Algebra és számítások = Algebra and calculations

Algebra es szamitasok, rovid osszefoglalo: Kutatasaink soran uj eredmenyeket ertunk el az algebra es szamitastudomany kovetkező teruletein: Grobner-bazisok, kombinatorikai alkalmazasok, algebrai geometria, kvantumszamitogepek, csoportokkal es reprezentaciokkal kapcsolatos szamitasok, automatak, felcsoportok, adatbazisok, adatbanyaszat. Nehany fontosabb eredmeny: ? Wilson tetelenek altalanositasa, uj bizonyitas Wilson es Frankl teteleire, Babai es Frankl egy sejtesenek igazolasa. Uj algoritmus a lexikografikus standard monomok szamitasara. A Sperner-lemma bonyolultsaganak vizsgalata. Kvantumszamitogep modellek vizsgalata. - Projektiv sokasagok divizorai kohomologiainak vizsgalata, a Briancon-Skoda tetel altalanositasa. - A Brauer?Nesbitt-tetel, valamint Landau-tetelenek altalanositasa. Szimmetrikus algebrak Cartan-invariansainak vizsgalata. Absztrakt gyokcsoportok es gyok arnyekterek vizsgalata. - Az RGCn-kommutativ felcsoportok kozul a regularisak, a deltafelcsoportok es a szubdirekt irreducibilisek leirasa. Kongruenciafelcserelhető permutacioazonossagos felcsoport medialitasanak bizonyitasa. Automatabővitesek vizsgalata. - Uj adatbanyaszati es webkeresesi algoritmusok kidolgozasa. | With the partial support of the present grant, we have achieved new results in the following fields of algebra and computer science: Grobner-bases, applications in combinatroics, algebraic geometry, quantum computation, computations related to groups and representations, automata, semigroups, databases, data mining. Some significant results: - Generalization of Wilson's theorem, new proof for theorems of Wilson and Frankl, proof of a conjecture of Babai and Frankl. New algorithm for the calculation of lexicographic standard monomials. Investigation of the complexity of the Sperner-lemma. Study of quanum computer models. - Investigation of cohomology of divisors of projective varieties. Generalization of the Briancon-Skoda theorem. - Generalization of the Brauer-Nesbitt theorem and of Landau's theorem. Investigations of the Cartan-invariants of symmetric algebras. Study of abstract root subgroups and of root shadow spaces. - Description of the regular, the deltasemigroups and of the subdirectly irreducible ones among the RGCn-commutativ semigroups. Congruence permutable semigroups satisfying a permutation identity, are medial. Automata extensions. - New algrorithms for data mining and for web search.