Nonnegatively and positively curved invariant metrics on circle bundles

We derive and study necessary and sufficient conditions for an -bundle to admit an invariant metric of positive or nonnegative sectional curvature. In case the total space has an invariant metric of nonnegative curvature and the base space is odd dimensional, we prove that the total space contains a flat totally geodesic immersed cylinder. We provide several examples, including a connection metric of nonnegative curvature on the trivial bundle that is not a product metric.

     

Subinverses of fuzzy matrices

A matrix operation is examined for fuzzy matrices and interesting properties of fuzzy matrices are obtained using the operation. Particularly some properties concerning subinverses and regularity of fuzzy matrices are given and the largest subinverse is shown by the properties. The properties are closely related to inverses of fuzzy matrices and fuzzy equations. Moreover fuzzy preorders are examined using the matrix operation and basic properties are obtained. The results are considered to be useful for the theory of fuzzy matrices.     

Structured nonnegative matrix factorization with applications to hidden Markov realization and clustering

In this paper, we study the structured nonnegative matrix factorization problem: given a square, nonnegative matrix P, decompose it as P=VAV^? with V and A nonnegative matrices and with the dimension of A as small as possible. We propose an iterative approach that minimizes the Kullback-Leibler divergence between P and VAV^? subject to the nonnegativity constraints on A and V with the dimension of A given. The approximate structured decomposition P?VAV^? is closely related to the approximate symmetric decomposition P?VV^?. It is shown that the approach for finding an approximate structured decomposition can be adapted to solve the symmetric decomposition problem approximately. Finally, we apply the nonnegative decomposition VAV^? to the hidden Markov realization problem and to the clustering of data vectors based on their distance matrix.     

A direct method for obtaining discrete relaxation spectra from creep data

A direct method for obtaining discrete relaxation spectra from creep data is proposed. The conversion of creep data to relaxation data is avoided and the discrete relaxation times are freely adjustable. The nonnegative least square method is used to generate nonnegative discrete relaxation strength. Received: 10 November 1999 Accepted: 22 September 2000     

Nonnegative functions in weighted hardy spaces

Let W be a nonnegative summable function whose logarithm is also summable with respect to the Lebesgue measure on the unit circle. For 0?<?p?<?∞ , H^p (W) denotes a weighted Hardy space on the unit circle. When W?≡?1, H ^p(W) is the usual Hardy space H^p . We are interested in H^p ( W)₊ the set of all nonnegative functions in H^p ( W). If p?≥?1/2, H^p ₊ consists of constant functions. However H^p ( W)₊ contains a nonconstant nonnegative function for some weight W. In this paper, if p?≥?1/2 we determine W and describe H^p ( W)₊ when the linear span of H^p ( W)₊ is of finite dimension. Moreover we show that the linear span of H^p (W)₊ is of infinite dimension for arbitrary weight W when 0?<?p?<?1/2.     

A note on the computation of the CP-rank

The purpose of this note is to address the computational question of determining whether or not a square nonnegative matrix (over the reals) is completely positive and finding its CP-rank when it is. We show that these questions can be resolved by finite algorithms and we provide (non-polynomial) complexity bounds on the number of arithmetic/Boolean operations that these algorithms require. We state several open questions including the existence of polynomial algorithms to resolve the above problems and availability of algorithms for addressing complete positivity over the rationals and over {0, 1} matrices.     

Calculation of discrete relaxation modulus and creep compliance

Müntz theorems from classical approximation theory are used to lay the mathematical foundations for discrete spectral representations of the relaxation modulus and creep conpliance. Algorithms that can automatically generate nonnegative discrete relaxation and retardation strengths are formulated by using the nonnegative least square method. Received: 30 September 1998 Accepted: 20 May 1999     

Convergence of the solution of a nonsymmetric matrix Riccati differential equation to its stable equilibrium solution

We consider the initial value problem for a nonsymmetric matrix Riccati differential equation, where the four coefficient matrices form an M-matrix. We show that for a wide range of initial values the Riccati differential equation has a global solution X(t) on [0,∞) and X(t) converges to the stable equilibrium solution as t goes to infinity.     

An efficient monotone projected Barzilai–Borwein method for nonnegative matrix factorization

In this paper, we present an efficient method for nonnegative matrix factorization based on the alternating nonnegative least squares framework. Our approach adopts a monotone projected Barzilai–Borwein (MPBB) method as an essential subroutine where the step length is determined without line search. The Lipschitz constant of the gradient is exploited to accelerate convergence. Global convergence of the proposed MPBB method is established. Numerical results are reported to demonstrate the efficiency of our algorithm.