Combinatorial representation and convex dimension of convex geometries

We develop a representation theory for convex geometries and meet distributive lattices in the spirit of Birkhoff's theorem characterizing distributive lattices. The results imply that every convex geometry on a set X has a canonical representation as a poset labelled by elements of X. These results are related to recent work of Korte and Lovász on antimatroids. We also compute the convex dimension of a convex geometry.Supported in part by NSF grant no. DMS-8501948.     

Nucleophilic Displacements in Alkylphosphonates: P–O Versus P–C Bond Cleavage. A Combined DFT and Experimental Study

Abstract

Potential energy surfaces for the nucleophilic displacements at phosphorus in dimethyl methyl, chloromethyl, dichloromethyl, and trichloromethyl phosphonates have been computed by DFT methods. The results reveal that sequential introduction of chlorine substituents at the methyl group in methyl phosphonates increases the stability of transition states and intermediates, which facilitates P–C bond cleavage. While nonsubstituted dimethyl methylphosphonate may undergo exclusive P–O bond cleavage, the trichlorinated analogue reacts exclusively via P–C bond dissociation to form dichlorocarbene, which was trapped by various olefins to form the corresponding gem-dichlorocyclopropanes.     

Counting the Interior Points of a Point Configuration

We prove a formula conjectured by Ahrens, Gordon, and McMahon for the number of interior points for a point configuration in \bf R ^d . Our method is to show that the formula can be interpreted as a sum of Euler characteristics of certain complexes associated with the point configuration, and then compute the homology of these complexes. This method extends to other examples of convex geometries. We sketch these applications, replicating an earlier result of Gordon, and proving a new result related to ordered sets. Received September 24, 1998, and in revised form July 6, 1999.     

Bifurcation of the Biot slow wave in a porous medium

The propagation of the Biot slow wave in a fluid-saturated porous medium at low frequencies is investigated by asymptotic methods. It is proven that the Biot wave has a bifurcation behavior depending on its wave number. The bifurcation occurs in a neighborhood of the critical value k(cr), which depends on the permeability of a medium and the viscosity of a fluid. The P2 wave is fully attenuated if its wave number is smaller than k(cr) and it becomes propagatory with wave numbers bigger than k(cr). Asymptotic formulas for the phase velocity and attenuation of the Biot wave are derived.     

Recent developments in consumer credit risk assessment

Consumer credit risk assessment involves the use of risk assessment tools to manage a borrower’s account from the time of pre-screening a potential application through to the management of the account during its life and possible write-off. The riskiness of lending to a credit applicant is usually estimated using a logistic regression model though researchers have considered many other types of classifier and whilst preliminary evidence suggest support vector machines seem to be the most accurate, data quality issues may prevent these laboratory based results from being achieved in practice. The training of a classifier on a sample of accepted applicants rather than on a sample representative of the applicant population seems not to result in bias though it does result in difficulties in setting the cut off. Profit scoring is a promising line of research and the Basel 2 accord has had profound implications for the way in which credit applicants are assessed and bank policies adopted.     

Weak mixing and anomalous kinetics along filamented surfaces

We consider chaotic properties of a particle in a square billiard with a horizontal bar in the middle. Such a system can model field-line windings of the merged surfaces. The system has weak-mixing properties with zero Lyapunov exponent and entropy, and it can be also interesting as an example of a system with intermediate chaotic properties, between the integrability and strong mixing. We show that the transport is anomalous and that its properties can be linked to the ergodic properties of continued fractions. The distribution of Poincare recurrences, distribution of the displacements, and the moments of the truncated distribution of the displacements are obtained. Connections between different exponents are found. It is shown that the distribution function of displacements and its truncated moments as a function of time exhibit log-periodic oscillations (modulations) with a universal period T(log)=pi(2)/12 ln 2. We note that similar results are valid for a family of billiard, particularly for billiards with square-in-square geometry. (c) 2001 American Institute of Physics.     

The Stochastically Subordinated Poisson Normal Process for Modelling Financial Assets

The method of stochastic subordination, or random time indexing, has been recently applied to Wiener process price processes to model financial returns. Previous emphasis in stochastic subordination models has involved explicitly identifying the subordinating process with an observable quantity such as number of trades. In contrast, the approach taken here does not depend on the specific identification of the subordinated time variable, but rather assumes a class of time models and estimates parameters from data. In addition, a simple Markov process is proposed for the characteristic parameter of the subordinating distribution to explain the significant autocorrelation of the squared returns. It is shown, in particular, that the proposed model, while containing only a few more parameters than the commonly used Wiener process models, fits selected financial time series particularly well, characterising the autocorrelation structure and heavy tails, as well as preserving the desirable self-similarity structure, and the existence of risk-neutral measures necessary for objective derivative valuation. Also, it will be shown that the model proposed fits financial times series data better than the popular generalised autoregressive conditional heteroscedasticity (GARCH) models. Additionally, this paper will develop a skew model by replacing the normal variates with Lévy stable variates.