An analytical study of the sensitivity of a discrete stochastic model for sales dynamics

We consider a discrete model for sales dynamics in the case of a stochastic model of the market. The model includes “fast” and “slow” components of the market situation described by a stochastic process of “white noise” type and the correlated stochastic process. By using an integral representation of the main characteristics of the Kalman filter, we obtain expressions for stochastic parameters of additional errors of the estimate that arise in the case where the characteristics of noises are inexact. We make an asymptotical analysis of these expressions and give recommendations for the price-forming strategy in the case of uncertainty of the market situation. Bibliography: 2 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 81, 1997, pp. 110–116.     

Test of symmetry of a one-dimensional distribution against positive biasedness

Summary Definitions of different strengths are given to the notion of ‘a positively biased random variable’. This notion is related to that of ‘a stochastically larger component of a two-dimensional random vector’, which was introduced previously by the authors. Properties of common rank tests of symmetry about zero against our specification of alternatives are studied in detail. The positive biasedness is extended to ‘positively more biased’. Test of symmetry of a two-dimensional random vector is also referred to.     

The effective action and the spectrum of collective modes in the antiferromagnetic phase of the three-band repulsive hubbard model

The paper is concerned with the application of path integration (successive integration over “fast” and “slow” variables and the Hubbard-Stratonovich transformation) to the antiferromagnetic state in the three-band, two-dimensional repulsive Hubbard model. It is shown that, in the lowest approximation, the model under consideration is equivalent to the half-filled single-band Hubbard model. The effective action for the collective Bose-fields, the critical temperature of the phase transition, and the spectra of excitations are obtained. Bibliography: 24 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 245, 1997, pp. 216–230. Translated by C. L. Malyshev.     

Quaternion Polar Representation with a Complex Modulus and Complex Argument Inspired by the Cayley-Dickson Form

We present a new polar representation of quaternions inspired by the Cayley-Dickson representation. In this new polar representation, a quaternion is represented by a pair of complex numbers as in the Cayley-Dickson form, but here these two complex numbers are a complex ‘modulus’ and a complex ‘argument’. As in the Cayley-Dickson form, the two complex numbers are in the same complex plane (using the same complex root of −1), but the complex phase is multiplied by a different complex root of −1 in the exponential function. We show how to calculate the ‘modulus’ and ‘argument’ from an arbitrary quaternion in Cartesian form.     

Constructing greedy linear extensions by interchanging chains

A natural way to prove that a particular linear extension of an ordered set is ‘optimal’ with respect to the ‘jump number’ is to transform this linear extension ‘canonically’ into one that is ‘optimal’. We treat a ‘greedy chain interchange’ transformation which has applications to ordered sets for which each ‘greedy’ linear extension is ‘optimal’.     

Subsonic and supersonic disturbances in the generation of surface waves in a liquid layer adjacent to a high-speed gas-stream

In an effort to shed further light upon the nature of “supersonic” disturbances as distinct from that of ‘subsonic’ disturbances in parallel compressible flows, this paper makes an investigation of the stability characteristics of the surface waves generated in a liquid layer adjacent to a high-speed gas-stream. It turns out that the nature of the surface waves generated in the liquid layer depends markedly upon the type of disturbances present in the high-speed gas-stream. For the case of ‘subsonic’ disturbances it is shown that the energy transfer from the gas stream to the surface waves is contributed predominantly by the Fourier component of the normal gas-pressure force-field in phase with the slope of the wavy surface. For the case of ‘supersonic’ disturbances, this energy transfer is shown to be predominantly due to the component of the pressure-field in phase with the surface-wave displacement and is related to the presence of travelling periodic waves in the gas-stream—this energy transfer is shown to promote always the growth of the surface waves.     

Universal properties of stable states of a free energy model with small parameters

We consider the problem of minimizing functionals of the form , for small , subject to the constraint . (Here is typically a double well potential.) We study structural properties of minimizers that are shared by all minimizers of the problem, for sufficiently small values of . It is shown that the average ‘mass’ and ‘energy’ of minimizers, over intervals of length , is almost uniformly distributed (depending on ), for all sufficiently small . Here is a constant depending on the degree of ‘uniformity’, but independent of . Received May 5, 1996 / Accepted October 28, 1996     

Experiments with an interactive paired comparison simplex method for molp problems

In this paper, an interactive paired comparison simplex based method formultiple objective linear programming (MOLP) problems is developed and compared to other interactive MOLP methods. The decision maker (DM)’s utility function is assumed to be unknown, but is an additive function of his known linearized objective functions. A test for ‘utility efficiency’ for MOLP problems is developed to reduce the number of efficient extreme points generated and the number of questions posed to the DM. The notion of ‘strength of preference ’ is developed for the assessment of the DM’s unknown utility function where he can express his preference for a pair of extreme points as ‘strong ’, ‘weak ’, or ‘almost indifferent ’. The problem of ‘inconsistency of the DM’ is formalized and its resolution is discussed. An example of the method and detailed computational results comparing it with other interactive MOLP methods are presented. Several performance measures for comparative evaluations of interactive multiple objective programming methods are also discussed. All rights reserved. This study, or parts thereof, may not be reproduced in any form without written permission of the authors.     

On Uniqueness of Maximal Coupling for Diffusion Processes with a Reflection

A maximal coupling of two diffusion processes makes two diffusion particles meet as early as possible. We study the uniqueness of maximal couplings under a sort of ‘reflection structure’ which ensures the existence of such couplings. In this framework, the uniqueness in the class of Markovian couplings holds for the Brownian motion on a Riemannian manifold whereas it fails in more singular cases. We also prove that a Kendall-Cranston coupling is maximal under the reflection structure.     

Convergence analysis of a conforming adaptive finite element method for an obstacle problem

The adaptive algorithm for the obstacle problem presented in this paper relies on the jump residual contributions of a standard explicit residual-based a posteriori error estimator. Each cycle of the adaptive loop consists of the steps ‘SOLVE’, ‘ESTIMATE’, ‘MARK’, and ‘REFINE’. The techniques from the unrestricted variational problem are modified for the convergence analysis to overcome the lack of Galerkin orthogonality. We establish R-linear convergence of the part of the energy above its minimal value, if there is appropriate control of the data oscillations. Surprisingly, the adaptive mesh-refinement algorithm is the same as in the unconstrained case of a linear PDE—in fact, there is no modification near the discrete free boundary necessary for R-linear convergence. The arguments are presented for a model obstacle problem with an affine obstacle χ and homogeneous Dirichlet boundary conditions. The proof of the discrete local efficiency is more involved than in the unconstrained case. Numerical results are given to illustrate the performance of the error estimator.     

Remarks on the Zeros of the Associated Legendre Functions with Integral Degree

We present some formulas for the computation of the zeros of the integral-degree associated Legendre functions with respect to the order. Work supported in part by the ‘Anillo Ecuaciones Asociadas a Reticulados’, financed by the World Bank through the ‘Programa Bicentenario de Ciencia y Tecnología’, and by the ‘Programa Reticulados y Ecuaciones’ of the Universidad de Talca.     

Roof duality,complementation and persistency in quadratic 0–1 optimization

The paper is concerned with the ‘primal’ problem of maximizing a given quadratic pseudo-boolean function. Four equivalent problems are discussed—the primal, the ‘complementation’, the ‘discrete Rhys LP’ and the ‘weighted stability problem of a SAM graph’. Each of them has a relaxation—the ‘roof dual’, the ‘quadratic complementation,’ the ‘continuous Rhys LP’ and the ‘fractional weighted stability problem of a SAM graph’. The main result is that the four gaps associated with the four relaxations are equal. Furthermore, a solution to any of these problems leads at once to solutions of the other three equivalent ones. The four relaxations can be solved in polynomial time by transforming them to a bipartite maximum flow problem. The optimal solutions of the ‘roof-dual’ define ‘best’ linear majorantsp(x) off, having the following persistency property: if theith coefficient inp is positive (negative) thenx _i=1 (0) in every optimum of the primal problem. Several characterizations are given for the case where these persistency results cannot be used to fix any variable of the primal. On the other hand, a class of gap-free functions (properly including the supermodular ones) is exhibited.     

Analysis and design of selection committees: a game theoretic secretary problem

Firms often delegate important decisions to committees which are set up specifically for that purpose; for example selection committees. We analyze the equilibrium behavior of a game in which committee members (the players) interview candidates sequentially, either hiring or going on to the next one. The players have differing evaluations of candidates (e.g. one cares about typing skills; the other about IT skills), which become their utilities if the candidate is hired. We then consider the optimal design (rules of the game) of such a committee, from the point of view of the firm. That is, which rules hire candidates which maximize the firm’s utility. Our committee game has a first round in which the members sequentially, by order of player number, say ‘yea’ or ‘nea’ to the candidate. If there are sufficient ‘yeas’ then she is tentatively hired; otherwise she is rejected. In the former case, members who said nea can veto the candidate in the second round. Thus the candidate is either hired, rejected, or vetoed. In the last case, the member casting a veto has one less to use on later candidates. We analyze equilibria where a player may say ‘yea’ to a candidate he would prefer not to hire, in order to force the other player to use up a valuable veto. We show that for the uniform candidate distribution there is a unique equilibrium and better candidates for the firm are hired when there are more vetoes. However we exhibit a candidate distribution where increasing the numbers of vetoes results in hiring worse candidates.     

Moment inequalities and central limit properties of isotropic convex bodies

The object of our investigations are isotropic convex bodies , centred at the origin and normed to volume one, in arbitrary dimensions. We show that a certain subset of these bodies – specified by bounds on the second and fourth moments – is invariant under forming ‘expanded joinsrsquo;. Considering a body K as above as a probability space and taking , we define random variables on K. It is known that for subclasses of isotropic convex bodies satisfying a ‘concentration of mass property’, the distributions of these random variables are close to Gaussian distributions, for high dimensions n and ‘most’ directions . We show that this ‘central limit property’, which is known to hold with respect to convergence in law, is also true with respect to -convergence and -convergence of the corresponding densities. Received: 21 March 2001 / in final form: 17 October 2001 / Published online: 4 April 2002     

Stochastically larger component of a random vector

Summary Definitions of different strengths are given to the notion of ‘a stochastically larger component of a two-dimensional random vector.’ Some of them reduce to the known definitions of stochastic order relationship when the components are stochastically independent. The definitions and the approach are related to nonparametric problems.     

On a class of second order variational problems with constraints

We study the structure of optimal solutions for a class of constrained, second order variational problems on bounded intervals. We show that, for intervals of length greater than some positive constant, the optimal solutions are bounded inC ¹ by a bound independent of the length of the interval. Furthermore, for sufficiently large intervals, the ‘mass’ and ‘energy’ of optimal solutions are almost uniformly distributed.     

Unions of slender groups

It is well-known that the class of slender groups is closed under extensions, arbitrary direct sums and subgroups. We show that it is also closed under taking unions of continuous well-ordered ascending chains of groups where each factor of consecutive groups is slender (Theorem 2.2). We also prove that the class of slender groups cannot be obtained from any set of slender groups by forming unions of chains of the mentioned kind and taking subgroups (Theorem 4.2). This generalizes a result by G?bel-Wald [11]. We observe that the same results hold if in the theorems ‘slender’ is replaced by ‘reduced torsion-free’ or ‘cotorsion-free’. Received: 30 June 2005     

Program Explanation and Higher-Order Properties

Our aim in this paper is to evaluate Frank Jackson and Philip Pettit’s ‘program explanation’ framework as an account of the autonomy of the special sciences. We argue that this framework can only explain the autonomy of a limited range of special science explanations. The reason for this limitation is that the framework overlooks a distinction between two kinds of properties, which we refer to as ‘higher-level’ and ‘higher-order’ properties. The program explanation framework can account for the autonomy of special science explanations that appeal to higher-level properties but it does not account for the autonomy of most of those explanations that appeal to higher-order properties.