Complexity of clique‐coloring odd‐hole‐free graphs

In this paper we investigate the problem of clique‐coloring, which consists in coloring the vertices of a graph in such a way that no monochromatic maximal clique appears, and we focus on odd‐hole‐free graphs. On the one hand we do not know any odd‐hole‐free graph that is not 3‐clique‐colorable, but on the other hand it is NP‐hard to decide if they are 2‐clique‐colorable, and we do not know if there exists any bound k₀ such that they are all k₀ ‐clique‐colorable. First we will prove that (odd hole, codiamond)‐free graphs are 2‐clique‐colorable. Then we will demonstrate that the complexity of 2‐clique‐coloring odd‐hole‐free graphs is actually Σ₂ P‐complete. Finally we will study the complexity of deciding whether or not a graph and all its subgraphs are 2‐clique‐colorable. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 139–156, 2009     

T‐joins intersecting small edge‐cuts in graphs

In an earlier paper 3 , we studied cycles in graphs that intersect all edge‐cuts of prescribed sizes. Passing to a more general setting, we examine the existence of T‐joins in grafts that intersect all edge‐cuts whose size is in a given set A ?{1,2,3}. In particular, we characterize all the contraction‐minimal grafts admitting no T‐joins that intersect all edge‐cuts of size 1 and 2. We also show that every 3‐edge‐connected graft admits a T‐join intersecting all 3‐edge‐cuts. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 64–71, 2007     

Two‐dimensional adaptive Fourier decomposition

One‐dimensional adaptive Fourier decomposition, abbreviated as 1‐D AFD, or AFD, is an adaptive representation of a physically realizable signal into a linear combination of parameterized Szegö and higher‐order Szegö kernels of the context. In the present paper, we study multi‐dimensional AFDs based on multivariate complex Hardy spaces theory. We proceed with two approaches of which one uses Product‐TM Systems; and the other uses Product‐Szegö Dictionaries. With the Product‐TM Systems approach, we prove that at each selection of a pair of parameters, the maximal energy may be attained, and, accordingly, we prove the convergence. With the Product‐Szegö dictionary approach, we show that pure greedy algorithm is applicable. We next introduce a new type of greedy algorithm, called Pre‐orthogonal Greedy Algorithm (P‐OGA). We prove its convergence and convergence rate estimation, allowing a weak‐type version of P‐OGA as well. The convergence rate estimation of the proposed P‐OGA evidences its advantage over orthogonal greedy algorithm (OGA). In the last part, we analyze P‐OGA in depth and introduce the concept P‐OGA‐Induced Complete Dictionary, abbreviated as Complete Dictionary. We show that with the Complete Dictionary P‐OGA is applicable to the Hardy H² space on 2‐torus. Copyright © 2016 John Wiley & Sons, Ltd.     

Transcendental lattices of some K 3‐surfaces

In a previous paper, [12], we described six families of K 3‐surfaces (over ?) with Picard‐number 19, and we identified surfaces with Picard‐number 20. In these notes we classify some of the surfaces by computing their transcendental lattices. Moreover, we show that the surfaces with Picard‐number 19 are birational to a Kummer surface which is the quotient of a non‐product type abelian surface by an involution. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some limit analysis of a three dimensional viscous compressible capillary model for plasma

In this study, we discuss some limit analysis of a viscous capillary model of plasma, which is expressed as a so‐called the compressible Navier‐Stokes‐Poisson‐Korteweg equation. First, the existence of global smooth solutions for the initial value problem to the compressible Navier‐Stokes‐Poisson‐Korteweg equation with a given Debye length λ and a given capillary coefficient κ is obtained. We also show the uniform estimates of global smooth solutions with respect to the Debye length λ and the capillary coefficient κ. Then, from Aubin lemma, we show that the unique smooth solution of the 3‐dimensional Navier‐Stokes‐Poisson‐Korteweg equations converges globally in time to the strong solution of the corresponding limit equations, as λ tends to zero, κ tends to zero, and λ and κ simultaneously tend to zero. Moreover, we also give the convergence rates of these limits for any given positive time one by one.     

A state‐dependent Markov‐modulated mechanism for generating events and stochastic models

In this paper, we introduce a versatile block‐structured state‐dependent event (BSDE) approach that provides a methodological tool to construct non‐homogeneous Markov‐modulated stochastic models. Alternatively, the BSDE approach can be used to construct even a part (e.g. the arrival process) of the model. To illustrate the usefulness of the BSDE approach, several arrival patterns as well as queueing and epidemic models are considered. In particular, we deal with a state‐dependent quasi‐birth‐and‐death process that gives a constructive generalization of the scalar birth‐and‐death process and the homogeneous quasi‐birth‐and‐death process. Copyright © 2009 John Wiley & Sons, Ltd.     

On evolutionary equations with material laws containing fractional integrals

A well‐posedness result for a time‐shift invariant class of evolutionary operator equations involving material laws with fractional time‐integrals of order α ? ]0, 1[ is considered. The fractional derivatives are defined via a function calculus for the (time‐)derivative established as a normal operator in a suitable L² type space. Employing causality, we show that the fractional derivatives thus obtained coincide with the Riemann‐Liouville fractional derivative. We exemplify our results by applications to a fractional Fokker‐Planck equation, equations describing super‐diffusion and sub‐diffusion processes, and a Kelvin‐Voigt type model in fractional visco‐elasticity. Moreover, we elaborate a suitable perspective to deal with initial boundary value problems. Copyright © 2014 John Wiley & Sons, Ltd.     

A new kind of nonlocal symmetry for the μ‐Camassa‐Holm equation with linear dispersion

The μ‐Camassa‐Holm equation with linear dispersion is a completely integrable model. In this paper, it is shown that this equation has quadratic pseudo‐potentials that allow us to construct pseudo‐potential–type nonlocal symmetries. As an application, we obtain its recursion operator by using this kind of nonlocal symmetry, and we construct a Darboux transformation for the μ‐Camassa‐Holm equation.     

On the parallel complexity of hierarchical clustering and CC‐complete problems

Complex data sets are often unmanageable unless they can be subdivided and simplified in an intelligent manner. Clustering is a technique that is used in data mining and scientific analysis for partitioning a data set into groups of similar or nearby items. Hierarchical clustering is an important and well‐studied clustering method involving both top‐down and bottom‐up subdivisions of data. In this article we address the parallel complexity of hierarchical clustering. We describe known sequential algorithms for top‐down and bottom‐up hierarchical clustering. The top‐down algorithm can be parallelized, and when there are n points to be clustered, we provide an O(log n)‐time, n²‐processor Crew Pram algorithm that computes the same output as its corresponding sequential algorithm. We define a natural decision problem based on bottom‐up hierarchical clustering, and add this HIERARCHICAL CLUSTERING PROBLEM (HCP) to the slowly growing list of CC‐complete problems, thereby showing that HCP is one of the computationally most difficult problems in the COMPARATOR CIRCUIT VALUE PROBLEM class. This class contains a variety of interesting problems, and now for the first time a problem from data mining as well. By proving that HCP is CC‐complete, we have demonstrated that HCP is very unlikely to have an NC algorithm. This result is in sharp contrast to the NC algorithm which we give for the top‐down sequential approach, and the result surprisingly shows that the parallel complexities of the top‐down and bottom‐up approaches are different, unless CC equals NC. In addition, we provide a compendium of all known CC‐complete problems. © 2008 Wiley Periodicals, Inc. Complexity, 2008     

Generalized nonisospectral super integrable hierarchies

For the Lie superalgebra B(0,1), we choose a set of basis matrices. Then we consider a linear combination of the basis matrices, which is exactly the spectral matrix of the spatial part for the super Ablowitz‐Kaup‐Newell‐Segur (AKNS) hierarchy. The compatible condition of the spatial and temporal spectral problems leads to the well‐known zero curvature equation. Here, when the spectral parameter is independent (dependent) of temporal parameter, we obtain isospectral (nonisospectral) super AKNS hierarchy. Furthermore, we derive the generalized nonisospectral super AKNS hierarchy (GNI‐SAKNS). As another example, similar method is successfully applied to the super Dirac hierarchy, and we obtain the generalized nonisospectral super Dirac hierarchy (GNI‐SD).     

Projective‐Planar Graphs with no K3, 4‐Minor. II

The authors previously published an iterative process to generate a class of projective‐planar K_{3, 4}‐free graphs called “patch graphs.” They also showed that any simple, almost 4‐connected, nonplanar, and projective‐planar graph that is K_{3, 4}‐free is a subgraph of a patch graph. In this article, we describe a simpler and more natural class of cubic K_{3, 4}‐free projective‐planar graphs that we call Möbius hyperladders. Furthermore, every simple, almost 4‐connected, nonplanar, and projective‐planar graph that is K_{3, 4}‐free is a minor of a Möbius hyperladder. As applications of these structures we determine the page number of patch graphs and of Möbius hyperladders.     

Finding a subdivision of a prescribed digraph of order 4

The problem of when a given digraph contains a subdivision of a fixed digraph F is considered. Bang‐Jensen et al. [4] laid out foundations for approaching this problem from the algorithmic point of view. In this article, we give further support to several open conjectures and speculations about algorithmic complexity of finding F‐subdivisions. In particular, up to five exceptions, we completely classify for which 4‐vertex digraphs F, the F‐subdivision problem is polynomial‐time solvable and for which it is NP‐complete. While all NP‐hardness proofs are made by reduction from some version of the 2‐linkage problem in digraphs, some of the polynomial‐time solvable cases involve relatively complicated algorithms.     

On the category of hyper MV‐algebras

In this paper we study the category of hyper MV‐algebras and we prove that it has a terminal object and a coequalizer. We show that Jia's construction can be modified to provide a free hyper MV‐algebra by a set. We use this to show that in the category of hyper MV‐algebras the monomorphisms are exactly the one‐to‐one homomorphisms. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Clique‐coloring some classes of odd‐hole‐free graphs

We consider the problem of clique‐coloring, that is coloring the vertices of a given graph such that no maximal clique of size at least 2 is monocolored. Whereas we do not know any odd‐hole‐free graph that is not 3‐clique‐colorable, the existence of a constant C such that any perfect graph is C‐clique‐colorable is an open problem. In this paper we solve this problem for some subclasses of odd‐hole‐free graphs: those that are diamond‐free and those that are bull‐free. We also prove the NP‐completeness of 2‐clique‐coloring K₄‐free perfect graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 233–249, 2006     

Line graphs of multigraphs and Hamilton‐connectedness of claw‐free graphs

We introduce a closure concept that turns a claw‐free graph into the line graph of a multigraph while preserving its (non‐)Hamilton‐connectedness. As an application, we show that every 7‐connected claw‐free graph is Hamilton‐connected, and we show that the well‐known conjecture by Matthews and Sumner (every 4‐connected claw‐free graph is hamiltonian) is equivalent with the statement that every 4‐connected claw‐free graph is Hamilton‐connected. Finally, we show a natural way to avoid the non‐uniqueness of a preimage of a line graph of a multigraph, and we prove that the closure operation is, in a sense, best possible. © 2010 Wiley Periodicals, Inc. J Graph Theory 66:152‐173, 2011     

Regular path decompositions of odd regular graphs

Kotzig asked in 1979 what are necessary and sufficient conditions for a d‐regular simple graph to admit a decomposition into paths of length d for odd d>3. For cubic graphs, the existence of a 1‐factor is both necessary and sufficient. Even more, each 1‐factor is extendable to a decomposition of the graph into paths of length 3 where the middle edges of the paths coincide with the 1‐factor. We conjecture that existence of a 1‐factor is indeed a sufficient condition for Kotzig's problem. For general odd regular graphs, most 1‐factors appear to be extendable and we show that for the family of simple 5‐regular graphs with no cycles of length 4, all 1‐factors are extendable. However, for d>3 we found infinite families of d‐regular simple graphs with non‐extendable 1‐factors. Few authors have studied the decompositions of general regular graphs. We present examples and open problems; in particular, we conjecture that in planar 5‐regular graphs all 1‐factors are extendable. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 114–128, 2010     

Global weak solutions for a periodic two‐component μ‐Camassa–Holm system

In this paper, we consider the global existence of weak solutions for a two‐component μ‐Camassa–Holm system in the periodic setting. Global existence for strong solutions to the system with smooth approximate initial value is derived. Then, we show that the limit of approximate solutions is a global‐in‐time weak solution of the two‐component μ‐Camassa–Holm system. Copyright © 2013 John Wiley & Sons, Ltd.     

Robust residual‐ and recovery‐based a posteriori error estimators for a multipoint flux mixed finite element discretization of interface problems

We consider a posteriori error estimation for a multipoint flux mixed finite element method for two‐dimensional elliptic interface problems. Within the class of modified quasi‐monotonically distributed coefficients, we derive a residual‐type a posteriori error estimator of the weighted sum of the scalar and flux errors which is robust with respect to the jumps of the coefficients. Moreover, we develop robust implicit and explicit recovery‐type estimators through gradient recovery in an H(curl)‐conforming finite element space. In particular, we apply a modified L² projection in the implicit recovery procedure so as to reduce the computational cost of the recovered gradient. Numerical experiments confirm the theoretical results.     

On (∈, ∈ ∨ q)‐fuzzy filters of R0‐algebras

In this paper, we introduce the notions of (∈, ∈ ∨ q)‐fuzzy filters and (∈, ∈ ∨ q)‐fuzzy Boolean (implicative) filters in R₀‐algebras and investigate some of their related properties. Some characterization theorems of these generalized fuzzy filters are derived. In particular, we prove that a fuzzy set in R₀‐algebras is an (∈, ∈ ∨ q)‐fuzzy Boolean filter if and only if it is an (∈, ∈ ∨ q)‐fuzzy implicative filter. Finally, we consider the concepts of implication‐based fuzzy Boolean (implicative) filters of R₀‐algebras (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

First‐passage percolation on a ladder graph,and the path cost in a VCG auction

This paper studies the time constant for first‐passage percolation, and the Vickrey‐Clarke‐Groves (VCG) payment, for the shortest path on a ladder graph (a width‐2 strip) with random edge costs, treating both in a unified way based on recursive distributional equations. For first‐passage percolation where the edge costs are independent Bernoulli random variables we find the time constant exactly; it is a rational function of the Bernoulli parameter. For first‐passage percolation where the edge costs are uniform random variables we present a reasonably efficient means for obtaining arbitrarily close upper and lower bounds. Using properties of Harris chains we also show that the incremental cost to advance through the medium has a unique stationary distribution, and we compute stochastic lower and upper bounds. We rely on no special properties of the uniform distribution: the same methods could be applied to any well‐behaved, bounded cost distribution. For the VCG payment, with Bernoulli‐distributed costs the payment for an n‐long ladder, divided by n, tends to an explicit rational function of the Bernoulli parameter. Again, our methods apply more generally. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 350‐364, 2011