Functional Characterizations of Basic Properties of Utility Representations

 Consider uncertain alternatives for which an event has two consequences (binary gambles, “gambles” for short) and over them an operation of joint receipt which need not be closed and may be non-commutative. The two structures are linked by a distributivity property called segregation and a preference order. Utility functions order nonnegative numbers to consequences and gambles. Utility representations describe how the utility of a gamble depends on the utilities of consequences and on the “weight” of the event (a number in [0,1], depending on the event). Functional characterizations give necessary and sufficient conditions, often in form of functional equations, for certain properties of representations. We first give a functional characterization of the often postulated event commutativity stating that two events can be interchanged in special composite gambles where one outcome is a consequence but the other is itself a gamble. A utility representation is separable if it is multiplicative for gambles with one consequence having 0 utility. We give three more specific characterizations of separable representations by segregation, by homogeneity and event commutativity, and by homogeneity and segregation, and show that in the last case event commutativity follows.     

Separable and additive representations of binary gambles of gains

Two approaches are taken to a new utility representation of binary gambles that is called “ratio rank-dependent utility.” Both are based on known axiomatizations of a ranked-additive representation of consequence pairs (x, y) in binary gambles (x, C; y) of gains with C held fixed and of a separable one of the special gambles (x, C; e), where e denotes the status quo. The axiomatized version imposes the condition of status-quo event commutativity to get a functional equation that leads to the result. The other assumes, but does not axiomatize, a separable representation of the (C; y) portion of the gamble. These assumptions lead to two difficult functional equations that are solved in the mathematical literature, but the former only under the assumption that the function is twice differentiable. Three behavioral conditions are shown to force this new utility representation to reduce to the standard rank-dependent utility one for gains. They are co-monotonic trade-off consistency, ranked bisymmetry, and segregation, the latter requiring the addition of an operation of joint receipt.     

Approximate equivalence in von Neumann algebras

One formulation of D. Voiculescu's theorem on approximate unitary equivalence is that two unital representations π and ρ of a separable C*-algebra are approximately unitarily equivalent if and only if rank οπ = rank ορ. We study the analog when the ranges of π and ρ are contained in a von Neumann algebra R, the unitaries inducing the approximate equivalence must come from R, and "rank" is replaced with "R -rank" (defined as the Murray-von Neumann equivalence of the range projection).     

State Space Formulas for a Solution of the Suboptimal Nehari Problem on the Unit Disc

We give state space formulas for a (“central”) solution of the suboptimal Nehari problem for functions defined on the unit disc and taking values in the space of bounded operators in separable Hilbert spaces. Instead of assuming exponential stability, we assume a weaker stability concept (the combination of input-, output- and input-output stability), which allows us to solve the problem for general H-infinity functions.      

L p solutions of refinement equations

In the recent characterizations of the L_p solution of the refinement equation in terms of the “p-norm joint spectral radius,” there are problems in choosing the initial function for iteration [3, 23], or in addition, requiring stability of the refinable function [13, 17]. In this article we overcome these difficulties and give a more complete characterization of this nature. The criterion is constructive and can be implemented. It can be used to describe the regularity of the solution without assuming stability. This has significant advantages over the previous work. The corresponding results for vector refinement equations are also discussed.     

Central elements in pseudoeffect algebras

We introduce the definition of pseudoorthoalgebras and discuss some relationships between orthomodular lattices and pseudoorthoalgebras. Then we study the conditions that a pseudoeffect algebra is isomorphic to an “internal direct product” of ideals generated by orthogonal principal elements. At last, we give some characterizations of central elements in pseudoeffect algebras.     

Dual effect free stochastic controls

In stochastic optimal control, a key issue is the fact that “solutions” are searched for in terms of “closed-loop control laws” over available information and, as a consequence, a major potential difficulty is the fact that present control may affect future available information. This is known as the “dual effect” of control. Our main result consists in characterizing the maximal set of closed-loop control laws containing open-loop ones and for which the information provided by observations closed with such a feedback remains fixed. We give more specific results in the two following cases: multi-agent systems and discrete time stochastic input-output systems with dynamic information structure.     

Singular hyperelliptic curves

Singular curves with a morphism of degree two onto a projective line should be classified into two types according as the equipped morphism is separable or not; we call a curve with a separable one a hyperelliptic curve of separable type, and the other a hyperelliptic curve of inseparable type. We give concrete expressions of a hyperelliptic curve of separable type by means of its global “equation” and a hyperelliptic curve of inseparable type by means of its local rings. Furthermore, we discuss about Weierstrass points of a hyperelliptic curve of inseparable type. Received: 26 March 1997 / Revised version: 21 May 1998     

H-convergence of multiplicable matrices

The aim of this paper is to characterize the H-limit of multiplicable matrices, that is matrices such that there exist with where and have specific dependence on coordinates. We prove that, under suitable assumptions, the H-limit is also multiplicable. We give examples generalizing both the “stratified case” of F. Murat and L. Tartar and the “isotropic factorizable case” of A. Marino and S. Spagnolo. Received June 9, 1997 / Accepted October 3, 1997     

Convex programming with single separable constraint and bounded variables

In this paper a minimization problem with convex objective function subject to a separable convex inequality constraint “≤” and bounded variables (box constraints) is considered. We propose an iterative algorithm for solving this problem based on line search and convergence of this algorithm is proved. At each iteration, a separable convex programming problem with the same constraint set is solved using Karush-Kuhn-Tucker conditions. Convex minimization problems subject to linear equality/ linear inequality “≥” constraint and bounds on the variables are also considered. Numerical illustration is included in support of theory.     

Relation of orbital integrals on SO(5) and PGL(2)

We relate the “Fourier” orbital integrals of corresponding spherical functions on thep-adic groups SO(5) and PGL(2). The correspondence is defined by a “lifting” of representations of these groups. This is a local “fundamental lemma” needed to compare the geometric sides of the global Fourier summation formulae (or relative trace formulae) on these two groups. This comparison leads to conclusions about a well known lifting of representations from PGL(2) to PGSp(4). This lifting produces counter examples to the Ramanujan conjecture.     

Sobolev classes of Banach space-valued functions and quasiconformal mappings

We give a definition for the class of Sobolev functions from a metric measure space into a Banach space. We give various characterizations of Sobolev classes and study the absolute continuity in measure of Sobolev mappings in the “borderline case”. We show under rather weak assumptions on the source space that quasisymmetric homeomorphisms belong to a Sobolev space of borderline degree; in particular, they are absolutely continuous. This leads to an analytic characterization of quasiconformal mappings between Ahlfors regular Loewner spaces akin to the classical Euclidean situation. As a consequence, we deduce that quasisymmetric maps respect the Cheeger differentials of Lipschitz functions on metric measure spaces with borderline Poincaré inequality. J. H. supported by NSF grant DMS9970427. P. K. supported by the Academy of Finland, project 39788. N. S. supported in part by Enterprise Ireland. J. T. T. supported by an NSF Postdoctoral Research Fellowship.     

Two Characterizations of Optimality in Dynamic Programming

It holds in great generality that a plan is optimal for a dynamic programming problem, if and only if it is “thrifty” and “equalizing.” An alternative characterization of an optimal plan, that applies in many economic models, is that the plan must satisfy an appropriate Euler equation and a transversality condition. Here we explore the connections between these two characterizations.     

The Geometry of Standard Deontic Logic

Whereas geometrical oppositions (logical squares and hexagons) have been so far investigated in many fields of modal logic (both abstract and applied), the oppositional geometrical side of “deontic logic” (the logic of “obligatory”, “forbidden”, “permitted”, . . .) has rather been neglected. Besides the classical “deontic square” (the deontic counterpart of Aristotle’s “logical square”), some interesting attempts have nevertheless been made to deepen the geometrical investigation of the deontic oppositions: Kalinowski (La logique des normes, PUF, Paris, 1972) has proposed a “deontic hexagon” as being the geometrical representation of standard deontic logic, whereas Joerden (jointly with Hruschka, in Archiv für Rechtsund Sozialphilosophie 73:1, 1987), McNamara (Mind 105:419, 1996) and Wessels (Die gute Samariterin. Zur Struktur der Supererogation, Walter de Gruyter, Berlin, 2002) have proposed some new “deontic polygons” for dealing with conservative extensions of standard deontic logic internalising the concept of “supererogation”. Since 2004 a new formal science of the geometrical oppositions inside logic has appeared, that is “n-opposition theory”, or “NOT”, which relies on the notion of “logical bi-simplex of dimension m” (m = n − 1). This theory has received a complete mathematical foundation in 2008, and since then several extensions. In this paper, by using it, we show that in standard deontic logic there are in fact many more oppositional deontic figures than Kalinowski’s unique “hexagon of norms” (more ones, and more complex ones, geometrically speaking: “deontic squares”, “deontic hexagons”, “deontic cubes”, . . ., “deontic tetraicosahedra”, . . .): the real geometry of the oppositions between deontic modalities is composed by the aforementioned structures (squares, hexagons, cubes, . . ., tetraicosahedra and hyper-tetraicosahedra), whose complete mathematical closure happens in fact to be a “deontic 5-dimensional hyper-tetraicosahedron” (an oppositional very regular solid).      

Local Linear Convergence for Alternating and Averaged Nonconvex Projections

The idea of a finite collection of closed sets having “linearly regular intersection” at a point is crucial in variational analysis. This central theoretical condition also has striking algorithmic consequences: in the case of two sets, one of which satisfies a further regularity condition (convexity or smoothness, for example), we prove that von Neumann’s method of “alternating projections” converges locally to a point in the intersection, at a linear rate associated with a modulus of regularity. As a consequence, in the case of several arbitrary closed sets having linearly regular intersection at some point, the method of “averaged projections” converges locally at a linear rate to a point in the intersection. Inexact versions of both algorithms also converge linearly. Research of A.S. Lewis supported in part by National Science Foundation Grant DMS-0504032. Research of D.R. Luke supported in part by National Science Foundation Grant DMS-0712796.     

A note on approximation of semigroups of contractions on Hilbert spaces

We show that every contractive C ₀-semigroup on a separable, infinite-dimensional Hilbert space X can be approximated by unitary C ₀-groups in the weak operator topology uniformly on compact subsets of ℝ₊. As a consequence we get a new characterization of a bounded H ^∞-calculus for the negatives of generators of bounded holomorphic semigroups. Applications of our results to the study of a topological structure of the set of (almost) weakly stable contractive C ₀-semigroups on X are also discussed. The author was partially supported by the Marie Curie “Transfer of Knowledge” programme, project “TODEQ”, and by a MNiSzW grant Nr. N201384834.     

Separable statistics derived from a given separable statistic by “restricting” the probability space

The problem of derivation of a separable statistic from another separable statistic is considered for the case of a distribution with atoms. This work was performed with the financial support of the International Science Foundation's “Cultural Initiative”. Translated fromStatisticheskie Metody Proverki i Otsenivaniya Gipotez, pp. 109–117, Perm, 1993.     

Reverse mathematics of separably closed sets

This paper contains a corrected proof that the statement “every non-empty closed subset of a compact complete separable metric space is separably closed” implies the arithmetical comprehension axiom of reverse mathematics.     

On a class of conformal metrics arising in the work of Seiberg and Witten

We examine a class of conformal metrics arising in the “N = 2 supersymmetric Yang-Mills theory” of Seiberg and Witten. We provide several alternate characterizations of this class of metrics and proceed to examine issues of existence and boundary behavior and to parameterize the collection of Seiberg-Witten metrics with isolated non-essential singularities on a fixed compact Riemann surface. In consequence of these results, the Riemann sphere does not admit a Seiberg-Witten metric, but for all there is a conformal metric on of regularity which is Seiberg-Witten off of a finite set. Received August 18, 1998