John disks,the Apollonian metric,and min-max properties

The main results of this paper are characterizations of John disks-the simply connected proper subdomains of the complex plane that satisfy a twisted double cone connectivity property. One of the characterizations of John disks is an analog of a result due to Gehring and Hag who found such a characterization for quasidisks. In both situations the geometric condition is an estimate for the domain’s hyperbolic metric in terms of its Apollonian metric. The other characterization is in terms of an arc min-max property.     

Bounded BCK‐algebras and their generated variety

In this paper we prove that the equational class generated by bounded BCK‐algebras is the variety generated by the class of finite simple bounded BCK‐algebras. To obtain these results we prove that every simple algebra in the equational class generated by bounded BCK‐algebras is also a relatively simple bounded BCK‐algebra. Moreover, we show that every simple bounded BCK‐algebra can be embedded into a simple integral commutative bounded residuated lattice. We extend our main results to some richer subreducts of the class of integral commutative bounded residuated lattices and to the involutive case. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

关于拟圆性质的评注

设D是扩充复平面R-2中的单连通Jordan真子域,D*=R-2\D-是D的外部.本文肯定并证明 了K.Hag在1999年提出的如下两个悬而未决的问题:(1)D是Cigar域当且仅当D是OLC域; (2)D是Turning域当且仅当D*是Cigar域.     

A Family of Partial Geometric Designs from Three‐Class Association Schemes

In this paper, we show that partial geometric designs can be constructed from certain three‐class association schemes and ternary linear codes with dual distance three. In particular, we obtain a family of partial geometric designs from the three‐class association schemes introduced by Kageyama, Saha, and Das in their article [“Reduction of the number of associate classes of hypercubic association schemes,” Ann Inst Statist Math 30 (1978)]. We also give a list of directed strongly regular graphs arising from the partial geometric designs obtained in this paper.     

Noncompact‐type Krasnoselskii fixed‐point theorems and their applications

In this paper, we first establish some user‐friendly versions of fixed‐point theorems for the sum of two operators in the setting that the involved operators are not necessarily compact and continuous. These fixed‐point results generalize, encompass, and complement a number of previously known generalizations of the Krasnoselskii fixed‐point theorem. Next, with these obtained fixed‐point results, we study the existence of solutions for a class of transport equations, the existence of global solutions for a class of Darboux problems on the first quadrant, the existence and/or uniqueness of periodic solutions for a class of difference equations, and the existence and/or uniqueness of solutions for some kind of perturbed Volterra‐type integral equations. Copyright © 2015 John Wiley & Sons, Ltd.     

Commutative rings whose ideals form an MV‐algebra

In this work we introduce a class of commutative rings whose defining condition is that its lattice of ideals, augmented with the ideal product, the semi‐ring of ideals, is isomorphic to an MV‐algebra. This class of rings coincides with the class of commutative rings which are direct sums of local Artinian chain rings with unit (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Q‐integral unicyclic,bicyclic and tricyclic graphs

A graph is called Q‐integral if its signless Laplacian spectrum consists of integers. In this paper, we characterize a class of k‐cyclic graphs whose second smallest signless Laplacian eigenvalue is less than one. Using this result we determine all the Q‐integral unicyclic, bicyclic and tricyclic graphs.     

On the potential in non‐Gaussian chain polymer models

In this paper, we investigate the potential for a class of non‐Gaussian processes so‐called generalized grey Brownian motion. We obtain a closed analytic form for the potential as an integral of the M‐Wright functions and the Green function. In particular, we recover the special cases of Brownian motion and fractional Brownian motion. In addition, we give the connection to a fractional partial differential equation and its the fundamental solution.     

Rough stochastic PDEs

In this article, we show how the theory of rough paths can be used to provide a notion of solution to a class of nonlinear stochastic PDEs of Burgers type that exhibit too‐high spatial roughness for classical analytical methods to apply. In fact, the class of SPDEs that we consider is genuinely ill‐posed in the sense that different approximations to the nonlinearity may converge to different limits. Using rough path theory, a pathwise notion of solution to these SPDEs is formulated, and we show that this yields a well‐posed problem that is stable under a large class of perturbations, including the approximation of the rough‐driving noise by a mollified version and the addition of hyperviscosity. We also show that under certain structural assumptions on the coefficients, the SPDEs under consideration generate a reversible Markov semigroup with respect to a diffusion measure that can be given explicitly. © 2011 Wiley Periodicals, Inc.     

Homoclinic orbits for Hamiltonian systems induced by impulses

In this paper, we consider a class of impulsive Hamiltonian systems with a p‐Laplacian operator. Under certain conditions, we establish the existence of homoclinic orbits by means of the mountain pass theorem and an approximation technique. In some special cases, the homoclinic orbits are induced by the impulses in the sense that the associated non‐impulsive systems admit no non‐trivial homoclinic orbits. Copyright © 2015 John Wiley & Sons, Ltd.     

Amenable Köthe co‐echelon algebras

In this note we investigate amenability properties in the class of the so‐called DF‐algebras. We also characterize, in terms of the defining sequence of weights, amenable Köthe co‐echelon algebras.     

Compact representation of the full Broyden class of quasi‐Newton updates

In this paper, we present the compact representation for matrices belonging to the Broyden class of quasi‐Newton updates, where each update may be either rank one or rank two. This work extends previous results solely for the restricted Broyden class of rank‐two updates. In this article, it is not assumed that the same Broyden update is used in each iteration; rather, different members of the Broyden class may be used in each iteration. Numerical experiments suggest that a practical implementation of the compact representation is able to accurately represent matrices belonging to the Broyden class of updates. Furthermore, we demonstrate how to compute the compact representation for the inverse of these matrices and a practical algorithm for solving linear systems with members of the Broyden class of updates. We demonstrate through numerical experiments that the proposed linear solver is able to efficiently solve linear systems with members of the Broyden class of matrices with high accuracy. As an immediate consequence of this work, it is now possible to efficiently compute the eigenvalues of any limited‐memory member of the Broyden class of matrices, allowing for the computation of condition numbers and the ability to perform sensitivity analysis.     

Numerical resolution of linear evolution multidimensional problems of second order in time

We present a new class of efficient time integrators for solving linear evolution multidimensional problems of second‐order in time named Fractional Step Runge‐Kutta‐Nyström methods (FSRKN). We show that these methods, combined with suitable spliting of the space differential operator and adequate space discretizations provide important advantages from the computational point of view, mainly parallelization facilities and reduction of computational complexity. In this article, we study in detail the consistency of such methods and we introduce an extension of the concept of R‐stability for Runge‐Kutta‐Nyström methods. We also present some numerical experiments showing the unconditional convergence of a third order method of this class applied to resolve one Initial Boundary Value Problem of second order in time. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 597–620, 2012     

Global existence and bounds for blow‐up time in a class of nonlinear pseudo‐parabolic equations with a memory term

In this article, we consider the initial boundary value problem for a class of nonlinear pseudo‐parabolic equations with a memory term: Under suitable assumptions, we obtain the local and global existence of the solution by Galerkin method. We prove finite‐time blow‐up of the solution for initial data at arbitrary energy level and obtain upper bounds for blow‐up time by using the concavity method. In addition, by means of differential inequality technique, we obtain a lower bound for blow‐up time of the solution if blow‐up occurs.     

Bounding the Clique‐Width of H‐Free Chordal Graphs

A graph is H‐free if it has no induced subgraph isomorphic to H. Brandstädt, Engelfriet, Le, and Lozin proved that the class of chordal graphs with independence number at most 3 has unbounded clique‐width. Brandstädt, Le, and Mosca erroneously claimed that the gem and co‐gem are the only two 1‐vertex P₄‐extensions H for which the class of H‐free chordal graphs has bounded clique‐width. In fact we prove that bull‐free chordal and co‐chair‐free chordal graphs have clique‐width at most 3 and 4, respectively. In particular, we find four new classes of H‐free chordal graphs of bounded clique‐width. Our main result, obtained by combining new and known results, provides a classification of all but two stubborn cases, that is, with two potential exceptions we determine all graphs H for which the class of H‐free chordal graphs has bounded clique‐width. We illustrate the usefulness of this classification for classifying other types of graph classes by proving that the class of ‐free graphs has bounded clique‐width via a reduction to K₄‐free chordal graphs. Finally, we give a complete classification of the (un)boundedness of clique‐width of H‐free weakly chordal graphs.     

On electromagnetic waves in complex linear media in nonsmooth domains

Electromagnetic wave propagation in more complex linear materials such as bi‐anisotropic media have come to a considerable attention within the last 15–20 years. We shall propose a general framework to approach a class of highly complex materials. Such problems have been extensively studied mostly in the time‐harmonic case. In this paper, we focus on the time‐dependent case. A well‐posedness result for a large class of media is obtained. We also analyze Drude–Born–Fedorov type media in nonsmooth domains of arbitrary topological genus within this framework. Copyright © 2012 John Wiley & Sons, Ltd.     

Partial Characterizations of 1‐Perfectly Orientable Graphs

We study the class of 1‐perfectly orientable graphs, that is, graphs having an orientation in which every out‐neighborhood induces a tournament. 1‐perfectly orientable graphs form a common generalization of chordal graphs and circular arc graphs. Even though they can be recognized in polynomial time, little is known about their structure. In this article, we develop several results on 1‐perfectly orientable graphs. In particular, we (i) give a characterization of 1‐perfectly orientable graphs in terms of edge clique covers, (ii) identify several graph transformations preserving the class of 1‐perfectly orientable graphs, (iii) exhibit an infinite family of minimal forbidden induced minors for the class of 1‐perfectly orientable graphs, and (iv) characterize the class of 1‐perfectly orientable graphs within the classes of cographs and of cobipartite graphs. The class of 1‐perfectly orientable cobipartite graphs coincides with the class of cobipartite circular arc graphs.     

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     

Factorization problem on the Hilbert‐Schmidt group and the Camassa‐Holm equation

In this paper we solve the Camassa‐Holm equation for a relatively large class of initial data by using a factorization problem on the Hilbert‐Schmidt group. © 2007 Wiley Periodicals, Inc.