The graphs of polytopes with involutory automorphisms

Steinitz’ theorem states that a graph is the graph of a 3-dimensional convex polytope if and only if it is planar and 3-connected. Grünbaum has shown that Steinitz’ proof can be modified to characterize the graphs of polytopes that are centrally symmetric or have a plane of symmetry. We show how to modify Steinitz’ proof to take care of the remaining involutory case—polytopes that are symmetric about a line. Research supported by NSF Grant GP-3470.     

Moral responsibility—What is all the fuss about?

Examination of several accounts regarding the nature of moral responsibility allows the extraction of a conceptual core common to all of them. Relying on that core conception of moral responsibility, the paper explores what human life without moral responsibility would be like. That exploration establishes that many robust forms of human relationship and nonmoral normativity could continue, absent moral responsibility, even if moral responsibility were abandoned on incompatibilist grounds. Much more importantly, it also establishes, contra Waller and Pereboom, that only some forms of morality—so-called “behavioral” forms—remain possible without moral responsibility. The paper argues that normative moral approaches that take into account agent intentions in order to assess the moral status of action cannot be applied without moral responsibility of agents. Thus, morality without responsibility needs to be behavioral, not consequentialist, as has often been thought.     

Covariant and dynamical reduction for principal bundle field theories

Reduction for field theories with symmetry can be done either covariantly—that is, on spacetime—or dynamically—that is, after spacetime is split into space and time. The purpose of this article is to show that these two reduction procedures are, in an appropriate sense, equivalent for a class of field theories whose fields take values in a principal bundle. One can think of this class of field theories as including examples such as a “sea of rigid bodies” with and appropriate interbody coupling potential.     

Hilbert’s early career: Encounters with allies and rivals

It seems to me that the mathematicians of today understand each other far too little and that they do not take an intense enough interest in one another. They also seem to know—so far as I can judge—too little of our classical authors (Klassiker); many, moreover, spend much effort working on dead ends. “ David Hilbert to Felix Klein, 24 July 1890     

Continuous Selections and Locally Pseudocompact Groups

We characterize locally pseudocompact groups by means of the selection theory. Our result is the selection version of the well-known Comfort—Ross theorem on pseudocompactness which states that a topological group is pseudocompact if and only its Stone—Čech compactification is a topological group.     

Equivalence between Nonlinear H ∞ Control Problems and Existence of Viscosity Solutions of Hamilton—Jacobi—Isaacs Equations rid="*" id="*" This work was written while the author was visiting the University of California at Santa Barbara. He was partially supported by Consiglio Nazionale delle Ricerche of Italy.

In this paper we extend to completely general nonlinear systems the result stating that the suboptimal control problem is solved if and only if the corresponding Hamilton—Jacobi—Isaacs (HJI) equation has a nonnegative (super)solution. This is well known for linear systems, using the Riccati equation instead of the HJI equation. We do this using the theory of differential games and viscosity solutions. Accepted 14 February 1997     

Generalization of Primal—Dual Interior-Point Methods to Convex Optimization Problems in Conic Form

We generalize primal—dual interior-point methods for linear programming (LP) problems to the convex optimization problems in conic form. Previously, the most comprehensive theory of symmetric primal—dual interior-point algorithms was given by Nesterov and Todd for feasible regions expressed as the intersection of a symmetric cone with an affine subspace. In our setting, we allow an arbitrary convex cone in place of the symmetric cone. Even though some of the impressive properties attained by Nesterov—Todd algorithms are impossible in this general setting of convex optimization problems, we show that essentially all primal—dual interior-point algorithms for LP can be extended easily to the general setting. We provide three frameworks for primal—dual algorithms, each framework corresponding to a different level of sophistication in the algorithms. As the level of sophistication increases, we demand better formulations of the feasible solution sets. Our algorithms, in return, attain provably better theoretical properties. We also make a very strong connection to quasi-Newton methods by expressing the square of the symmetric primal—dual linear transformation (the so-called scaling) as a quasi-Newton update in the case of the least sophisticated framework. August 25, 1999. Final version received: March 7, 2001.     

Heteroclinic cycles in lattice dynamical systems

A criterion of spatial chaos occurring in lattice dynamical systems—heteroclinic cycle—is discussed It is proved that if the system has asymptotically stable heteroclinic cycle, then it has asymptotically stable homoclinic point which implies spatial chaos. Project supported by the National Natural Science Foundation of China.     

Trees and the bireflection property

The group of automorphisms of a tree (partially ordered set where the set of predecessors of an element is well ordered) with no infinite levels enjoys the property that every member is a product of two elements of order ≦2. It is shown that this property—called the bireflection property—fails for some trees having infinite levels. In fact, every subtree of a treeT has the the bireflection property if and only if the tree of all zero-one sequences of length ≦ω with finitely many ones is not embeddable inT.     

Embedding Levy families into Banach spaces

We prove that if a metric probability space with a usual concentration property embeds into a finite dimensional Banach space X, then X has a Euclidean subspace of a proportional dimension. In particular this yields a new characterization of weak cotype 2. We also find optimal lower estimates on embeddings of metric spaces with concentration properties into , generalizing estimates of Bourgain—Lindenstrauss—Milman, Carl—Pajor and Gluskin. Submitted: February 2001, Revised: August 2001.     

Completely reducible Lie subalgebras

Let G be a connected and reductive group over the algebraically closed field K. J-P. Serre has introduced the notion of a G-completely reducible subgroup H ⊂ G. In this paper, we give a notion of G-complete reducibility—G-cr for short—for Lie subalgebras of Lie(G), and we show that if the closed subgroup H ⊂ G is G-cr, then Lie(H) is G-cr as well.     

Arithmetically Cohen Macaulay embeddings of reducible curves

Here we study the arithmetically Cohen—Macaulay non-special embeddings of reducible connected complete algebraic curves. We allow to take “general” the irreducible components and the line bundle (with fixed multidegree) and in this case we obtain sharp results.     

Convergence Analysis of an Inexact Potential Reduction Method for Convex Quadratic Programming

We analyze the convergence of an infeasible inexact potential reduction method for quadratic programming problems. We show that the convergence of this method is achieved if the residual of the KKT system satisfies a bound related to the duality gap. This result suggests stopping criteria for inner iterations that can be used to adapt the accuracy of the computed direction to the quality of the potential reduction iterate in order to achieve computational efficiency. This research was partially supported by the Italian MIUR, Project FIRB—Large Scale Nonlinear Optimization # RBNE01WBBB and Project PRIN—Innovative Problems and Methods in Nonlinear Optimization # 2005017083.     

Some remarks on conjugate closed groups

Let G be a group. In this note we define conjugate closed groups, which are briefly called CC — Groups. These groups form a proper subclass of T — Groups. We prove that if G = Z(G) × H, then G is conjugate closed if and only if H is conjugate closed. We also show that a finite group G is semisimple, conjugate closed and perfect if and only if it is a direct product of non-abelian and simple groups.     

Weak Convergence of Varying Measures and Hermite—Padé Orthogonal Polynomials

It is known that the common denominator of the Hermite—Padé approximants of a mixed Angelesco—Nikishin system shares orthogonality relations with respect to each function in the system. It is less well known that they also satisfy full orthogonality with respect to a varying measure. This problem motivates our interest in extending the class of varying measures with respect to which weak asymptotics of orthogonal polynomials takes place. In particular, for the case of a Nikishin system, we prove weak asymptotics of the corresponding varying measures. October 23, 1997. Date revised: September 23, 1998. Date accepted: November 10, 1998.     

A double-dual characterization of separable Banach spaces containingl 1

It is proved that a separable Banach spaceB contains a subspace isomorphic tol ¹ if (and only if) there exists an element inB**, the double-dual ofB, which is not a weak* limit of a sequence of elements inB. ConsequentlyB contains an isomorph ofl ¹ if (and only if) the cardinality ofB** is greater than that of the continuum. The research of the second-named author was partially supported by NSF — GP — 30798X1     

Viscosity Solutions of an Infinite-Dimensional Black—Scholes—Barenblatt Equation

We study an infinite-dimensional Black—Scholes—Barenblatt equation which is a Hamilton—Jacobi—Bellman equation that is related to option pricing in the Musiela model of interest rate dynamics. We prove the existence and uniqueness of viscosity solutions of the Black—Scholes—Barenblatt equation and discuss their stochastic optimal control interpretation. We also show that in some cases the solution can be locally uniformly approximated by solutions of suitable finite-dimensional Hamilton—Jacobi—Bellman equations.     

Superstable operators on Banach spaces

It has been shown by W. Arendt—C.J.K. Batty and Yu.I. Lyubich— V.Q. Phong that the powers of a linear contraction on a reflexive Banach space converge strongly to zero if the boundary spectrum is countable and contains no eigenvalues. In this paper we characterize the countability of the boundary spectrum through a stronger convergence property in terms of ultrapower extensions. This paper is part of a research project supported by the Deutsche Forschungsgemeinschaft DFG     

Quasi-stationary Distributions for a Class of Discrete-time Markov Chains

This paper is concerned with the circumstances under which a discrete-time absorbing Markov chain has a quasi-stationary distribution. We showed in a previous paper that a pure birth-death process with an absorbing bottom state has a quasi-stationary distribution—actually an infinite family of quasi-stationary distributions— if and only if absorption is certain and the chain is geometrically transient. If we widen the setting by allowing absorption in one step (killing) from any state, the two conditions are still necessary, but no longer sufficient. We show that the birth–death-type of behaviour prevails as long as the number of states in which killing can occur is finite. But if there are infinitely many such states, and if the chain is geometrically transient and absorption certain, then there may be 0, 1, or infinitely many quasi-stationary distributions. Examples of each type of behaviour are presented. We also survey and supplement the theory of quasi-stationary distributions for discrete-time Markov chains in general.