A convergence theorem of Rosen’s gradient projection method

Since Rosen’s gradient projection method was published in 1960, a rigorous convergence proof of his method has remained an open question. A convergence theorem is given in this paper. Part of this author’s work was done while he studied at the Department of Mathematics, University of California at Santa Barbara, and was supported by the National Science Foundation under Grant No. MCS83-14977. Part of this author’s work was done while he visited the Computer Science Department, University of Minnesota, Minneapolis, and was supported by the National Science Foundation under Grant No. MCS81-01214.     

Best simultaneous approximation to totally bounded sequences in Banach spaces

This paper is concerned with the problem of best weighted simultaneous approximations to totally bounded sequences in Banach spaces. Characterization results from convex sets in Banach spaces are established under the assumption that the Banach space is uniformly smooth. The first author is supported in part by Scientific Research Fund of Hunan Provincial Education Department (Grant No. 06C651); the second author is supported in part by the National Natural Science Foundation of China (Grant Nos. 10671175, 10731060) and Program for New Century Excellent Talents in University; the third author is supported in part by Projects MTM2006-13997-C02-01 and FQM-127 of Spain     

Perturbation of operators and applications to frame theory

A celebrated classical result states that an operator U on a Banach space is invertible if it is close enough to the identity operator I in the sense that ‖I−U‖<1. Here we show that U actually is invertible under a much weaker condition. As an application we prove new theorems concerning stability offrames (and frame-like decompositions) under perturbation in both Hilbert spaces and Banach spaces. The first named author is partially supported by grants from the U.S. National Science Foundation (grant no. NSF DMS-9201357), the Danish Natural Science Research Council (Grant no. 9401598), and grants from the University of Missouri System Research Board, and the MU Research Council. The second named author thanks the University of Missouri for its hospitality during a visit, where the first draft of the paper was written.     

On the geometric character of stress in continuum mechanics

This paper shows that the stress field in the classical theory of continuum mechanics may be taken to be a covector-valued differential two-form. The balance laws and other fundamental laws of continuum mechanics may be neatly rewritten in terms of this geometric stress. A geometrically attractive and covariant derivation of the balance laws from the principle of energy balance in terms of this stress is presented. Jerrold G. Marsden: Research partially supported by the California Institute of Technology and NSF-ITR Grant ACI-0204932.     

On the complexity of an optimal routing tree problem

A routing tree for a set of tasks is a decision tree which assigns the tasks to their destinations according to the features of the tasks. A weighted routing tree is one with costs attached to each link of the tree. Links of the same feature have the same cost. It is proved that the problem of finding a routing tree of the minimum cost for a given set of tasks of two features is NP-complete.This research was supported in part by theNSF grantsDCR-8501226 andDCR-8696135. Part of this work was done while the first author was at the Mathematical Sciences Research Institute, Berkeley, California, and while the second author was at the Department of Mathematics, University of California, Santa Barbara, California.     

Reserch problem

This section appears in nearly every issue. Contributions are invited, and should be sent to R. C. Thompson, Department of Mathematics, University of California at Santa Barbara, Santa Barbara, CA. 93106, USA.     

Centralisers of spaces of symmetric tensor products and applications

We show that the centraliser of the space of n-fold symmetric injective tensors, n≥2, on a real Banach space is trivial. With a geometric condition on the set of extreme points of its dual, the space of integral polynomials we obtain the same result for complex Banach spaces. We give some applications of this results to centralisers of spaces of homogeneous polynomials and complex Banach spaces. In addition, we derive a Banach-Stone Theorem for spaces of vector-valued approximable polynomials. This project was supported in part by Enterprise Ireland, International Collaboration Grant – 2004 (IC/2004/009). The second author was also partially supported by PIP 5272,UBACYTX108 and PICT 03-15033     

On strongly exposed points and frechet differentiability

Using properties of convex functionals, it is shown that closed and bounded convex sets in a class of Banach spaces which includes separable conjugate spaces are the closed convex hulls of their strongly exposed points. This research was supported by the National Research Council of Canada, Grant A-3999.     

The Fermat rule for multifunctions on Banach spaces

Using variational analysis, we study vector optimization problems with objectives being closed multifunctions on Banach spaces or in Asplund spaces. In particular, in terms of the coderivatives, we present Fermat’s rules as necessary conditions for an optimal solution of the above problems. As applications, we also provide some necessary conditions (in terms of Clarke’s normal cones or the limiting normal cones) for Pareto efficient points.This research was supported by a postdoctoral fellowship scheme (CUHK) and an Earmarked Grant from the Research Grant Council of Hong Kong. Research of the first author was also supported by the National Natural Science Foundation of P. R. China (Grant No. 10361008) and the Natural Science Foundation of Yunnan Province, P. R. China (Grant No. 2003A002M).     

The isoperimetric quotient and some classical Banach spaces

We obtain estimates for projection bodies of unit balls of certain Banach spaces. Some of these estimates involve the isoperimetric quotient. Research supported by NSF Grant DMS 86-02395.     

Linear pursuit-evasion games with bounded state spaces

The problem of linear pursuit-evasion games with bounded state spaces is considered. Some sufficient conditions for optimality are established, and an example is given.This research was carried out while the author was a Visiting Associate Research Engineer at the University of California at Berkeley. The research was supported by the Office of Naval Research, Grant No. N00014-69-A-0200-1012. The author would like to express his gratitude to Professor G. Leitmann for discussions and for making possible his visit at Berkeley.     

Convergence of Newton‘’s Method and Uniqueness of the Solution of Equations in Banach SpacesⅡ

Some results on convergence of Newton‘s method in Banach spaces are established under the assumption that the derivative of the opderators satisfies the radius or center Lipschitz condition with a weak L average.     

Lower bounds for weak Byzantine agreement

Lower bounds for Byzantine agreement obtained by Dolev and Strong are shown, by a simpler method, also to be true for weak Byzantine agreement.This work was done while the author was studying at University of California, Santa Barbara, in 1984.     

Controllability of Second-Order Differential and Integro-Differential Inclusions in Banach Spaces

In this paper, the controllability of second-order differential and integro-differential inclusions in Banach spaces are investigated. Two new results are obtained by using the theory of strongly continuous cosine families and a fixed point theorem for multivalued maps. Communicated by F. Zirilli Research supported by NNSF of China Grant 10571078, by NSF of Gansu Province of China Grant ZS011-A25-007-Z, and by the Teaching and Research Award Program for Outstanding Young Teacher in Higher Education Institutions, Ministry of Education of China.     

Research Problem

This Section appears in nearly every issue. Send submissions to R.C. Thompson, at the Algebra Institute, University, University of California, Santa Barbara, CA, 93106, U.S.A.     

Duality and Penalization in Optimization via an Augmented Lagrangian Function with Applications

This paper aims to establish duality and exact penalization results for the primal problem of minimizing an extended real-valued function in a reflexive Banach space in terms of a valley-at-0 augmented Lagrangian function. It is shown that every weak limit point of a sequence of optimal solutions generated by the valley-at-0 augmented Lagrangian problems is a solution of the original problem. A zero duality gap property and an exact penalization representation between the primal problem and the valley-at-0 augmented Lagrangian dual problem are obtained. These results are then applied to an inequality and equality constrained optimization problem in infinite-dimensional spaces and variational problems in Sobolev spaces, respectively. The first author was supported by the Research Committee of Hong Kong Polytechnic University, by Grant 10571174 from the National Natural Science Foundation of China and Grant 08KJB11009 from the Jiangsu Education Committee of China. The second author was supported by Grant BQ771 from the Research Grants Council of Hong Kong. We are grateful to the referees for useful suggestions which have contributed to the final presentation of the paper.     

A weak stochastic integral in Banach space with application to a linear stochastic differential equation

Cylindrical Wiener processes in real separable Banach spaces are defined, and an approximation theorem involving scalar Wiener processes is given for such processes. A weak stochastic integral for Banach spaces involving a cylindrical Wiener process as integrator and an operator-valued stochastic process as integrand is defined. Basic properties of this integral are stated and proved.A class of linear, time-invariant, stochastic differential equations in real, separable, reflexive Banach spaces is formulated in such fashion that a solution of the equation is a cylindrical process. An existence and uniqueness theorem is proved. A stochastic version of the problem of heat conduction in a ring provides an example.Research supported by National Science Foundation under Grant No. ECS-8005960.     

Research problem

This section appears in nearly every issue. Submissions should be sent to R. C. Thompson, Mathematics Department, University of California. Santa Barbara, CA. 93106, USA.     

Weak convergence to a Markov process: The martingale approach

Summary In this article, we obtain some sufficient conditions for weak convergence of a sequence of processes {X _n } toX, whenX arises as a solution to a well posed martingale problem. These conditions are tailored for application to the case when the state space for the processesX _n ,X is infinite dimensional. The usefulness of these conditions is illustrated by deriving Donsker's invariance principle for Hilbert space valued random variables. Also, continuous dependence of Hilbert space valued diffusions on diffusion and drift coefficients is proved.Research supported by National Board for Higher Mathematics, Bombay, IndiaPart of the work was done at University of California, Santa Barbara, USA     

Harmonic maps between complex projective spaces

Using the DoCarmo-Wallach theory, we classify (homogeneous) polynomial harmonic maps of complex projective spaces into spheres and complex projective spaces in terms of finite dimensional moduli spaces. We make use of representation theory of the (special) unitary group to give, for a spherical range, the exact dimension and, for complex projective spaces, a lower bound of the dimension of the moduli spaces.Supported in part by NSF Grant DMS 8603172.Research done in part while the third named author was on leave at University of California, Berkeley.