Authors:M. Fedorova Abstract: Each action of a finitely generated group on a set uniquely defines a labelled directed graph called the Schreier graph of the action. Schreier graphs are used mainly as a tool to establish geometrical and dynamical properties of corresponding group actions. In particilar, they are widely used in order to check amenability of different classed of groups. In the present paper Schreier graphs are utilized to construct new examples of faithful actions of free products of groups. Using Schreier graphs of group actions a sufficient condition for a group action to be faithful is presented. This result is applied to finite automaton actions on spaces of words i.e. actions defined by finite automata over finite alphabets. It is shown how to construct new faithful automaton presentations of groups upon given such a presentation. As an example a new countable series of faithful finite automaton presentations of free products of finite groups is constructed. The obtained results can be regarded as another way to construct new faithful actions of groups as soon as at least one such an action is provided. PubDate: 2018-01-03 Issue No:Vol. 9 (2018)

Authors:Ya. O. Baranetskij, P. I. Kalenyuk, L. I. Kolyasa, M. I. Kopach Abstract: In this paper, the problem with boundary nonself-adjoint conditions for a differential-operator equations of the order $2n$ with involution is studied. Spectral properties of operator of the problem is investigated.By analogy of separation of variables the nonlocal problem for the differential-operator equation of the even order is reduced to a sequence $ \{L_{k}\}_{k=1}^{\infty}$ of operators of boundary value problems for ordinary differential equations of even order. It is established that each element $L_{k}$, of this sequence, is an isospectral perturbation of the self-adjoint operator $L_{0,k}$ of the boundary value problem for some linear differential equation of order 2n.We construct a commutative group of transformation operators whose elements reflect the system $V(L_{0,k})$ of the eigenfunctions of the operator $L_{0,k}$ in the system $V(L_{k})$ of the eigenfunctions of the operators $L_{k}$. The eigenfunctions of the operator $L$ of the boundary value problem for a differential equation with involution are obtained as the result of the action of some specially constructed operator on eigenfunctions of the sequence of operators $L_{0,k}.$The conditions under which the system of eigenfunctions of operator $L$ the studied problem is a Riesz basis is established. PubDate: 2018-01-02 Issue No:Vol. 9 (2018)

Authors:R. I. Dmytryshyn Abstract: In this paper, we consider the problem of convergence of an important type of multidimensional generalization of continued fractions, the branched continued fractions with independent variables. These fractions are an efficient apparatus for the approximation of multivariable functions, which are represented by multiple power series. We have established the effective criterion of absolute convergence of branched continued fractions of the special form in the case when the partial numerators are complex numbers and partial denominators are equal to one. This result is a multidimensional analog of the Worpitzky's criterion for continued fractions. We have investigated the polycircular domain of uniform convergence for multidimensional C-fractions with independent variables in the case of nonnegative coefficients of this fraction. PubDate: 2018-01-02 Issue No:Vol. 9 (2018)

Authors:J. Jinto, K. A. Germina, P. Shaini Abstract: A distance compatible set labeling (dcsl) of a connected graph $G$ is an injective set assignment $f : V(G) \rightarrow 2^{X},$ $X$ being a non empty ground set, such that the corresponding induced function $f^{\oplus} :E(G) \rightarrow 2^{X}\setminus \{\phi\}$ given by $f^{\oplus}(uv)= f(u)\oplus f(v)$ satisfies $ f^{\oplus}(uv) = k_{(u,v)}^{f}d_{G}(u,v) $ for every pair of distinct vertices $u, v \in V(G),$ where $d_{G}(u,v)$ denotes the path distance between $u$ and $v$ and $k_{(u,v)}^{f}$ is a constant, not necessarily an integer, depending on the pair of vertices $u,v$ chosen. $G$ is distance compatible set labeled (dcsl) graph if it admits a dcsl. A dcsl $f$ of a $(p, q)$-graph $G$ is dispersive if the constants of proportionality $k^f_{(u,v)}$ with respect to $f, u \neq v, u, v \in V(G)$ are all distinct and $G$ is dispersible if it admits a dispersive dcsl. In this paper we proved that all paths and graphs with diameter less than or equal to $2$ are dispersible.

Authors:V. Kaladevi, R Murugesan, K. Pattabiraman Abstract: A topological index of a graph is a parameter related to the graph; it does not depend on labeling or pictorial representation of the graph. Graph operations plays a vital role to analyze the structure and properties of a large graph which is derived from the smaller graphs. The Zagreb indices are the important topological indices found to have the applications in Quantitative Structure Property Relationship(QSPR) and Quantitative Structure Activity Relationship(QSAR) studies as well. There are various study of different versions of Zagreb indices. One of the most important Zagreb indices is the reformulated Zagreb index which is used in QSPR study. In this paper, we obtain the first reformulated Zagreb indices of some derived graphs such as double graph, extended double graph, thorn graph, subdivision vertex corona graph, subdivision graph and triangle parallel graph. In addition, we compute the first reformulated Zagreb indices of two important transformation graphs such as the generalized transformation graph and generalized Mycielskian graph. PubDate: 2018-01-02 Issue No:Vol. 9 (2018)

Authors:V. A. Litovchenko Abstract: Because of the parabolic instability of the Shilov systems to change their coefficients, the definition parabolicity of Shilov for systems with time-dependent $t$ coefficients, unlike the definition parabolicity of Petrovsky, is formulated by imposing conditions on the matricant of corresponding dual by Fourier system. For parabolic systems by Petrovsky with time-dependent coefficients, these conditions are the property of a matricant, which follows directly from the definition of parabolicity. In connection with this, the question of the wealth of the class Shilov systems with time-dependent coefficients is important.

A new class of linear parabolic systems with partial derivatives to the first order by the time $t$ with time-dependent coefficients is considered in this work. It covers the class by Petrovsky systems with time-dependent younger coefficients. A main part of differential expression of each such system is parabolic (by Shilov) expression with constant coefficients. The fundamental solution of the Cauchy problem for systems of this class is constructed by the Fourier transform method. Also proved their parabolicity by Shilov. Only the structure of the system and the conditions on the eigenvalues of the matrix symbol were used. First of all, this class characterizes the wealth by Shilov class of systems with time-dependents coefficients.

Also it is given a general method for investigating a fundamental solution of the Cauchy problem for Shilov parabolic systems with positive genus, which is the development of the well-known method of Y.I. Zhitomirskii. PubDate: 2018-01-02 Issue No:Vol. 9 (2018)

Authors:Yu. Trukhan, O. Mulyava Abstract: According to M.L. Mogra, T.R. Reddy and O.P. Juneja an analytic in ${\mathbb D_0}=\{z: 0< z <1\}$ function $f(z)=\frac{1}{z}+\sum_{n=1}^{\infty}f_n z^{n}$ is said to be meromorphically starlike of order $\alpha\in [0,\,1)$ and type $\beta\in (0,\,1]$ if $ zf'(z)+f(z) <\beta zf'(z)+(2\alpha-1)f(z) , \, z\in {\mathbb D_0}. $ Here we investigate conditions on complex parameters $\beta_0,\,\beta_1,\,\gamma_0,\,\gamma_1,\,\gamma_2$, under which the differential equation of S. Shah $z^2 w''+(\beta_0 z^2+\beta_1 z) w'+(\gamma_0 z^2+\gamma_1 z+\gamma_2)w=0$ has meromorphically starlike solutions of order $\alpha\in [0,\,1)$ and type $\beta\in (0,\,1]$. Beside the main case $n+\gamma_2\not=0, \, n\ge 1,$ cases $\gamma_2=-1$ and $\gamma_2=-2$ are considered. Also the possibility of the existence of the solutions of the form $f(z)=\frac{1}{z}+\sum_{n=1}^{m}f_n z^{n}, \, m\ge 2,$ is studied. In addition we call an analytic in ${\mathbb D_0}$ function $f(z)=\frac{1}{z}+\sum_{n=1}^{\infty}f_n z^{n}$ meromorphically convex of order $\alpha\in [0,1)$ and type $\beta\in (0,1]$ if $ zf''(z)+2f'(z) <\beta zf''(z)+2\alpha f'(z) , \, z\in {\mathbb D_0}$ and investigate sufficient conditions on parameters $\beta_0,\,\beta_1,\,\gamma_0,$ $\gamma_1,\,\gamma_2$ under which the differential equation of S. Shah has meromorphically convex solutions of order $\alpha\in [0,\,1)$ and type $\beta\in (0,\,1]$. The same cases as for the meromorphically starlike solutions are considered. PubDate: 2018-01-02 Issue No:Vol. 9 (2018)

Authors:E. Prajisha, P. Shaini Abstract: The concept of $FG$- coupled fixed point introduced recently is a generalization of coupled fixed point introduced by Guo and Lakshmikantham. A point $(x,y)\in X\times X$ is said to be a coupled fixed point of the mapping $F: X\times X \rightarrow X$ if $F(x,y)=x$ and $F(y,x)=y$, where $X$ is a non empty set. In this paper, we introduce $FG$- coupled fixed point in cone metric spaces for the mappings $F:X\times Y \rightarrow X$ and $G:Y\times X\rightarrow Y$ and establish some $FG$- coupled fixed point theorems for various mappings such as contraction type mappings, Kannan type mappings and Chatterjea type mappings. All the theorems assure the uniqueness of $FG$- coupled fixed point. Our results generalize several results in literature, mainly the coupled fixed point theorems established by Sabetghadam et al. for various contraction type mappings. An example is provided to substantiate the main theorem.

Authors:S. M. Sangurlu, D. Turkoglu Abstract: The Banach contraction principle is the most important result. This principle has many applications and some authors was interested in this principle in various metric spaces as Brianciari.

The author initiated the notion of the generalized metric space as a generalization of a metric space by replacing the triangle inequality by a more general inequality, $d(x,y)\leq d(x,u)+d(u,v)+d(v,y)$ for all pairwise distinct points $x,y,u,v$ of $X$. As such, any metric space is a generalized metric space but the converse is not true. He proved the Banach fixed point theorem in such a space. Some authors proved different types of fixed point theorems by extending the Banach's result. Wardowski introduced a new contraction which generalizes the Banach contraction. He using a mapping $F: \mathbb{R}^{+} \rightarrow \mathbb{R}$ introduced a new type of contraction called $F$-contraction and proved a new fixed point theorem concerning $F$-contraction. In this paper, we have dealt with $F-$ contraction and $F-$weak contraction in complete generalized metric spaces. We prove some results for $F-$ contraction and $F-$weak contraction and we show that the existence and uniqueness of fixed point for satisfying $F-$contraction and $F$-weak contraction in complete generalized metric spaces. Some examples are supplied in order to support the useability of our results. The obtained result is an extension and a generalization of many existing results in the literature. PubDate: 2018-01-02 Issue No:Vol. 9 (2018)

Authors:M. M. Sheremeta Abstract: Let $\gamma$ be a positive continuous on $[0,\,+\infty)$ function increasing to $+\infty$ and $f$ and $g$ be arbitrary entire functions of positive lower order and finite order.

In order that for $$\lim\limits_{r\to+\infty} \frac{\ln\ln\,M_{f(g)}(r)}{\ln\ln\,M_f(\exp\{\gamma(r)\})}=+\infty, \quad M_f(r)=\max\{ f(z) :\, z =r\}, $$ it is necessary and sufficient that $(\ln\,\gamma(r))/(\ln\,r)\to 0$ as $r\to+\infty$. This statement is an answer to the question posed by A.P. Singh and M.S. Baloria in 1991.

Also in order that $$ \lim\limits_{r\to+\infty}\frac{\ln\ln\,M_F(r)} {\ln\ln\,M_f(\exp\{\gamma(r)\})}=0,\quad F(z)=f(g(z)), $$ it is necessary and sufficient that $(\ln\,\gamma(r))/(\ln\,r)\to \infty$ as $r\to+\infty$. PubDate: 2018-01-02 Issue No:Vol. 9 (2018)

Authors:M. D. Siddiqi, A. Haseeb, M. Ahmad Abstract: In the present paper, we study a new class of submanifolds of a generalized Quasi-Sasakian manifold, called skew semi-invariant submanifold. We obtain integrability conditions of the distributions on a skew semi-invariant submanifold and also find the condition for a skew semi-invariant submanifold of a generalized Quasi-Sasakian manifold to be mixed totally geodesic. Also it is shown that a skew semi-invariant submanifold of a generalized Quasi-Sasakian manifold will be anti-invariant if and only if $A_{\xi}=0$; and the submanifold will be skew semi-invariant submanifold if $\nabla w=0$. The equivalence relations for the skew semi-invariant submanifold of a generalized Quasi-Sasakian manifold are given. Furthermore, we have proved that a skew semi-invariant $\xi^\perp$-submanifold of a normal almost contact metric manifold and a generalized Quasi-Sasakian manifold with non-trivial invariant distribution is $CR$-manifold. An example of dimension 5 is given to show that a skew semi-invariant $\xi^\perp$ submanifold is a $CR$-structure on the manifold. PubDate: 2018-01-02 Issue No:Vol. 9 (2018)

Authors:T. V. Vasylyshyn Abstract: It is known that every complex-valued homomorphism of the Fr\'{e}chet algebra $H_{bs}(L_\infty)$ of all entire symmetric functions of bounded type on the complex Banach space $L_\infty$ is a point-evaluation functional $\delta_x$ (defined by $\delta_x(f) = f(x)$ for $f \in H_{bs}(L_\infty)$) at some point $x \in L_\infty.$ Therefore, the spectrum (the set of all continuous complex-valued homomorphisms) $M_{bs}$ of the algebra $H_{bs}(L_\infty)$ is one-to-one with the quotient set $L_\infty/_\sim,$ where an equivalence relation "$\sim$'' on $L_\infty$ is defined by $x\sim y \Leftrightarrow \delta_x = \delta_y.$ Consequently, $M_{bs}$ can be endowed with the quotient topology. On the other hand, $M_{bs}$ has a natural representation as a set of sequences which endowed with the coordinate-wise addition and the quotient topology forms an Abelian topological group. We show that the topology on $M_{bs}$ is metrizable and it is induced by the metric $d(\xi, \eta) = \sup_{n\in\mathbb{N}}\sqrt[n]{ \xi_n-\eta_n },$ where $\xi = \{\xi_n\}_{n=1}^\infty,\eta = \{\eta_n\}_{n=1}^\infty \in M_{bs}.$ PubDate: 2018-01-02 Issue No:Vol. 9 (2018)