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Abstract: In the first part of this paper the existence, uniqueness and almost sure polynomial stability of solutions for pantograph stochastic differential equations are considered, under nonlinear growth conditions. The results are obtained using the idea from (Mao and Rassias, 2005) [10], where stochastic differential equation with constant delay are considered. However, the presence of the unbounded delay in stochastic pantograph differential equations required certain modification of that idea. Moreover, the convergence in probability of the appropriate Euler–Maruyama solution is proved under the same nonlinear growth conditions. Adding the linear growth condition, we show that the almost sure polynomial stability of the Euler–Maruyama solution implies the almost sure polynomial stability of the exact solution. This part of the paper represents the extension of the idea from (Wu et al., 2010) [17]. The main novelty in this part of the paper is also related to the treatment of the unbounded delay in pantograph stochastic differential equations. [Copyright &y& Elsevier]

Abstract: In this paper we study the limit cycles of two families of differential systems in the plane. These systems are obtained by polynomial perturbations with arbitrary degree on the second component of the standard linear center. The classes under consideration are polynomial generalizations of certain canonical form of a Kukles system with an invariant ellipse, previously studied in the literature. We provide, in both cases, an accurate upper bound of the maximum number of limit cycles that the perturbed system can have bifurcating from the periodic orbits of the linear center, using the averaging theory of first, second and third order. These upper bounds are presented in terms of the degree of the respective systems. Moreover, the existence of a weak focus with the highest order is also studied. [Copyright &y& Elsevier]

Abstract: In Suzuki (1998) [7] Suzuki gave a classification of association schemes with multiple Q-polynomial structures, allowing for one exceptional case which has five classes. In this paper, we rule out the existence of this case. Hence Suzukiʼs theorem mirrors exactly the well-known counterpart for association schemes with multiple P-polynomial structures, a result due to Eiichi Bannai and Etsuko Bannai in 1980. [Copyright &y& Elsevier]

Abstract: Let be positive integers. A -edge-colored graph is -e.c. or -existentially closed if for any disjoint sets of vertices with , there is a vertex not in such that all edges from this vertex to the set are colored by the -th color. In this paper, we give an explicit construction of a -e.c. graph of polynomial order. [Copyright &y& Elsevier]

Abstract: Let be the maximum possible minimum Hamming distance of a linear [] code over . Tables of best known linear codes exist for small fields. In this paper, linear codes over are constructed for k up to 6. The codes constructed are from the class of quasi-cyclic codes. The number of circulant matrices over is enumerated. In addition, the minimum distance of the extended quadratic residue code of length 44 is determined. [Copyright &y& Elsevier]

Abstract: Interval Markov Chains (IMC), or Markov Chains with probability intervals in the transition matrix, are the base of a classic specification theory for probabilistic systems . The standard semantics of IMCs assigns to a specification the set of all Markov Chains that satisfy its interval constraints. The theory then provides operators for deciding emptiness of conjunction and refinement (entailment) for such specifications. In this paper, we study complexity of several problems for IMCs, that stem from compositional modeling methodologies. In particular, we close the complexity gap for thorough refinement of two IMCs and for deciding the existence of a common implementation for an unbounded number of IMCs, showing that these problems are EXPTIME-complete. We discuss suitable notions of determinism for specifications, and show that for deterministic IMCs the syntactic refinement operators are complete with respect to model inclusion. Finally, we show that deciding consistency (emptiness) for an IMC is polynomial and that existence of common implementation can be established in polynomial time for any constant number of IMCs. [Copyright &y& Elsevier]