On a number of poisson matrices in Bang-Bang representations for 3 × 3 embeddable matrices

We give a counterexample to the Strong Bang-Bang Conjecture according to which any 3 × 3 embeddable matrix can be expressed as a product of six Poisson matrices. We exhibit a 3 × 3 embeddable matrix which can be expressed as a product of seven but not six Poisson matrices. We show that an embeddable 3 × 3 matrix P with det $\text{P ≥}\text{1}\text{8}$ can be expressed as a product of at most six Poisson matrices and give necessary and sufficient conditions for a 3 × 3 stochastic matrix P with det $\text{P ≥}\text{1}\text{8}$ to be embeddable. For an embeddable 3 × 3 matrix P with det $\text{P <}\text{1}\text{8}$ we give a new bound for the number of Poisson matrices in its Bang-Bang representation.     

A problem of Mirsky concerning nonsingular doubly stochastic matrices

The paper presents a new constructive proof of a theorem of Hardy, Littlewood, and Polya relating vector majorization and doubly stochastic matrices. Conditions on the vectors which guarantee that the corresponding matrices will be direct sums are given. These two results are applied to solve the problem, posed by Mirsky, of characterizing those majorization relations for which there is a corresponding doubly stochastic matrix which is nonsingular.     

The community problem for Stieltjes matrices

The communality problem in factor analysis is that of reducing the diagonal elements of a correlation matrix so that the resulting matrix will be positive semidefinite and of minimum rank. The problem is well studied, but no effective solution procedures have been devised. In this paper, we propose a variant problem and give an algorithm for its solution. We prove that a solution to this problem also solves the communality problem if the correlation matrix is Stieltjes.     

A structure of the Bang-Bang representation for 3×3 embeddable matrices

Summary Necessary and sufficient conditions are given for a 3×3 stochastic matrix to be embeddable by 6 elementary stochastic matrices (Poisson matrices). For a 3×3 embeddable matrix, a structure of the minimal Bang-Bang representation, i.e. the one that contains the smallest number of elementary matrices, is obtained. Based on the minimal Bang-Bang representation an algorithm for determining the embeddability of a 3×3 stochastic matrix is given.I would like to thank Søren Johansen for helpful comments and stimulating discussions on the subject of this paper     

Estimates for eigenvalues of stochastic matrices

It is well-known that the eigenvalues of stochastic matrices lie in the unit circle and at least one of them has the value one. Let {1, r 2 , ··· , r N } be the eigenvalues of stochastic matrix X of size N × N . We will present in this paper a simple necessary and sufficient condition for X such that |r j | < 1, j = 2, ··· , N . Moreover, such condition can be very quickly examined by using some search algorithms from graph theory.     

The stochastic goodwill problem

Stochastic control problems related to optimal advertising under uncertainty are considered. In particular, we determine the optimal strategies for the problem of maximizing the utility of goodwill at launch time and minimizing the disutility of a stream of advertising costs that extends until the launch time for some classes of stochastic perturbations of the classical Nerlove–Arrow dynamics. We also consider some generalizations such as problems with constrained budget and with discretionary launching.     

The stochastic trim-loss problem

The cutting stock problem (CSP) is one of the most fascinating problems in operations research. The problem aims at determining the optimal plan to cut a number of parts of various length from an inventory of standard-size material so to satisfy the customers demands. The deterministic CSP ignores the uncertain nature of the demands thus typically providing recommendations that may result in overproduction or in profit loss. This paper proposes a stochastic version of the CSP which explicitly takes into account uncertainty. Using a scenario-based approach, we develop a two-stage stochastic programming formulation. The highly non-convex nature of the model together with its huge size prevent the application of standard software. We use a solution approach designed to exploit the specific problem structure. Encouraging preliminary computational results are provided.     

The efficiency criteria problem for stochastic processes

This paper is concerned with the problem of finding a suitable (asymptotic) efficiency criterion for inference concerning parameters of stochastic processes. Special attention is aid to conditional exponential families of stochastic processes and to three tests based on the maximum likelihood estimate as well as to the likelihood ratio test. A contiguity calculation is used to show that a previously suggested criterion is inadequate and itself provides a partial solution to the problem. A heuristic argument is also put forward to support a proposition implying the optimality of the maximum likelihood estimate in a certain sense. Two examples which illustrate the theory are discussed.     

The asymptotic number of integer stochastic matrices

Let Hⁿ_r be the number of n × n matrices, with nonnegative integer elements, all of whose row and column sums are equal to some prescribed integer r. Similarly, let Aⁿ_r be the number of n × n (0.1) matrices with common row and column sum r. An asymptotic formula for Hⁿ_r is stated and proved, the method of proof being essentially elementary. A simple modification of the proof yields an analogous asymptotic formula for Aⁿ_r. The latter agrees with a result of O'Neil, obtained by a completely different method.     

Essentially doubly stochastic matrices

In this paper we extend the general theory of essentially doubly stochastic (e.d.s.) matrices begun in earlier papers in this series. We complete the investigation in one direction by characterizing all of the algebra isomorphisms between the algebra of e.d.s. matrices of order n over a field F,E_n(F), and the total algebra of matrices of order n - 1over F,M_n-1(F) We then develop some of the theory when Fis a field with an involution. We show that for any e,f§Fof norm 1,e≠f every e.d.s. matrix in E_n(F) is a unique e.d.s. sum of an e.d.s. e-hermitian matrix and an e.d.s. f-hermitian matrix in E_n(F) Next, we completely determine the cases for which there exists an above-mentioned matrix algebra isomorphism preserving adjoints. Finally, we consider cogredience in E_n(F) and show that when such an adjoint-preserving isomorphism exists and char M_n(F) two e.d.s. e-hermitian matrices which are cogredient in M_n(F) are also cogredient in E_n(F). Using this result, we obtain simple canonical forms for cogredience of e.d.s. e-hermitian matrices in E_n(F) when Fsatisfies special conditions. This ncludes the e.d.s. skew-symmetric matrices, where the involution is trivial and E = -1.     

Power symmetric stochastic matrices

Stochastic matrices A which satisfy the equation A^T=A^p are characterized for integral values of p > 1.     

A maximum problem for matrices

The following maximum problem is considered: To find among all contractions T on an n-dimensional Hilbert space whose spectral radius does not exceed a given number p< 1, the operator T for which |Tⁿ| is maximum. A matrix T of Toeplitz type is constructed for which this maximum is attained.     

Bounds for eigenvalues of certain stochastic matrices

We consider the class of stochastic matrices M generated in the following way from graphs: if G is an undirected connected graph on n vertices with adjacency matrix A, we form M from A by dividing the entries in each row of A by their row sum. Being stochastic, M has the eigenvalue λ=1 and possibly also an eigenvalue λ=-1. We prove that the remaining eigenvalues of M lie in the disk ¦λ¦?1–n^-3, and show by examples that the order of magnitude of this estimate is best possible. In these examples, G has a bar-bell structure, in which n/3 of the vertices are arranged along a line, with n/3 vertices fully interconnected at each end. We also obtain better bounds when either the diameter of G or the maximal degree of a vertex is restricted.     

Ergodicity for products of infinite stochastic matrices

A sufficient condition ensuring weak ergodicity asr of productsP _m,r ={p _ij ^(m,r) }=P _m+1 P _m+2 P _m+r formed from a sequence {P _k } of infinite stochastic matrices each of which contains no zero column, is given. The condition framed in terms of a generalization of Birkhoff's coefficient of ergodicity to such matrices, ensures also thatp _is ^(m,r) /p _js ^(m,r) 1 asr uniformlyiss, for fixedi, j, m. The result, which relies partly on work of Gibert and Mukherjea,⁽⁴⁾ also generalizes a classical result of Kolmogorov.⁽⁶⁾ A corresponding discussion is given for backwards products.Forms part of results announced at the conference 50 years after Doeblin: Developments in the theory of Markov chains, Markov processes and sums of random variables held at Blaubeuren, Germany, November 2–7, 1991.