Authors:A. R. Moghaddamfar; S. Navid Salehy; S. Nima Salehy Pages: 15 - 28 Abstract: Abstract We first consider the companion matrix associated with the characteristic polynomial of a linear recurrence relation, and we investigate its powers. Next, we introduce a new matrix associated with a given linear recurrence sequence, and we get a factorization of this matrix. Finally, we give several applications of our results. Actually, we obtain some identities concerning Fibonacci, Lucas, Pell, and Jacobsthal numbers using matrix theory. PubDate: 2018-01-01 DOI: 10.1007/s40840-015-0216-z Issue No:Vol. 41, No. 1 (2018)

Authors:Zhiqiang Wei Pages: 49 - 61 Abstract: Abstract In this paper, we investigate the cross-coupled Camassa–Holm system. First, we prove local well-posedness for the cross-coupled Camassa–Holm equations in \(H^s ({\mathbb {R}})\) with \(s > 3/2\) . Then, a blow-up criterion is established. PubDate: 2018-01-01 DOI: 10.1007/s40840-015-0218-x Issue No:Vol. 41, No. 1 (2018)

Authors:Hongwu Wu Pages: 63 - 79 Abstract: Abstract This paper is concerned with the distribution of zeros of solutions of first-order linear advanced dynamic equations. By introducing a class of sequences involving iterates of the argument, some estimates for the distances between consecutive zeros of solutions are obtained on arbitrary time scales. Our results improve those for differential equations, and provide new results on the distribution of zeros of solutions for arbitrary time scales. Some examples are given to illustrate our results. PubDate: 2018-01-01 DOI: 10.1007/s40840-015-0238-6 Issue No:Vol. 41, No. 1 (2018)

Authors:Zaihong Jiang; Mingxuan Zhu Pages: 105 - 122 Abstract: Abstract In this paper, we consider regularity criteria for the 3D generalized MHD and Hall-MHD systems with fractional dissipative terms. Some scaling invariant regularity criteria are established for the two systems. Global regularity for the Hall-MHD equation is also proved for the case \(\alpha \ge \frac{5}{4}, \beta \ge \frac{7}{4}\) . PubDate: 2018-01-01 DOI: 10.1007/s40840-015-0243-9 Issue No:Vol. 41, No. 1 (2018)

Authors:Krzysztof Bartoszek; Małgorzata Pułka Pages: 159 - 173 Abstract: Abstract This paper is devoted to the study of the problem of prevalence in the class of quadratic stochastic operators acting on the \(L^{1}\) space for the uniform topology. We obtain that the set of norm quasi-mixing quadratic stochastic operators is a dense and open set in the topology induced by a very natural metric. This shows the typical long-term behaviour of iterates of quadratic stochastic operators. PubDate: 2018-01-01 DOI: 10.1007/s40840-015-0245-7 Issue No:Vol. 41, No. 1 (2018)

Authors:R. Sharma; R. Kumar; R. Saini; G. Kapoor Pages: 175 - 190 Abstract: Abstract We prove some inequalities involving fourth central moment of a random variable that takes values in a given finite interval. Both discrete and continuous cases are considered. Bounds for the spread are obtained when a given \(n\times n\) complex matrix has real eigenvalues. Likewise, we discuss bounds for the spans of polynomial equations. PubDate: 2018-01-01 DOI: 10.1007/s40840-015-0267-1 Issue No:Vol. 41, No. 1 (2018)

Authors:Andrzej Matraś; Artur Siemaszko Pages: 231 - 248 Abstract: Abstract The distant graph \(G=G({\mathbb {P}}(Z), \vartriangle )\) of the projective line over the ring of integers is considered. The shortest path problem in this graph is solved by use of Klein’s geometric interpretation of Euclidean continued fractions. In case the minimal path is non-unique, all the possible splitting are described which allows us to give necessary and sufficient conditions for existence of a unique shortest path. PubDate: 2018-01-01 DOI: 10.1007/s40840-015-0273-3 Issue No:Vol. 41, No. 1 (2018)

Authors:Dehong Ji Pages: 249 - 263 Abstract: Abstract In this paper, we present some new results concerning positive solutions for the singular fractional boundary value problem with p-Laplacian. By imposing some suitable conditions on the nonlinear term f, existence results of positive solutions are obtained. The proof is based upon theory of Leray–Schauder degree. The interesting point is the nonlinear term f(t, u) may be singular at \(u=0\) . PubDate: 2018-01-01 DOI: 10.1007/s40840-015-0276-0 Issue No:Vol. 41, No. 1 (2018)

Authors:Mamadou Abdoul Diop; Khalil Ezzinbi; Mamadou Moustapha Mbaye Pages: 287 - 310 Abstract: Abstract In this work, we establish a new concept of square-mean pseudo almost periodic and automorphic processes using the measure theory. We use the \({\mu }\!-\) ergodic process to define the spaces of \({\mu }\!-\) pseudo almost periodic and automorphic processes in the square-mean sense. We establish many interesting results on those spaces like completeness and composition theorems. Then, we study the existence, the uniqueness, and the stability of the square-mean \({\mu }\!-\) pseudo almost periodic and automorphic solutions of the stochastic evolution equation. We provide an example to illustrate ours results. PubDate: 2018-01-01 DOI: 10.1007/s40840-015-0278-y Issue No:Vol. 41, No. 1 (2018)

Authors:Flavia-Corina Mitroi-Symeonidis; Nicuşor Minculete Pages: 311 - 319 Abstract: Abstract In this note we describe some results concerning upper and lower bounds for the Jensen functional. We use several known and new results to shed light on the concept of a strongly convex function. PubDate: 2018-01-01 DOI: 10.1007/s40840-015-0293-z Issue No:Vol. 41, No. 1 (2018)

Authors:H. Amraei; H. R. Maimani; M. R. Pournaki; A. Zaeembashi Pages: 321 - 334 Abstract: Abstract Let R be a finite commutative ring with nonzero identity and denote its Jacobson radical by J(R). The Jacobson graph of R is the graph in which the vertex set is \(R\setminus J(R)\) , and two distinct vertices x and y are adjacent if and only if \(1-xy\) is not a unit in R. In this paper, up to isomorphism, we classify the rings R whose Jacobson graphs are toroidal. PubDate: 2018-01-01 DOI: 10.1007/s40840-015-0294-y Issue No:Vol. 41, No. 1 (2018)

Authors:Olcay Karaatlı; Refik Keskin Pages: 335 - 353 Abstract: Abstract Let P be a nonzero integer and let \((U_{n})\) and \((V_{n})\) denote Lucas sequences of first and second kind defined by \(U_{0}=0, U_{1}=1; V_{0}=2, V_{1}=P;\) and \(U_{n+1}=PU_{n}+U_{n-1},V_{n+1}=PV_{n}+V_{n-1}\) for \( n\ge 1.\) In this study, when P is odd, we show that the equation \( U_{n} =7\square \) has only the solution \((n,P)=(2,7\square )\) when 7 P and the equation \(V_{n}=7\square \) has only the solution \( (n,P)=(1,7\square )\) when 7 P or \((n,P)=(4,1)\) when \(P^{2}\equiv 1( \text{ mod } 7).\) In addition, we show that the equation \(V_{n}=7V_{m}\square \) has a solution if and only if \(P^{2}=-3+7\square \) and \( (n,m)=(3,1).\) Moreover, we show that the equation \(U_{n}=7U_{m} \square \) has only the solution \((n,m,P,\square )=(8,4,1,1)\) when P is odd. PubDate: 2018-01-01 DOI: 10.1007/s40840-015-0295-x Issue No:Vol. 41, No. 1 (2018)

Authors:R. Rabaoui; A. Saddi Pages: 371 - 392 Abstract: Abstract For a positive integer m, a bounded linear operator T on a Hilbert space \(\mathbb {H}\) is called an (A, m)-isometry, if \(\Theta ^{(m)}_{A}(T) =\sum _{k=0}^{m}(-1)^{m-k}{m\atopwithdelims ()k}T^{*k}AT^{k}=0\) , where A is a positive (semi-definite) operator. In this paper we give a characterization of (A, m)-isometric and strict (A, m)-isometric unilateral weighted shifts in terms of their weight sequences, respectively. Moreover, we characterize (A, 2)-expansive unilateral weighted shifts (i.e. operators satisfying \(\Theta ^{(2)}_{A}(T)\le 0\) ). PubDate: 2018-01-01 DOI: 10.1007/s40840-016-0307-5 Issue No:Vol. 41, No. 1 (2018)

Authors:Hui Jiang; Xueliang Li; Yingying Zhang; Yan Zhao Pages: 415 - 425 Abstract: Abstract A path in a vertex-colored graph is a vertex-proper path if any two internal adjacent vertices differ in color. A vertex-colored graph is proper vertex k-connected if any two vertices of the graph are connected by k disjoint vertex-proper paths of the graph. For a k-connected graph G, the proper vertex k-connection number of G, denoted by \(pvc_{k}(G)\) , is defined as the smallest number of colors required to make G proper vertex k-connected. A vertex-colored graph is strong proper vertex-connected, if for any two vertices u, v of the graph, there exists a vertex-proper u-v geodesic. For a connected graph G, the strong proper vertex-connection number of G, denoted by spvc(G), is the smallest number of colors required to make G strong proper vertex-connected. These concepts are inspired by the concepts of rainbow vertex k-connection number \(rvc_k(G)\) , strong rainbow vertex-connection number srvc(G), and proper k-connection number \(pc_k(G)\) of a k-connected graph G. Firstly, we determine the value of pvc(G) for general graphs and \(pvc_k(G)\) for some specific graphs. We also compare the values of \(pvc_k(G)\) and \(pc_k(G)\) . Then, sharp bounds of spvc(G) are given for a connected graph G of order n, that is, \(0\le spvc(G)\le n-2\) . Moreover, we characterize the graphs of order n such that \(spvc(G)=n-2,n-3\) , respectively. Finally, we study the relationship among the three vertex-coloring parameters, namely, \(spvc(G), \ srvc(G)\) , and the chromatic number \(\chi (G)\) of a connected graph G. PubDate: 2018-01-01 DOI: 10.1007/s40840-015-0271-5 Issue No:Vol. 41, No. 1 (2018)

Authors:K. Ratnavelu; M. Kalpana; P. Balasubramaniam Pages: 491 - 505 Abstract: Abstract The synthetic genetic regulatory networks (GRNs) prove to be a powerful tool in studying gene regulation processes in living organisms. In order to take vagueness into consideration, fuzzy theory is incorporated in the GRNs and we found a novel system, namely fuzzy genetic regulatory networks (FGRNs). By applying the Homeomorphism theorem, we demonstrated an existence, uniqueness, and asymptotic stability analysis of the equilibrium point of FGRNs with time delay in the leakage term and unbounded distributed delays. By utilizing Lyapunov functional method and the linear matrix inequality (LMI) techniques, some new and useful criteria of the FGRNs with respect to the equilibrium point are derived. The derived criteria are of the form of LMI, and hence they can be verified easily. Finally, illustrative example with simulations are demonstrated to prove the effectiveness of the proposed results. PubDate: 2018-01-01 DOI: 10.1007/s40840-016-0427-y Issue No:Vol. 41, No. 1 (2018)

Authors:Priyanka Sangal; A. Swaminathan Pages: 507 - 521 Abstract: Abstract In this work, sufficient conditions on the sequence \(\{a_n\}\) are obtained to guarantee the starlikeness, close-to-convexity and convex in the direction of imaginary axis of the analytic function \(f(z)=z+\displaystyle \sum \nolimits _{n=2}^{\infty }a_nz^n\) in the unit disc \(\mathbb {D}.\) These results are obtained using the positivity technique of trigonometric sum as a tool. These coefficient conditions are extended to the triplet \((a,\,b,\,c)\) to ensure that the normalized Gaussian hypergeometric function \(zF(a,\,b;\,c;\,z)\) is starlike. Examples are provided to compare the obtained conditions with the existing results in the literature. PubDate: 2018-01-01 DOI: 10.1007/s40840-016-0420-5 Issue No:Vol. 41, No. 1 (2018)

Authors:N. E. Cho; B. Kowalczyk; O. S. Kwon; A. Lecko; Y. J. Sim Pages: 523 - 535 Abstract: Abstract In the present paper, sharp estimates of some determinants over the class \(\mathcal {S}^*(\alpha ),\ \alpha \in [0,1)\) , of analytic functions f such that \({{\mathrm{Re}}}(zf'(z)/f(z))>\alpha ,\) \(z\in {{\mathbb {D}}}:=\left\lbrace z \in {{\mathbb {C}}} : z <1 \right\rbrace \) , are computed. PubDate: 2018-01-01 DOI: 10.1007/s40840-017-0476-x Issue No:Vol. 41, No. 1 (2018)

Authors:Shih-Sen Chang; Ching-Feng Wen; Jen-Chih Yao Abstract: Abstract The purpose of this paper is by using a generalized forward–backward splitting method to propose an iterative algorithm for finding a common element of the set of solutions to a system of quasi-variational inclusions with accretive mappings and the set of fixed points for a \(\lambda \) -strictly pseudo-contractive mapping in Banach spaces. Some strong convergence theorems of the sequence generated by the algorithm are proved. The results presented in the paper extend and improve some recent results. As applications, we utilize our results to study the approximation problem of solutions to a system of variational inequalities, accretive variational inequality problem and convex minimization problem in Banach spaces. PubDate: 2018-01-08 DOI: 10.1007/s40840-017-0599-0

Authors:Xinlei Wang; Dein Wong Abstract: Abstract In a recent paper, Das introduced the graph \(\mathcal {I}n(\mathbb {V})\) , called subspace inclusion graph on a finite dimensional vector space \(\mathbb {V}\) , where the vertex set is the collection of nontrivial proper subspaces of \(\mathbb {V}\) and two vertices are adjacent if one is properly contained in another. Das studied the diameter, girth, clique number and chromatic number of \(\mathcal {I}n(\mathbb {V})\) when the base field is arbitrary, and he also studied some other properties of \(\mathcal {I}n(\mathbb {V})\) when the base field is finite. At the end of the above paper, the author posed the open problem of determining the automorphisms of \(\mathcal {I}n(\mathbb {V})\) . In this paper, we give the answer to the open problem. PubDate: 2018-01-04 DOI: 10.1007/s40840-017-0597-2

Authors:Reza Aminizadeh; Hamid Rasouli; Abolfazl Tehranian Abstract: Abstract In this paper, the notion of (short) quasi-exact sequence of S-acts is introduced. We study the behaviour of quasi-exact sequences in regard to some algebraic properties of S-acts including principal weak injectivity, principal weak flatness, regularity and torsion freeness. Moreover, some results concerning commutative diagrams of modules are generalized to acts over monoids. PubDate: 2018-01-03 DOI: 10.1007/s40840-017-0596-3