Symmetric Hamilton cycle decompositions of complete graphs minus a 1‐factor

Let n≥2 be an integer. The complete graph K_n with a 1‐factor F removed has a decomposition into Hamilton cycles if and only if n is even. We show that K_n−F has a decomposition into Hamilton cycles which are symmetric with respect to the 1‐factor F if and only if n≡2, 4 mod 8. We also show that the complete bipartite graph K_{n, n} has a symmetric Hamilton cycle decomposition if and only if n is even, and that if F is a 1‐factor of K_{n, n}, then K_{n, n}−F has a symmetric Hamilton cycle decomposition if and only if n is odd. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:1‐15, 2010     

Least‐squares solutions of matrix inverse problem for bi‐symmetric matrices with a submatrix constraint

An n × n real matrix A = (a_ij)_{n × n} is called bi‐symmetric matrix if A is both symmetric and per‐symmetric, that is, a_ij = a_ji and a_ij = a_n+1?1,n+1?i (i, j = 1, 2,..., n). This paper is mainly concerned with finding the least‐squares bi‐symmetric solutions of matrix inverse problem AX = B with a submatrix constraint, where X and B are given matrices of suitable sizes. Moreover, in the corresponding solution set, the analytical expression of the optimal approximation solution to a given matrix A^* is derived. A direct method for finding the optimal approximation solution is described in detail, and three numerical examples are provided to show the validity of our algorithm. Copyright © 2007 John Wiley & Sons, Ltd.     

The recognition of symmetric latin squares

A latin square S is isotopic to another latin square S′ if S′ can be obtained from S by permuting the row indices, the column indices and the symbols in S. Because the three permutations used above may all be different, a latin square which is isotopic to a symmetric latin square need not be symmetric. We call the problem of determining whether a latin square is isotopic to a symmetric latin square the symmetry recognition problem. It is the purpose of this article to give a solution to this problem. For this purpose we will introduce a cocycle corresponding to a latin square which transforms very simply under isotopy, and we show this cocycle contains all the information needed to determine whether a latin square is isotopic to a symmetric latin square. Our results relate to 1‐factorizations of the complete graph on n + 1 vertices, K_{n + 1}. There is a well known construction which can be used to make an n × n latin square from a 1‐factorization on n + 1 vertices. The symmetric idempotent latin squares are exactly the latin squares that result from this construction. The idempotent recognition problem is simple for symmetric latin squares, so our results enable us to recognize exactly which latin squares arise from 1‐factorizations of K_{n + 1}. As an example we show that the patterned starter 1‐factorization for the group G gives rise to a latin square which is in the main class of the Cayley latin square for G if and only if G is abelian. Hence, every non‐abelian group gives rise to two latin squares in different main classes. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 291–300, 2008     

Symmetries that latin squares inherit from 1‐factorizations

A 1‐factorization of a graph is a decomposition of the graph into edge disjoint perfect matchings. There is a well‐known method, which we call the ??‐construction, for building a 1‐factorization of K_n,n from a 1‐factorization of K_{n + 1}. The 1‐factorization of K_n,n can be written as a latin square of order n. The ??‐construction has been used, among other things, to make perfect 1‐factorizations, subsquare‐free latin squares, and atomic latin squares. This paper studies the relationship between the factorizations involved in the ??‐construction. In particular, we show how symmetries (automorphisms) of the starting factorization are inherited as symmetries by the end product, either as automorphisms of the factorization or as autotopies of the latin square. Suppose that the ??‐construction produces a latin square L from a 1‐factorization F of K_{n + 1}. We show that the main class of L determines the isomorphism class of F, although the converse is false. We also prove a number of restrictions on the symmetries (autotopies and paratopies) which L may possess, many of which are simple consequences of the fact that L must be symmetric (in the usual matrix sense) and idempotent. In some circumstances, these restrictions are tight enough to ensure that L has trivial autotopy group. Finally, we give a cubic time algorithm for deciding whether a main class of latin squares contains any square derived from the ??‐construction. The algorithm also detects symmetric squares and totally symmetric squares (latin squares that equal their six conjugates). © 2005 Wiley Periodicals, Inc. J Combin Designs 13: 157–172, 2005.     

On the linearity of regression

Summary A stochastic process X={X _t :tT| is called spherically generated if for each random vector , there exist a random vector Y=(Y₁,..., Y _m) with a spherical (radially symmetric) distribution and a matrix A such that X is distributed as AY. X is said to have the linear regression property if (X ₀¦X ₁,..., X _n) is a linear function of X ₁,..., X _n whenever the X _j's are elements of the linear span of X. It is shown that providing the linear span of X has dimension larger than two, then X has the linear regression property if and only if it is spherically generated. The class of symmetric stable processes which are spherically generated is shown to coincide with the class of socalled sub-Gaussian processes, characterizing those stable processes having the linear regression property.This research was supported by a grant from the University of Wisconsin-Milwaukee     

Metric trees of generalized roundness one

Every finite metric tree has generalized roundness strictly greater than one. On the other hand, some countable metric trees have generalized roundness precisely one. The purpose of this paper is to identify several large classes of countable metric trees that have generalized roundness precisely one. At the outset we consider spherically symmetric trees endowed with the usual path metric (SSTs). Using a simple geometric argument we show how to determine reasonable upper bounds on the generalized roundness of finite SSTs that depend only on the downward degree sequence of the tree in question. By considering limits, it follows that if the downward degree sequence (d ₀, d ₁, d ₂, . . .) of an SST (T, ρ) satisfies ${|\{j \, | \, d_{j} > 1 \}| = \aleph_{0}}${|\{j \, | \, d_{j} > 1 \}| = \aleph_{0}} , then (T, ρ) has generalized roundness one. In particular, all complete n-ary trees of depth ∞ (n ≥ 2), all k-regular trees (k ≥ 3) and all inductive limits of Cantor trees are seen to have generalized roundness one. The remainder of the paper deals with two classes of countable metric trees of generalized roundness one whose members are not, in general, spherically symmetric. The first such class of trees are merely required to spread out at a sufficient rate (with a restriction on the number of leaves) and the second such class of trees resemble infinite combs. It remains an intriguing problem to completely classify countable metric trees of generalized roundness one.     

The generalized centro‐symmetric and least squares generalized centro‐symmetric solutions of the matrix equation AYB + CYTD = E

An n×n real matrix P is said to be a symmetric orthogonal matrix if P = P^?1 = P^T. An n × n real matrix Y is called a generalized centro‐symmetric with respect to P, if Y = PYP. It is obvious that every matrix is also a generalized centro‐symmetric matrix with respect to I. In this work by extending the conjugate gradient approach, two iterative methods are proposed for solving the linear matrix equation and the minimum Frobenius norm residual problem over the generalized centro‐symmetric Y, respectively. By the first (second) algorithm for any initial generalized centro‐symmetric matrix, a generalized centro‐symmetric solution (least squares generalized centro‐symmetric solution) can be obtained within a finite number of iterations in the absence of round‐off errors, and the least Frobenius norm generalized centro‐symmetric solution (the minimal Frobenius norm least squares generalized centro‐symmetric solution) can be derived by choosing a special kind of initial generalized centro‐symmetric matrices. We also obtain the optimal approximation generalized centro‐symmetric solution to a given generalized centro‐symmetric matrix Y₀ in the solution set of the matrix equation (minimum Frobenius norm residual problem). Finally, some numerical examples are presented to support the theoretical results of this paper. Copyright © 2011 John Wiley & Sons, Ltd.     

Asymptotic Behavior of Spherically Symmetric Marginally Trapped Tubes

We give conditions on a general stress-energy tensor T _αβ in a spherically symmetric black hole spacetime which are sufficient to guarantee that the black hole will contain a (spherically symmetric) marginally trapped tube which is eventually achronal, connected, and asymptotic to the event horizon. Price law decay per se is not required for this asymptotic result, and in this general setting, such decay only implies that the marginally trapped tube has finite length with respect to the induced metric. We do, however, impose a smallness condition (B1) which one may obtain in practice by imposing decay on the T _vv component of the stress-energy tensor along the event horizon. We give two applications of the theorem to self-gravitating Higgs field spacetimes, one using weak Price law decay, the other certain strong smallness and monotonicity assumptions. Submitted: August 6, 2007. Accepted: June 6, 2008.     

Minimax estimation of the mean of spherically symmetric distributions under general quadratic loss

For X one observation on a p-dimensional (p ≥ 4) spherically symmetric (s.s.) distribution about θ, minimax estimators whose risks dominate the risk of X (the best invariant procedure) are found with respect to general quadratic loss, L(δ, θ) = (δ − θ)′ D(δ − θ) where D is a known p × p positive definite matrix. For C a p × p known positive definite matrix, conditions are given under which estimators of the form δ_a,r,C,D(X) = (I − (ar(|X|²)) D^−1/2CD^1/2 |X|⁻²)X are minimax with smaller risk than X. For the problem of estimating the mean when n observations X₁, X₂, …, X_n are taken on a p-dimensional s.s. distribution about θ, any spherically symmetric translation invariant estimator, δ(X₁, X₂, …, X_n), with have a s.s. distribution about θ. Among the estimators which have these properties are best invariant estimators, sample means and maximum likelihood estimators. Moreover, under certain conditions, improved robust estimators can be found.     

Blowup for nonlinear wave equations describing boson stars

We consider the nonlinear wave equation modeling the dynamics of (pseudorelativistic) boson stars. For spherically symmetric initial data, u₀(x) ∈ C (?³), with negative energy, we prove blowup of u(t, x) in the H^1/2‐norm within a finite time. Physically this phenomenon describes the onset of “gravitational collapse” of a boson star. We also study blowup in external, spherically symmetric potentials, and we consider more general Hartree‐type nonlinearities. As an application, we exhibit instability of ground state solitary waves at rest if m = 0. © 2007 Wiley Periodicals, Inc.     

Zonal polynomials for wreath products

The pair of groups, symmetric group S _2n and hyperoctohedral group H _n , form a Gelfand pair. The characteristic map is a mapping from the graded algebra generated by the zonal spherical functions of (S _2n ,H _n ) into the ring of symmetric functions. The images of the zonal spherical functions under this map are called the zonal polynomials. A wreath product generalization of the Gelfand pair (S _2n ,H _n ) is discussed in this paper. Then a multi-partition versions of the theory is constructed. The multi-partition version of zonal polynomials are products of zonal polynomials and Schur functions and are obtained from a characteristic map from the graded Hecke algebra into a multipartition version of the ring of symmetric functions. Dedicated to Professor Eiichi Bannai on his 60th birthday.     

Constructing orthogonal pandiagonal Latin squares and panmagic squares from modular n‐queens solutions

In this article, we show how to construct pairs of orthogonal pandiagonal Latin squares and panmagic squares from certain types of modular n‐queens solutions. We prove that when these modular n‐queens solutions are symmetric, the panmagic squares thus constructed will be associative, where for an n × n associative magic square A = (a_ij), for all i and j it holds that a_ij + an?i?1,n?j?1 = c for a fixed c. We further show how to construct orthogonal Latin squares whose modular difference diagonals are Latin from any modular n‐queens solution. As well, we analyze constructing orthogonal pandiagonal Latin squares from particular classes of non‐linear modular n‐queens solutions. These pandiagonal Latin squares are not row cyclic, giving a partial solution to a problem of Hedayat. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 221–234, 2007     

Interior normal lengths on convex configurations

We present a simple method to derive matricial representations of exponentials operators exp [G_n ],where G _n is a nx nmatrix. We verify that our results reproduce the simple case where G _n is a 2 x 2 matrix and give new relations in the three‐dimensional case. We also discuss a matricial representation for the related Green's function when G _n , for n = 2 and 3 is a real symmetric matrix.     

Uniform boundness of global solutions for a n-dimensional spherically symmetric combustion model

This paper concerns a n-dimensional spherically symmetric model for the combustion of a viscous, compressible, radiative-reactive gas with a chemical kinetics equation. Under suitable assumptions, we establish some uniform-in-time estimates of global solutions to this model which improve some known results.     

The solvability conditions for inverse eigenproblem of symmetric and anti‐persymmetric matrices and its approximation

The problem of generating a matrix A with specified eigen‐pair, where A is a symmetric and anti‐persymmetric matrix, is presented. An existence theorem is given and proved. A general expression of such a matrix is provided. We denote the set of such matrices by ??????_Eⁿ. The optimal approximation problem associated with ??????_Eⁿ is discussed, that is: to find the nearest matrix to a given matrix A* by A∈??????_Eⁿ. The existence and uniqueness of the optimal approximation problem is proved and the expression is provided for this nearest matrix. Copyright © 2002 John Wiley & Sons, Ltd.     

Nondegenerate families of convex cones and convex polytopes

This paper describes relations between convex polytopes and certain families of convex cones in R ⁿ .The purpose is to use known properties of convex cones in order to solve Helly type problems for convex sets in R ⁿ or for spherically convex sets in S _n , the n-dimensional unit sphere. These results are strongly related to Gale diagrams.     

On symmetric basic sequences in lorentz sequence spaces II

It is shown that if {y _n} is a block of type I of a symmetric basis {x _n} in a Banach spaceX, then {y _n} is equivalent to {x _n} if and only if the closed linear span [y _n] of {y _n} is complemented inX. The result is used to study the symmetric basic sequences of the dual space of a Lorentz sequence spaced(a, p). Let {x _n,f _n} be the unit vector basis ofd(a, p), for 1≤p<+∞. It is shown that every infinite-dimensional subspace ofd(a, p) (respectively, [f _n] has a complemented subspace isomorphic tol _p (respectively,l _q, 1/p+1/q=1 when 1<p<+∞ andc ₀ whenp=1) and numerous other results on complemented subspaces ofd(a, p) and [f _n] are obtained. We also obtain necessary and sufficient conditions such that [f _n] have exactly two non-equivalent symmetric basic sequences. Finally, we exhibit a Banach spaceX with symmetric basis {x _n} such that every symmetric block basic sequence of {x _n} spans a complemented subspace inX butX is not isomorphic to eitherc ₀ orl _p, 1≤p<+∞.     

On the cut‐off phenomenon for the transitivity of randomly generated subgroups

Consider K ≥ 2 independent copies of the random walk on the symmetric group S_N starting from the identity and generated by the products of either independent uniform transpositions or independent uniform neighbor transpositions. At any time $n\in \mathbb{N}$, let G_n be the subgroup of S_N generated by the K positions of the chains. In the uniform transposition model, we prove that there is a cut‐off phenomenon at time N ln(N)/(2K) for the non‐existence of fixed point of G_n and for the transitivity of G_n, thus showing that these properties occur before the chains have reached equilibrium. In the uniform neighbor transposition model, a transition for the non‐existence of a fixed point of G_n appears at time of order $N^{1+\frac{2}{K}}$ (at least for K ≥ 3), but there is no cut‐off phenomenon. In the latter model, we recover a cut‐off phenomenon for the non‐existence of a fixed point at a time proportional to N by allowing the number K to be proportional to ln(N). The main tools of the proofs are spectral analysis and coupling techniques. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Cyclic Tableaux and Symmetric Functions

We introduce the notion of cyclic tableaux and develop involutions for Waring's formulas expressing the power sum symmetric function p_n in terms of the elementary symmetric function e_n and the homogeneous symmetric function h_n. The coefficients appearing in Waring's formulas are shown to be a cyclic analog of the multinomial coefficients, a fact that seems to have been neglected before. Our involutions also spell out the duality between these two forms of Waring's formulas, which turns out to be exactly the “duality between sets and multisets.” We also present an involution for permutations in cycle notation, leading to probably the simplest combinatorial interpretation of the Möbius function of the partition lattice and a purely combinatorial treatment of the fundamental theorem on symmetric functions. This paper is motivated by Chebyshev polynomials in connection with Waring's formula in two variables.     

Approximation Numbers of Identity Operators between Symmetric Sequence Spaces

We prove asymptotic formulas for the behavior of approximation quantities of identity operators between symmetric sequence spaces. These formulas extend recent results of Defant, Masty o, and Michels for identities l_pⁿF_n with an n-dimensional symmetric normed space F_n with p-concavity conditions on F_n and 1p2. We consider the general case of identities E_nF_n with weak assumptions on the asymptotic behavior of the fundamental sequences of the n-dimensional symmetric spaces E_n and F_n. We give applications to Lorentz and Orlicz sequence spaces, again considerably generalizing results of Pietsch, Defant, Masty o, and Michels.