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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.

In this paper we prove Hyres-Ulam-Rassias stability of an additive functional equation in random normed spaces. The concept of Hyers-Ulam- Rassias stability originated from Th.M. Rassias stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297-300.

In this paper concept of self and strongly function chainable sets is introduced for topological spaces. Results proved in (8) are extended to self and strongly function chainable sets. Two characterizations of strongly function chainable sets have been established in the paper. It is shown that the space is connected if and only if it is function chainable provided the function is one to one.

Qualitative properties as blow up, decay bounds and extinction in finite time for solutions to linear and nonlinear parabolic problems have received much attention in the recent literature. We refer to the papers [1], [4], [5] and the references therein. We also mention the results of Galaktionov-Mitidieri-Pohoazev ([3], [13]) on blow-up and global existence for different classes of parabolic problems, including higher order parabolic equations. The blow-up phenomena of solutions to various nonlinear problems, particularly for parabolic systems, have been investigated by different authors. For results in this area, the reader can reference the book [19] and the survey paper

The concept of Lie-recurrence in a Finsler space was introduced by the second author [8] of the present paper in 1982.The Lie-recurrence in a Riemannian space was studied by K. L. Duggal [3] in 1992 but he used the term curvature inheriting symmetry in place of Lie-recurrence. K. L. Duggal also applied the theory to the study of fluid space time. Since then both the terms (Lie-recurrence and curvature inheriting symmetry) are in use. The present authors [11], Shivalika Saxena and P. N. Pandey [12], [13], C. K. Mishra and Gautam Lodhi [1] studied a Lie-recurrence (curvature inheriting symmetry) in a Finsler space and discussed the possibilities for contra and concurrent vector fields to generate Lie-recurrence. The present paper deals with a Lie-recurrence generated by a special concircular vector field and such Lie-recurrence is termed as a special concircular Lie-recurrence. We obtain certain results related to a special concircular Lie-recurrence in a general Finsler space as well as in birecurrent and bisymmetric Finsler spaces. AMS Subject Classification: 53B40

The concept of polar composite polynomials was introduced by Zaheer and Alam (1) and was studied in quite a detail in the same paper. Some further applications were obtained by the authors in their subsequent articles (2, 3). In the present paper the author applies the results of (1) to find the real values of certain partial fraction sums.

Recently an algebra based on propositional calculi was introdued by Tamilarasi and Mekalai in the year 2010 known as TM−algebras, see (6). In our paper (1) we introduced the notion of derivation on TM-algebras. In this paper, we introduce the notion of symmetric bi-derivation on TM-algebras and study some of its properties.

This paper introduces a new Estimator for multicollinearly matrix data and autocorrelated errors. We purpose two Stages Ridge Estimator(TR) for the multiple linear regression model, which suffers from both problems au- tocorrelation (AR(1)) and multicollinearity. After adjusting this with the ordi- nary ridge regression estimator (ORR), we use a mixed method to apply the two stages least squares procedure (TS). We also derive some statistical properties of this biased estimator and the paper is achieved by an application example.

In this paper, we introduce a new concept of super edge-magic sequence (SEMS) of a super edge-magic graph (SEMG) with p vertices and q edges. The super edge-magic sequence of natural numbers is denoted by , 1 ≤ i ≤ q. This sequence need not to be monotonic. In this track, we also drive some families of super edge-magic graphs from fabrication of new super edge-magic sequences by considering additional parameter. We complete this paper by discussing the special case like monotonic sequences related to the super edge-magic sequence.

Our aim in this paper is to prove a strong convergence theorem by three-step iteration scheme with respect to contractive condition in a Banach space. Our results obtained in this paper is new extension as well as refinement of previous known results. AMS Subject Classification: 47H05, 47H10, 49M05

The classification of the dihedral folding tessellations of the sphere and the plane whose prototiles are a kite and an equilateral triangle were obtained in [C. Avelino and A. Santos, Spherical and planar folding tessellations by kites and equilateral triangles, Australasian Journal of Combinatorics, 53 (2012), 109–125.]. Recently, this classification was extended to isosceles triangles so that the classification of spherical folding tesselations by kites and isosceles triangles in three cases of adjacency was presented in [C. Avelino and A. Santos, Spherical Folding Tessellations by Kites and Isosceles Triangles: a case of adjacency, Mathematical Communications, 19 (2014), 1–28.; C. Avelino and A. Santos, Spherical Folding Tessellations by Kites and Isosceles Triangles II, International Journal of Pure and Applied Mathematics, 85 (2013), 45–67.; C.Avelino and A.Santos, Spherical Folding Tessellations by Kites and Isosceles Triangles III, submitted.]. In this paper we finalize this classification presenting all the dihedral folding tessellations of the sphere by kites and isosceles triangles in the remaining three cases of adjacency, that consists of five sporadic isolated tilings. A list containing these tilings including its combinatorial structure is presented at the end of this paper.

In this paper, a new one parameter family of iterative methods with eighth-order of convergence for solving nonlinear equations is presented and analyzed. This new family of iterative methods is obtained by composing an iterative method proposed by Chun (3) with Newton's method and approxi- mating the first-appeared derivative in the last step by a combination of already evaluated function values. The proposed family is optimal since its efficiency index is 8 1/4 ≈ 1.6818. The convergence analysis of the new family is studied in this paper. Several numerical examples are presented to illustrate the efficiency and accuracy of the family.

In this paper an enumeration technique for solving linear frac- tional fuzzy set covering problem is defined. Set covering problems belong to the class of 0-1 integer programming problems that are NP-complete. Many applications arises having the set covering problems, switching theory, testing of VLSI circuits and line balancing often take on a set covering structure. Linear fractional set covering problems involving coefficients in t objective function with some lack of precision are usual. To solve them several approaches have been proposed. In this paper a solution algorithm to fuzzy linear fractional set covering problem is suggested, in order to defuzzify the problem the concept of vector ranking function is presented further for obtaining efficient solution to the problem, a lexicographic approach is used. A linearization technique is used to obtain the optimal solution for crisp linear fractional set covering problem. An illustrative example is included to demonstrate the correctness of the proposed solution algorithm.

In this paper concept of e− chainability in topological spaces has been introduced using continuity. Several well known results prove perquisites to results established in the paper. In case of metric spaces the equivalence of e − chainability and function − e − chainability is also established in the paper. AMS Subject Classification: 54A99

Many mathematicians investigated in papers various types of integer matrices the entries of which satisfy a second order recurrence. Some of the authors used methods leading to obtain real or complex factorizations of the Fibonacci or the Lucas numbers. Civciv (2008) computed the determinant of a five-diagonal matrix with the Fibonacci numbers as its entries. His result is given more generally and completely in this paper. It is showed that the determinant of a matrix, the entries of which are the Gibonacci numbers, is related to the values of the Fibonacci polynomial. Calculations are done by using the eigenvalues of the given matrix.

This paper divides the set of natural numbers in six equivalence classes and determines two of them as candidate to include all prime numbers. Concerning the even numbers themselves, these were divided into three subsets using a basic cell (6n − 2, 6n and 6n + 2). Based on the aforementioned tools, this paper proposes a deterministic process of finding all pairs (p,q) of odd numbers (composites and primes) of natural numbers ≥ 3 whose sum (p+q) is equal to a given even natural number 2n ≥6. Based on this procedure and also relying on the distribution of primes in the set N of natural numbers, a closed analytical formula is proposed for the estimation of the number of primes that satisfy Goldbach's conjecture for positive integers ≥6.

*Information theory is concerned with the analysis of Communication System, the purpose of which is the transmission of information from one point to another. The various parts of the communication system consists of a source which produces message to be communicated, the transmitter converts the message in to the signals for transmission through the channel, the channel is the medium over which the signals are transmitted, the receiver extract the original message from the output of the channel for delivering to the destination. The theory is an attempt to construct a mathematical model for each of the parts of the communication system. C.E. Shannon an electrical engineer published a Paper in 1948 ’A Mathematical Theory of Communication ’ which gave birth to this new Discipline. In this paper the concept of production and transmission of Information was defined quantitatively by Shannon. In this lecture the mathematical foundation of Information Measure or Shannon’s Entropy is discussed An application of this entropy in coding theory is also discussed showing that how much data can be compressed. Finally two procedures for encoding are described showing that how the code word lengths can be reduced.

In 1975, Rhoades (9) generalized the classes of, operators of lp type and operators of Cesaro type by introducing an arbitrary infinite matrix A = (ank) using approximation numbers of a bounded linear operator. In the same paper Rhoades has proved that for each fixed matrix A satisfying the condition |an,2k−1| + |an,2k| ≤ M|ank| on the matrix A = (ank) for each n,k = 1,2,··· and each p,0 < p ≤ ∞ the set of A−p type operators is a linear space and raised an open question whether this condition is necessary to be a linear space? In this paper we have answered the question in negation. Further, we have introduced and studied the class A (s) −p of s-type cesp operators using s-number sequence and Cesaro sequence spaces. We have also shown that the class A (s) − p forms a quasi-Banach operator ideal. Moreover, the inclusion

As the fourth paper of our series of papers concerned with ax- iomatic differential geometry, this paper is devoted to the general Jacobi iden- tity supporting the Jacobi identity of vector fields. The general Jacobi identity can be regarded as one of the few fundamental results belonging properly to smootheology.

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.

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.