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Let $M_C=\begin{pmatrix} A & C \\ 0 & B \\ \end{pmatrix}\in\LB(\x,\y)$ be be an upper triangulate Banach spaceoperator. The relationship between the spectra of $M_C$ and $M_0,$ and theirvarious distinguished parts, has been studied by a large number of authors inthe recent past. This paper brings forth the important role played by SVEP,the {\it single-valued extension property,} in the study of some of these relations. In this work, we prove necessary and sufficient conditions of implication of the type $M_0$ satisfies property $(w)$ $\Leftrightarrow$ $M_C$ satisfies property $(w)$ to hold. Moreover, we explore certain conditions on $T\in\LB(\hh)$ and $S\in\LB(\K)$ so that the direct sum $T\oplus S$ obeys property $(w)$, where $\hh$ and $\K$ are Hilbert spaces.

We extend and generalize the results of Xh. Z. Krasniqi [Acta Comment. Univ.Tartu. Math. 17 (2013), 89-101] and the authors [Acta Comment. Univ. Tartu.Math. 13 (2009), 11-24], [Proc. Estonian Acad. Sci. 2018, 67, 1, 50--60] aswell the jont paper with M. Kubiak [Journal of Inequalities and Applications(2018) 2018:92]. We consider the modified conjugate function $\widetilde{f}%_{r}$ for $2\pi /\rho $--periodic function $f$ . Moreover, the measure ofapproximations depends on \textbf{\ }$\mathbf{\rho }$\textbf{ - }differencesof the entries of matrices defined the method of summability.

In the paper, the authors present several explicit formulas for the $(p,q,r)$-Tribonacci polynomials and generalized Tribonacci sequences in terms of the Hessenberg determinants and, consequently, derive several explicit formulas for the Tribonacci numbers and polynomials, the Tribonacci--Lucas numbers, the Perrin numbers, the Padovan (Cordonnier) numbers, the Van der Laan numbers, the Narayana numbers, the third order Jacobsthal numbers, and the third order Jacobsthal--Lucas numbers in terms of special Hessenberg determinants.

This paper studies the following two porous medium equations with singular boundary conditions. First, we obtain that finite time quenching on the boundary, as well as $k_{t}\ $blows up at the same finite time\ and lower bound estimates of the quenching time of the equation $k_{t}=(k^{n})_{xx}+(1-k)^{-\alpha }$,\ $(x,t)\in (0,L)\times (0,T)\ $with $(k^{n})_{x}\left( 0,t\right) =0$, $\ (k^{n})_{x}\left( L,t\right)=(1-k(L,t))^{-\beta }$,$\ t\in (0,T)\ $and initial function $k\left(x,0\right) =k_{0}\left( x\right) $,$\ x\in \lbrack 0,L]\ $where $n>1$, $\alpha \ $and $\beta \ $and positive constants. Second, we obtain that finite time quenching on the boundary, as well as $k_{t}\ $blows up at the same finite time\ and a local existence result by the help of steady state of the equation $k_{t}=(k^{n})_{xx}$,\ $(x,t)\in (0,L)\times (0,T)\ $with $(k^{n})_{x}\left( 0,t\right) =(1-k(0,t))^{-\alpha }$, $\ (k^{n})_{x}\left(L, t\right) =(1-k(L,t))^{-\beta }$,$\ t\in (0,T)\ $and initial function $k\left( x,0\right) =k_{0}\left( x\right) $,$\ x\in \lbrack 0,L]\ $where $n>1$, $\alpha \ $and $\beta \ $and positive constants.

Let $L$ be an $n$-Lie algebra over a field $\F$. In this paper, we introduce the notion of non-abelian tensor square $L\otimes L$ of $L$ and define the central ideal $L\square L$ of it. Using techniques from group theory and Lie algebras, we show that that $L\square L\cong L^{ab}\square L^{ab}$. Also, we establish the short exact sequence\[0\lra\M(L)\lra\frac{L\otimes L}{L\square L}\lra L^2\lra0\]and apply it to compute an upper bound for the dimension of non-abelian tensor square of $L$.

Given a ring $R$, an ideal $I$ of $R$, and an element $a\in I$, we say $a=\lambda b_1\cdots b_k$ is a $\tau_I$-factorization of $a$ if $\lambda$ is any unit and $b_1\equiv\cdots\equiv b_k\pmod{I}$. In this paper, we investigate the $\tau_I$-atomicity of PIDs with ideals where $R/I$ has size four.

The aim of this research paper is to evaluate fifty double integrals invoving generalized hypergeometric function (25 each) in the form of\begin{align*}\int_{0}^{1}\int_{0}^{1} & x^{c-1}y^{c+\al-1} (1-x)^{\al- 1}(1-y)^{\be-1}\, (1-xy)^{c+\ell-\al-\be+1}\;\\ &\times \;{}_3F_2 \left[\begin{array}{c}a,\,\,\,\,\,b,\,\,\,\,\,2c+\ell+ 1 \\ \frac{1}{2}(a+b+i+1),\,\,2c+j \end{array}; xy\right]\,dxdy\end{align*}and\begin{align*}\int_{0}^{1}\int_{0}^{1} & x^{c+\ell}y^{c+\ell+\al} (1-x)^{\al-1}(1-y)^{\be-1}\, (1-xy)^{c- \al-\be}\;\\ &\times \;{}_3F_2 \left[\begin{array}{c}a,\,\,\,\,\,b,\,\,\,\,\,2c+\ell+ 1 \\ \frac{1}{2}(a+b+i+1),\,\,2c+j \end{array}; 1-xy\right]\,dxdy\end{align*}in the most general form for any $\ell \in \mathbb{Z}$ and $i, j = 0, \pm 1, \pm2$.The results are derived with the help of generalization of Edwards's well known double integral due to Kim, {\it et al.} and generalized classical Watson's summation theorem obtained earlier by Lavoie, {\it et al}.More than one hundred ineteresting special cases have also been obtained.

A restrained $k$-rainbow dominating function (R$k$RDF) of a graph $G$ is a function $f$ from the vertex set $V(G)$ to the set of all subsets of the set $\{1,2,\ldots,k\}$ such that for any vertex $v \in V (G)$ with $f(v) = \emptyset$ the conditions $\bigcup_{u \in N(v)} f(u)=\{1,2,\ldots,k\}$ and $|N(v)\cap \{u\in V\mid f(u)=\emptyset\}|\ge 1$ are fulfilled, where $N(v)$ is the open neighborhood of $v$. The weight of a restrained $k$-rainbow dominating function is the value $w(f)=\sum_{v\in V}|f (v)|$. The minimum weight of a restrained $k$-rainbow dominating function of $G$ is called the restrained $k$-rainbow domination number of $G$, denoted by $\gamma_{rrk}(G)$. The restrained $k$-rainbow bondage number $b_{rrk}(G)$ of a graph $G$ with maximum degree at least two is the minimum cardinality of all sets $F \subseteq E(G)$ for which $\gamma_{rrk}(G-F) > \gamma_{rrk}(G)$. In this paper, we initiate the study of the restrained $k$-rainbow bondage number in graphs and we present some sharp bounds for $b_{rr2}(G)$. In addition, we determine the restrained 2-rainbow bondage number of some classes of graphs.

In this paper we give higher order sufficient optimality conditions for a fractional vector optimization problem over cones, using higher order cone-convex functions. A higher order Schaible type dual program is formulated over cones.Weak, strong and converse duality results are established by using the higher order cone convex and other related functions.

In this paper, we present some oscillation criteria for the second order half-linear neutral delay differential equation with ``maxima" of the from\begin{equation*}\left(r(t)((x(t)+p(t)x(\tau(t)))')^{\alpha}\right)'+q(t) \max_{[\sigma(t),\;t]}x^{\alpha}(s)=0\end{equation*}under the condition $\int_{t_0}^{\infty}\frac{1}{r^{1/ \alpha}(t)}dt

Let $D$ be a finite simple digraph with vertex set $V(D)$. A twin signed Roman dominating function on the digraph $D$ is a function $f:V(D)\rightarrow\{-1,1,2\}$ satisfying the conditions that (i) $\sum_{x\in N^-[v]}f(x)\ge 1$ and $\sum_{x\in N^+[v]}f(x)\ge 1$ for each $v\in V(D)$, where $N^-[v]$ (resp. $N^+[v]$) consists of $v$ and all in-neighbors (resp. out-neighbors) of $v$, and (ii) every vertex $u$ for which $f(u)=-1$ has an in-neighbor $v$ and an out-neighbor $w$ for which $f(v)=f(w)=2$. A set $\{f_1,f_2,\ldots,f_d\}$ of distinct twin signed Roman dominating functions on $D$ with the property that $\sum_{i=1}^df_i(v)\le 1$ for each $v\in V(D)$, is called a twin signed Roman dominating family (of functions) on $D$. The maximum number of functions in a twin signed Roman dominating family on $D$ is the twin signed Roman domatic number of $D$, denoted by $d_{sR}^*(D)$. In this paper, we initiate the study of the twin signed Roman domatic number in digraphs and we present some sharp bounds on $d_{sR}^*(D)$. In addition, we determine the twin signed Roman domatic number of some classes of digraphs.

Let $G=(V,E)$ be a finite and simple graph of order $n$ and maximum degree $\Delta$. A signed strong Roman dominating function (abbreviated SStRDF) on a graph $G$ is a function $f:V\to \{-1,1,2,\ldots,\lceil\frac{\Delta}{2}\rceil+1\}$ satisfying the conditions that (i) for every vertex $v$ of $G$, $\sum_{u\in N[v]} f(u)\ge 1$, where $N[v]$ is the closed neighborhood of $v$ and (ii) every vertex $v$ for which $f(v)=-1$ is adjacent to at least one vertex $u$ for which $f(u)\ge 1+\lceil\frac{1}{2}|N(u)\cap V_{-1}|\rceil$, where $V_{-1}=\{v\in V \mid f(v)=-1\}$. The minimum of the values $\sum_{v\in V} f(v)$, taken over all signed strong Roman dominating functions $f$ of $G$, is called the signed strong Roman domination number of $G$ and is denoted by $\gamma_{ssR}(G)$. In this paper we initiate the study of the signed strong Roman domination in graphs and present some (sharp) bounds for this parameter.

A set $S\subseteq V(G)$ is a liar's dominating set ($lds$) of graph $G$ if $|N_G[v]\cap S|\geq 2$ for every $v\in V(G)$ and $|(N_G[u]\cup N_G[v])\cap S|\geq 3$ for any two distinct vertices $u,v \in V(G)$. The liar's domination number of $G$, denoted by $\gamma_{LR}(G)$, is the smallest cardinality of a liar's dominating set of $G$. In this paper we study the concept of liar's domination in the join, corona, and lexicographic product of graphs.

Let $D$ be a finite simple digraph with vertex set $V(D)$ and arc set $A(D)$. A twin signed Roman dominating function (TSRDF) on the digraph $D$ is a function $f:V(D)\rightarrow\{-1,1,2\}$ satisfying the conditions that (i) $\sum_{x\in N^-[v]}f(x)\ge 1$ and $\sum_{x\in N^+[v]}f(x)\ge 1$ for each $v\in V(D)$, where $N^-[v]$ (resp. $N^+[v]$) consists of $v$ and all in-neighbors (resp. out-neighbors) of $v$, and (ii) every vertex $u$ for which $f(u)=-1$ has an in-neighbor $v$ and an out-neighbor $w$ for which $f(v)=f(w)=2$. The weight of an TSRDF $f$ is $\omega(f)=\sum_{v\in V(D)}f(v)$. The twin signed Roman domination number $\gamma_{sR}^*(D)$ of $D$ is the minimum weight of an TSRDF on $D$. In this paper, we initiate the study of twin signed Roman domination in digraphs and we present some sharp bounds on $\gamma_{sR}^*(D)$. In addition, we determine the twin signed Roman domination number of some classes of digraphs.

The vertex-edge Wiener index of a simple connected graph $G$ is defined as the sum of distances between vertices and edges of $G$. The vertex-edge Wiener polynomial of $G$ is a generating function whose first derivative is a $q-$analog of the vertex-edge Wiener index. Two possible distances $D_1(u, e|G)$ and $D_2(u, e|G)$ between a vertex $u$ and an edge $e$ of $G$ can be considered and corresponding to them, the first and second vertex-edge Wiener indices of $G$, and the first and second vertex-edge Wiener polynomials of $G$ are introduced. In this paper, we study the behavior of these indices and polynomials under the join and corona product of graphs. Results are applied for some classes of graphs such as suspensions, bottlenecks, and thorny graphs.