Triangles in 3-connected matroids

A collection F of 3-connected matroids is triangle-rounded if, whenever M is a 3-connected matroid having a minor in F, and T is a 3-element circuit of M, then M has a minor which uses T and is isomorphic to a member of F. An efficient theorem for testing a collection of matroids for this property is presented. This test is used to obtain several results including the following extension of a result of Asano, Nishizeki, and Seymour. Let T be a 3-element circuit of a 3-connected binary nonregular matroid M with at least eight elements. Then M has a minor using T that is isomorphic to S₈ or the generalized parallel connection across T of F₇ and M(K₄).     

Products of involutory matrices. I

It is shown that, for every integer ≥1 and every field F, each n×n matrix over F of determinant ±1 is the product of four involutory matrices over F. Products of three ×n involutory matrices over F are characterized for the special cases where n≤4 or F has prime order ≤5. It is also shown for every field F that every matrix over F of determinant ±1 having no more than two nontrivial invariant factors is a product of three involutory matrices over F.     

Convexoid and generalized derivations

Let E,F be two Banach spaces and let S be a symmetric norm ideal of L(E,F). For AL(F) and BL(E) the generalized derivation δ_S,A,B is the operator on S that sends X to AX−XB. A bounded linear operator is said to be convexoid if its (algebraic) numerical range coincides with the convex hull of its spectrum. We show that δ_S,A,B is convexoid if and only if A and B are convexoid.     

Random packing by ρ-connected ρ-regular graphs

A graph G is packable by the graph F if its edges can be partitioned into copies of F. If deleting the edges of any F-packable subgraph from G leaves an F-packable graph, then G is randomly F-packable. If G is F-packable but not randomly F-packable then G is F-forbidden. The minimal F-forbidden graphs provide a characterization of randomly F-packable graphs. We show that for each ρ-connected ρ-regular graph F with ρ > 1, there is a set (F) of minimal F-forbidden graphs of a simple form, such that any other minimal F-forbidden graph can be obtained from a graph in (F) by a process of identifying vertices and removing copies of F. When F is a connected strongly edge-transitive graph having more than one edge (such as a cycle or hypercube), there is only one graph in (F).     

Limit cycles of generalized Liénard equations

The generalized Liénard equations of the form:

where F, g, and h are polynomials, are examined. It has been found that the results given by Blows, Lloyd and Lynch [1–5] for Liénard equations hold also for the generalized systems. A new result is also presented within this article.     

Linear mappings which are rank-k nonincreasing, II

We prove the following result. Let F be an infinite field of characteristic other than two. Let k be a positive integer. Let S_n(F) denote the space of all n × n symmetric matrices with entries in F, and let T:S_n(F)→S_n(F) be a linear operator. Suppose that T is rank-k nonincreasing and its image contains a matrix with rank higher than K. Then, there exist λεF and PεF^n,n such that T(A)=λPAP^t for all AεS_n(F). λ can be chosen to be 1 if F is algebraically closed and ±1 if F=R, the real field.     

On modeling failure and repair times in stochastic models of manufacturing systems using generalized exponential distributions

Failures of machines have a significant effect on the behavior of manufacturing systems. As a result it is important to model this phenomenon. Many queueing models of manufacturing systems do incorporate the unreliability of the machines. Most models assume that the times to failure and the times to repair of each machine are exponentially distributed (or geometrically distributed in the case of discrete-time models). However, exponential distributions do not always accurately represent actual distributions encountered in real manufacturing systems. In this paper, we propose to model failure and repair time distributions bygeneralized exponential (GE) distributions (orgeneralized geometric distributions in the case of a discretetime model). The GE distribution can be used to approximate distributions with any coefficient of variation greater than one. The main contribution of the paper is to show that queueing models in which failure and repair times are represented by GE distributions can be analyzed with the same complexity as if these distributions were exponential. Indeed, we show that failures and repair times represented by GE distributions can (under certain assumptions) be equivalently represented by exponential distributions.This work was performed while the author was visiting the Laboratory for Manufacturing and Productivity, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.     

Noncentral matrix quadratic forms of the skew elliptical variables

In this paper, the noncentral matrix quadratic forms of the skew elliptical variables are studied. A family of the matrix variate noncentral generalized Dirichlet distributions is introduced as the extension of the noncentral Wishart distributions, the Dirichlet distributions and the noncentral generalized Dirichlet distributions. Main distributional properties are investigated. These include probability density and closure property under linear transformation and marginalization, the joint distribution of the sub-matrices of the matrix quadratic forms in the skew elliptical variables and the moment generating functions and Bartlett's decomposition of the matrix quadratic forms in the skew normal variables. Two versions of the noncentral Cochran's Theorem for the matrix variate skew normal distributions are obtained, providing sufficient and necessary conditions for the quadratic forms in the skew normal variables to have the matrix variate noncentral generalized Dirichlet distributions. Applications include the properties of the least squares estimation in multivariate linear model and the robustness property of the Wilk's likelihood ratio statistic in the family of the matrix variate skew elliptical distributions.     

A characterization on n-critical economical generalized tic-tac-toe games

There is a so called generalized tic-tac-toe game playing on a finite set X with winning sets A₁, A₂,…, A_m. Two players, F and S, take in turn a previous untaken vertex of X, with F going first. The one who takes all the vertices of some winning set first wins the game. Erd s and Selfridge proved that if |A₁|=|A₂|==|A_m|=n and m<2ⁿ⁻¹, then the game is a draw. This result is best possible in the sense that once m=2ⁿ⁻¹, then there is a family A₁, A₂,…, A_m so that F can win. In this paper we characterize all those sets A₁,…, A_2n−1 so that F can win in exactly n moves. We also get similar result in the biased games.     

The generalized Gompertz distribution

This paper deals with a new generalization of the exponential, Gompertz, and generalized exponential distributions. This distribution is called the generalized Gompertz distribution (GGD). The main advantage of this new distribution is that it has increasing or constant or decreasing or bathtub curve failure rate depending upon the shape parameter. This property makes GGD is very useful in survival analysis. Some statistical properties such as moments, mode, and quantiles are derived. The failure rate function is also derived. The maximum likelihood estimators of the parameters are derived using a simulations study. Real data set is used to determine whether the GGD is better than other well-known distributions in modeling lifetime data or not.