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 survey of geodesic paths on 3D surfaces

This survey gives a brief overview of theoretically and practically relevant algorithms to compute geodesic paths and distances on three-dimensional surfaces. The survey focuses on three-dimensional polyhedral surfaces. The goal of this survey is to identify the most relevant open problems, both theoretical and practical.     

Some theoretical challenges in digital geometry: A perspective

In recent years image analysis has become a research field of exceptional significance, due to its relevance to real life problems in important societal and governmental sectors, such as medicine, defense, and security. The explicit purpose of the present Perspective is to suggest a number of strategic objectives for theoretical research, with an emphasis on the combinatorial approach in image analysis. Most of the proposed objectives relate to the need to make the theoretical foundations of combinatorial image analysis better integrated within a number of well-established subjects of theoretical computer science and discrete applied mathematics, such as the theory of algorithms and problem complexity, combinatorial optimization and polyhedral combinatorics, integer and linear programming, and computational geometry.     

Notes on “Ideal 0, 1 matrices” by Cornuéjols and Novick

In 1994, Cornuéjols and Novick published a classification of ideal and minimally non-ideal circulant clutters. One of their main results for doing so relates contractions of these clutters, simple directed cycles in an appropriate graph, and algebraic conditions. The purpose of this paper is twofold: to correct a small inaccuracy of the necessity of the algebraic conditions in the original proof, and to show that these algebraic conditions are actually sufficient, by giving a constructive proof of the existence of cycles.     

Some properties of a smoothing process

We consider the approximation family {f_? } introduced by Guillaume and Seeger in the case of affine functions. We generalise the framework to the case of convex functions and give some further properties.     

Periodicity and repetitions in parameterized strings

One of the most beautiful and useful notions in the Mathematical Theory of Strings is that of a Period, i.e., an initial piece of a given string that can generate that string by repeating itself at regular intervals. Periods have an elegant mathematical structure and a wealth of applications [F. Mignosi and A. Restivo, Periodicity, Algebraic Combinatorics on Words, in: M. Lothaire (Ed.), Cambridge University Press, Cambridge, pp. 237–274, 2002]. At the hearth of their theory, there are two Periodicity Lemmas: one due to Lyndon and Schutzenberger [The equation a^M=b^Nc^P in a free group, Michigan Math. J. 9 (1962) 289–298], referred to as the Weak Version, and the other due to Fine and Wilf [Uniqueness theorems for periodic functions, Proc. Amer. Math. Soc. 16 (1965) 109–114]. In this paper, we investigate the notion of periodicity and the closely related one of repetition in connection with parameterized strings as introduced by Baker [Parameterized pattern matching: algorithms and applications, J. Comput. System Sci. 52(1) (1996) 28–42; Parameterized duplication in strings: algorithms and an application to software maintenance, SIAM J. Comput. 26(5) (1997) 1343–1362]. In such strings, the notion of pairwise match or “equivalence” of symbols is more relaxed than the usual one, in that it rests on some mapping, rather than identity, of symbols. It seems natural to try and extend notions of periods and periodicities to encompass parameterized strings. However, we know of no previous attempt in this direction. Our preliminary investigation yields results as follows. For periodicity, we get (a) a generalization of the Weak Version of the Periodicity Lemma for parameterized strings, showing that it is essential that the two mappings inducing the periodicity must commute; (b) a proof that an analogous of the Lemma by Fine and Wilf [Uniqueness theorems for periodic functions, Proc. Amer. Math. Soc. 16 (1965) 109–114] cannot hold for parameterized strings, even if the mappings inducing the periodicity “commute”, in a sense to be specified below; (c) a proof that parameterized strings over an alphabet of at least three letters may have a set of periods which differ from those of any binary string of the same length—whereby the parameterized analog of a classic result by Guibas and Odlyzko [String overlaps, pattern matching, and nontransitive games, J. Combin. Theory Ser. A 30 (1981) 183–208] cannot hold. We also derive necessary and sufficient conditions characterizing parameterized repetitions, which are patterns of length at least twice that of the period, and show how the notion of root differs from the standard case, and highlight some of the implications on extending algorithmic criteria previously adopted for string searching, detection of repetitions and the likes. Finally, as a corollary of our main results, we also show that binary parameterized strings behave much in the same way as non-parameterized ones with respect to periodicity and repetitions, while there is a substantial difference for strings over alphabets of at least three symbols.     

Unbordered factors and Lyndon words

A primitive word w is a Lyndon word if w is minimal among all its conjugates with respect to some lexicographic order. A word w is bordered if there is a nonempty word u such that w=uvu for some word v. A right extension of a word w of length n is a word wu where all factors longer than n are bordered. A right extension wu of w is called trivial if there exists a positive integer k such that w^k=uv for some word v.We prove that Lyndon words have only trivial right extensions. Moreover, we give a conjecture which characterizes a property of every word w which has a nontrivial right extension of length 2|w|-2.     

Polyhedrality in pieces

The aim of this paper is to present two tools,  and , that make the task of finding equivalent polyhedral norms on certain Banach spaces easier and more transparent. The hypotheses of both tools are based on countable decompositions, either of the unit sphere _SX$S_X$ or of certain subsets of the dual ball _BX?$B_{X^{?}}$ of a given Banach space X. The sufficient conditions of Theorem 4 are shown to be necessary in the separable case. Using Theorem 7, we can unify two known results regarding the polyhedral renorming of certain C(K)$C(K)$ spaces, and spaces having an (uncountable) unconditional basis. New examples of spaces having equivalent polyhedral norms are given in the final section.     

The isometries of the cut, metric and hypermetric cones

We show that the symmetry groups of the cut cone Cut_n and the metric cone Met_n both consist of the isometries induced by the permutations on , that is, for n ≥ 5. For n = 4 we have . This result can be extended to cones containing the cuts as extreme rays and for which the triangle inequalities are facet-inducing. For instance, for n ≥ 5, where Hyp_n denotes the hypermetric cone.