Perturbation analysis for the trace quotient problem

Given a symmetric matrix B?∈?? ^m×m and a symmetric and positive-definite matrix W?∈?? ^m×m , maximizing the ratio trace(V ^? BV)/trace(V ^? WV) with respect to V?∈?? ^m×? (??≤?m) subject to the orthogonal constraint V ^? V?=?I _? is called the trace quotient problem or the trace ratio problem (TRP). TRP arises originally from the linear discriminant analysis (LDA), which is a popular approach for feature extraction and dimension reduction. It has been known that TRP is equivalent to a nonlinear extreme eigenvalue problem and very efficient method has been proposed to find a global optimal solution successfully. The matrices B and W arising in LDA are constructed from samples, and thereby are contaminated by noises and errors. In this article, we perform a perturbation analysis for TRP assuming the original B and W are perturbed. The upper perturbation bounds of both the global optimal value and the set of global optimal solutions are derived, and numerical investigation is carried out to illustrate these perturbation estimates.     

A note on utumi quotient rings

Let R be a ring and ρ a right ideal of R with zero right annihilate. Then ρ and R have the same left Utumi quotient ring. We study the lifting properties of GPIs and some chain conditions inherited by such right ideals. Next, we prove a generalization of Chatters’ theorem. Precisely, we show that if R is a right nonsingular ring with finite right Goldie dimension and possesses a right ideal ρ such that both ρ and l _R(ρ) are PI-rings, then the right Utumi quotient ring of R is also a Pi-ring.     

A note on the Waring-Goldbach problem

The focus of this paper will be the extension of the Waring-Goldbach problem to all sufficiently large integers, without congruence restrictions. By reintroducing the effect of small primes, we are able to consider questions which more naturally resemble Waring's problem and the Goldbach conjecture. We extend the results of S.S. Pillai by considering the problem without the use of zero as an addend and we give a small improvement on the number of additional terms required.     

A note on the complementarity problem

We show by an example that, in a complementarity problem where the given map is continuous and monotone on the nonnegative orthant, the existence of a feasible solution is not sufficient to guarantee existence of a solution to the complementarity problem.The author thanks Professor S. Karamardian and Dr. J. More for helpful discussions regarding this note.     

A note on the syzygy problem

In the theorem of Evans and Griffith on the syzygy problem. The restriction on the ring being Cohen Macaulay domain is completely eliminated.     

A note on Fermat's problem

The General Fermat Problem asks for the minimum of the weighted sum of distances fromm points inn-space. Dozens of papers have been written on variants of this problem and most of them have merely reproduced known results. This note calls attention to the work of Weiszfeld in 1937, who may have been the first to propose an iterative algorithm. Although the same algorithm has been rediscovered at least three times, there seems to be no completely correct treatment of its properties in the literature. Such a treatment, including a proof of convergence, is the sole object of this note. Other aspects of the problem are given scant attention.This paper was written while the author was Science Faculty Fellow of the National Science Foundation at the London School of Economics.     

A note on Chern-Yamabe problem

We propose a flow to study the Chern-Yamabe problem and discuss the long time existence of the flow. In the balanced case we show that the Chern-Yamabe problem is the Euler-Lagrange equation of some functional. The monotonicity of the functional along the flow is derived. We also show that the functional is not bounded from below.     

A note on the abstract cauchy problem

LetA be the infinitesimal generator of aC ₀ semigroup in a Banach spaceE. We obtain necessary conditions for a solution of the Cauchy problem {fx112-1} to be classical for arbitrary ϕ εC([0,T]) andf εE.     

A note on the rotating chain problem

For a non-linear boundary value problem describing the shape of a heavy rotating chain a priori estimates of the maximal deflection are obtained. Also, for the transformed chain problem an extremum Variational principle and error estimates are presented.

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Zusammenfassung Für eine nichtlineare Differentialgleichung, die die Form einer schwer rotierenden Kette beschreibt, werden a priori Abschätzungen der maximalen Ausbiegung erhalten. Dabei werden für ein transformiertes Ketten-Problem ein extremales Variationsprinzip und die Feller Abschätzung angegeben.

     

A note on the three color problem

It is shown that a planar graph withouti-circuits, 4 i 9, is 3-colorable. This result strengthens the result obtained by H.L. Abbott and B. Zhou.The author's research was partially supported by the Office of Naval Research, Grant number N00014-92-J-1965.     

A note on the Weber location problem

In this note, we collect some interesting and useful results about the Weber problem. We investigate an accelerated Weiszfeld procedure which increases the step size and find a formula for the step size that empirically produces the fastest convergence rate. We also derive an estimate for the optimal cost of the system.     

A note on the circle containment problem

We extend a result due to Bárányet al. and prove the following theorem: given any setS ofn points in the plane, there are pointsx andy inS, such that every circle that containsx andy contains at least [5/84(n – 2)] other points ofS.     

A note on the space charge problem

A partial differential equation that governs unipolar ionizationin a doubly-connected region is studied. This is often knownas the space charge problem. It is shown that if the potentialis smooth enough then neither the field nor the charge vanishesin the region.