Search

In this paper, we continue the program initiated by I. Beck's now classical paper concerning zero-divisor graphs of commutative rings. After the success of much research regarding zero-divisor graphs, many authors have turned their attention to studying divisor graphs of non-zero elements in integral domains. This inspired the so called irreducible divisor graph of an integral domain studied by J. Coykendall and J. Maney. Factorization in rings with zero-divisors is considerably more complicated than integral domains and has been widely studied recently. We find that many of the same techniques can be extended to rings with zero-divisors. In this article, we construct several distinct irreducible divisor graphs of a commutative ring with zero-divisors. This allows us to use graph theoretic properties to help characterize finite factorization properties of commutative rings, and conversely.

In this paper, we are concerned with the existence of solutions of the following multi-point boundary value problem consisting of the higher-order differential equations $ x^{(n)}(t)=f(t,x(t),x'(t),\cdots,x^{(n-1)}(t))+e(t),\;\;0 and the following multi-point boundary value conditions $ \begin{array}{ll} x^{(i)}(0)=0\;\;for\;i=0,1,\cdots,n-3,\\ x^{(n-2)}(0)=\alpha x^{(n-1)}(\xi),\;\;x^{(n-1)}(1)=\beta x^{(n-2)}(\eta),\end{array} \eqno{(\ast\ast)} $ Sufficient conditions for the existence of at least one solution of the BVP$ (\ast) $ and $ (\ast\ast) $ at resonance are established. This paper is directly motivated by Liu and Yu [India J. Pure Appl. Math., 33(4)(2002)475-494] and Qi [Acta Math. Appl. Sinica, 17(2)(2001)271-278].

Let $ R $ be an integral domain and $ \alpha $ an anti-integral element of degree $ d $ over $ R $. In the paper [3] we give a condition for $ \alpha^2-a$ to be a unit of $ R[\alpha] $. In this paper we will generalize the result to an arbitrary positive integer $n$ and give a condition, in terms of the ideal $ I_{[\alpha]}^{n}D(\sqrt[n]{a}) $ of $ R $, for $ \alpha^{n}-a$ to be a unit of $ R[\alpha] $.

Let $S$ be a subsemigroup which contains 0 of a torsion-free abelian (additive) group. Then $S$ is called a grading monoid (or a $g$-monoid). The group $ \{s-s'|s,s'\in S\}$ is called the quotient group of $S$, and is denored by $q(S)$. Let $R$ be a commutative ring. The total quotient ring of $R$ is denoted by $q(R)$. Throught the paper, we assume that a $g$-monoid properly contains $ \{0\}$. A commutative ring is called a ring, and a non-zero-divisor of a ring is called a regular element of the ring. We consider integral elements over the semigroup ring $ R[X;S]$ of $S$ over $R$. Let $S$ be a $g$-monoid with quotient group $G$. If $ n\alpha\in S$ for an element $ \alpha$ of $G$ and a natural number $n$ implies $ \alpha\in S$, then $S$ is called an integrally closed semigroup. We know the following fact: ${\bf Theorem~1}$ ([G2, Corollary 12.11]). Let $D$ be an integral domain and $S$ a $g$-monoid. Then $D[X;S]$ is integrally closed if and only if $D$ is an integrally closed domain and $S$ is an integrally closed semigroup. Let $R$ be a ring. In this paper, we show that conditions for $R[X;S]$ to be integrally closed reduce to conditions for the polynomial ring of an indeterminate over a reduced total quotient ring to be integrally closed (Theorem 15). Clearly the quotient field of an integral domain is a von Neumann regular ring. Assume that $q(R)$ is a von Neumann regular ring. We show that $R[X;S]$ is integrally closed if and only if $R$ is integrally closed and $S$ is integrally closed (Theorem 20). Let $G$ be a $g$-monoid which is a group. If $R$ is a subring of the ring $T$ which is integrally closed in $T$, we show that $R[X;G]$ is integrally closed in $T[X;S]$ (Theorem 13). Finally, let $S$ be sub-$g$-monoid of a totally ordered abelian group. Let $R$ be a subring of the ring $T$ which is integrally closed in $T$. If $g$ and $h$ are elements of $T[X;S]$ with $h$ monic and $gh\in R[X;S]$, we show that $g\in R[X;S]$ (Theorem 24).

In this paper, we introduced new subclasses $UCV_S(\alpha,\beta)$ and $SP_S(\alpha,\beta)$ which are sub classes of $UCV(\alpha,\beta)$ and $SP(\alpha, \beta)$ and studied the $T$ - $\delta$ neighbourhoods of functions in these classes. The results obtained in this paper generalizes the recent results of Parvatham and Premabai[5], Ram Reddy and Thirupathi Reddy[6,7], Padmanabhan[4], and Ronning[8].

For a simple connected graph $G$, the convex linear combinations $D_{\alpha}(G)$ of \ $Tr(G)$ and $D(G)$ is defined as $D_{\alpha}(G)=\alpha Tr(G)+(1-\alpha)D(G)$, $0\leq \alpha\leq 1$. As $D_{0}(G)=D(G)$, $2D_{\frac{1}{2}}(G)=D^{Q}(G)$, $D_{1}(G)=Tr(G)$ and $D_{\alpha}(G)-D_{\beta}(G)=(\alpha-\beta)D^{L}(G)$, this matrix reduces to merging the distance spectral and distance signless Laplacian spectral theories. In this paper, we study the spectral properties of the generalized distance matrix $D_{\alpha}(G)$. We obtain some lower and upper bounds for the generalized distance spectral radius, involving different graph parameters and characterize the extremal graphs. Further, we obtain upper and lower bounds for the maximal and minimal entries of the $ p $-norm normalized Perron vector corresponding to spectral radius $ \partial(G) $ of the generalized distance matrix $D_{\alpha}(G)$ and characterize the extremal graphs.

The Babai numbers and the upper chromatic number are parameters that can be assigned to any metric space. They can, therefore, be assigned to any connected simple graph. In this paper we make progress in the theory of the first Babai number and the upper chromatic number in the simple graph setting, with emphasis on graphs of diameter 2.

In this paper, we show that every continuous linear operator from $H(\Omega_w\times \Omega_z)$ to $H(\Omega_w\times \Omega_{\zeta})$ has an integral representation with a kernel function $M(z, w, \zeta)$. We give two sufficient conditions on $M(z, w, \zeta)$ to ensure that its corresponding operator preserves Polya property. We also prove that a continuous linear operator from $H(\Omega_w\times \Omega_z)$ to $H(\Omega_w\times \Omega_{\zeta})$ either preserves the Polya property for all functions with that property or does not preserve the Polya property for any function.

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.

In this paper, we first prove that if f be a non-constant meromorphic function, all of whose zeros have multiplicity at least $k$, then f^{n}f^{(k)}-a has at least m+1 distinct zeros, where $k(\geq2),m(\geq1),n(\geq m+1)$ are three integers, and $a\in \mathbb{C}\cup\setminus\{0\}$.Also, in relation to this result, a normality criteria is given, which extends the related result.

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.

Let $R$ be a commutative ring with unity. The annihilator ideal graph of $R$, denoted by $\Gamma _{\mathrm{Ann}} (R) $, is a graph whose vertices are all non-trivial ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if$ I \cap \mathrm{Ann} _{R} (J) \neq \lbrace 0\rbrace $ or $J \cap \mathrm{Ann} _{R} (I) \neq \lbrace 0\rbrace $.In this paper, all rings with planar annihilator ideal graphs are classified.Furthermore, we show that all annihilator ideal graphs are perfect. Among other results, it is proved that if $\Gamma _{\mathrm{Ann}} (R) $ is a tree, then $\Gamma _{\mathrm{Ann}} (R) $ is star.

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.

In this paper, the partial inverse nodal problem for differential pencils with real-valued coefficients on a finite interval \([0,1]\) was studied. The authors showed that the coefficients \((q_{0}(x),q_{1}(x),h,H_0)\) of the differential pencil \(L_0\) can be uniquely determined by partial nodal data on the right(or, left) arbitrary subinterval \([a,b]\) of \([0,1].\) Finally, an example was given to verify the validity of the reconstruction algorithm for this inverse nodal problem.

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.

Suppose $p$ and $q$ are odd prime numbers. In this paper, the connected Cayley graph of groups of order $3pq$, for primes $p$ and $q$, are investigated and all connected normal $\frac{1}{2}-$arc-transitive Cayley graphs of group of these orders will be classified.

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.

J-L. Li and Y-L. Li \cite{LL2007} gave the following Redheffer's type inequality; \begin{equation*} \frac{ 1 -\left( \frac{x}{\pi} \right)^2 }{ \sqrt{1 + 3 \left( \frac{x}{\pi} \right)^4}} > \frac{\sin{x}}{x} \end{equation*} holds for $0 < x < \pi$, where the constant $3$ is the best possible. In this paper, we establish two inequalities extended the constant part of the above inequality.

An edge-regular graph is a regular graph in which, for some $\lambda$, any two adjacent vertices have exactly $\lambda$ common neighbors. This paper is about the existence and structure of edge-regular graphs with $\lambda =1$ and about edge-regular graphs with $\lambda >1$ which have local neighborhood structure analogous to that of the edge-regular graphs with $\lambda =1$.

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 $D=(V,A)$ be a finite and simple digraph. A Roman dominating function on $D$ is a labeling $f:V (D)\rightarrow \{0, 1, 2\}$ such that every vertex with label 0 has an in-neighbor with label 2. The weight of an RDF $f$ is the value $\omega(f)=\sum_{v\in V}f (v)$. The minimum weight of a Roman dominating function on a digraph $D$ is called the Roman domination number, denoted by $\gamma_{R}(D)$. The Roman bondage number $b_{R}(D)$ of a digraph $D$ with maximum out-degree at least two is the minimum cardinality of all sets $A'\subseteq A$ for which $\gamma_{R}(D-A')>\gamma_R(D)$. In this paper, we initiate the study of the Roman bondage number of a digraph. We determine the Roman bondage number in several classes of digraphs and give some sharp bounds.

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.

In this paper, a Second degree generalized Jacobi Iteration method for solving system of linear equations, $Ax=b$ and discuss about the optimal values $a_{1}$ and $b_{1}$ in terms of spectral radius about for the convergence of SDGJ method of $x^{(n+1)}=b_{1}[D_{m}^{-1}(L_{m}+U_{m})x^{(n)}+k_{1m}]-a_{1}x^{(n-1)}.$ Few numerical examples are considered to show that the effective of the Second degree Generalized Jacobi Iteration method (SDGJ) in comparison with FDJ, FDGJ, SDJ.

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.

Let $Q({\bf{x}}) = Q(x_1 ,x_2 ,\ldots,x_n )$ be a quadratic form with integer coefficients, $p$ be an odd prime and $\left\| \bf{x} \right\| = \max _i \left| {x_i } \right|.$ A solution of the congruence $Q({\mathbf{x}}) \equiv {\mathbf{0}}\;(\bmod\; p^2 )$ is said to be a primitive solution if $p\nmid x_i $ for some $i$. In this paper, we seek to obtain primitive solutions of this congruence in small rectangular boxes of the type $ \mathcal{B} = \{ {\mathbf{x}} \in \mathbb{Z}^n : |x_i| \le M_i ,\;1 \leqslant i \leqslant n\} $ where for $1 \le i \le l$ we have $M_i \le p$, while for $i>l$ we have $M_i>p$. In particular, we show that if $n \ge 4$, $n$ even, $l \le \frac n2-2$, and $Q$ is nonsingular $\pmod p$, then there exists a primitive solution with $x_i = 0$, $1 \le i \le l$, and $|x_i| \le 2^{\frac {4n+3}{n-l}} p^{\frac n{n-l}} +1$, for $l

In this paper some Hadamard-type inequalities for product of convex functions of $2-$variables on the co-ordinates are given.

In this paper, by using the theory of generalized convexity functions we introduce and prove monotonicity of sequences of the forms $$ \left\{\left(\prod\limits_{k=1}^nf\left({a_k\over a_n}\right)\right)^{1/n}\right\},\quad \left\{\left(\prod\limits_{k=1}^nf\left({\varphi(k)\over\varphi(n)}\right)\right)^{1/\varphi(n)}\right\}, $$ $$ \left\{{1\over n}\sum_{k=1}^nf\left({a_n\over a_k}\right)\right\}\quad\text{or}\quad \left\{{1\over\varphi(n)}\sum_{k=1}^nf\left({\varphi(n)\over\varphi(k)}\right)\right\}, $$ where $f$ belongs to the classes of $AG$-convex (concave), $HA$-convex (concave), or $HG$-convex (concave) functions defined on suitable intervals, $\{a_n\}$ is a given sequence and $\varphi$ is a given function that satisfy some preset conditions. As a consequence, we obtain some generalizations of Alzer type inequalities.

In the present paper, the real fixed points of one parameter family ${\cal T}=\{f_{\lambda}(x)$$=\lambda\frac{x}{b^{x}-1}\; \text{and}\;f_{\lambda}(0)=\frac{\lambda}{\ln b} : \lambda>0, x \in \mathbb{R},b>0, b\neq 1\}$ are investigated. Further, the nature of these fixed points of $f_{\lambda}(x)$ are shown for $b>0$ except $b=1$.

The revised Szeged index is a molecular structure descriptor equal to the sum of products $[n_u(e) + \frac{n_0(e)}{2}][n_v(e) + \frac{n_0(e)}{2}]$ over all edges $e = uv$ of the molecular graph $G$, where $n_0(e)$ is the number of vertices equidistant from $u$ and $v$, $n_u(e)$ is the number of vertices closer to $u$ than $v$ and $n_v(e)$ is defined analogously. The adjacency matrix of a graph weighted in this way is called its revised Szeged matrix and the set of its eigenvalues is the revised Szeged spectrum of $G$. In this paper some new results on the revised Szeged spectrum of graphs are presented.

A 1-vertex magic vertex labeling of a graph $G$ with $p$ vertices is defined as a bijection $f$ from the vertices to the integers $1, 2, \ldots, p$ with the property that there is a constant $k$ such that at any vertex $x$, $\sum_{y \in N(x)} f(y) = k$, where $N(x)$ is the set of vertices adjacent to $x$. In this paper we introduce 1-vertex bimagic vertex labeling of a graph $G$ and obtain the necessary condition for a graph to be 1-vertex bimagic. We exhibit the same type of labeling for some class of graphs and give some general results.

The edge dominating graph $E_{D}(G)$ of a graph $G=(V,E)$ is a graph with $V(E_{D}(G))=E(G)\cup S(G)$, where $S(G)$ is the set of all minimal edge dominating sets of $G$ with two vertices $u,v\in V(E_{D}(G))$ adjacent if $u\in E$ and $v$ is a minimal edge dominating set of $G$ containing $u$. In this paper, we establish the bounds on order and size, diameter and vertex(edge)connectivity.

The purpose of the present paper is to introduce two new classes $HS_p(\alpha)$ and $HC_p(\alpha)$ of $p$-harmonic mappings together with their corresponding subclasses $HS^0_p(\alpha)$ and $HC^0_p(\alpha)$. We prove that the mappings in $HS_p(\alpha)$ and $HC_p(\alpha)$ are univalent and sense-preserving in $U$ and obtain extreme points of $HS^0_p(\alpha)$ and $HC^0_p(\alpha)$, $HS_p(\alpha)\cap T_p$ and $HC_p(\alpha)\cap T_p$ are determined, where $T_p$ denotes the set of $p$-harmonic mapping with non negative coefficients. Finally, we establish the existence of the neighborhoods of mappings in $HC_p(\alpha)$. Relevant connections of the results presented here with various known results are briefly indicated.

This article is concerned with a pair of second-order symmetric duals in the context of non-differentiable multiobjective fractional programming problems. We establish the weak and strong duality theorems for the new pair of dual models. Discussion on some special cases shows that results in this paper extend previous work in this area.

In this paper, we discuss the inverse problem for Sturm- Liouville operators with boundary conditions having fractional linear function of spectral parameter on the finite interval $[0, 1].$ Using Weyl m-function techniques, we establish a uniqueness theorem. i.e., If q(x) is prescribed on $[0,\frac{1}{2}+\alpha]$ for some $\alpha\in [0,1),$ then the potential $q(x)$ on the interval $[0, 1]$ and fractional linear function $\frac{a_2\lambda+b_2}{c_2\lambda+d_2}$ of the boundary condition are uniquely determined by a subset $S\subset \sigma (L)$ and fractional linear function $\frac{a_1\lambda+b_1}{c_1\lambda+d_1}$ of the boundary condition.

Let $G$ be an abelian group and let $R$ be a commutative $G$-graded super-ring (briefly, graded super-ring) with unity $1\not=0$. We say that $a\in h(R)$, where $h(R)$ is the set of homogeneous elements in $R$, is {\it weakly prime} to a graded superideal $I$ of $R$ if $0\not=ra\in I$, where $r\in h(R)$, then $r\in I$. If $\nu(I)$ is the set of homogeneous elements in $R$ that are not weakly prime to $I$, then we define $I$ to be weakly primal if $P=\bigoplus_{g\in G}(\nu(I)\cap R_g^0+\nu(I)\cap R_g^1)\cup\{0\}$ forms a graded superideal of $R$. In this paper we study weakly primal graded superideals of $R$. Moreover, we classify the relationship among the families of weakly prime graded superideals, primal and weakly primal graded superideals of $R$.

In this paper, the class of free and cyclic canonical $(m,n)-$ary hypermodules over Krasner $(m,n)-$ary hyperrings is defined. Free canonical $(m,n)-$ary hypermodules are a generalization of free canonical hypermodules and a generalization of free modules. Also, several properties are found. In addition, we introduce the concepts of a free basis and a free $(m,n)$-hypermodules as a free object in the category of $(m,n)$-hypermodules and prove some results in this respect. Finally, we obtain some results and relations among a finitely generated torsion free and a free $(m,n)$-hypermodule.

A graph is said to be a \textit{sum graph} if there exists a set $S$ of positive integers as its vertex set with two vertices adjacent whenever their sum is in $S$. An integral sum graph is defined just as the sum graph, the difference being that the label set $S$ is a subset of $Z$ instead of set of positive integers. The sum number of a given graph $G$ is defined as the smallest number of isolated vertices which when added to $G$ results in a sum graph. The integral sum number of $G$ is analogous. In this paper, we mainly prove that any connected graph $G$ of order $n$ with at least three vertices of degree $(n-1)$ is not an integral sum graph. We characterise the integral sum graph $G$ of order $n$ having exactly two vertices of degree $(n-1)$ each and hence give an alternative proof for the existence theorem of sum graphs.

A blender is an indispensable concept presented by Bonatti and Diaz [3] to study high-dimensional $C^1$-robust transitive dynamics around heterodimensional cycles. In this paper, we present a certain Henon-like family of real quadratic diffeomorphisms on $\mathbb{R}^3$, which exhibits an phase transition from non-normal horseshoes to blenders. It can be observable from a rapidly jump of topological dimension for some projected stable segments in some characteristic region of $\mathbb{R}^3$.

Rook placements and rook polynomials have been studied by mathematicians since the early 1970's. Since then many relationships between rook placements and other subjects have been discovered (cf. [1], [6-15]). In [2] and [3], K. Ding introduced the rook length polynomials and the $ \gamma - $compatible rook length polynomials. In [3] and [4], he used these polynomials to establish a connection between rook placements and algebraic geometry for the first time. In this paper, we give explicit formulas for the $ \gamma - $compatible rook length polynomials in more general cases than considered in [3]. In particular, we generalize the formula for the rook length polynomial in the parabolic case in [2] to the $ \gamma -$compatible rook length polynomial.

Let $X subset V$ be a set of vertices in a graph $G = (V, E)$. The boundary $B(X)$ of $X$ is defined to be the set of vertices in $V-X$ dominated by vertices in $X$, that is, $B(X) = (V-X) cap N(X)$. The differential $ partial(X)$ of $X$ equals the value $ partial(X) = |B(X)| - |X|$. The differential of a graph $G$ is defined as $ partial(G) = max { partial(X) | X subset V }$. It is easy to see that for any graph $G$ having vertices of maximum degree $ Delta(G)$, $ partial(G) geq Delta (G) -1$. In this paper we characterize the classes of unicyclic graphs, split graphs, grid graphs, $k$-regular graphs, for $k leq 4$, and bipartite graphs for which $ partial(G) = Delta(G)-1$. We also determine the value of $ partial(T)$ for any complete binary tree $T$.

Suppose $G$ is a multiplicatively written abelian $p$-group, where $p$ is a prime, and $F$ is a field of arbitrary characteristic. The main results in this paper are that none of the Sylow $p$-group of all normalized units $S(FG)$ in the group ring $FG$ and its quotient group $S(FG)/G$ cannot be Prufer groups. This contrasts a classical conjecture for which $S(FG)/G$ is a direct factor of a direct sum of generalized Prufer groups whenever $F$ is a perfect field of characteristic $p$.

Let $T$ be a bounded linear operator acting on a complex Hilbert space $ \mathcal{H} $. In this paper, we show that if $A$ is quasi-class $A$, $ B^* $ is invertible quasi-class $A$, $X$ is a Hilbert-Schmidt operator, $AX=XB$ and $ \left\Vert |A^*| \right\Vert \left\Vert |B|^{-1} \right\Vert \leq 1 $, then $ A^* X = X B^* $.

A set ${S\subseteq V}$ is said to be a complementary nil dominating set of a graph $G$ if it is a dominating set and its complement ${V-S}$ is not a dominating set for $G$. The minimum cardinality of a $cnd$-set is called the complementary nil domination number of $G$ and is denoted by ${\gamma}_{\rm cnd}(G)$. In this paper some results on the complementary nil domination number are obtained.