a metric space (X,d) is a ring if and only if every subset A ? X has one of the following properties: ? A is Bourbaki-bounded, i.e., every uniformly continuous function on X is bounded on A. ? A contains an infinite uniformly isolated subset, i.e., there exist ? > 0 and an infinite subset F ? A such that d(a,x) ? ? for every a ? F, x ? X n \{a}.

For a Tychonoff space $X$, we denote by $(C(X), \tau_k, \tau_p)$ the bitopological space of all real-valued continuous functions on $X$ where $\tau_k$ is the compact-open topology and $\tau_p$ is the topology of pointwise convergence. In papers [5, 6, 13] variations of selective separability and tightness in $(C(X), \tau_k, \tau_p)$ were investigated. In this paper we continued to study the selective properties and the corresponding topological games in the space $(C(X), \tau_k, \tau_p)$., Comment: 10 pages. arXiv admin note: text overlap with arXiv:1805.04363

Let k be a nonzero complex number. In this paper we consider k - circulant matrices with arithmetic sequence and investigate the eigenvalues, determinants and Euclidean norms of such matrices. Also, for k=1 , the inverses of such ( invertible ) matrices are obtained (in a way different from the way presented in the paper: M. Bahsi and S. Solak , On the Circulant Matrices with Arithmetic Sequence , Int. J. Contemp . Math. Sci. 5 (25) (2010), 1213-1222, and the Moore-Penrose inverses of such (singular) matrices are derived.

Let $H_1$, $H_2$, $H_3$ be real Hilbert spaces, let $A:H_1\rightarrow H_3$, $B:H_2\rightarrow H_3$ be two bounded linear operators. The general split common fixed-point problem under consideration in this paper is to $$\text{find}\ \ x \in \cap_{i=1}^p F(U_i),\ \ y \in \cap_{j=1}^r F(T_j)\ \ \text{such that}\ \ Ax = By,\eqno{(1)}$$ where $p$, $r\geq 1$ are integers, $U_i:H_1\rightarrow H_1$ $(1\leq i\leq p)$ and $T_j:H_2\rightarrow H_2$ $(1\leq j\leq r)$ are quasi-nonexpansive mappings with nonempty common fixed-point sets $\cap_{i=1}^pF(U_i)=\cap_{i=1}^p\{x\in H_1:U_ix=x\}$ and $\cap_{j=1}^rF(T_j)=\cap_{j=1}^r\{x\in H_2:T_jx=x\}$. Note that, the above problem (1) allows asymmetric and partial relations between the variables $x$ and $y$. If $H_2=H_3$ and $B=I$, then the general split common fixed-point problem (1) reduces to the general split common fixed-point problem proposed by Censor and Segal $\cite{C}$. In this paper, we introduce simultaneous parallel and cyclic algorithms for the general split common fixed-point problems (1). We introduce a way of selecting the stepsizes such that the implementation of our algorithms does not need any prior information about the operator norms. We prove the weak convergence of the proposed algorithms and apply the proposed algorithms to the multiple-set split feasibility problems. Our results improve and extend the corresponding results announced by many others.

Suppose $R$ is an associative ring with identity 1. In this paper, We give the necessary and sufficient conditions for the existence and explicit representations of the group inverses of the following two classes block matrices. (i) $M=\left(\begin{array}{cc}AX+YBA (ii) $M=\left(\begin{array}{cc}A&B\\C&D\end{array}\right)$, where $A^{\sharp}$ exists, $A^{\pi}B=0$ and $(D-CA^{\sharp}B)^{\sharp}$ exists.\\ The paper's results generalize some relative results of Li et al. (J. Harbin Engineering Univ., 34, 658-661, 2013), Cao et al. (Appl. Math. Comput., 217, 10271-10277, 2011), Bu et al. (Appl. Math. Comput., 215, 132-139, 2009), Deng et al. (Linear Algebra Appl, 435, 2766-2783, 2011), and Liu et al. (J. Appl. Math. http://dx.doi.org/10.1155 /2013/247028). Some examples are given to illustrate our results.

The purpose of this paper is to prove a general fixed point theorem for mappings involving almost altering distances and satisfying a new type of common limit range property which generalize the results from Theorem 2.9 [19]. In the last part of the paper, as applications, some fixed point results for mappings satisfying contractive conditions of integral type for almost contractive mappings for ?-contractive mappings and (?,?) - weak contractive mappings in metric spaces are obtained.

In this paper, the complete moment convergence and the integrability of the supremum for weighted sums of negatively orthant dependent (NOD, in short) random variables are presented. As applications, the complete convergence and the Marcinkiewicz-Zygmund type strong law of large numbers for NODrandom variables are obtained. The results established in the paper generalize some corresponding ones for independent random variables and negatively associated random variables.

Recently Srivastava et al. (26) introduced the incomplete Pochhammer symbols by means of the incomplete gamma functions (s; x) and (s; x), and defined incomplete hypergeometric functions whose a number of interesting and fundamental properties and characteristics have been investigated. Further, Cetinkaya (6) introduced the incomplete second Appell hypergeometric functions and studied many interesting and fundamental properties and characteristics. In this paper, motivated by the above- mentioned works, we introduce two incomplete Srivastava's triple hypergeometric functions H and H by using the incomplete Pochhammer symbols and investigate certain properties, for example, their various integral representations, derivative formula, reduction formula and recurrence relation. Various (known or new) special cases and consequences of the results presented here are also considered. 1. Introduction, Definitions and Preliminaries Throughout this paper, N, Z , and C denote the sets of positive integers, negative integers, complex numbers, respectively, and N0 := N(f0g and Z 0 := Z (f0g. The familiar incomplete Gamma functions (s; x) and (s; x) defined by

The aim of this paper is to continue our investigations in line of our recent paper, Savas [24] and [26]. We introduce the notion of AI-double statistical convergence which includes the new summabilitymethods studied in [24] and [23] as special cases and make certain observations on this new and moregeneral summability method. © 2016, University of Nis. All rights reserved.

Since a locally finite topological structure plays an important role in the fields of pure and applied topology, the paper studies a special kind of locally finite spaces, so called a space set topology (for brevity, SST) and further, proves that an SST is an Alexandroff space satisfying the separation axiom T0. Unlike a point set topology, since each element of an SST is a space, the present paper names the topology by the space set topology. Besides, for a connected topological space (X,T) with |X| = 2 the axioms T0, semi-T1/2 and T1/2 are proved to be equivalent to each other. Furthermore, the paper shows that an SST can be used for studying both continuous and digital spaces so that it plays a crucial role in both classical and digital topology, combinatorial, discrete and computational geometry. In addition, a connected SST can be a good example showing that the separation axiom semi-T1/2 does not imply T1/2.

In this paper we illustrate a tight interplay between homotopy theory and combinatorics within toric topology by explicitly calculating homotopy and combinatorial invariants of toric objects associated with the dodecahedron. In particular, we calculate the cohomology ring of the (complex and real) moment-angle manifolds over the dodecahedron, and of a certain quasitoric manifold and of a related small cover. We finish by studying Massey products in the cohomology ring of moment-angle manifolds over the dodecahedron and how the existence of nontrivial Massey products influences the behaviour of the Poincar? series of the corresponding Pontryagin algebra.

A strong triangle blocking arrangement is a geometric arrangement of some line segments in a triangle with certain intersection properties. It turns out that they are closely related to blocking sets. Our aim in this paper is to prove a classification theorem for strong triangle blocking arrangements. As an application, we obtain a new proof of the result of Ackerman, Buchin, Knauer, Pinchasi and Rote which says that $n$ points in general position cannot be blocked by $n-1$ points, unless $n = 2,4$. We also conjecture an extremal variant of the blocking points problem., Comment: 22 pages, 16 figures

In this paper, we study the generating functions for the number of visible levels in compositions of n and set partitions of [n].

The Mostar index of a graph G is defined as the sum of absolute values of the differences between nu and nv over all edges e = uv of G, where nu(e) and nv(e) are respectively, the number of vertices of G lying closer to vertex u than to vertex v and the number of vertices of G lying closer to vertex v than to vertex u. A cactus is a graph in which any two cycles have at most one common vertex. In this paper, we determine all the n-vertex cacti with the largest Mostar index, and we give a sharp upper bound of the Mostar index for cacti of order n with k cycles, and characterize all the cacti that achieve this bound.

In this paper, we present a new generalization to compute determinants and inverses of r-circulant matrices Qn = circr ((b/a)?(2)/2 q1,(b/a)?(3)/2 q2,..., (b/a)?(n+1)/2 qn) and Ln = circr ((b/a)?(1)/2 l1,(b/a)?(2)/2 l2,..., (b/a)?(n)/2 ln) whose entries are the biperiodic Fibonacci and the biperiodic Lucas numbers, respectively. Also, we express determinants of the matrices Qn and Ln by using only the biperiodic Fibonacci and the biperiodic Lucas numbers.

In $n$-dimensional hyperbolic space $\mathbf{H}^n$ $(n\ge2)$ there are $3$-types of spheres (balls): the sphere, horosphere and hypersphere. If $n=2,3$ we know an universal upper bound of the ball packing densities, where each ball volume is related to the volume of the corresponding Dirichlet-Voronoi (D-V) cell. E.g. in $\mathbf{H}^3$ the densest horoball packing is derived from the $\{3,3,6\}$ Coxeter tiling consisting of ideal regular simplices $T_{reg}^\infty$ with dihedral angles $\frac{\pi}{3}$. The density of this packing is $\delta_3^\infty\approx 0.85328$ and this provides a very rough upper bound for the ball packing densities as well. However, there are no "essential" results regarding the "classical" ball packings with congruent balls, and for ball coverings either. The goal of this paper to find the extremal ball arrangements in $\mathbf{H}^3$ with "classical balls". We consider only periodic congruent ball arrangements (for simplicity) related to the generalized, so-called {\it complete Coxeter orthoschemes} and their extended groups. In Theorems 1.1-1.2 we formulate also conjectures for the densest ball packing with density $0.77147\dots$ and the loosest ball covering with density $1.36893\dots$, respectively. Both are related with the extended Coxeter group $(5, 3, 5)$ and the so-called hyperbolic football manifold (look at Fig.~3). These facts can have important relations with fullerens in crystallography., Comment: 23 pages, 3 figures

Let D be a digraph. A dominating set of D is the subset S of V(D) such that each vertex in V(D)?S is an out-neighbor of a vertex in S. The minimum cardinality of a dominating set of G is denoted by ?(D). We let D?denote the reverse of D. In [Discrete Math. 197/198 (1999) 179-183], Chartrand, Harary and Yue proved that every connected digraph D of order n ? 2 satisfies ?(D)+ ?(D?) ? 4n 3 and characterized the digraphs D attaining the equality. In this paper, we pose a reduction of the determining problem for (D)+(D?) using the total domination concept. As a corollary of such a reduction and known results, we give new bounds for (D)+(D?) and an alternative proof of Chartrand-Harary-Yue theorem.

In this paper, we extend the proximal point algorithm proposed by Chang et al.[8] for total asymptotically nonexpansive mapping in CAT(0) spaces. We also demonstrate the ?-convergence and strong convergence to a common element of the set of minimizers of a convex function and the set of fixed points of the Ces?ro type mean of total asymptotically nonexpansive mappings in CAT(0) spaces.

In this paper, we define the adjacency-Jacobsthal-Hurwitz sequences of the first and second kind. Then we give the exponential, combinatorial, permanental and determinantal representations and the Binet formulas of the adjacency-Jacobsthal-Hurwitz numbers of the first and second kind by the aid of the generating functions and the generating matrices of the sequences defined.

In this paper, by using some families of special numbers and polynomials with their generating functions and functional equations, we derive many new identities and relations related to these numbers and polynomials. These results are associated with well-known numbers and polynomials such as Euler numbers, Stirling numbers of the second kind, central factorial numbers and array polynomials. Furthermore, by using higher-order partial differential equations, we derive some combinatorial sums and identities. Finally, we give two computation algorithms for Euler numbers and central factorial numbers.

A 2-rainbow dominating function (2RDF) of a graph G is a function f : V(G) ? P({1,2}) such that for each v ? V(G) with f (v) = ? we have Uu?N(v) f (u) = {1,2}. For a 2RDF f of a graph G, the weight w(f) of f is defined as w(f)=?v?V(G)?f(v)?. The minimum weight over all 2RDFs of G is called the 2-rainbow domination number of G, which is denoted by ?r2(G). A subset S of vertices of a graph G without isolated vertices, is a total dominating set of G if every vertex in V(G) has a neighbor in S. The total domination number ?t(G) is the minimum cardinality of a total dominating set of G. Chellali, Haynes and Hedetniemi conjectured that ?t(G)? ?r2(G) [M. Chellali, T.W. Haynes and S.T. Hedetniemi, Bounds on weak Roman and 2-rainbow domination numbers, Discrete Appl. Math. 178 (2014), 27-32.], and later Furuya confirmed the conjecture [M. Furuya, A note on total domination and 2-rainbow domination in graphs, Discrete Appl. Math. 184 (2015), 229-230.]. In this paper, we provide a constructive characterization of trees T with ?r2(T) = ?t(T).

For a bounded linear operator T on a complex Hilbert space and n ? N, T is said to be n-normal if T*Tn = TnT*. In this paper we show that if T is a 2-normal operator and satisfies ?(T) ? (-?(T)) ? {0}, then T is isoloid and ?(T) = ?a(T). Under the same assumption, we show that if z and w are distinct eigenvalues of T, then ker(T-z)? ker(T-w). And if non-zero number z ? C is an isolated point of ?(T), then we show that ker(T-z) is a reducing subspace for T. We show that if T is a 2-normal operator satisfying ?(T) ?(-?(T)) = 0, then Weyl?s theorem holds for T. Similarly, we show spectral properties of n-normal operators under similar assumption. Finally, we introduce (n,m)-normal operators and show some properties of this kind of operators.

In this paper, the authors study a complete moment convergence result for Sung?s type weighted sums of ?*-mixing random variables. This result extends and improves the corresponding theorem of Sung [S.H. Sung, Complete convergence for weighted sums of ?*-mixing random variables, Discrete Dyn. Nat. Soc. 2010 (2010), Article ID 630608, 13 pages].

In this paper, we obtain the spectrum of signed complete and complete bipartite graphs whose negative edges form a matching. Moreover, we construct a family of signed complete graphs having symmetric spectrum.

The edge Szeged index and edge-vertex Szeged index of a graph are defined as $Sz_{e}(G)=\sum\limits_{uv\in E(G)}m_{u}(uv|G)m_{v}(uv|G)$ and $Sz_{ev}(G)=\frac{1}{2} \sum\limits_{uv \in E(G)}[n_{u}(uv|G)m_{v}(uv|G)+n_{v}(uv|G)m_{u}(uv|G)],$ respectively, where $m_{u}(uv|G)$ (resp., $m_{v}(uv|G)$) is the number of edges whose distance to vertex $u$ (resp., $v$) is smaller than the distance to vertex $v$ (resp., $u$), and $n_{u}(uv|G)$ (resp., $n_{v}(uv|G)$) is the number of vertices whose distance to vertex $u$ (resp., $v$) is smaller than the distance to vertex $v$ (resp., $u$), respectively. A cactus is a graph in which any two cycles have at most one common vertex. In this paper, the lower bounds of edge Szeged index and edge-vertex Szeged index for cacti with order $n$ and $k$ cycles are determined, and all the graphs that achieve the lower bounds are identified., 12 pages, 5 figures

Let $G^{\sigma}$ be an oriented graph and $S(G^{\sigma})$ be its skew-adjacency matrix, where $G$ is called the underlying graph of $G^{\sigma}$. The skew-rank of $G^{\sigma}$, denoted by $sr(G^{\sigma})$, is the rank of $S(G^{\sigma})$. Denote by $d(G)=|E(G)|-|V(G)|+\theta(G)$ the dimension of cycle spaces of $G$, where $|E(G)|$, $|V(G)|$ and $\theta(G)$ are the edge number, vertex number and the number of connected components of $G$, respectively. Recently, Wong, Ma and Tian [European J. Combin. 54 (2016) 76--86] proved that $sr(G^{\sigma})\leq r(G)+2d(G)$ for an oriented graph $G^{\sigma}$, where $r(G)$ is the rank of the adjacency matrix of $G$, and characterized the graphs whose skew-rank attain the upper bound. However, the problem of the lower bound of $sr(G^{\sigma})$ of an oriented graph $G^{\sigma}$ in terms of $r(G)$ and $d(G)$ of its underlying graph $G$ is left open till now. In this paper, we prove that $sr(G^{\sigma})\geq r(G)-2d(G)$ for an oriented graph $G^{\sigma}$ and characterize the graphs whose skew-rank attain the lower bound., Comment: 12 pages

A vertex triple (u, v, w) with v adjacent to both u and w is called a 2-geodesic if u ? w and u,w are not adjacent. A graph ? is said to be 2-geodesic-transitive if its automorphism group is transitive on both arcs and 2-geodesics. In this paper, a complete classification of 2-geodesic-transitive graphs is given which are neighbor cubic or neighbor tetravalent.

In this work, we introduce and investigate a subclass Hh,p ?m(?,?) of analytic and bi-univalent functions when both f(z) and f-1(z) are m-fold symmetric in the open unit disk U. Moreover, we find upper bounds for the initial coefficients |am+1| and |a2m+1| for functions belonging to this subclass Hh,p ?m(?,?). The results presented in this paper would generalize and improve those that were given in several recent works.

The purpose of this paper is the study of hypergroups associated with hypergraphs. In this regard, we construct a hypergroupoid by defining a hyperoperation on the set of degrees of vertices of a hypergraph. We will see that the constructed hypergroupoid is always an Hv-group. We will investigate some conditions to have a hypergroup.

In this paper, we generalize some matrix inequalities involving the matrix power means and Karcher mean of positive definite matrices. Among other inequalities, it is shown that if A = (A1,...,An) is an n-tuple of positive definite matrices such that 0 < m ? Ai ? M (i = 1,...,n) for some scalars m < M and ? = (w1,...,wn) is a weight vector with wi ? 0 and ?n,i=1 wi=1, then ?p (?n,i=1 wiAi)? ?p?p(Pt(?,A)) and ?p (?n,i=1 wiAi) ? ?p?p(?(?,A)), where p > 0,? = max {(M+m)2/4Mm,(M+m)2/42p Mm}, ? is a positive unital linear map and t ? [-1,1]\{0}.

Let $G=(V,\,E)$ be a simple connected graph of order $n$ with $m$ edges. Also let $e_G(v_i)$ be the eccentricity of a vertex $v_i$ in $G$. We can assume that $e_G(v_1)\geq e_G(v_2)\geq \cdots\geq e_G(v_{n-1})\geq e_G(v_n).$ The average eccentricity of a graph $G$ is the mean value of eccentricities of vertices of $G$, $$avec(G)=\frac{1}{n}\,\sum\limits^n_{i=1}\,e_G(v_i)\,.$$ Let $\gamma=\gamma_G$ be the largest positive integer such that $$e_G(v_{\gamma_G})\geq avec(G).$$ In this paper, we study on the value of $\gamma_G$ of a graph $G$. For any tree $T$ of order $n$, we prove that $2\leq \gamma_T\leq n-1$ and characterize the extremal graphs. Moreover, we prove that for any graph $G$ of order $n$, $2\leq \gamma_G\leq n$ and characterize the extremal graphs. Finally some Nordhaus-Gaddum type results are obtained on $\gamma_G$ of general graphs $G$.

In this paper we will investigate the problem of finding the open-locating-dominating number for some classes of planar graphs - convex polytopes. We considered Dn, Tn, Bn, Cn, En and Rn classes of convex polytopes known from the literature. The exact values of open-locating-dominating number for Dn and Rn polytopes are presented, along with the upper bounds for Tn, Bn, Cn, and En polytopes.

Let T,S : A U B ? A U B be mappings such that T(A) ? B,T(B)? A and S(A) ? A,S(B)?B. Then the pair (T,S) of mappings defined on A[B is called cyclic-noncyclic pair, where A and B are two nonempty subsets of a metric space (X,d). A coincidence best proximity point p ? A U B for such a pair of mappings (T,S) is a point such that d(Sp,Tp) = dist(A,B). In this paper, we study the existence and convergence of coincidence best proximity points in the setting of convex metric spaces. We also present an application of one of our results to an integral equation.

In this paper, the concepts of C-precontinuous posets, quasi C-precontinuous posets and meet Cprecontinuous posets are introduced. The main results are: (1) A complete semilattice P is C-precontinuous (resp., quasi C-precontinuous) if and only if its normal completion is a C-continuous lattice (resp., quasi C-continuous lattice); (2) A poset is both quasi C-precontinuous and Frink quasicontinuous if and only if it is generalized completely continuous; (3) A complete semilattice is meet C-precontinuous if and only if its normal completion is meet C-continuous; (4) A poset is both quasi C-precontinuous and meet C-precontinuous if and only if it is C-precontinuous.

In this paper, Cline's formula for the well-known generalized inverses such as Drazin inverse, pseudo Drazin inverse and generalized Drazin inverse is extended to the case when $\left\{ \begin{aligned} acd=dbd \ \\ dba=aca. \\ \end{aligned} \right.$ Also, applications are given to some interesting Banach space operator properties like algebraic, meromorphic, polaroidness and B-Fredholmness.

In this paper we study the existence of positive solutions for the fractional $p$-Laplacian boundary value problem \[\left\{\aligned & D_{0+}^\beta (\phi_p (D_{0+}^\alpha u(t)))=f(t,u(t)), t\in (0,1),\\ & u(0)=u'(0)=0, u'(1)=au'(\xi), D_{0+}^\alpha u(0)=0, D_{0+}^\alpha u(1)= b D_{0+}^\alpha u(\eta), \endaligned \right.\] where $2 1$, $\phi_p^{-1}=\phi_q$, $1/p+1/q=1$, $0

In this paper, we will define the exponent of convergence for double sequences and in that terminology, we will prove three new theorems in theory of selection principles for double sequences

In this paper, we prove Sherman like inequalities for convex sequences and nondecreasing convex functions. Thus we develop some results by S. Wu and L. Debnath [19]. In consequence, we derive discrete versions for convex sequences of Petrovic and Giaccardi?s inequalities. As applications, we establish some generalizatons of Fej?r inequality for convex sequences. We also study inequalities of Hermite-Hadamard type. Thus we extend some recent results of Latreuch and Bela?di [8]. In our considerations we use some matrix methods based on column stochastic and doubly stochastic matrices.

For a simple graph G, the Q-eigenvalues are the eigenvalues of the signless Laplacian matrix Q of G. A Q-eigenvalue is said to be a Q-main eigenvalue if it admits a corresponding eigenvector non orthogonal to the all-one vector, or alternatively if the sum of its component entries is non-zero. In the literature the trees, unicyclic, bicyclic and tricyclic graphs with exactly two Q-main eigenvalues have been recently identied. In this paper we continue these investigations by identifying the trees with exactly three Q-main eigenvalues, where one of them is zero.

Let $R$ be a commutative ring with identity and $X$ be a Tychonoff space. An ideal $I$ of $R$ is Von Neumann regular (briefly, regular) if for every $a\in I$, there exists $b\in R$ such that $a=a^2b$. In the present paper, we obtain the general form of a regular ideal in $C(X)$ which is $O^A$, for some closed subset $A$ of ${\beta X}$, for which $ A^c\cap X\subseteq (P(X))^{\circ }$, where $P(X)$ is the set of all $P$-points of $X$. We show that the ideals and subrings such as $C_K(X)$, $C_\psi(X)$, $C_\infty(X)$, $Soc_mC(X)$ and $M^{\beta X\backslash X}$ are regular if and only if they are equal to the socle of $C(X)$. We carry further the study of maximal regular ideal, for instance, it is shown that for a vast class of topological spaces (we call them $OPD$-spaces) the maximal regular ideal is $O^{X\backslash I(X)}$, where $I(X)$ is the set of nonisolated points of $X$. Also, for this class, the socle of $C(X)$ is the maximal regular ideal if and only if $I(X)$ contains no infinite closed set. We also show that $C(X)$ contains an ideal which is both essential and regular if and only if $(P(X))^{\circ }$ is dense in $X$. Finally it is shown that, for semiprimitive rings pure ideals are of the form $O^A$ which $A$ is a closed subset of $Max(R)$, also a $P$-point of $X=Max(R)$ is introduced and it is shown that the maximal regular ideal of an arbitrary ring $R$ is $O^{X\backslash P(X)}$, which $P(X)$ is the set of $P$-points of $X=Max(R)$

This paper presents two kinds of symmetric tridiagonal plus paw form (hereafter TPPF) matrices, which are the combination of tridiagonal matrices and bordered diagonal matrices. In particular, we exploit the interlacing properties of their eigenvalues. On this basis, the inverse eigenvalue problems for the two kinds of symmetric TPPF matrices are to construct these matrices from the minimal and the maximal eigenvalues of all their leading principal submatrices respectively. The necessary and sufficient conditions for the solvability of the problems are derived. Finally, numerical algorithms and some examples of the results developed here are given.

The q-digamma function ?q(x) and the q-polygamma functions ?(r)q(x), r ? N = {1,2,...} are defined for all x > 0 and 0 < q < 1. In this paper, the neutrices and the neutrix limit are used to define the q-digamma function q(x) and the q-polygamma functions ?(r)q(x), r ? N for all x ? R. Moreover, further results are given.

In this paper, we consider the problem of planarity and outerplanarity of iterated line graphs of the unit, unitary and total graphs when R is a finite commutative ring. We give a full characterization of all these graphs with respect to their planarity and outerplanarity indexes.

For a given symmetric orthogonal matrix R, i.e., RT = R, R2 = I, a matrix A ? Cnxn is termed Hermitian R-conjugate matrix if A = AH, RAR = ?. In this paper, an iterative method is constructed for finding the Hermitian R-conjugate solutions of general coupled Sylvester matrix equations. Convergence analysis shows that when the considered matrix equations have a unique solution group then the proposed method is always convergent for any initial Hermitian R-conjugate matrix group under a loose restriction on the convergent factor. Furthermore, the optimal convergent factor is derived. Finally, two numerical examples are given to demonstrate the theoretical results and effectiveness.

A mapping $f: X \rightarrow Y $ is statistically sequence covering map if whenever a sequence $\lbrace y_n \rbrace$ convergent to $y$ in $Y,$ there is a sequence $\lbrace x_n \rbrace$ statistically converges to $x$ in $X$ with each $x_n \in f^{-1}(y_n) $ and $ x \in f^{-1}(y).$ In this paper, we introduce the concept of statistically sequence covering map which is a generalization of sequence covering map and discuss the relation with covering maps by some examples. Using this concept, we prove that every closed and statistically sequence-covering image of a metric space is metrizable. Also, we give characterizations of statistically sequence covering compact images of spaces with a weaker metric topology.

In the present paper we are interested in the study of the distance Laplacian eigenvalues of a connected graph with fixed order n and chromatic number x. We prove lower bounds on the distance Laplacian spectral radius in terms of n and x. We also prove results related to the distribution of the distance Laplacian eigenvalues with respect to the values of the chromatic number x. For some of the results, we characterize the extremal graphs, for others, we give examples of extremal graphs.

The sequence of random variables ${\{X_{n}}\}_{n\in \mathbb{N}}$ is said to be weighted modulus $\alpha\beta$-statistically convergent in probability to a random variable $X$ \cite{Ghosal2014(3)} if for any $\varepsilon, \delta>0,$ $$\underset{n\rightarrow \infty}{\lim}\frac{1}{T_{\alpha\beta(n)}}~ |\{k \leq T_{\alpha\beta(n)}: t_{k}\phi(P(|X_{k}-X|\geq \varepsilon))\geq \delta\}|=0,$$ where $\phi$ be a modulus function and $\{t_{n}\}_{n\in \mathbb{N}}$ be a sequence of real numbers such that $\displaystyle{\liminf_{n\rightarrow \infty}} t_n>0$ and $T_{\alpha\beta(n)}=\displaystyle{\sum_{k\in [\alpha_{n},\beta_{n}]}} t_k $, $\forall ~n\in \mathbb{N}.$ In this paper we study a related concept of convergence in which the value $\frac{1}{T_{\alpha\beta(n)}}$ is replaced by $\frac{1}{C_n},$ for some sequence of real numbers $\{C_n\}_{n\in \mathbb{N}}$ : $C_{n}>0,~ \forall~ n\in \mathbb{N},$ $\underset{n\rightarrow \infty}{\lim}C_n=\infty$ and $\underset{n\rightarrow \infty}{\limsup} \frac{C_n}{T_{\alpha\beta(n)}} 0,$ then quasi-weighted modulus $\alpha\beta$-statistical convergence in probability and weighted modulus $\alpha\beta$-statistical convergence in probability are equivalent except the condition {\textbf{$\underset{n\rightarrow \infty}{\liminf} \frac{C_n}{T_{\alpha\beta(n)}}=0$}}. So our main objective is to interpret the above exceptional condition and produce a relational behavior of above mention four convergences.

Let G? be a signed graph with the underlying graph G and with sign function ? : E(G) ? {?}. In this paper, we characterize the signed graphs with two distinct eigenvalues whose underlying graphs are triangle-free. Also, we classify all 3-regular and 4-regular signed graphs whose underlying graphs are triangle-free and give their adjacency matrices as well.

In this paper, we introduce regularly generated sequences for double sequence of real numbers, and obtain some Tauberian theorems for (C; 1; 1) summability method using the concept of regularly generated sequence.