Improving the high order nonlinearity lower bound for Boolean functions with given algebraic immunity

Algebraic immunity is a recently introduced cryptographic parameter for Boolean functions used in stream ciphers. If pAI(f) and pAI(f⊕1) are the minimum degree of all annihilators of f and f⊕1 respectively, the algebraic immunity AI(f) is defined as the minimum of the two values. Several relations between the new parameter and old ones, like the degree, the r-th order nonlinearity and the weight of the Boolean function, have been proposed over the last few years.In this paper, we improve the existing lower bounds of the r-th order nonlinearity of a Boolean function f with given algebraic immunity. More precisely, we introduce the notion of complementary algebraic immunity defined as the maximum of pAI(f) and pAI(f⊕1). The value of can be computed as part of the calculation of AI(f), with no extra computational cost. We show that by taking advantage of all the available information from the computation of AI(f), that is both AI(f) and , the bound is tighter than all known lower bounds, where only the algebraic immunity AI(f) is used.     

Global existence for the one-dimensional second order semilinear hyperbolic equations

The aim of the paper is to study necessary and sufficient conditions for the existence of the global solution of the one-dimensional semilinear equation appearing in the boundary value problems of gas dynamics. We investigate the Cauchy problem for such equation in the domain where the operator is weakly hyperbolic. We obtain the necessary condition for the existence of the self-similar solutions for the semilinear Gellerstedt-type equation. The approach used in the paper is based on the fundamental solution of the linear Gellerstedt operator and the Lp-Lq estimates.     

Boundary value problem for abstract first order differential equation with integral condition

The present paper is devoted to the study of a boundary value problem for abstract first order linear differential equation with integral boundary conditions. We obtain necessary and sufficient conditions for the unique solvability and well-posedness. We also study the Fredholm solvability. Finally, we obtain a result of the stability of solution with respect to small perturbation.     

Global bifurcation for a class of degenerate elliptic equations with variable exponents

We are concerned with the following nonlinear problem

     

Infinitely many homoclinic orbits for the second order Hamiltonian systems with general potentials

In this paper we study the following nonperiodic second order Hamiltonian systems

     

The Independence Number for De Bruijn networks and Kautz networks

Let D be a directed graph; the (l,ω)-Independence Number of graph D, denoted by α_l,ω(D), is an important performance parameter for interconnection networks. De Bruijn networks and Kautz networks, denoted by B(d,n) and K(d,n) respectively, are versatile and efficient topological structures of interconnection networks. For l=1,2,…,n, this paper shows that α_l,d−1(B(d,n))=dⁿ,α_l,d−1(K(d,n))=α_l,d(K(d,n))=dⁿ+dⁿ⁻¹ if d≥3 and n≤d−2. In particular, the paper shows the exact value of the Independence Number for B(d,1) and B(d,2) for any d. For the generalized situation, the paper obtains a lower bound α_l,d−1(B(d,n))≥d² if n≥3 and d≥5.     

On the complexity of the multicut problem in bounded tree-width graphs and digraphs

Given an edge- or vertex-weighted graph or digraph and a list of source-sink pairs, the minimum multicut problem consists in selecting a minimum weight set of edges or vertices whose removal leaves no path from each source to the corresponding sink. This is a classical NP-hard problem, and we show that the edge version becomes tractable in bounded tree-width graphs if the number of source-sink pairs is fixed, but remains NP-hard in directed acyclic graphs and APX-hard in bounded tree-width and bounded degree unweighted digraphs. The vertex version, although tractable in trees, is proved to be NP-hard in unweighted cacti of bounded degree and bounded path-width.     

The first and second Zagreb indices of some graph operations

In this paper some exact expressions for the first and second Zagreb indices of graph operations containing the Cartesian product, composition, join, disjunction and symmetric difference of graphs will be presented. We apply some of our results to compute the Zagreb indices of arbitrary C₄ tube, C₄ torus and q-multi-walled polyhex nanotorus.     

On island sequences of labelings with a condition at distance two

An L(2,1)-labeling of a graph G is a function f from the vertex set of G to the set of nonnegative integers such that |f(x)−f(y)|≥2 if d(x,y)=1, and |f(x)−f(y)|≥1 if d(x,y)=2, where d(x,y) denotes the distance between the pair of vertices x,y. The lambda number of G, denoted λ(G), is the minimum range of labels used over all L(2,1)-labelings of G. An L(2,1)-labeling of G which achieves the range λ(G) is referred to as a λ-labeling. A hole of an L(2,1)-labeling is an unused integer within the range of integers used. The hole index of G, denoted ρ(G), is the minimum number of holes taken over all its λ-labelings. An island of a given λ-labeling of G with ρ(G) holes is a maximal set of consecutive integers used by the labeling. Georges and Mauro [J.P. Georges, D.W. Mauro, On the structure of graphs with non-surjective L(2,1)-labelings, SIAM J. Discrete Math. 19 (2005) 208-223] inquired about the existence of a connected graph G with ρ(G)≥1 possessing two λ-labelings with different ordered sequences of island cardinalities. This paper provides an infinite family of such graphs together with their lambda numbers and hole indices. Key to our discussion is the determination of the path covering number of certain 2-sparse graphs, that is, graphs containing no pair of adjacent vertices of degree greater than 2.     

Twisted first homology groups of the automorphism group of a free group

The automorphism group and outer automorphism group of a free group F_n of rank n act on the abelianized group H of F_n and the dual group H^* of H. The twisted first homology groups of and with coefficients in H and H^* are calculated.     

Complexity results on restricted instances of a paint shop problem for words

We study the following problem: an instance is a word with every letter occurring twice. A solution is a 2-coloring of its letters such that the two occurrences of every letter are colored with different colors. The goal is to minimize the number of color changes between adjacent letters.This is a special case of the paint shop problem for words, which was previously shown to be NP-complete. We show that this special case is also NP-complete and even APX-hard. Furthermore, derive lower bounds for this problem and discuss a transformation into matroid theory enabling us to solve some specific instances within polynomial time.     

The Dirichlet problem of higher order quasilinear elliptic equation

The paper is concerned with the Dirichlet problem of higher order quasilinear elliptic equation:

     

Solution techniques for the Large Set Covering Problem

Given a finite set E and a family F={E₁,…,E_m} of subsets of E such that F covers E, the famous unicost set covering problem (USCP) is to determine the smallest possible subset of F that also covers E. We study in this paper a variant, called the Large Set Covering Problem (LSCP), which differs from the USCP in that E and the subsets E_i are not given in extension because they are very large sets that are possibly infinite. We propose three exact algorithms for solving the LSCP. Two of them determine minimal covers, while the third one produces minimum covers. Heuristic versions of these algorithms are also proposed and analysed. We then give several procedures for the computation of a lower bound on the minimum size of a cover. We finally present algorithms for finding the largest possible subset of F that does not cover E. We also show that a particular case of the LSCP is to determine irreducible infeasible sets in inconsistent constraint satisfaction problems. All concepts presented in the paper are illustrated on the k-colouring problem which is formulated as a constraint satisfaction problem.     

An improved randomized approximation algorithm for maximum triangle packing

This paper deals with the maximum triangle packing problem. For this problem, Hassin and Rubinstein gave a randomized polynomial-time approximation algorithm that achieves an expected ratio of for any constant ?>0. By modifying their algorithm, we obtain a new randomized polynomial-time approximation algorithm for the problem which achieves an expected ratio of 0.5257(1−?) for any constant ?>0.     

Global attractor for a nonlinear viscoelastic model

In this paper we establish the existence of global attractors of a semiflow and a semigroup for a nonlinear 1-d viscoelasticity in two frameworks. We exploit the properties of the analytic semigroup to show the compactness for the semiflow and semigroup generated by the weak global solutions.     

On the packing chromatic number of some lattices

For a positive integer k, a k-packing in a graph G is a subset A of vertices such that the distance between any two distinct vertices from A is more than k. The packing chromatic number of G is the smallest integer m such that the vertex set of G can be partitioned as V₁,V₂,…,V_m where V_i is an i-packing for each i. It is proved that the planar triangular lattice T and the three-dimensional integer lattice Z³ do not have finite packing chromatic numbers.     

Dilation equations and Markov operators

Using the theory of Markov operators, we give a new proof of the known fact saying that for every positive integers N and k>1, and for every nonnegative reals c₀,…,cN satisfying the first sum rule the dilation equation

     

Well-posedness of a class of perturbed optimization problems in Banach spaces

Let X be a Banach space and Z a nonempty subset of X. Let J:Z→R be a lower semicontinuous function bounded from below and p?1. This paper is concerned with the perturbed optimization problem of finding z₀∈Z such that ‖x−z₀p^‖+J(z₀)=infz∈Z{‖x−zp^‖+J(z)}, which is denoted by minJ(x,Z). The notions of the J-strictly convex with respect to Z and of the Kadec with respect to Z are introduced and used in the present paper. It is proved that if X is a Kadec Banach space with respect to Z and Z is a closed relatively boundedly weakly compact subset, then the set of all x∈X for which every minimizing sequence of the problem minJ(x,Z) has a converging subsequence is a dense Gδ-subset of X?Z₀, where Z₀ is the set of all points z∈Z such that z is a solution of the problem minJ(z,Z). If additionally p>1 and X is J-strictly convex with respect to Z, then the set of all x∈X for which the problem minJ(x,Z) is well-posed is a dense Gδ-subset of X?Z₀.     

Finding monotone paths in edge-ordered graphs

An edge-ordering of a graph G=(V,E) is a one-to-one function f from E to a subset of the set of positive integers. A path P in G is called an f-ascent if f increases along the edge sequence of P. The heighth(f) of f is the maximum length of an f-ascent in G.In this paper we deal with computational problems concerning finding ascents in graphs. We prove that for a given edge-ordering f of a graph G the problem of determining the value of h(f) is NP-hard. In particular, the problem of deciding whether there is an f-ascent containing all the vertices of G is NP-complete. We also study several variants of this problem, discuss randomized and deterministic approaches and provide an algorithm for the finding of ascents of order at least k in graphs of order n in running time O(4^kn^O(1)).