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.     

More on extremal ranks of the matrix expressions A − BX ± X*B* with statistical applications

Through a Hermitian‐type (skew‐Hermitian‐type) singular value decomposition for pair of matrices (A, B) introduced by Zha (Linear Algebra Appl. 1996; 240 :199–205), where A is Hermitian (skew‐Hermitian), we show how to find a Hermitian (skew‐Hermitian) matrix X such that the matrix expressions A ? BX ± X^*B^* achieve their maximal and minimal possible ranks, respectively. For the consistent matrix equations BX ± X^*B^* = A, we give general solutions through the two kinds of generalized singular value decompositions. As applications to the general linear model {y, Xβ, σ²V}, we discuss the existence of a symmetric matrix G such that Gy is the weighted least‐squares estimator and the best linear unbiased estimator of Xβ, respectively. Copyright © 2007 John Wiley & Sons, Ltd.     

Defining Sets and Critical Sets in (0,1)‐Matrices

If D is a partially filled‐in (0, 1)‐matrix with a unique completion to a (0, 1)‐matrix M (with prescribed row and column sums), we say that D is a defining set for M. If the removal of any entry of D destroys this property (i.e. at least two completions become possible), we say that D is a critical set for M. In this note, we show that the complement of a critical set for a (0, 1)‐matrix M is a defining set for M. We also study the possible sizes (number of filled‐in cells) of defining sets for square matrices M with uniform row and column sums, which are also frequency squares. In particular, we show that when the matrix is of even order 2m and the row and column sums are all equal to m, the smallest possible size of a critical set is precisely m². We give the exact structure of critical sets with this property. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 21: 253–266, 2013     

Least‐squares solutions and least‐rank solutions of the matrix equation AXA* = B and their relations

A Hermitian matrix X is called a least‐squares solution of the inconsistent matrix equation AXA^* = B, where B is Hermitian. A^* denotes the conjugate transpose of A if it minimizes the F‐norm of B ? AXA^*; it is called a least‐rank solution of AXA^* = B if it minimizes the rank of B ? AXA^*. In this paper, we study these two types of solutions by using generalized inverses of matrices and some matrix decompositions. In particular, we derive necessary and sufficient conditions for the two types of solutions to coincide. Copyright © 2012 John Wiley & Sons, Ltd.     

Blended kernel approximation in the ℋ︁‐matrix techniques

Several types of ??‐matrices were shown to provide a data‐sparse approximation of non‐local (integral) operators in FEM and BEM applications. The general construction is applied to the operators with asymptotically smooth kernel function provided that the Galerkin ansatz space has a hierarchical structure. The new class of ??‐matrices is based on the so‐called blended FE and polynomial approximations of the kernel function and leads to matrix blocks with a tensor‐product of block‐Toeplitz (block‐circulant) and rank‐k matrices. This requires the translation (rotation) invariance of the kernel combined with the corresponding tensor‐product grids. The approach allows for the fast evaluation of volume/boundary integral operators with possibly non‐smooth kernels defined on canonical domains/manifolds in the FEM/BEM applications. (Here and in the following, we call domains canonical if they are obtained by translation or rotation of one of their parts, e.g. parallelepiped, cylinder, sphere, etc.) In particular, we provide the error and complexity analysis for blended expansions to the Helmholtz kernel. Copyright © 2002 John Wiley & Sons, Ltd.     

On realization of the Kreĭn–Langer class Nκ of matrix‐valued functions in Pontryagin spaces

In this paper the realization problems for the Kre?n–Langer class N_κ of matrix‐valued functions are being considered. We found the criterion when a given matrix‐valued function from the class N_κ can be realized as linear‐fractional transformation of the transfer function of canonical conservative system of the M. Livsic type (Brodskii–Livsic rigged operator colligation) with the main operator acting on a rigged Pontryagin space Π_κ with indefinite metric. We specify three subclasses of the class N_κ (R) of all realizable matrix‐valued functions that correspond to different properties of a realizing system, in particular, when the domains of the main operator of a system and its conjugate coincide, when the domain of the hermitian part of a main operator is dense in Π_κ . Alternatively we show that the class N_κ (R) can be realized as transfer matrix‐functions of some canonical impedance systems with self‐adjoint main operators in rigged spaces Π_κ . The case of scalar functions of the class N_κ (R) is considered in details and some examples are presented. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Directional ‐matrix compression for high‐frequency problems

Standard numerical algorithms, such as the fast multipole method or ‐matrix schemes, rely on low‐rank approximations of the underlying kernel function. For high‐frequency problems, the ranks grow rapidly as the mesh is refined, and standard techniques are no longer attractive. Directional compression techniques solve this problem by using decompositions based on plane waves. Taking advantage of hierarchical relations between these waves' directions, an efficient approximation is obtained. This paper is dedicated to directional ‐matrices that employ local low‐rank approximations to handle directional representations efficiently. The key result is an algorithm that takes an arbitrary matrix and finds a quasi‐optimal approximation of this matrix as a directional ‐matrix using a prescribed block tree. The algorithm can reach any given accuracy, and the approximation requires only units of storage, where n is the matrix dimension, κ is the wave number, and k is the local rank. In particular, we have a complexity of if κ is constant and for high‐frequency problems characterized by κ²～n. Because the algorithm can be applied to arbitrary matrices, it can serve as the foundation of fast techniques for constructing preconditioners.     

Robust stability and H∞ control for nonlinear discrete‐time switched systems with interval time‐varying delay

This paper considers the problems of the robust stability analysis and H_∞ controller synthesis for uncertain discrete‐time switched systems with interval time‐varying delay and nonlinear disturbances. Based on the system transformation and by introducing a switched Lyapunov‐Krasovskii functional, the novel sufficient conditions, which guarantee that the uncertain discrete‐time switched system is robust asymptotically stable are obtained in terms of linear matrix inequalities. Then, the robust H_∞ control synthesis via switched state feedback is studied for a class of discrete‐time switched systems with uncertainties and nonlinear disturbances. We designed a switched state feedback controller to stabilize asymptotically discrete‐time switched systems with interval time‐varying delay and H_∞ disturbance attenuation level based on matrix inequality conditions. Examples are provided to illustrate the advantage and effectiveness of the proposed method.     

A structure‐preserving doubling algorithm for Lur'e equations

We introduce a numerical method for the numerical solution of the Lur'e equations, a system of matrix equations that arises, for instance, in linear‐quadratic infinite time horizon optimal control. We focus on small‐scale, dense problems. Via a Cayley transformation, the problem is transformed to the discrete‐time case, and the structural infinite eigenvalues of the associated matrix pencil are deflated. The deflated problem is associated with a symplectic pencil with several Jordan blocks of eigenvalue 1 and even size, which arise from the nontrivial Kronecker chains at infinity of the original problem. For the solution of this modified problem, we use the structure‐preserving doubling algorithm. Implementation issues such as the choice of the parameter γ in the Cayley transform are discussed. The most interesting feature of this method, with respect to the competing approaches, is the absence of arbitrary rank decisions, which may be ill‐posed and numerically troublesome. The numerical examples presented confirm the effectiveness of this method. Copyright © 2015 John Wiley & Sons, Ltd.     

Darboux transformation and Rogue waves of the Kundu–nonlinear Schrödinger equation

In this paper, the Darboux transformation of the Kundu–nonlinear Schrödinger equation is derived and generalized to the matrix of n‐fold Darboux transformation. From known solution Q, the determinant representation of n‐th new solutions of Q^[n] are obtained by the n‐fold Darboux transformation. Then soliton solutions and positon solutions are generated from trivial seed solutions, breather solutions and rogue wave solutions that are obtained from periodic seed solutions. After that, the higher order rogue wave solutions of the Kundu–nonlinear Schrödinger equation are given. We show that free parameters in eigenfunctions can adjust the patterns of the higher order rogue waves. Meanwhile, the third‐order rogue waves are given explicitly. Copyright © 2014 John Wiley & Sons, Ltd.     

Every 3‐connected claw‐free Z8‐free graph is Hamiltonian

A biclique of a graph G is a maximal induced complete bipartite subgraph of G. Given a graph G, the biclique matrix of G is a {0,1,?1} matrix having one row for each biclique and one column for each vertex of G, and such that a pair of 1, ?1 entries in a same row corresponds exactly to adjacent vertices in the corresponding biclique. We describe a characterization of biclique matrices, in similar terms as those employed in Gilmore's characterization of clique matrices. On the other hand, the biclique graph of a graph is the intersection graph of the bicliques of G. Using the concept of biclique matrices, we describe a Krausz‐type characterization of biclique graphs. Finally, we show that every induced P₃ of a biclique graph must be included in a diamond or in a 3‐fan and we also characterize biclique graphs of bipartite graphs. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 1–16, 2010     

‐differential equations for ‐classical polynomials and ‐Jacobi–Stirling numbers

We introduce, characterise and provide a combinatorial interpretation for the so‐called q‐Jacobi–Stirling numbers. This study is motivated by their key role in the (reciprocal) expansion of any power of a second order q‐differential operator having the q‐classical polynomials as eigenfunctions in terms of other even order operators, which we explicitly construct in this work. The results here obtained can be viewed as the q‐version of those given by Everitt et al. and by the first author, whilst the combinatorics of this new set of numbers is a q‐version of the Jacobi–Stirling numbers given by Gelineau and the second author.     

Finding paths between 3‐colorings

Given a 3‐colorable graph G together with two proper vertex 3‐colorings α and β of G, consider the following question: is it possible to transform α into β by recoloring vertices of G one at a time, making sure that all intermediate colorings are proper 3‐colorings? We prove that this question is answerable in polynomial time. We do so by characterizing the instances G, α, β where the transformation is possible; the proof of this characterization is via an algorithm that either finds a sequence of recolorings between α and β, or exhibits a structure which proves that no such sequence exists. In the case that a sequence of recolorings does exist, the algorithm uses O(|V(G)|²) recoloring steps and in many cases returns a shortest sequence of recolorings. We also exhibit a class of instances G, α, β that require Ω(|V(G)|²) recoloring steps. © 2010 Wiley Periodicals, Inc. J Graph Theory 67: 69‐82, 2011     

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     

Oscillation analysis of θ‐methods for the Nicholson's blowflies model

This paper is concerned with the oscillation of numerical solution for the Nicholson's blowflies model. Using two kinds of θ‐methods, namely, the linear θ‐method and the one‐leg θ‐method, several conditions under which the numerical solution oscillates are derived. Moreover, it is shown that every non‐oscillatory numerical solution tends to equilibrium point of the original continuous‐time model. Finally, numerical experiments are provided to illustrate the analytical results. Copyright © 2015 John Wiley & Sons, Ltd.     

2‐connected 7‐coverings of 3‐connected graphs on surfaces

An m‐covering of a graph G is a spanning subgraph of G with maximum degree at most m. In this paper, we shall show that every 3‐connected graph on a surface with Euler genus k ≥ 2 with sufficiently large representativity has a 2‐connected 7‐covering with at most 6k ? 12 vertices of degree 7. We also construct, for every surface F² with Euler genus k ≥ 2, a 3‐connected graph G on F² with arbitrarily large representativity each of whose 2‐connected 7‐coverings contains at least 6k ? 12 vertices of degree 7. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 26–36, 2003     

Self‐Intersection Local Time for 𝒮′(ℝd)‐Ornstein‐Uhlenbeck Processes Arising from Immigration Systems

Fluctuation limits of an immigration branching particle system and an immigration branching measure‐valued process yield different types of 𝒮′(ℝ^d)‐valued Ornstein‐Uhlenbeck processes whose covariances are given in terms of an excessive measure for the underlying motion in Rd, which is taken to be a symmetric α‐stable process. In this paper we prove existence and path continuity results for the self‐intersection local time of these Ornstein‐Uhlenbeck processes. The results depend on relationships between the dimension d and the parameter α.     

A note on the complex semi‐definite matrix Procrustes problem

This note outlines an algorithm for solving the complex ‘matrix Procrustes problem’. This is a least‐squares approximation over the cone of positive semi‐definite Hermitian matrices, which has a number of applications in the areas of Optimization, Signal Processing and Control. The work generalizes the method of Allwright (SIAM J. Control Optim. 1988; 26 (3):537–556), who obtained a numerical solution to the real‐valued version of the problem. It is shown that, subject to an appropriate rank assumption, the complex problem can be formulated in a real setting using a matrix‐dilation technique, for which the method of Allwright is applicable. However, this transformation results in an over‐parametrization of the problem and, therefore, convergence to the optimal solution is slow. Here, an alternative algorithm is developed for solving the complex problem, which exploits fully the special structure of the dilated matrix. The advantages of the modified algorithm are demonstrated via a numerical example. Copyright © 2007 John Wiley & Sons, Ltd.     

Groups and C*‐Algebras from a 4‐Dimensional Anzai Flow

The rotation flow on the circle T gives a concrete representation of the irrational rotation algebra, which is an in finite dimensional simple quotient of the group C*‐algebra of the discrete Heisenberg group H₃ analogously certain 2‐ and 3‐dimensional Anzai flows on T ² and T ³are known to give concrete representations of the corresponding quotients of the group C*‐algebras of the groups H₄ and H_5,5. Considered here is the (minimal, effective) 4‐dimensional Anzai flow F = (ℤ, T ⁴) generated by the homeomorphism (y, x, w, v) ↦ (λy, yx, xw, wv); a group H_6,10 is determined by F the faithful in finite dimensional simple quotients of whose group C*‐algebra C*‐(H_6,10 have concrete representations given by F. Furthermore, the rest of the infinite dimensional simple quotients of C*‐(H_6,10 are identified and displayed as C*‐crossed products generated by minimal effective actions and also as matrix algebras over simple C*‐algebras from groups of lower dimension; these lower dimensional groups are H₃ and subgroups of H₄ and H_5,5.