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The paper deals with a special class of neutral functional differential equations (FDE): the so called FDE with iterated delay. They are special equations of superneutral type. Equations with iterated delay had been introduced and studied by Prof. D.D. Bainov (1933-2011) and the author of this paper in the period 1972-1978. As a result a number of papers in this area had been published and the term "FDE with iterated delay" was accepted in the literature.

In this paper, Infinite series solutions of prey-predator system with intraspecies competition is obtained using the Adomian decomposition method. The approximated solu- tions are shown graphically. The results reveals that Adomian decomposition method is an altenative method of solving systems of non-linear equations. The software used in this paper is Mathematica 10.

In the area of membrane computing, a new computability model, called P system is a highly distributed and parallel theoretical computing model. Generating two dimensional picture languages is one of the applications of P systems. Ceterachi et al. began a study on connecting the two areas, membrane computing and array grammars. Triangular tile pasting system and the Triangular Tile pasting P system models have been introduced for pattern generation. This paper introduce the k-order TTPS and ETTPS. Also this paper has extended the theory of Triangular tile pasting P system to generate two dimensional picture languages. A comparative study between triangular tile pasting system and Triangular tile pasting P system is made. Comparison is also made with the array generating Petri nets.

The first three Zagreb indices of a graph G denoted, M1(G),M2(G) and M3(G), are well known. Equally well known is the eccentricity connectivity index denoted, ξ(G). In this paper we derive closed formula for these indices for thorn cycles, thorn star graphs and thorn complete graphs, respectively. Same is repeated for the eccentricity connectivity index. The further aim of the paper is to emphasize the subtle difference between mathematical induction and immediate induction as equally valid techniques of proof. Valid immediate induction requires the rarely utilised property of many graphs namely, well-defineness. AMS Subject Classification: 05C07, 05C38, 05C75, 05C85

In this paper we consider stochastic stability, namely, asymptotic stability in mean, asymptotic mean-square stability, asymptotic stability a.s., exponential p-stability and stability with stochastic Lyapunov fucntion for vector stochastic differential equations. We apply these results to the stochastic epidemic models induced by bacteriophages in the marine bacteria populations. The novelty of the paper consists of new stability results for vector SDE and their applications to a new stochastic epidemic model.

In this paper we consider the Gerber-Shiu penalty function in the compound binomial risk model with time-correlated claims. It is assumed that each main claim will induce a by-claim but the occurrence of the by-claim may be delayed with a certain probability. Formulas for the probability generating function of the penalty function are obtained, together with the expression for the penalty function with zero initial surplus. Then we show that the penalty function satisfies a defective renewal equation. When the claims follow geometric distributions, explicit expression for the Gerber-Shiu function is derived. Numerical example is provided to illustrate the application of the result discussed in the paper.

Manuscript version. All journals published by Academic Publications, Ltd. are open access journals. Open access journals are scholarly journals that are available on-line to the reader - without any limitations for download full text of all published articles. https://ijpam.eu/instr.html Link to publishers version: https://ijpam.eu/contents/2015-103-4/12/12.pdf Ordinary univariate Bernstein polynomials can be represented in matrix form using factor matrices. In this paper we present the definition and basic properties of such factor matrices extended from the univariate case to the general case of arbitrary number of variables by using barycentric coordinates in the hyper-simplices of respective dimension. The main results in the paper are related to the design of an iterative algorithm for fast convex computation of multivariate Bernstein polynomials based on sparse-matrix factorization. In the process of derivation of this algorithm, we investigate some properties of the factorization, including symmetry, commutativity and differentiability of the factor matrices, and address the relevance of this factorization to the de Casteljau algorithm for evaluating curves and surfaces on B´ezier form. A set of representative examples is provided, including a geometric interpretation of the de Casteljau algorithm, and representation by factor matrices of multivariate surfaces and their derivatives in B´ezier form. Another new result is the observation that inverting the order of steps of a part of the new factorization algorithm provides a new, matrix-based, algebraic representation of a multivariate generalization of a special case of the de Boor-Cox computational algorithm.

By the classification theorem by F. Oort and J. Tate [6], any group scheme of prime order is isomorphic to a group scheme Ga,b under the suitable choice of a and b. We computed the torsors for some kinds of group schemes Ga,b in [8], which is a joint work with T. Sekiguchi, as in the following way: denote by p a prime number and by m = φ(p−1) the value of the Euler function φ. Suppose p is a prime ideal lying over p (which splits completely in Z[ζ]), where ζ is a primitive (p − 1)-st root of the unity. In case p is principal, the sequence 0 → μp,B → G m m,B p −→ Gm,B → 0 is exact, and the Galois descent of μp,B is isomorphic to Ga,b under the suitable choice of a and b, thus one can compute the torsors for this kinds of group schemes. The non-principal case is solved by Y. Koide [3] by using our method. The aim of this paper is to study some group schemes of order a power of a prime number. In section from 1 to 3, we would like to review the main result of the papers [6] by F. Oort and J. Tate, [4] by Y. Koide and T. Sekiguchi, and [8] by T. Sekiguchi and Y. Toda. In section 4, we give our main result, namely, the torsor for the Galois descent of μpn,B.

This paper presents the development of a new variant of two-stage Geometric Mean (GM) method for solving 2D elliptic equation. The proposed iterative scheme, called the Skewed Modified Geometric Mean (SkMGM), is derived from finite difference approximation discretized on a skewed grid with red-black ordering. The skewed finite difference scheme combined with redblack ordering is shown to be suitable for parallel implementation. The developed SkMGM scheme is compared with the other methods on the standard grid to confirm the effectiveness of the proposed method in terms of computational complexity and execution time. It is shown that the new proposed method gives the least number of iterations and fastest execution time compared to the other tested methods.

In this paper, an analytical technique, namely the new iterative method (NIM), is applied to obtain an approximate analytical solution of the nonlinear Harry-Dym equation which is often used in the theory of solitons. The rapid convergence of the method results in qualitatively accurate solutions in relatively few iterations; this is obvious upon comparing the obtained analytical solutions with the exact solutions. Our results indicate that NIM is highly accurate and efficient, therefore can be considered a very useful and valuable method.

In this paper we study a class of continuous functions satisfying a certain Zyg-mund condition dependent on a parameter γ > 0. It shown that the modulus of continuity of such functions is O(δ(log 1/δ)1-γ) if ∈ (0, 1) and O(δ(log log 1/δ )) if γ = 1. Moreover, these functions are differentiable if γ > 1. These results extend the results in literatures [4], [5].

In this paper, we solved the forced perturbed Korteweg-de vries (FpKdV) with variable coefficient arises in nonlinear wave propagation in an elastic tube filled with a sym- metrical stenosis filled with a viscous fluid by two numerical methods, namely method of lines and pseudospectral method. We then compared both numerical solution with its progressive wave solution. Both methods solve FpKdV equation with maximum absolute errors of 10-2.

This paper deals with the monoid PF_N of all partial transformations on N preserving a zig-zag order on N. We determine the relative rank of PF_N modulo a set containing all idempotents and all surjections in PF_N. Moreover, we show that all transformations in PFN with finite rank can be generated by the idempotents with finite rank and the full transformation \gamma_0 with infinite rank, where \gamma_0 maps each natural number n to n + 2.

We consider a simple motion pursuit differential game of three pursuers and one evader with equal dynamic possibilities in the plane. Geometric constraints are imposed on the controls of players and the control set of each player is unit circle centered at the origin. Pursuit is said to be completed if the state of at least one of the pursuers coincides with that of evader. In this paper, we construct a new effective strategy for the pursuers. We prove that new strategy has advantage over the P-strategy.

Stochastic volatility (SV) model is widely applied in the extension of the constant volatility in Black-Scholes option pricing.In this paper, we extend the SV model driven by fractional Brownian motion (FBM). A crucial problem in its application is how the unknown parameters in the model are to be estimated. We propose the innovation algorithm, and follow by the maximum likelihood estimation approach, which enables us to derive the estimators of parameters involved in this model.We will also present the simulation outcomes to illustrate the efficiency and reliability of the proposed method.

In this paper, we investigate the relations between soft L-fuzzy topogenous orders and soft L-fuzzy pre-uniform spaces. We give their examples.

In this paper, we introduce the notions of R-weakly commuting and its variants for multiplicative expansive mappings in multiplicative metric spaces and prove common fixed point theorems for these mappings. Moreover, we prove a common fixed point theorem for weakly reciprocally continuous mappings. We also provide examples in support of our results. AMS Subject Classification: 47H10, 54H25

Let G = (V,E) be a simple graph on the vertex set V . In a graph G, A set S ⊆ V is a dominating set of G if every vertex in V S is adjacent to some vertex in S. The domination number of a graph Gγ(G)] is the minimum size of a dominating set of vertices in G. The bondage number of a graph G[Bdγ(G)] is the cardinality of a smallest set of edges whose removal results in a graph with domination number larger than that of G. In this paper we establish domination number and the bondage number for some families of graphs. AMS Subject Classification: 05C69

This paper presents new implementation of reliable iterative method proposed by Temimi and Ansari namely (TAM) for analytic and numerical solutions of Duffing equations in two types: damping and undamping equations. The solution is obtained in the series form that converge to the exact solution with easily computed components. Several problems are solved to demonstrate the efficiency of the proposed methods without any restrictive assumptions for nonlinear terms. However, for some examples numerical calculations evidenced by tables and figures for the analysis of the maximal error remainder values.The software used for the calculations in this study was MATHEMATICA r 9.0.

In this paper, we study the zero distributions on the derivatives of q-shift dif- ference polynomials of meromorphic functions with zero order and obtain two theorems that extend results of (3).

Uniform structure on the T-valued spaces are defined using different set of axioms and basic terms. A different approach to fuzzy uniformities, in terms of T-covers was introduced which seems the most natural one. The purpose of this paper is to present another characterization of fuzzy uniformities in the style of Weil that we call it TWeil uniformity. We formulate and investigate a definition of entourage uniformity alternative to that one of Hutton. It is expressed in terms of coproduct of fuzzy spaces. We identify this new notion with the existing ones by proving the isomorphism between the respective categories. AMS Subject Classification: 54A40, 06D20

In this paper, we introduce a new viscosity approximation method of nonexpan- sive mappings in Hilbert spaces. The strong convergence theorems of the rules are proved under certain assumptions imposed on the sequence of parameters. Moreover, we give appli- cations of the purposed viscosity method.

In this paper, we obtain a Suzuki type unique common coupled fixed point theorem by using ψ −φ contraction in partially ordered multiplicative metric spaces. We also give an example to illustrate our main theorem.

In this paper, we introduce a new finite element on the triangle using P 1 non- conforming element enriched with incomplete P 2 conforming element. The resultant finite element approximation is piecewise quadratic discontinuous along the edges except at the mid points of the edges. The new finite element is unisolvent and satisfies the interpolation estimates.

In this paper, we introduce the concept of coupled fixed point in partially ordered multiplicative metric space by proving some theorems for the existence and uniqueness of coupled fixed point. Also, we discuss the application to the existence and uniqueness of solution for a periodic boundary value problem.

Hanging modified theta graphs are obtained from attaching a modified theta graph to each teeth of comb graph Pai ⊙ L1, where Pai denotes a path on ai vertices. In this paper it is proved that symmetric hanging modified theta graphs and non-decreasing symmetric hanging modified theta graphs are graceful. AMS Subject Classification: 05C78

In this paper, we present a new iteration for set contraction mappings in a Banach space. In further, we introduce the concept of set stability for these class of mappings. Finally, we present some results on the existence of fixed points for power set contraction mappings, these results include in special case many results existing in literature like Darbo and generalized Darbo fixed point theorems. AMS Subject Classification: 47H10, 47H08

In this paper, we introduce the double power iteration method witch can be seen as an extension of the classical power iteration in the sense that we calculate the two dominants eigenvalues at each stage. This work aims to propose a solution of slow convergence problem to the power iteration method and the calculation of the second dominant eigenvalue. We develop a parallel iterative procedure for the calculation of eigenvalues of a given matrix and we can expressed this method as a Quadrant Interlocking Factorization (QIF) which introduced by (7) and studied by (8)-(9) in other works.

The purpose of this paper is to introduce the notions of several types of semicompactness in double fuzzy topological spaces known as generalized double fuzzy semicompactness, generalized double fuzzy nearly semi-compactness and generalized double fuzzy weakly semi-compactness. Some interesting properties and characterizations of the new concepts are introduced and studied. AMS Subject Classification: 54A40, 54D30, 03E72

In this paper, we present the sufficient criteria for the existence of solutions for perturbed fractional differential equations and inclusions with generalized Riemann-Liouville fractional integral boundary conditions. We make use of a nonlinear alternative, which deals with the sum of completely continuous and contractive single-valued or multi-valued operators, to obtain the desired results.

In this paper, we investigate the properties of fuzzy soft sets and fuzzy soft maps in stsc-quantales. We define a L-fuzzy (K,E)-soft topology and a L-fuzzy (K,E)-soft closure spaces as a Hohle's sense. We study the relations between L-fuzzy (K,E)-soft topologies and L-fuzzy (K,E)-soft closure spaces. We give their examples.

In this paper we investigate fuzzy fractional integral inclusions under compactness type conditions. We prove the existence of solutions when the right-hand side is almost upper semicontinuous. We also show that the solution set is connected. Finally, an application to fuzzy fractional differential inclusions is given.

In this paper, we consider the Hyers-Ulam stability of Cauchy type functional equations of module left (m,n)-derivations and generalized module left (m,n)-derivations in complete metric spaces.

In this paper, we obtain the solutions of a generalizedi-mixed difference equa- tion in closed, finite and infinite multi-series forms. By equating closed form with multi-series type solutions of thei-mixed difference equation, we obtain the values of certain types of infinite and finite multi-series. Suitable examples are provided to illustrate the main results.

This paper deals with two-person non-zero sum games with interval payoffs. Graphical method to find a mixed strategy equilibrium is adapted to interval bimatrix games. In addition, interval bimatrix games Nash equilibrium is attained by graphical method. Nu- merical examples are also illustrated.

In this paper, we proved the some common fixed point results for compatible mappings of types in multiplicative metric spaces. AMS Subject Classification: 47H10, 54H25

In this paper, we developed a non-Gaussian Wick calculus based on the theory of hypecomplex systems L1(Q,dm(x)). Using the Delsarte charactersn(x), we introduce a �-Wick product, a �-Hermite transform on the space of generalized functions Hq and discuss their properties. By means of the usual properties of complex analytic functions, we proved the characterization theorem for Hq. Moreover, we setup a framework to study the stochastic

In this paper a technique of controlling chaotic behavior of a three species food chain model with Sokol-Howell functional response by involving intraspecific competition is discussed. The model is proved to be dissipative in the positive octant. Controlling conditions of chaotic behavior of the system are carried out. Stability analysis of the equilibrium points is analyzed. Lyaponuv exponents and bifurcation diagrams are used to study the complex dynamics of the model. The impact of the intraspecific competition in the dynamics of the foodchain model with Sokol-Howell response is investigated theoretically as well as numerically.

In this paper, we present a modified new sixth-order fixed point iterative method for solving nonlinear functional equations and analyzed. The modified new sixth-order fixed point iterative method has convergence of order six and efficiency index 1.8171 which is larger than most of the existing methods and the methods discussed in Table 1. The modified new sixth-order fixed point iterative method converges faster than the methods discussed in Tables 1-6. The comparison tables demonstrate the faster convergence of the modified new sixth-order fixed point method.

In this paper, based on linear ranking functions we investigate some properties for the fuzzy number linear programming problems and its dual with the relationships between them. AMS Subject Classification: 90C05, 90C46, 90C70

In this paper, we introduce the notion of L-fuzzy (K,E)-soft preclosure spaces. We investigate two L-fuzzy (K,E)-soft preclosure spaces induced by an L-fuzzy (K,E)-soft quasi-uniform space. Also, we study the relationship among L-fuzzy (K,E)-soft preclosure spaces, L-fuzzy (K,E)-soft topological spaces and L-fuzzy (K,E)-soft quasi-uniform space. Finally, we give their examples.

This paper presents a general solitary solution of a class of nonlinear time- fractional partial differential equations by Adomian decomposition method(ADM). This class of nonlinear time-fractional partial differential equations include a lot of standard nonlinear partial differential equations in mathematical physics. The solitary solution obtained by ADM is a general solitary solution and admit you investigate the solution for different initial condi- tions and different �( � is the order of derivative respect to time 0 < � ≤ 1). Also the solution subject to the especial initial conditions and � = 1 reduce to the solution of standard partial differential equation. Additionally, it use the fractional derivative of Caputo sense.

In this paper, we shall study completely continuous maps which leave a cone invariant in an ordered Banach space. A generalisation of some results of Krasnosel'skii and Amann, where the map are supposed to be monotone, will be given.

In this paper, we prove strong and weak convergence results of modified Ishikawa iterative process to common fixed points of two nearly lipschitzian mappings in the framework of hyperbolic spaces.Our results extend and improve results of (4) and (11) from uniformly convex Banach spaces and CAT (0) spaces to hyperbolic spaces.

Rishivarman and Parthasarathy (2013) have constructed a map from the direct product of two copies of the set of ordinary elliptic curves with short Weierstrass equation over F25 to non-negative real numbers (plus infinity), and they have claimed that the map is a metric. In this paper we point out that the proof of the claim contains several flaws. Even worse, we shall show that the map is not a metric.

In the present paper, we study evolute offsets of a non-developable ruled surface in Euclidean 3-space E and classify the evolute offset with constant Gaussian curvature and constant mean curvature. In last section, we investigate linear Weingarten evolute offsets in E . A linear Weingarten surface is the surface having a linear equation between the Gaussian curvature and the mean curvature of a surface. AMS Subject Classification: 53C30, 53B25

In this paper, we prove common fixed point results for compatible mappings and its variants satisfying expansive mappings in multiplicative metric spaces. We also give examples in support of our results. AMS Subject Classification: 47H10, 54H25

In this paper, we introduce the notions of soft L-fuzzy pre-uniformities and soft L-fuzzy topogenous orders in complete residuated lattices. We study some relationships between previous spaces. As main results, we investigate the soft ^ ^ �

In this paper three types of strong vertices are introduced.A necessary and sufficient conditions for a vertex to be strong have been given.Some of the properties of �- strong,�-strong and -strong vertices are discussed.Also the concept of vertex order colouring is introduced.An algorithm for vertex order colouring is also given at the end of the chapter.

In this paper, we prove Kannan - type fixed point theorem for two self mappings in non-normal cone pentagonal metric spaces. Our results extend and improve the recent results announced by many authors.