Exact solitary and periodic wave solutions for a generalized nonlinear Schrödinger equation

The generalized nonlinear Schrödinger equation (GNLS) iu_t + u_xx + βu²u + γu⁴u + iα (u²u)_x + iτ(u²)_xu = 0 is studied. Using the bifurcation of travelling waves of this equation, some exact solitary wave solutions were obtained in [Wang W, Sun J,Chen G, Bifurcation, Exact solutions and nonsmooth behavior of solitary waves in the generalized nonlinear Schrödinger equation. Int J Bifucat Chaos 2005:3295–305.]. In this paper, more explicit exact solitary wave solutions and some new smooth periodic wave solutions are obtained.     

Inscribing an axially symmetric polygon and other approximation algorithms for planar convex sets

Given a planar convex set C, we give sublinear approximation algorithms to determine approximations of the largest axially symmetric convex set S contained in C, and the smallest such set S^′ that contains C. More precisely, for any >0, we find an axially symmetric convex polygon QC with area |Q|>(1−)|S| and we find an axially symmetric convex polygon Q^′ containing C with area |Q^′|<(1+)|S^′|. We assume that C is given in a data structure that allows to answer the following two types of query in time T_C: given a direction u, find an extreme point of C in direction u, and given a line ℓ, find C∩ℓ. For instance, if C is a convex n-gon and its vertices are given in a sorted array, then T_C=O(logn). Then we can find Q and Q^′ in time O(^−1/2T_C+^−3/2). Using these techniques, we can also find approximations to the perimeter, area, diameter, width, smallest enclosing rectangle and smallest enclosing circle of C in time O(^−1/2T_C).     

A Schrödinger singular perturbation problem

Consider the equation −ε²Δu_ε + q(x)u_ε = f(u_ε) in , u(∞) < ∞, ε = const > 0. Under what assumptions on q(x) and f(u) can one prove that the solution u_ε exists and lim_ε→0u_ε = u(x), where u(x) solves the limiting problem q(x)u = f(u)? These are the questions discussed in the paper.     

On the Fibonacci origin of the internal symmetries of super strings and 5-Brane in 11 dimensions

El Naschie recently showed that the exceptional Lie symmetry group E12 together with the compactified Klein modular curve SL(2,7)_c gives E12 +  SL(2,7)_c  = 685 + 339 = 1024. (See CS& F (2008) doi: 10.1016/j.chaos.2008.08.005). The same result is found for Dim E8E8 = 496 when added to the number of states of the 5-Branes in 11-dimensions model, namely 528. The present work gives the Fibonacci explanation for all these remarkable results. We conclude that the Fibonacci growth law is not only fundamental in biology and econometrics but also in high energy physics as exemplified by El Naschie’s fractal-Cantorian spacetime theory.     

On the Common Substring Alignment Problem

The Common Substring Alignment Problem is defined as follows: Given a set of one or more strings S₁, S₂ … S_c and a target string T, Y is a common substring of all strings S_i, that is, S_i = B_iYF_i. The goal is to compute the similarity of all strings S_i with T, without computing the part of Y again and again. Using the classical dynamic programming tables, each appearance of Y in a source string would require the computation of all the values in a dynamic programming table of size O(nℓ) where ℓ is the size of Y. Here we describe an algorithm which is composed of an encoding stage and an alignment stage. During the first stage, a data structure is constructed which encodes the comparison of Y with T. Then, during the alignment stage, for each comparison of a source S_i with T, the pre-compiled data structure is used to speed up the part of Y. We show how to reduce the O(nℓ) alignment work, for each appearance of the common substring Y in a source string, to O(n)-at the cost of O(nℓ) encoding work, which is executed only once.     

Sparse Dynamic Programming for Longest Common Subsequence from Fragments

Sparse Dynamic Programming has emerged as an essential tool for the design of efficient algorithms for optimization problems coming from such diverse areas as computer science, computational biology, and speech recognition. We provide a new sparse dynamic programming technique that extends the Hunt–Szymanski paradigm for the computation of the longest common subsequence (LCS) and apply it to solve the LCS from Fragments problem: given a pair of strings X and Y (of length n and m, respectively) and a set M of matching substrings of X and Y, find the longest common subsequence based only on the symbol correspondences induced by the substrings. This problem arises in an application to analysis of software systems. Our algorithm solves the problem in O(|M| log |M|) time using balanced trees, or O(|M| log log min(|M|, nm/|M|)) time using Johnson's version of Flat Trees. These bounds apply for two cost measures. The algorithm can also be adapted to finding the usual LCS in O((m + n) log |Σ| + |M| log |M|) time using balanced trees or O((m + n) log |Σ| + |M| log log min (|M|, nm/|M|)) time using Johnson's version of Flat Trees, where M is the set of maximal matches between substrings of X and Y and Σ is the alphabet. These bounds improve on those of the original Hunt–Szymanski algorithm while retaining the overall approach.     

Generating orthogonal matrix polynomials satisfying second order differential equations from a trio of triangular matrices

The method developed in [A.J. Durán, F.A. Grünbaum, Orthogonal matrix polynomials satisfying second order differential equations, Int. Math. Res. Not. 10 (2004) 461–484] led us to consider matrix polynomials that are orthogonal with respect to weight matrices W(t) of the form , , and (1−t)^α(1+t)^βT(t)T^*(t), with T satisfying T^′=(2Bt+A)T, T(0)=I, T^′=(A+B/t)T, T(1)=I, and T^′(t)=(−A/(1−t)+B/(1+t))T, T(0)=I, respectively. Here A and B are in general two non-commuting matrices. We are interested in sequences of orthogonal polynomials (P_n)_n which also satisfy a second order differential equation with differential coefficients that are matrix polynomials F₂, F₁ and F₀ (independent of n) of degrees not bigger than 2, 1 and 0 respectively. To proceed further and find situations where these second order differential equations hold, we only dealt with the case when one of the matrices A or B vanishes.The purpose of this paper is to show a method which allows us to deal with the case when A, B and F₀ are simultaneously triangularizable (but without making any commutativity assumption).     

Enveloppes convexes des processus gaussiens

Let X={X(t), t[0,1]} be a process on [0,1] and V_X=Conv{(t,x)t[0,1], x=X(t)} be the convex hull of its path.The structure of the set ext(V_X) of extreme points of V_X is studied. For a Gaussian process X with stationary increments it is proved that:

• The set ext(V_X) is negligible if X is non-differentiable.

• If X is absolutely continuous process and its derivative X′ is continuous but non-differentiable, then ext(V_X) is also negligible and moreover it is a Cantor set.

It is proved also that these properties are stable under the transformations of the type Y(t)=f(X(t)), if f is a sufficiently smooth function.     

The rainbow number of matchings in regular bipartite graphs

Given a graph G and a subgraph H of G, let rb(G,H) be the minimum number r for which any edge-coloring of G with r colors has a rainbow subgraph H. The number rb(G,H) is called the rainbow number of H with respect to G. Denote as mK₂ a matching of size m and as B_n,k the set of all the k-regular bipartite graphs with bipartition (X,Y) such that X=Y=n and k≤n. Let k,m,n be given positive integers, where k≥3, m≥2 and n>3(m−1). We show that for every GB_n,k, rb(G,mK₂)=k(m−2)+2. We also determine the rainbow numbers of matchings in paths and cycles.     

Splitting Off Edges within a Specified Subset Preserving the Edge-Connectivity of the Graph

Splitting off a pair su, sv of edges in a graph G means the operation that deletes su and sv and adds a new edge uv. Given a graph G = (V + s, E) which is k-edge-connected (k ≥ 2) between vertices of V and a specified subset R  V, first we consider the problem of finding a longest possible sequence of disjoint pairs of edges sx, sy, (x ,y  R) which can be split off preserving k-edge-connectivity in V. If R = V and d(s) is even then a well-known theorem of Lovász asserts that a complete R-splitting exists, that is, all the edges connecting s to R can be split off in pairs. This is not the case in general. We characterize the graphs possessing a complete R-splitting and give a formula for the length of a longest R-splitting sequence. Motivated by the connection between splitting off results and connectivity augmentation problems we also investigate the following problem that we call the split completion problem: given G and R as above, find a smallest set F of new edges incident to s such that G′ = (V + s, E + F) has a complete R-splitting. We give a min-max formula for F as well as a polynomial algorithm to find a smallest F. As a corollary we show a polynomial algorithm which finds a solution of size at most k/2 + 1 more than the optimum for the following augmentation problem, raised in [[2]]: given a graph H = (V, E), an integer k ≥ 2, and a set R  V, find a smallest set F′ of new edges for which H′ = (V, E + F′) is k-edge-connected and no edge of F′ crosses R.     

On Equivalence of Moduli of Smoothness

It is known that iffW^k_p, thenω_m(f, t)_ptω_m−1(f′, t)_p…. Its inverse with any constants independent offis not true in general. Hu and Yu proved that the inverse holds true for splinesSwith equally spaced knots, thusω_m(S, t)_ptω_m−1(S′, t)_pt²ω_m−2(S″, t)_p…. In this paper, we extend their results to splines with any given knot sequence, and further to principal shift-invariant spaces and wavelets under certain conditions. Applications are given at the end of the paper.     

On Polynomials over Prime Fields Taking Only Two Values on the Multiplicative Group

Let p>2 be a prime, denote by F_p the field with F_p=p, and let F^*_p=F_p\{0}. We prove that if fεF_p[x] and f takes only two values on F^*_p, then (excluding some exceptional cases) the degree of f is at least (p−1).     

On the chromatic number of random graphs

In this paper we consider the classical Erdős–Rényi model of random graphs G_n,p. We show that for p=p(n)n^−3/4−δ, for any fixed δ>0, the chromatic number χ(G_n,p) is a.a.s. ℓ, ℓ+1, or ℓ+2, where ℓ is the maximum integer satisfying 2(ℓ−1)log(ℓ−1)p(n−1).     

Mass Points of Measures and Orthogonal Polynomials on the Unit Circle

Orthogonal polynomials on the unit circle are completely determined by their reflection coefficients through the Szeg recurrences. We assume that the reflection coefficients tend to some complex number a with 0<a<1. The orthogonality measure μ then lives essentially on the arc {e^it :αt2π−α} where sin with α(0,π). Under the certain rate of convergence it was proved in (Golinskii et al. (J. Approx. Theory96 (1999), 1–32)) that μ has no mass points inside this arc. We show that this result is sharp in a sense. We also examine the case of the whole unit circle and some examples of singular continuous measures given by their reflection coefficients.     

Faster algorithms for string matching with k mismatches

The string matching with mismatches problem is that of finding the number of mismatches between a pattern P of length m and every length m substring of the text T. Currently, the fastest algorithms for this problem are the following. The Galil–Giancarlo algorithm finds all locations where the pattern has at most k errors (where k is part of the input) in time O(nk). The Abrahamson algorithm finds the number of mismatches at every location in time . We present an algorithm that is faster than both. Our algorithm finds all locations where the pattern has at most k errors in time . We also show an algorithm that solves the above problem in time O((n+(nk³)/m)logk).     

Algebras Generated by Reciprocals of Linear Forms

Let Δ be a finite set of nonzero linear forms in several variables with coefficients in a field K of characteristic zero. Consider the K-algebra C(Δ) of rational functions generated by {1/α  α  Δ}. Then the ring ∂(V) of differential operators with constant coefficients naturally acts on C(Δ). We study the graded ∂(V)-module structure of C(Δ). We especially find standard systems of minimal generators and a combinatorial formula for the Poincaré series of C(Δ). Our proofs are based on a theorem by Brion–Vergne [4] and results by Orlik–Terao [9].     

Character sets of strings

Given a string S over a finite alphabet Σ, the character set (also called the fingerprint) of a substring S^′ of S is the subset CΣ of the symbols occurring in S^′. The study of the character sets of all the substrings of a given string (or a given collection of strings) appears in several domains such as rule induction for natural language processing or comparative genomics. Several computational problems concerning the character sets of a string arise from these applications, especially:

(1) Output all the maximal locations of substrings having a given character set.

(2) Output for each character set C occurring in a given string (or a given collection of strings) all the maximal locations of C.

Denoting by n the total length of the considered string or collection of strings, we solve the first problem in Θ(n) time using Θ(n) space. We present two algorithms solving the second problem. The first one runs in Θ(n²) time using Θ(n) space. The second algorithm has Θ(n|Σ|log|Σ|) time and Θ(n) space complexity and is an adaptation of an algorithm by Amir et al. [A. Amir, A. Apostolico, G.M. Landau, G. Satta, Efficient text fingerprinting via Parikh mapping, J. Discrete Algorithms 26 (2003) 1–13].     

Sparse hypergraphs and pebble game algorithms

A hypergraph G=(V,E) is (k,ℓ)-sparse if no subset V^′V spans more than k|V^′|−ℓ hyperedges. We characterize (k,ℓ)-sparse hypergraphs in terms of graph theoretic, matroidal and algorithmic properties. We extend several well-known theorems of Haas, Lovász, Nash-Williams, Tutte, and White and Whiteley, linking arboricity of graphs to certain counts on the number of edges. We also address the problem of finding lower-dimensional representations of sparse hypergraphs, and identify a critical behavior in terms of the sparsity parameters k and ℓ. Our constructions extend the pebble games of Lee and Streinu [A. Lee, I. Streinu, Pebble game algorithms and sparse graphs, Discrete Math. 308 (8) (2008) 1425–1437] from graphs to hypergraphs.     

A unified and generalized set of shrinkage bounds on minimax Stein estimates

Consider the problem of estimating the mean vector θ of a random variable X in , with a spherically symmetric density f(x−θ²), under loss δ−θ². We give an increasing sequence of bounds on the shrinkage constant of Stein-type estimators depending on properties of f(t) that unify and extend several classical bounds from the literature. The basic way to view the conditions on f(t) is that the distribution of X arises as the projection of a spherically symmetric vector (X,U) in . A second way is that f(t) satisfies (−1)^jf^(j)(t)≥0 for 0≤j≤ℓ and that (−1)^ℓf^(ℓ)(t) is non-increasing where k=2(ℓ+1). The case ℓ=0 (k=2) corresponds to unimodality, while the case ℓ=k=∞ corresponds to complete monotonicity of f(t) (or equivalently that f(x−θ²) is a scale mixture of normals). The bounds on the minimax shrinkage constant in this paper agree with the classical bounds in the literature for the case of spherical symmetry, spherical symmetry and unimodality, and scale mixtures of normals. However, they extend these bounds to an increasing sequence (in k or ℓ) of minimax bounds.