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Convex Analysisの二,三の進展について

From the Publisher:
Discrete geometry investigates combinatorial properties of configurations of geometric objects. To a working mathematician or computer scientist, it offers sophisticated results and techniques of great diversity and it is a foundation for fields such as computational geometry or combinatorial optimization.
This book is primarily a textbook introduction to various areas of discrete geometry. In each area, it explains several key results and methods, in an accessible and concrete manner. It also contains more advanced material in separate sections and thus it can serve as a collection of surveys in several narrower subfields. The main topics include: basics on convex sets, convex polytopes, and hyperplane arrangements; combinatorial complexity of geometric configurations; intersection patterns and transversals of convex sets; geometric Ramsey-type results; polyhedral combinatorics and
high-dimensional convexity; and lastly, embeddings of finite metric spaces into normed spaces.
Jiri Matousek is Professor of Computer Science at Charles University in Prague. His research has contributed to several of the considered areas and to their algorithmic applications. This is his third book.

Lectures on Discrete Geometry

Groupes et algébres de lie

Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the “bible of computer algebra”, gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany oneor two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

Modern computer algebra

Note: Professor Pach's number: [045]; Also in: Facsimile edition: World Classics in Mathematics Series, vol. 28, China Science Press, Beijing, 2006. Mimeographed from 100 Research Problems in Discrete Geometry (with W. O. J. Moser). Reference DCG-BOOK-2008-001 URL: http://www.math.nyu.edu/~pach/research_problems.html Record created on 2008-11-18, modified on 2017-05-12

Research Problems in Discrete Geometry

Norm-Graphs

For everyt>1 and positiven we construct explicit examples of graphsG with |V (G)|=n, |E(G)|≥ct·n2−1/t which do not contain a complete bipartite graghKt,t!+1 This establishes the exact order of magnitude of the Turan numbers ex (n, Kt,s) for any fixedt and alls≥t!+1, improving over the previous probabilistic lower bounds for such pairs (t, s). The construction relies on elementary facts from commutative algebra.

Norm-graphs and bipartite turán numbers

http://unipub.lib.uni-corvinus.hu/729/1/BozokiFulopRonyai2010_manu.pdf

On optimal completion of incomplete pairwise comparison matrices

We propose a new deterministic method of factoring polynomials over finite fields. Assuming the Generalized Riemann Hypothesis (GRH), we obtain, in polynomial time, the factorization of any polynomial with a bounded number of irreducible factors. Other consequences include a polynomial time algorithm to find a nontrivial factor of any completely splitting even degree polynomial when a quadratic nonresidue in the field is given.

Lajos Rónyai

Papers

Norm-Graphs

Norm-graphs and bipartite turán numbers

On optimal completion of incomplete pairwise comparison matrices

Factoring polynomials over finite fields

Computing the structure of finite algebras