The class of microarray games and the relevance index for genes

Nowadays, microarray technology is available to generate a huge amount of information on gene expression. This information must be statistically processed and analyzed, in particular, to identify those genes which are useful for the diagnosis and prognosis of specific diseases. We discuss the possibility of applying game-theoretical tools, like the Shapley value, to the analysis of gene expression data. Via a “truncation” technique, we build a coalitional game whose aim is to stress the relevance (“sufficiency”) of groups of genes for the specific disease we are interested in. The Shapley value of this game is used to select those genes which deserve further investigation. To justify the use of the Shapley value in this context, we axiomatically characterize it using properties with a genetic interpretation. The authors are grateful to two anonymous referees for their extremely helpful comments. An earlier version of this paper was presented at the VI Spanish Meeting on Game Theory and Practice, July 12–14, 2004, Elche, Spain. S. Moretti gratefully acknowledges the financial support of the EU project NewGeneris, European Union 6th FP (FOOD-CT-2005-016320).     

Multidimensional Lipschitz global optimization based on efficient diagonal partitions

This is a summary of the author’s PhD thesis, supervised by Yaroslav D. Sergeyev and defended on May 5, 2006, at the University of Rome “La Sapienza”. The thesis is written in English and is available from the author upon request. In this work, the global optimization problem of a multidimensional “black-box” function satisfying the Lipschitz condition over a hyperinterval with an unknown Lipschitz constant is considered. The objective function is assumed hard to evaluate. A new efficient diagonal scheme for constructing fast algorithms for solving this problem is examined and illustrated by developing several powerful global optimization methods. A deep theoretical study is performed which highlights the benefit of the approach introduced over traditionally used diagonal algorithms. Theoretical conclusions are confirmed by results of extensive numerical experiments.      

Maximally Clustered Elements and Schubert Varieties

We introduce and study a class of “maximally clustered” elements for simply laced Coxeter groups. Such elements include as a special case the freely braided elements of Green and the author, which in turn constitute a superset of the iji-avoiding elements of Fan. We show that any reduced expression for a maximally clustered element is short-braid equivalent to a “contracted” expression, which can be characterized in terms of certain subwords called “braid clusters”. We establish some properties of contracted reduced expressions and apply these to the study of Schubert varieties in the simply laced setting. Specifically, we give a smoothness criterion for Schubert varieties indexed by maximally clustered elements. Received December 30, 2005     

Yang-Baxter invariants for line configurations

We study line configurations in 3-space by means of “line diagrams”, projections into a plane with an indication of over and under crossing at the vertices. If we orient such a diagram, we can associate a “contracted tensor”T with it in the same spirit as is done in Knot Theory. We give conditions to makeT independent of the orientation, and invariant under isotopy. The Yang-Baxter equation is one such condition. Afterwards we restrict ourselves to Yang-Baxter invariants with a topological state model, and give some new invariants for line isotopy.     

Exact and heuristic algorithms for routing problems

This is a summary of the authors PhD thesis supervised by Daniele Vigo and defended on 30 March 2010, at the Università di Bologna. The thesis is written in English and is available from the author upon request. Several rich routing problems attaining to the transportation area have been studied. “Simple” algorithms have been proposed to solve them, both exact and heuristic, producing high quality solutions and transferrable methods.     

Pseudoradial Spaces: Results and Problems

In this paper it is given a survey of principal results (old and new) concerning the class of pseudoradial spaces. In this class cardinal invariants and their inequalities are considered. The behaviour of pseudoradial spaces under the operations of taking topological products and subspaces are examined and a typical proof is given. A particular attention is dedicated to the so called “small cardinals” in connection with pseudoradiality. Pseudoradiality of 2^{ω 2} is also examined. It is proved that pseudoradiality can be ω¹ productive for spaces of weight at most ω². Finally, several open problems are presented. This work was supported by the National Group “Real Analysis, Measure Theory with Applications to Economy” of the Italian Ministery of Education, University and Research.     

Impact of replenishment rules with endogenous lead times on supply chain performance

This text summarizes the PhD thesis of Robert Boute, obtained at the Katholieke Universiteit Leuven (Belgium) under supervision of Marc Lambrecht. This doctoral dissertation in the field of Supply Chain Management demonstrates that significant cost reductions can be obtained for both the retailer and the manufacturer when they align their replenishment policy. Such a collaboration strategy goes far beyond “information sharing”. In this summary, we present the research model, the general outline of the thesis and the methodology used. The PhD thesis, written in English, is available from the author upon request.      

Space structures: Theory and applications. 3

This paper is the third part of the book Space Structures: Theory and Applications. This part consists of an introduction and two chapters: “P²-Topological Spaces” and “Structured Sets.” __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 352, 2008, pp. 7–93.     

Probabilistic methods

An overview of the “probabilistic method” in Combinatorial Analysis and Graph Theory is given.     

The Concavity Assumption on Felicities and Asymptotic Dynamics in the RSS Model

An analysis of the RSS model in mathematical economics involves the study of an infinite-horizon variational problem in discrete time. Under the assumption that the felicity function is upper semicontinuous and “supported” at the value of the maximally-sustainable level of a production good, we report a generalization of results on the equivalence, existence and asymptotic convergence of optimal trajectories in this model. We consider two parametric specifications, and under the second, identify a “symmetry” condition on the zeroes of a “discrepancy function” underlying the objective function that proves to be necessary and sufficient for the asymptotic convergence of good programs. With a concave objective function, as is standard in the antecedent literature, we show that the symmetry condition reduces to an equivalent “non-interiority” condition.     

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.     

Inverse singularly perturbed problems of the convection-diffusion type in quadrangular curvilinear domains

We have constructed an algorithm for the asymptotic approximation of the solutions of inverse singularly perturbed boundary-value problems of the convection-diffusion type with unknown diffusion coefficient, depending on the coordinates of a quadrangular curvilinear domain of filtration. The case of sufficient smoothness and consistency of the overdetermination, initial, and boundary conditions is considered. Unlike the construction of an algorithm for the solution of similar problems in doubly connected domains, here, in the corresponding relations, there appear corrections taking into account the influence of “lateral sources of pollution.” With the help of this algorithm, we have carried out a computer experiment, the results of which confirm the well-known fact of “strong sensitivity” of the model to assignment of the overdetermination condition. In particular, we have revealed the specific character of influence of this condition on the required diffusion coefficient depending on the filtration velocity.     

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.     

The Grand Riemann Hypothesis

These lectures are a continuation of Bombieri’s series “The classical Theory of Zeta and L-Functions” (in this volume). Naturally there is some overlap between his and our presentations. My aim is to formulate the Riemann Hypothesis “GRH” in its most general setting and to demonstrate its importance and power as well as to indicate some of the progress that has been made around these conjectures. A particular theme being that a number of the striking applications of the GRH have been proven unconditionally by establishing suitably strong approximations thereof.     

Models and algorithms for the reconfiguration of distributed wireless switching systems

This paper is a summary of the author’s PhD thesis entitled “Models and algorithms for the reconfiguration of wireless switching systems”. The thesis deals with the study of a strongly NP-hard resource-constrained scheduling problem arising from the telecommunication industry. This work was supervised by Jacques Carlier and Dritan Nace, both from Université de Technologie de Compiègne, and carried out while the author was a System Architect within Nortel GSM Access R&D organization. The thesis, which is written in both French and English, has been defended on 29 March 2007 and is available by email request to the author. This research was supported in part by Association Nationale de la Recherche Technique grant CIFRE-121/2004.     

A note on Furstenberg’s filtering problem

This note gives a positive answer to an old question in elementary probability theory that arose in Furstenberg’s seminal article “Disjointness in Ergodic Theory.” As a consequence, Furstenberg’s filtering theorem holds without any integrability assumption.     

Notes on piecewise-Koszul algebras

The relationships between piecewise-Koszul algebras and other “Koszul-type” algebras are discussed. The Yoneda-Ext algebra and the dual algebra of a piecewise-Koszul algebra are studied, and a sufficient condition for the dual algebra A ^! to be piecewise-Koszul is given. Finally, by studying the trivial extension algebras of the path algebras of Dynkin quivers in bipartite orientation, we give explicit constructions for piecewise-Koszul algebras with arbitrary “period” and piecewise-Koszul algebras with arbitrary “jump-degree”.     

Visualisation of “High p,Small n” data

Development of methods for visualisation of high-dimensional data where the number of observations, n, is small compared to the number of variables, p, is of increasing importance. One major application is the burgeoning field of microarray (gene expression) experiments. Because of their high cost, the number of chips (n) is O(10 − 10²) while the number (p) of genes (including expressed sequence tags) on each chip is O(10³ − 10⁴). Based on synthetic data simulated in accord with current biological interpretation of microarray data, we have adapted the biplot that simultaneously plots the genes and the chips to display relevant experimental information. Other ordination techniques are also useful for visually exploring microarray data. The biological information that can be revealed by applying these exploratory, visual techniques is illustrated using data from gene expression experiments. When ordination methods, or dimension reduction methods such as PCA and its many variants, are used, in association with gene selection methods, it is well known that “selection bias” can result. We show an application of bootstrap methodology to ordination methods that can be used to account for this bias. Such methods are invaluable when visualization methods are used for pattern recognition, such as when identifying previously unknown sub-classes of tumours in molecular classification. A colour version of the paper is available at: DOI:. The sample numbers shown on the plots can also be used for identifying the different classes if a colour version is not available. The sample numbers for the ALL B-cells are 1, 4, 5, 7, 8, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, and 27 respectively. Those for the ALL T-Cells are 2, 3, 6, 9, 10, 11, 14 and 23, and for the AML the samples are 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38.     

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.