Discretizing dynamic programs

Discretizing certain discrete-time, uncountable-state dynamic programs such that the respective solutions to a sequence of discretized versions converge uniformly to the solution of the original problem is shown. Via a random perturbation device, we apply our general approach to a separable nonconvex program.This paper was completed while the author was Visiting Professor of Management Science at the University of Chicago, Chicago, Illinois.     

Total risk aversion and the pricing of options

The objective of this paper is to derive the price of a call option in a stable market for amarket maker who is totally risk averse in the sense that he has a utility function with infinite index of absolute risk aversion. We prove that in the case when the borrowing rateR is the same as the lending rater the option price must be the Black—Scholes price with rateR = r. In the more interesting caseR > r, we prove that two option pricing functions are necessary and sufficient for making a market in the option. These two functions are the Black—choles prices with rater orR. Which price to use at each time will depend on the circumstances (buying or selling) and the capitalization of the market maker. We determine which price to use at each time and the optimal riskless portfolio.Partially supported by a grant, No. AFOSR-86-0202, from the Air Force Office for Scientific Research and a grant from Loyola University of Chicago.Partially supported by Grant No. AFOSR-86-0202 from the Air Force Office for Scientific Research, a grant from the National Science Foundation, and a grant from Loyola University of Chicago.     

One Year at Swarthmore: “When Laos Was Invaded,Nobody Budged”

Abstract

If You're So Smart: The Narrative of Economic Expertise by Donald McCloskey. Chicago: University of Chicago Press, 1990, ix + 180 pages, references, index, $17.95 hardbound.     

Jacobson - rings and hilbert algebras with polynomial identities

Summary We consider n-tuples of m × m matrices as zeroes of non-commutative polynomials in n-variables and establish an analogue of the classical Hilbert-Nullstellensatz. We study then finitely generated non-commutative algebras over Jacobson rings and obtain results conpletely analogous with the commutative tehory. In memory of Guido Castelnuovo, in the recurrence of the first centenary of his birth. Done partly under NSF Senior Foreign Scientist Fellowship at the Univ. of Chicago. The second named author has been supported by an A.R.O. grant n^o DA-ARO-D-31-124-G 501 at the Univ. of Chicago.     

An algorithm for minimizing a differentiable function subject to box constraints and errors

We consider the problem of minimizing a differentiable function ofn parameters, with upper and lower bounds on the parameters. The motivation for this work comes from the optimization of the design of transient electrical circuits. In such optimization, the parameters are circuit elements, the bound constraints keep these parameters physically meaningful, and both the function and gradient evaluations contain errors. We describe a quasi-Newton algorithm for such problems. This algorithm handles the box constraints directly and approximates the given function locally by nonsingular quadratic functions. Numerical tests indicate that the algorithm can tolerate the errors, if the errors in the function and gradient are of the same relative size.This paper was presented at the SIAM National Meeting, Chicago, Illinois, 1976.This research was sponsored in part by the Air Force Office of Scientific Research (AFSC), United States Air Force, under Contract No. F44620-76-C-0022.     

Numerical evaluation of gauge invariants for <Emphasis Type="Italic">a</Emphasis>-gauge solutions in open string field theory

We explore the vacuum structure in the bosonic open string field theory expanded near an identity-based solution parameterized by a (≥ −1/2). Analyzing the expanded theory using the level-truncation approximation up to the level 20, we find that the theory has the tachyon vacuum solution for a ≥ −1/2. We also find that at a = −1/2, there exists an unstable vacuum solution in the expanded theory and the solution is expected to be the perturbative open string vacuum. These results reasonably support the hypothesis that the identity-based solution is a trivial pure gauge configuration for a > −1/2, but it can be regarded as the tachyon vacuum solution at a = −1/2.     

An Ω(n 4/3) lower bound on the randomized complexity of graph properties

We improve King's (n ^5/4) lower bound on the randomized decision tree complexity of monotone graph properties to (n ^4/3). The proof follows Yao's approach and improves it in a different direction from King's. At the heart of the proof are a duality argument combined with a new packing lemma for bipartite graphs.The paper was written while the author was a graduate student at the University of Chicago and was completed at M.I.T. The work was supported in part by NSF under GRANT number NSF 5-27561, the Air Force under Contract OSR-86-0076 and by DIMACS (Center for Discret Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center-NSF-STC88-09648.     

Symmetry Properties of Positive Solutions of Parabolic Equations on ℝ N : II. Entire Solutions

We consider nonautonomous quasilinear parabolic equations satisfying certain symmetry conditions. We prove that each positive bounded solution u on ? ^N  × (?∞, T) decaying to zero at spatial infinity uniformly with respect to time is radially symmetric around some origin in ? ^N . The origin depends on the solution but is independent of time. We also consider the linearized equation along u and prove that each bounded (positive or not) solution is a linear combination of a radially symmetric solution and (nonsymmetric) spatial derivatives of u. Theorems on reflectional symmetry are also given.     

The structure of admissible points with respect to cone dominance

We study the set of admissible (Pareto-optimal) points of a closed, convex setX when preferences are described by a convex, but not necessarily closed, cone. Assuming that the preference cone is strictly supported and making mild assumptions about the recession directions ofX, we extend a representation theorem of Arrow, Barankin, and Blackwell by showing that all admissible points are either limit points of certainstrictly admissible alternatives or translations of such limit points by rays in the closure of the preference cone. We also show that the set of strictly admissible points is connected, as is the full set of admissible points.Relaxing the convexity assumption imposed uponX, we also consider local properties of admissible points in terms of Kuhn-Tucker type characterizations. We specify necessary and sufficient conditions for an element ofX to be a Kuhn-Tucker point, conditions which, in addition, provide local characterizations of strictly admissible points.Several results from this paper were presented in less general form at the National ORSA/TIMS Meeting, Chicago, Illinois, 1975.This research was supported, in part, by the United States Army Research Office (Durham), Grant No. DAAG-29-76-C-0064, and by the Office of Naval Research, Grant No. N00014-67-A-0244-0076. The research of the second author was partially conducted at the Center for Operations Research and Econometrics (CORE), Université Catholique de Louvain, Heverlee, Belgium.The authors are indebted to A. Assad for several helpful discussions and to A. Weiczorek for his careful reading of an earlier version of this paper.     

More analysis of double hashing

In [8], a deep and elegant analysis shows that double hashing is asymptotically equivalent to the ideal uniform hashing up to a load factor of about 0.319. In this paper we show how a randomization technique can be used to develop a surprisingly simple proof of the result that this equivalence holds for load factors arbitrarily close to 1.An earlier version of the paper was presented at the20th Annual ACM Symposium on Theory of Computing, Chicago, IL, May 1988.Supported by National Science Foundation Grants DCR 85-09667 and CCR 89-12063 at the University of California at IrvinePartially supported by National Science Foundation Grant DCR 85-09667 at the University of California at Irvine     

On packing bipartite graphs

G andH, two simple graphs, can be packed ifG is isomorphic to a subgraph of , the complement ofH. A theorem of Catlin, Spencer and Sauer gives a sufficient condition for the existence of packing in terms of the product of the maximal degrees ofG andH. We improve this theorem for bipartite graphs. Our condition involves products of a maximum degree with an average degree. Our relaxed condition still guarantees a packing of the two bipartite graphs.the paper was written while the authors were graduate students at the University of Chicago and was completed while the first author was at M.I.T. The work of the first author was supported in part by the Air Force under Contract OSR-86-0076 and by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center-NSF-STC88-09648. The work of the second author was supported in part by NSF grant CCR-8706518.     

Book Vignettes

Macarow , Leo , New Math—or New Myth? Chicago, Illinois: Greenview Publications, 1981, Pp. iv + 76, Lewis , R., and Tagg .     

Stochastic control with exit time and constraints,application to small time attainability of sets

We study the existence of a solution of controlled stochastic differential equations remaining in a given set of constraints at any time smaller than the exit time of a given open set. We also investigate the small time attainability of a given closed set K, i.e., the property that, for all arbitrary small time horizon T and for all initial condition in a sufficiently small neighborhood of K, there exists a solution to the controlled stochastic differential equation which reaches K before T.     

A Modified Regularization Method

We consider the Cauchy problem for the systems of the first-order ordinary differential equations with a small parameter E of degree q at the derivative. We study the possibility of solving this problem by means of the regularization method of the Lomov theory of singular perturbations. We show that, for q>1, an application of the procedure by Lomov leads only to the trivial solution to the problem in the class of resonance-free solutions. We suggest and describe a modification of the procedure which enables us to construct a nontrivial solution to the problem in the space of resonance-free solutions.     

Optimal Regularity and Long-Time Behavior of Solutions for the Westervelt Equation

We investigate an initial-boundary value problem for the quasilinear Westervelt equation which models the propagation of sound in fluidic media. We prove that, if the initial data are sufficiently small and regular, then there exists a unique global solution with optimal L _p -regularity. We show furthermore that the solution converges to zero at an exponential rate as time tends to infinity. Our techniques are based on maximal L _p -regularity for abstract quasilinear parabolic equations.     

Critical points with lack of compactness and singular dynamical systems

Sunto Si prova l'esistenza di punti critici di funzionali che non verificano la condizione (PS).I teoremi astratti vengono applicati per trovare soluzioni periodiche di sistemi dinamici con potenziali sia limitati sia con singolarità.

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This work has been done during a visit at the Math. Dept. of the University of Chicago. The Authors whish to thank for the kind hospitality.

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Supported by Ministero Pubblica Istruzione, 40%, Gruppo Nazionale «Calcolo delle Variazioni».     

Toward a solution to the high Weissenberg number problem

We present results of a numerical study of high Weissenberg number flows using the method of Trebotich, Colella and Miller, J. Comput. Phys. vol. 205 (2005). We consider benchmark viscoelastic flow of an Oldroyd-B fluid in four to one sudden contractions for We = 1, 10, Re = 1. We conclude that (i) underlying wave behavior in the solution for flows at critical and supercritical elastic Mach number prevents the solution from reaching a steady-state; and (ii) high resolution is needed to resolve structure in the solution downstream from the re-entrant corner in order to determine convergence. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Book Reviews

Book reviewed in this article: Buros, Oscar K. (editor), Highland Park, New Jersey: The Gryphon Press, 1975. Mathematics Tests and Reviews Science In the World Around Us by William C. Vergara. 1975 Guidebook to Michigan Taxes. Chicago: Commerce Clearing House, Inc.     

On the nucleolus of NTU-games defined by multiple objective linear programs

In this article we derive a class of cooperative games with non-transferable utility from multiple objective linear programs. This is done in order to introduce the nucleolus, a solution concept from cooperative game theory, as a solution to multiple objective linear problems.We show that the nucleolus of such a game is a singleton, which is characterized by inclusion in the least core and the reduced game property. Furthermore the nucleolus satisfies efficiency, anonymity and strategic equivalence.We also present a polynomially bounded algorithm for computation of the nucleolus. Letn be the number of objective functions. The nucleolus is obtained by solving at most2n linear programs. Initially the ideal point is computed by solvingn linear programs. Then a sequence of at mostn linear programs is solved, and the nucleolus is obtained as the unique solution of the last program.Financial support from Nordic Academy for Advanced Study (NorFA) is gratefully acknowledged. Part of this work was done during autumn 1993 at Institute of Finance and Management Science, Norwegian School of Economics and Business Administration.