Arbitrary Finite Set Research Articles

The Banach-Tarski theorem on polygons, and the cancellation law

The Banach-Tarski theorem on polygons in R 2 {{\mathbf {R}}^2} implies that two polygons are equidecomposable if and only if they are equidissectable. The possibility of strengthening this result in various ways is investigated. We show that if two polytopes in R d {{\mathbf {R}}^d} are equidecomposable under a finite set of isometries which generates a discrete group, then they are equidissectable using the same isometries. We then give a simple example in R {\mathbf {R}} showing that this is not true for arbitrary finite sets of isometries. A modification of this example is used to answer a question of S. Wagon concerning the cancellation law.

Multicriteria efficiency with arbitrary finite sets and cyclic preferences

It is shown that when X is an arbitrary finite subset of an n-factor product set and preference relations on each factor or criterion are assumed only to be asymmetric, efficient (undominated) points always exist in the set P of probability distributions on X when the preference relations are extended to probability distributions on the factors according to SSB utility theory

A method for exact calculation of the stardiscrepancy of plane sets applied to the sequences of Hammersley

A method is given to calculate exactly the stardiscrepancy of arbitrary finite plane sets. Using this method the stardiscrepancy of the sequences of Hammersley is obtained. The recursive structure of these sets allows for a proof by induction.

Properties of n-dimensional triangulations

This paper establishes a number of mathematical results relevant to the problem of constructing a triangulation, i.e., a simplical tessellation of the convex hull of an arbitrary finite set of points in n-space. The principal results of the present paper are: (1) a set of n + 2 points in n-space may be triangulated in at most 2 different ways; (2) the 'sphere test' defined in this paper selects a preferred one of these two triangulations; (3) a set of parameters is defined that permits the characterization and enumeration of all sets on n + 2 points in n-space that are significantly different from the point of view of their possible triangulations; and (4) the local sphere test induces a global sphere test property for a triangulation.

Laplace-Gauss Integrals, Gaussian Measure Asymptotic Behaviour and Probabilities of Moderate Deviations

There is given a general Gaussian measure representation on arbitrary finite dimensional Borel sets. This representation reflects B. Cavalieri’s and E. Torricelli’s "indivisible method" in a modern language. Based upon it, assertions are derived about the Gaussian measure asymptotic behaviour on Borel sets whose distance from the origin tends to infinity. Also two specific multivariate moderate deviation limit theorems for sums of i.i.d. random vectors are deduced.

The extent of non-conglomerability of finitely additive probabilities

An arbitrary finitely additive probability can be decomposed uniquely into a convex combination of a countably additive probability and a purely finitely additive (PFA) one. The coefficient of the PFA probability is an upper bound on the extent to which conglomerability may fail in a finitely additive probability with that decomposition. If the probability is defined on a σ-field, the bound is sharp. Hence, non-conglomerability (or equivalently non-disintegrability) characterizes finitely as opposed to countably additive probability. Nonetheless, there exists a PFA probability which is simultaneously conglomerable over an arbitrary finite set of partitions. Neither conglomerability nor non-conglomerability in a given partition is closed under convex combinations. But the convex combination of PFA ultrafilter probabilities, each of which cannot be made conglomerable in a common margin, is singular with respect to any finitely additive probability that is conglomerable in that margin.

Extended continued fractions and energies of the anharmonic oscillators

We describe the analytic solution to the Schrödinger eigenvalue problem for the class of the central potentials V(r)=∑δ∈Zaδrδ, where a−2>−1/4, amax δ >0, Z is an arbitrary finite set of the integer or rational exponents, −2≤δ1<δ2<⋅⋅⋅<δI, and the couplings aδ satisfy only one auxiliary (formal, ‘‘superconfinement’’) restriction of the type aδI−1 >0. The formalism is based on an analysis of the asymptotic behavior of the explicit regular solution ψ(r) and issues in the formulation of the ‘‘secular’’ equation 1/L1(E)=0 which determines the binding energies. The main result is the rigorous construction of L1(E) as a generalized (‘‘extended’’) and convergent continued fraction. The method cannot be applied to the aδI−1 <0 cases—this disproves the closely related Hill-determinant approach as conjectured recently by Singh et al. for the simplest potentials with Z={−2,2,4,6} and Z={−2,−1,1,2}.

Inverse problems for dynamical systems

In this paper, we consider the problem of reconstructing a structurally stable vector field from its global phase portrait or from certain local pieces of the phase portrait. We show that for arbitrary finite sets of points and of simple closed curves, we can construct a structurally stable system having the elements of those sets as its critical points and limit cycles, and a prescribed topological type in a neighborhood of each of them. Given a global, structurally stable phase portrait, we can construct a structurally stable polynomial system which is topologically equivalent to it. Some of our results were originally announced in [ 28 1. The general theory of dynamical systems orginated in the study of solution curves for systems of ordinary differential equations in a phase space. This qualitative theory had its first major development in the work of Poincare on the topology of integral curves. A typical problem is to give a geometric description of the solutions of a particular system, or class of systems, with emphasis on the location of special features such as critical points and periodic orbits. An inverse problem, then, is the construction of a system of equations given geometric properties of the solutions which may be in the form of a “picture” of the flow (a schematic diagram of phase space) or a list of critical orbits defined analytically. A method for solving such problems is potentially useful in building models in the various fields in which the theory of dynamical systems is now being applied. It could provide a basic model to which statistical identification techniques [ 141 could then be applied. There is also an intrinsic interest in methods for constructing polynomial systems of prescribed type which is related to the lack of any progress toward a solution of Hilbert’s sixteenth problem. The recent appearance of a counterexample [27] to the Petrovsky-Landis conjecture for quadratic systems leaves the Problem completely open. One of the earliest results of this kind was obtained by Forster in 1938

On the boundary of a finite set of points in the plane

In this paper a definition of the boundary of a finite set of points in the plane is given. It is based on the concept of the density of such a set of points. An algorithm is given for finding an approximation to such a boundary. They are well-known definitions of connectedness, boundary, hole, and cluster for a finite set of points in the plane (see, e.g., A. Rosenfeld, Amer. Math. Monthly86 1979, 621–630; Inform. and Contr.39 1978, 19–34) and algorithms for constructing these objects (see, e.g., A Rosenfeld and A. C. Kak, Digital Picture Processing, Academic Press, New York 1976). These definitions have been given for subsets of a grid of points. This paper attempts to define these objects for an arbitrary finite set of points.

On Pole Assignment for a Class of Infinite Dimensional Linear Systems

Let $\{ {A,B} \}$ be an infinite dimensional linear system where A is normal with compact resolvent and the input space is finite dimensional. Then the controllability of $\{ {A,B} \}$ implies the possibility of assigning an arbitrary finite set of poles to the transfer function of the closed loop system formed by means of suitable linear state feedback. Thus such systems can always be stabilized through state feedback.

A general possibility theorem for group decision rules with pareto-transitivity

This chapter discusses different types of domain restrictions. We begin by analyzing various qualitative conditions on preference profiles. Value-restricted preferences (with single-peaked preferences as one of its subcases), limited agreement as well as antagonistic and dichotomous preferences are relatively easy to interpret. In our view, the property of single-peakedness stands out in particular. It proves to be a central restriction under majority voting. However, it also plays an essential role in the context of strategy-proof voting rules (which is the topic of another chapter in this Handbook). Furthermore, we consider quantitative or number-specific requirements on the distribution of voters over different preference orderings, and we shall see that some of those requirements are logically related to the qualitative conditions such as extremal restriction and value-restricted preferences. While the latter restrictions are requirements on combinations of individual orderings, the domains of individual orderings that admit n-person nondictatorial social welfare functions à la Arrow result from restrictions on permissible preferences for individuals. While the first five sections study the aggregation problem within the framework of arbitrary finite sets of discrete alternatives, the final section discusses continuous choice rules; the alternatives are assumed to be n-dimensional vectors in Euclidean space. Contractibility as a condition on the topological space of preferences proves to be necessary and sufficient for the existence of continuous aggregation rules.

Rational choice and polynomial measurement models

Abstract Rational choice behavior is defined in a manner that subsumes rational choice theory as part of general measurement theory. The special class of polynomial measurement problems are defined, and their solutions are reduced to solving polynomial inequalities. An algebraic criterion is given for the solvability of arbitrary finite sets of polynomial inequalities, resolving a conjecture of Tversky.

Convex stochastic dominance with finite consequence sets

Stochastic dominance is a notion in expected-utility decision theory which has been developed to facilitate the analysis of risky or uncertain decision alternatives when the full form of the decision maker's von Neumann-Morgenstern utility function on the consequence space X is not completely specified. For example, if f and g are probability functions on X which correspond to two risky alternatives, then f first-degree stochastically dominates g if, for every consequence x in X, the chance of getting a consequence that is preferred to x is as great under f as under g. When this is true, the expected utility of f must be as great as the expected utility of g. Most work in stochastic dominance has been based on increasing utility functions on X with X an interval on the real line. The present paper, following [1], formulates appropriate notions of first-degree and second-degree stochastic dominance when X is an arbitrary finite set. The only ‘structure’ imposed on X arises from the decision maker's preferences. It is shown how typical analyses with stochastic dominance can be enriched by applying the notion to convex combinations of probability functions. The potential applications of convex stochastic dominance include analyses of simple-majority voting on risky alternatives when voters have similar preference orders on the consequences.

Exact integral positivity and crossing constraints on S and P wave pion-pion scattering amplitudes

Abstract S- and P-wave pion-pion scattering amplitudes are constrained by the fact that they have to be compatible with crossing symmetry and the positivity of the absorptive parts of all partial waves. The construction of the necessary and sufficient conditions ensuring the compatibility with positivity and an arbitrary finite set of Balachandran-Nuyts crossing conditions is described. Two sets of constraints are derived explicitly. The first set contains inequalities involving integrals of the S-waves over the unphysical interval 0⩽ s ⩽4 m π 2 . The inequalities of the second set relate integrals of S-waves over the unphysical interval to integrals of S- and P-wave absorptive parts extended to the physical region.

A note on Ken-Ichi Inada's “majority rule and rationality”

This chapter discusses different types of domain restrictions. We begin by analyzing various qualitative conditions on preference profiles. Value-restricted preferences (with single-peaked preferences as one of its subcases), limited agreement as well as antagonistic and dichotomous preferences are relatively easy to interpret. In our view, the property of single-peakedness stands out in particular. It proves to be a central restriction under majority voting. However, it also plays an essential role in the context of strategy-proof voting rules (which is the topic of another chapter in this Handbook). Furthermore, we consider quantitative or number-specific requirements on the distribution of voters over different preference orderings, and we shall see that some of those requirements are logically related to the qualitative conditions such as extremal restriction and value-restricted preferences. While the latter restrictions are requirements on combinations of individual orderings, the domains of individual orderings that admit n-person nondictatorial social welfare functions à la Arrow result from restrictions on permissible preferences for individuals. While the first five sections study the aggregation problem within the framework of arbitrary finite sets of discrete alternatives, the final section discusses continuous choice rules; the alternatives are assumed to be n-dimensional vectors in Euclidean space. Contractibility as a condition on the topological space of preferences proves to be necessary and sufficient for the existence of continuous aggregation rules.

Markov random fields and gibbs random fields

Spitzer has shown that every Markov random field (MRF) is a Gibbs random field (GRF) and vice versa when (i) both are translation invariant, (ii) the MRF is of first order, and (iii) the GRF is defined by a binary, nearest neighbor potential. In both cases, the field (iv) is defined onZ v, and (v) at anyxeZv, takes on one of two states. The current paper shows that a MRF is a GRF and vice versa even when (i)−(v) are relaxed, i.e., even if one relaxes translation invariance, replaces first order bykth order, allows for many states and replaces finite domains of Zv by arbitrary finite sets. This is achieved at the expense of using a many body rather than a pair potential, which turns out to be natural even in the classical (nearest neighbor) case when Zv is replaced by a triangular lattice.

The Synthesis of Multivalued Cellular Cascades and the Decomposability of Group Functions

This note is largely a generalization of the Elspas-Stone theory about the realization of group functions from Xm into H. In the paper of Elspas-Stone only the case X = {0,1} was treated, whereas we consider an arbitrary finite set X. We have shown that the most important results of [1] hold for this general case.

A general algorithm for nonnegative quadrature formulas

A general algorithm is presented for determining numerical integration formulas exact for an arbitrary finite set of continuous functions defined on a compact set, involving nonnegative combinations of function values at a finite number of points in the set. Examples are given.