Detection of C2‐singularities using uniform spline approximations

For the detection of C²‐singularities, we present lower estimates for the error in Schoenberg variation‐diminishing spline approximation with equidistant knots in terms of the classical second‐order modulus of smoothness. To this end, we investigate the behaviour of the iterates of the Schoenberg operator. In addition, we show an upper bound of the second‐order derivative of these iterative approximations. Finally, we provide an example of how to detect singularities based on the decay rate of the approximation error.     

Detecting change points in the stress‐strength reliability P(X < Y)

We address the statistical problem of detecting change points in the stress‐strength reliability R=P(X<Y) in a sequence of paired variables (X,Y). Without specifying their underlying distributions, we embed this nonparametric problem into a parametric framework and apply the maximum likelihood method via a dynamic programming approach to determine the locations of the change points in R. Under some mild conditions, we show the consistency and asymptotic properties of the procedure to locate the change points. Simulation experiments reveal that, in comparison with existing parametric and nonparametric change‐point detection methods, our proposed method performs well in detecting both single and multiple change points in R in terms of the accuracy of the location estimation and the computation time. Applications to real data demonstrate the usefulness of our proposed methodology for detecting the change points in the stress‐strength reliability R. Supplementary materials are available online.     

Modularity of Erdős‐Rényi random graphs

For a given graph G, each partition of the vertices has a modularity score, with higher values indicating that the partition better captures community structure in G. The modularity q^?(G) of the graph G is defined to be the maximum over all vertex partitions of the modularity score, and satisfies 0 ≤ q^?(G)<1. Modularity is at the heart of the most popular algorithms for community detection. We investigate the behaviour of the modularity of the Erd?s‐Rényi random graph G_n,p with n vertices and edge‐probability p. Two key findings are that the modularity is 1+o(1) with high probability (whp) for np up to 1+o(1) and no further; and when np ≥ 1 and p is bounded below 1, it has order (np)^?1/2 whp, in accord with a conjecture by Reichardt and Bornholdt in 2006. We also show that the modularity of a graph is robust to changes in a few edges, in contrast to the sensitivity of optimal vertex partitions.     

Statistical methods for automatic crack detection based on vibrothermography sequence‐of‐images data

Vibrothermography is a relatively new nondestructive evaluation technique for finding cracks through frictional heat generated from crack surface vibrations under external excitations. The vibrothermography inspection method provides a sequence of infrared images as the output. We use a matched filter technique to increase the signal‐to‐noise ratio of the sequence‐of‐images data. An automatic crack detection criterion based on the features extracted from the matched filter output greatly increases the sensitivity of the vibrothermography inspection method. In this paper, we develop a three‐dimensional matched filter for the sequence‐of‐images data, which presents the statistical analysis for the matched filter output, and evaluate the probability of detection. Our results show the crack detection criterion based on the matched filter output provides an improved detection capability. Copyright © 2010 John Wiley & Sons, Ltd.     

Multisignal histogram‐based islanding detection using neuro‐fuzzy algorithm

Islanding is an important concern for grid‐connected distributed resources due to personnel and equipment safety issues. Several techniques based on passive and active detection schemes have been proposed previously. Although passive schemes have a large nondetection zone (NDZ), concerns have been raised about active methods because of their degrading effect on power quality. Reliably detecting this condition is regarded by many as an ongoing challenge because existing methods are not entirely satisfactory. This article proposes a new integrated histogram analysis method using a neuro‐fuzzy approach for islanding detection in grid‐connected wind turbines. The main objective of the proposed approach is to reduce the NDZ to as close as possible to zero and to maintain the output power quality unchanged. In addition, this technique can also overcome the problem of setting detection thresholds which is inherent in existing techniques. The method proposed in this study has a small NDZ and is capable of detecting islanding accurately within the minimum standard time. Moreover, for those regions which require better visualization, the proposed approach can serve as an efficient aid for better detecting grid‐power disconnection. © 2014 Wiley Periodicals, Inc. Complexity 21: 195–205, 2015     

Four‐terminal reducibility and projective‐planar wye‐delta‐wye‐reducible graphs

A graph is YΔY‐reducible if it can be reduced to a vertex by a sequence of series‐parallel reductions and YΔY‐transformations. Terminals are distinguished vertices, that cannot be deleted by reductions and transformations. In this article, we show that four‐terminal planar graphs are YΔY‐reducible when at least three of the vertices lie on the same face. Using this result, we characterize YΔY‐reducible projective‐planar graphs. We also consider terminals in projective‐planar graphs, and establish that graphs of crossing‐number one are YΔY‐reducible. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 83–93, 2000     

Hyper‐Archimedean BL‐algebras are MV‐algebras

Generalizations of Boolean elements of a BL‐algebra L are studied. By utilizing the MV‐center MV(L) of L, it is reproved that an element x ∈ L is Boolean iff x ∨ x * = 1 . L is called semi‐Boolean if for all x ∈ L, x * is Boolean. An MV‐algebra L is semi‐Boolean iff L is a Boolean algebra. A BL‐algebra L is semi‐Boolean iff L is an SBL‐algebra. A BL‐algebra L is called hyper‐Archimedean if for all x ∈ L, xⁿ is Boolean for some finite n ≥ 1. It is proved that hyper‐Archimedean BL‐algebras are MV‐algebras. The study has application in mathematical fuzzy logics whose Lindenbaum algebras are MV‐algebras or BL‐algebras. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Sufficient conditions for graphs to be λ′‐optimal,super‐edge‐connected,and maximally edge‐connected

The restricted‐edge‐connectivity of a graph G, denoted by λ′(G), is defined as the minimum cardinality over all edge‐cuts S of G, where G‐S contains no isolated vertices. The graph G is called λ′‐optimal, if λ′(G) = ξ(G), where ξ(G) is the minimum edge‐degree in G. A graph is super‐edge‐connected, if every minimum edge‐cut consists of edges adjacent to a vertex of minimum degree. In this paper, we present sufficient conditions for arbitrary, triangle‐free, and bipartite graphs to be λ′‐optimal, as well as conditions depending on the clique number. These conditions imply super‐edge‐connectivity, if δ (G) ≥ 3, and the equality of edge‐connectivity and minimum degree. Different examples will show that these conditions are best possible and independent of other results in this area. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 228–246, 2005     

Tetravalent half‐edge‐transitive graphs and non‐normal Cayley graphs

Let X be a vertex‐transitive graph, that is, the automorphism group Aut(X) of X is transitive on the vertex set of X. The graph X is said to be symmetric if Aut(X) is transitive on the arc set of X. suppose that Aut(X) has two orbits of the same length on the arc set of X. Then X is said to be half‐arc‐transitive or half‐edge‐transitive if Aut(X) has one or two orbits on the edge set of X, respectively. Stabilizers of symmetric and half‐arc‐transitive graphs have been investigated by many authors. For example, see Tutte [Canad J Math 11 (1959), 621–624] and Conder and Maru?i? [J Combin Theory Ser B 88 (2003), 67–76]. It is trivial to construct connected tetravalent symmetric graphs with arbitrarily large stabilizers, and by Maru?i? [Discrete Math 299 (2005), 180–193], connected tetravalent half‐arc‐transitive graphs can have arbitrarily large stabilizers. In this article, we show that connected tetravalent half‐edge‐transitive graphs can also have arbitrarily large stabilizers. A Cayley graph Cay(G, S) on a group G is said to be normal if the right regular representation R(G) of G is normal in Aut(Cay(G, S)). There are only a few known examples of connected tetravalent non‐normal Cayley graphs on non‐abelian simple groups. In this article, we give a sufficient condition for non‐normal Cayley graphs and by using the condition, infinitely many connected tetravalent non‐normal Cayley graphs are constructed. As an application, all connected tetravalent non‐normal Cayley graphs on the alternating group A₆ are determined. © 2011 Wiley Periodicals, Inc. J Graph Theory     

3‐Factor‐Criticality of Vertex‐Transitive Graphs

A graph of order n is p ‐factor‐critical, where p is an integer of the same parity as n, if the removal of any set of p vertices results in a graph with a perfect matching. 1‐factor‐critical graphs and 2‐factor‐critical graphs are factor‐critical graphs and bicritical graphs, respectively. It is well known that every connected vertex‐transitive graph of odd order is factor‐critical and every connected nonbipartite vertex‐transitive graph of even order is bicritical. In this article, we show that a simple connected vertex‐transitive graph of odd order at least five is 3‐factor‐critical if and only if it is not a cycle.     

Some 3‐connected 4‐edge‐critical non‐Hamiltonian graphs

Let γ(G) be the domination number of graph G, thus a graph G is k‐edge‐critical if γ (G) = k, and for every nonadjacent pair of vertices u and υ, γ(G + uυ) = k?1. In Chapter 16 of the book “Domination in Graphs—Advanced Topics,” D. Sumner cites a conjecture of E. Wojcicka under the form “3‐connected 4‐critical graphs are Hamiltonian and perhaps, in general (i.e., for any k ≥ 4), (k?1)‐connected, k‐edge‐critical graphs are Hamiltonian.” In this paper, we prove that the conjecture is not true for k = 4 by constructing a class of 3‐connected 4‐edge‐critical non‐Hamiltonian graphs. © 2005 Wiley Periodicals, Inc.     

All (k;g)‐cages are k‐edge‐connected

A (k;g)‐cage is a k‐regular graph with girth g and with the least possible number of vertices. In this paper, we prove that (k;g)‐cages are k‐edge‐connected if g is even. Earlier, Wang, Xu, and Wang proved that (k;g)‐cages are k‐edge‐connected if g is odd. Combining our results, we conclude that the (k;g)‐cages are k‐edge‐connected. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 219–227, 2005     

Time‐weighted blow‐up rates and pointwise profile for single‐point blow‐up solutions in reaction–diffusion equations

This paper deals with asymptotic behavior for blow‐up solutions to time‐weighted reaction–diffusion equations u_t=Δu+e^αtv^p and v_t=Δv+e^βtu^q, subject to homogeneous Dirichlet boundary. The time‐weighted blow‐up rates are defined and obtained by ways of the scaling or auxiliary‐function methods for all α, . Aiding by key inequalities between components of solutions, we give lower pointwise blow‐up profiles for single‐point blow‐up solutions. We also study the solutions of the system with variable exponents instead of constant ones, where blow‐up rates and new blow‐up versus global existence criteria are obtained. Time‐weighted functions influence critical Fujita exponent, critical Fujita coefficient and formulae of blow‐up rates, but they do not limit the order of time‐weighted blow‐up rates and pointwise profile near blow‐up time. Copyright © 2017 John Wiley & Sons, Ltd.     

Self‐dual and self‐petrie‐dual regular maps

Regular maps are cellular decompositions of surfaces with the “highest level of symmetry”, not necessarily orientation‐preserving. Such maps can be identified with three‐generator presentations of groups G of the form G = 〈a, b, c|a² = b² = c² = (ab)^k = (bc)^m = (ca)² = … = 1〉; the positive integers k and m are the face length and the vertex degree of the map. A regular map (G;a, b, c) is self‐dual if the assignment b?b, c?a and a?c extends to an automorphism of G, and self‐Petrie‐dual if G admits an automorphism fixing b and c and interchanging a with ca. In this note we show that for infinitely many numbers k there exist finite, self‐dual and self‐Petrie‐dual regular maps of vertex degree and face length equal to k. We also prove that no such map with odd vertex degree is a normal Cayley map. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory 69:152‐159, 2012     

H‐self‐adjoint matrices. Application to radiosity

In computer graphics, in the radiosity context, a linear system Φx=b must be solved and there exists a diagonal positive matrix H such that H Φ is symmetric. In this article, we extend this property to complex matrices: we are interested in matrices which lead to Hermitian matrices under premultiplication by a Hermitian positive‐definite matrix H. We shall prove that these matrices are self‐adjoint with respect to a particular innerproduct defined on ?ⁿ. As a result, like Hermitian matrices, they have real eigenvalues and they are diagonalizable. We shall also show how to extend the Courant–Fisher theorem to this class of matrices. Finally, we shall give a new preconditioning matrix which really improves the convergence speed of the conjugate gradient method used for solving the radiosity problem. Copyright © 2002 John Wiley & Sons, Ltd.     

On the bounded quasi‐degrees of c.e. sets

We study the degree structure of bQ‐reducibility and we prove that for any noncomputable c.e. incomplete bQ‐degree a, there exists a nonspeedable bQ‐degree incomparable with it. The structure $\mathcal {D}_{\mbox{bs}}$ of the $\mbox{bs}$‐degrees is not elementary equivalent neither to the structure of the $\mbox{be}$‐degrees nor to the structure of the $\mbox{e}$‐degrees. If c.e. degrees a and b form a minimal pair in the c.e. bQ‐degrees, then a and b form a minimal pair in the bQ‐degrees. Also, for every simple set S there is a noncomputable nonspeedable set A which is bQ‐incomparable with S and bQ‐degrees of S and A does not form a minimal pair.     

Spanning k‐arc‐strong subdigraphs with few arcs in k‐arc‐strong tournaments

Given a k‐arc‐strong tournament T, we estimate the minimum number of arcs possible in a k‐arc‐strong spanning subdigraph of T. We give a construction which shows that for each k ≥ 2, there are tournaments T on n vertices such that every k‐arc‐strong spanning subdigraph of T contains at least arcs. In fact, the tournaments in our construction have the property that every spanning subdigraph with minimum in‐ and out‐degree at least k has arcs. This is best possible since it can be shown that every k‐arc‐strong tournament contains a spanning subdigraph with minimum in‐ and out‐degree at least k and no more than arcs. As our main result we prove that every k‐arc‐strong tournament contains a spanning k‐arc‐strong subdigraph with no more than arcs. We conjecture that for every k‐arc‐strong tournament T, the minimum number of arcs in a k‐arc‐strong spanning subdigraph of T is equal to the minimum number of arcs in a spanning subdigraph of T with the property that every vertex has in‐ and out‐degree at least k. We also discuss the implications of our results on related problems and conjectures. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 265–284, 2004     

Weak‐quasi‐Stone algebras

In this paper we shall introduce the variety WQS of weak‐quasi‐Stone algebras as a generalization of the variety QS of quasi‐Stone algebras introduced in [9]. We shall apply the Priestley duality developed in [4] for the variety N of ¬‐lattices to give a duality for WQS. We prove that a weak‐quasi‐Stone algebra is characterized by a property of the set of its regular elements, as well by mean of some principal lattice congruences. We will also determine the simple and subdirectly irreducible algebras (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Claw‐free circular‐perfect graphs

The circular chromatic number of a graph is a well‐studied refinement of the chromatic number. Circular‐perfect graphs form a superclass of perfect graphs defined by means of this more general coloring concept. This article studies claw‐free circular‐perfect graphs. First, we prove that if G is a connected claw‐free circular‐perfect graph with χ(G)>ω(G), then min{α(G), ω(G)}=2. We use this result to design a polynomial time algorithm that computes the circular chromatic number of claw‐free circular‐perfect graphs. A consequence of the strong perfect graph theorem is that minimal imperfect graphs G have min{α(G), ω(G)}=2. In contrast to this result, it is shown in Z. Pan and X. Zhu [European J Combin 29(4) (2008), 1055–1063] that minimal circular‐imperfect graphs G can have arbitrarily large independence number and arbitrarily large clique number. In this article, we prove that claw‐free minimal circular‐imperfect graphs G have min{α(G), ω(G)}≤3. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 163–172, 2010