Topological Properties of Generalized Context Structures

Práce je zaměřena na vzájemnou interakci několika odvětví matematiky. Hlavní myšlenkou práce bylo najít závislosti, vztahy a analogie mezi nimi. První část práce se týká vztahu mezi formální pojmovou analýzou, topologií a parciálními metrikami. Formální kontext je velice obecná matematická struktura, která může reprezentovat ostatní matematické struktury v jednotné a sjednocené formě. Přirozeným způsobem bychom mohli reprezentovat informaci podobně jako v tabulce, reprezentující formální kontext (s respektem ke všem množinově-teoretickým omezením) a generovat určité topologie na množinách atributů a objektů. V druhé části studujeme především pretopologické systémy jako speciální případ formálních kontextů. Od topologických systémů se pretopologické systémy liší především obecnější uspořádanou strukturou na množině atributů, reprezentujících zobecněné otevřené množiny. Vlastnosti tohoto uspořádání podstatně ovlivňují chování celé struktury a proto mu věnujeme zvláštní pozornost v závěru kapitoly, kde se mj. zabýváme konstrukcí analogie de Grootova duálu, včetně jeho iterovaných vlastností. Třetí část práce je zasvěcena struktuře framework, která má přirozenou strukturu formálního kontextu. Framework se skládá ze dvojice množin, z nichž první je množina míst a druhá obsahuje jistý systém podmnožin první množiny, aniž by bylo vyžadováno splnění nějakých axiómů. Struktura je opatřena jednoduchou konstrukcí duality, umožňující přepínání mezi klasickým, bodově-množinovým přístupem, podobně jako v topologii a bezbodovou reprezentací topologických vztahů. V závěru navrhujeme a studujeme, jak aproximovat libovolný framework pomocí usměrněného souboru konečných frameworků z hlediska generované topologie. V poslední části práce používáme metody obecné topologie ke korekci a zlepšení jednoho ze základních teorémů teorie her. Dokázali jsme mimo jiné, že pro hru v normální formě, v níž má i-tý hráč spojitou výherní funkci a množina jeho strategií je skoro-kompaktní, má tento hráč nedominovanou strategii. Kromě tohoto výsledku v poslední a předposlední kapitole ukazujeme, že teorie her přirozeným způsobem generuje velmi obecné, například nehausdorffovské topologické a kontextové struktury, čímž posouvá tradiční chápání reality neobvyklým směrem.This work is focused on the interaction of several branches of mathematics. The main idea was to nd dependencies, relationships and analogies between them. First part of the work is concerned to the relationship between Formal Concept Analysis, General Topology and Partial Metrics. A formal context is a very general mathematical structure that can represent other mathematical structures in a unied form. In a natural way, we could represent an information in a cross-table-like view of a formal context (fully respecting all set-theoretical limitations) and generate a topology on an attribute and object sets. In the second part the we study especially the pretopological systems as a special case of the formal contexts. They dier from topological systems especially by a more general poset structure of the set of attributes, representing the generalized open sets. Since the properties of this order structure are essential for the behavior of the whole structure, we pay them a special attention at the end of the chapter. Among others, we construct and study an analogue of the de Groot dual for posets, including its iteration properties. The third part is devoted to a mathematical structure called framework that has a contextual nature. A framework consists of two sets, rst one is a set of places, and the second one is a family of some its subsets, without the necessity of any external axioms to be fullled. The structure is equipped with a simple duality construction, allowing to switch between the classical point-set representation (like in topological spaces) and the point-less representation of topological relationships. At the end of the chapter, we suggest and study how a framework could be approximated by a directed family of nite frameworks from the point of view of the generated topology. In the last part the general topology methods were used to correct and improve one of the fundamental theorems in the game theory. It was showed that in a normal form game if i-th player has a continuous utility function and if the set of his strategies is almost-compact then he has an undominated strategy. In addition to this result, in the last two chapters we show that game theory naturally generates very general, for instance non-Hausdor topological and context structures, which shifts the traditional perception of reality in unexpected direction.

Matrix Bases for Star Products: a Review

We review the matrix bases for a family of noncommutative $\star$ products based on a Weyl map. These products include the Moyal product, as well as the Wick-Voros products and other translation invariant ones. We also review the derivation of Lie algebra type star products, with adapted matrix bases. We discuss the uses of these matrix bases for field theory, fuzzy spaces and emergent gravity

Courbure discrète : théorie et applications

International audienceThe present volume contains the proceedings of the 2013 Meeting on discrete curvature, held at CIRM, Luminy, France. The aim of this meeting was to bring together researchers from various backgrounds, ranging from mathematics to computer science, with a focus on both theory and applications. With 27 invited talks and 8 posters, the conference attracted 70 researchers from all over the world. The challenge of finding a common ground on the topic of discrete curvature was met with success, and these proceedings are a testimony of this wor

The Student-Project Allocation Problem: structure and algorithms

In this thesis we study the Student-Project Allocation problem (SPA), which is a matching problem based on the allocation of students to projects and lecturers. Students have preferences over projects, where each project is offered by one lecturer; whilst lecturers have preferences over students, or over the projects that they offer. We seek stable matchings of students to projects, which guarantee that no student and lecturer have an incentive to deviate from the matching by forming a private arrangement involving some project. We present new structural and algorithmic results for four problems related to SPA . We begin by characterising the stable matchings in an instance of the Student-Project Allocation problem with Lecturer preferences over Students (SPA-S) where the preferences are strictly ordered, in the special case that for each student in the instance, all of the projects in her preference list are offered by different lecturers. We achieve this characterisation by showing that, under this restriction, the set of stable matchings in an instance of SPA-S is a distributive lattice with respect to a natural dominance relation. Next, we study a variant of SPA - S where the preferences may involve ties — the Student- Project Allocation problem with Lecturer preferences over Students with Ties (SPA-ST). The presence of ties in the preference lists gives rise to three different concepts of stability, namely, weak stability, strong stability, and super-stability. We investigate stable matchings under the super-stability (respectively strong stability) concept. We present the first polynomial-time algorithm to find a super-stable (respectively strongly stable) matching or to report that no such matching exists, given an instance of SPA-ST . We also prove some structural results concerning the set of super-stable (respectively strongly stable) matchings in a given instance of SPA - ST . Further, we present results obtained from an empirical evaluation of our algorithms based on randomly-generated SPA-ST instances. Moving away from variants of SPA with lecturer preferences over students, we study the Student-Project Allocation problem with lecturer preferences over Projects (SPA-P). In this context it is known that stable matchings can have different sizes and the problem of finding a maximum size stable matching, denoted MAX-SPA-P , is NP-hard. There are two known approximation algorithms for MAX-SPA-P , with performance guarantees 2 and 3/2 . We show that MAX-SPA-P is polynomial-time solvable if there is only one lecturer involved, and NP-hard to approximate within some constant c > 1 if there are two lecturers involved. We also show that this problem remains NP-hard if each preference list is of length at most 3, with an arbitrary number of lecturers. We then describe an Integer Programming (IP) model to enable MAX-SPA-P to be solved optimally in the general case. Following this, we present results arising from an empirical evaluation that investigates how the solutions produced by the approximation algorithms compare to optimal solutions obtained from the IP model, with respect to the size of the stable matchings constructed, on instances that are both randomly-generated and derived from real datasets