Self-similar rogue waves in an inhomogeneous generalized nonlinear Schrödinger equation

We present an explicit analytical form of first and second order rogue waves for distributive nonlinear Schrödinger equation (NLSE) by mapping it to standard NLSE through similarity transformation. Upon obtaining the rogue wave solutions, we study the propagation of rogue waves through a periodically distributed system for the two cases when Wronskian of dispersion and nonlinearity is (i) zero, (ii) not equal to zero. For the former case, we discuss a mechanism to control their propagation and for the latter case we depict the interesting features of rogue waves as they propagate through dispersion increasing and decreasing fiber.     

Characterization of dynamical systems under noise using recurrence networks: Application to simulated and EEG data

In this letter, we study the influence of observational noise on recurrence network (RN) measures, the global clustering coefficient (C) and average path length (L  ) using the Rössler system and propose the application of RN measures to analyze the structural properties of electroencephalographic (EEG) data. We find that for an appropriate recurrence rate (RR>0.02$RR>0.02$) the influence of noise on C can be minimized while L is independent of RR for increasing levels of noise. Indications of structural complexity were found for healthy EEG, but to a lesser extent than epileptic EEG. Furthermore, C performed better than L in case of epileptic EEG. Our results show that RN measures can provide insights into the structural properties of EEG in normal and pathological states.     

Shortened recurrence relations for Bernoulli numbers

Starting with two little-known results of Saalschütz, we derive a number of general recurrence relations for Bernoulli numbers. These relations involve an arbitrarily small number of terms and have Stirling numbers of both kinds as coefficients. As special cases we obtain explicit formulas for Bernoulli numbers, as well as several known identities.     

Matrix differential equations and scalar polynomials satisfying higher order recursions

We show that any scalar differential operator with a family of polynomials as its common eigenfunctions leads canonically to a matrix differential operator with the same property. The construction of the corresponding family of matrix valued polynomials has been studied in [A. Durán, A generalization of Favard's theorem for polynomials satisfying a recurrence relation, J. Approx. Theory 74 (1993) 83-109; A. Durán, On orthogonal polynomials with respect to a positive definite matrix of measures, Canad. J. Math. 47 (1995) 88-112; A. Durán, W. van Assche, Orthogonal matrix polynomials and higher order recurrence relations, Linear Algebra Appl. 219 (1995) 261-280] but the existence of a differential operator having them as common eigenfunctions had not been considered. This correspondence goes only one way and most matrix valued situations do not arise in this fashion. We illustrate this general construction with a few examples. In the case of some families of scalar valued polynomials introduced in [F.A. Grünbaum, L. Haine, Bispectral Darboux transformations: An extension of the Krall polynomials, Int. Math. Res. Not. 8 (1997) 359-392] we take a first look at the algebra of all matrix differential operators that share these common eigenfunctions and uncover a number of phenomena that are new to the matrix valued case.     

Recurrence and strong stochastic persistence of a stochastic single-species model

A stochastic generalized logistic model is considered in this paper. The condition of the existence of its stationary distribution is generalized. Recurrence and strong stochastic persistence are obtained. Finally some numerical simulations are carried out to support our results.     

On the relation between the full Kostant-Toda lattice and multiple orthogonal polynomials

The correspondence between a high-order non-symmetric difference operator with complex coefficients and the evolution of an operator defined by a Lax pair is established. The solution of the discrete dynamical system is studied, giving explicit expressions for the resolvent function and, under some conditions, the representation of the vector of functionals, associated with the solution for our integrable systems. The method of investigation is based on the evolutions of the matrical moments.     

Recurrences of strange attractors

The transitions from or to strange nonchaotic attractors are investigated by recurrence plot-based methods. The techniques used here take into account the recurrence times and the fact that trajectories on strange nonchaotic attractors (SNAs) synchronize. The performance of these techniques is shown for the Heagy-Hammel transition to SNAs and for the fractalization transition to SNAs for which other usual nonlinear analysis tools are not successful.      

A modification of Cauchy's method for quadratic equations

The plan of this paper is to obtain one-point iterative methods with any R-order of convergence, when they are applied to approximate solutions of quadratic equations in Banach spaces. To do this, we consider real Cauchy's method and, under certain natural modifications, it is extended to Banach spaces. Some applications are also provided.     

Counting labeled claw-free cubic graphs by connectivity

Claw-free cubic graphs are counted with given connectedness and order. Tables are provided for claw-free cubic graphs with given connectedness. This builds on methods for counting general cubic graphs by connectivity previously developed by Chae, Palmer, and Robinson, and on the earlier enumeration of all claw-free cubic graphs by McKay, Palmer, Read, and Robinson.     

Applications of the complexity space to the General Probabilistic Divide and Conquer Algorithms

Schellekens [M. Schellekens, The Smyth completion: A common foundation for denotational semantics and complexity analysis, in: Proc. MFPS 11, in: Electron. Notes Theor. Comput. Sci., vol. 1, 1995, pp. 535-556], and Romaguera and Schellekens [S. Romaguera, M. Schellekens, Quasi-metric properties of complexity spaces, Topology Appl. 98 (1999) 311-322] introduced a topological foundation to obtain complexity results through the application of Semantic techniques to Divide and Conquer Algorithms. This involved the fact that the complexity (quasi-metric) space is Smyth complete and the use of a version of the Banach fixed point theorem and improver functionals. To further bridge the gap between Semantics and Complexity, we show here that these techniques of analysis, based on the theory of complexity spaces, extend to General Probabilistic Divide and Conquer schema discussed by Flajolet [P. Flajolet, Analytic analysis of algorithms, in: W. Kuich (Ed.), 19th Internat. Colloq. ICALP'92, Vienna, July 1992; Automata, Languages and Programming, in: Lecture Notes in Comput. Sci., vol. 623, 1992, pp. 186-210]. In particular, we obtain a general method which is useful to show that for several recurrence equations based on the recursive structure of General Probabilistic Divide and Conquer Algorithms, the associated functionals have a unique fixed point which is the solution for the corresponding recurrence equation.