Many countries are reporting of difficulties in exact education at schools: mathematics, informatics, physics etc. Various methods are proposed to awaken and preserve students’ interest in these disciplines. Among them, the simplification, accent on applications, avoiding of argumentation (especially in mathematics) etc. must be mentioned. As one of reasons for these approaches the growing amount of knowledge/skills to be acquired at school is often mentioned. In this paper we consider one of the possibilities to integrate partially teaching of important chapters of discrete mathematics and informatics not reducing the high educational standards. The approach is based on the identification and mastering general combinatorial principles underlying many topics in both disciplines. A special attention in the paper is given to the so-called “pigeonhole principle” and its generalizations. In folklore, this principle is usually formulated in the following way: “if there are n + 1 rabbits in n cages, you can find a cage with at least two rabbits in it“. Examples of appearances of this principle both in mathematics and in computer science are considered.

Thesis (MSc.)--University of Stellenbosch, 2000. / ENGLISH ABSTRACT: The notion of a classical graph theoretic Ramsey number is generalized by assuming that both the original graph whose edges are arbitrarily bicoloured and the monochromatic subgraphs to be forced are complete, balanced, multipartite graphs, instead of complete graphs as in the standard definition. Some small multipartite Ramsey numbers are found, while upper- and lower bounds are established for others. Analytic arguments as well as computer searches are used. / AFRIKAANSE OPSOMMING: Die klassieke grafiek-teoretiese definisie van ’n Ramsey getal word veralgemeen deur te aanvaar dat beide die oorspronklike grafiek, waarvan die lyne willekeurig met twee kleure gekleur word en die gesogte subgrafieke almal volledige, gebalanseerde, veelledige grafieke is, anders as in die standaard definisie. Klein veelledige Ramsey getalle word gevind, terwyl bo- en ondergrense vir ander daargestel word. Analitiese argumente en rekenaarsoektogte word gebruik.

When it comes to moral political endeavors, the good eventually achieved is never identical to the good initially pursued. This is true if for no other reason than the fact that time passes with every new political moment. We move from goods intended to goods achieved. At the same time such movement is not simply a product of the temporal character of our lives. Occupying a middle ground between accounts of human agency as wholly determined or wholly ambiguous there is a sense in which any exercise of the will is fundamentally indeterminate. Said more simply, we are contingent beings. Thus, while it may be possible to speak conceptually of a determinate or atemporal political good, the possibility of a moral political endeavor – that is, a purposive movement toward some political good – rests upon the inescapably contingent and temporal character of our lives. If political endeavors are never entirely under (or out of) our control and always take shape temporally then it is important to insist that the discrepancy between intended goods and actual goods need not be interpreted negatively. That is to say, the indeterminate character of our moral lives need not be seen as a tragic disruption to what would otherwise be seamless political existence. Rather, the indeterminacy is a deliberate (read: good) feature of created existence in time. This allows for recognition of a structure to political morality. Agents seize the opportunity afforded by contingency to pursue identified political goods with purpose and direction. At the same time moral pursuits are always highly conditioned by contingencies of delimited authority, responsibilities of representation, demands of process, etc. The constantly changing political landscape perpetually requires both reactive and anticipatory adjustments of the political good in sight. If contingency and temporality shape and limit any political pursuit of the good, then a chief task of political theology is to illuminate the theological significance of those features of created existence. Political theology bears the burden of articulating the divine origin and purpose of the structures which make political morality possible. In this way contingency comes into view not as an incidental feature of humanity but as the gift of a good creator making possible faithful creaturely response. Similarly, political goods take shape not merely in time but in a particular time between creation and eschaton. This thesis is a study in the theological significance of indeterminacy and temporality in the pursuit of political goods by way of an analysis of the political writings of 20th century moral theologian Paul Ramsey. His reflections on the unique moral structure of political actions provide the theological and analytical resources to animate such a study. Close attention to his work pursues an understanding of how theological language describes, interprets and accounts for the nature of political morality and the function that such descriptions have in defining and shaping concepts of the political good.

Notre travail ici concerne certaines pistes pour des constructions nouveaux groupes, et en particulier de contre-exemples à la conjecture de Cherlin-Zilber. On parvient à trouver une réponse pour la stabilité de groupes CSA existentiellement clos. On exhibe un mot de groupe en deux variables qui a la propriété d’indépendance par rapport à la classe de groupes hyperboliques sans torsion. On en déduit que l’équation correspondante donne la propriété d’indépendance des groupes CSA existentiellement clos, ce qui en particulier implique leur instabilité. En outre, on prouve que les équations, et en particulier les ensembles définissables sans quantificateurs, définissent des ensembles stables dans les boules bornées des produits libres de groupes, en utilisant la version finie du théorème de Ramsey. Enfin, on introduit certains groupes construits comme tours particulières de produits libres et d’extensions HNN. / Our work here relates to certain routes for the construction of new groups, and in particular, of counter-examples to the Cherlin-Zilber conjecture. We managed to find an answer for the stability of existentially closed CSA- groups. We build a group word in two variables that has the independence property relatively to the class of torsion-free hyperbolic groups. We deduced that the corresponding equation gives the independence property of existentially closed CSA groups which in turn implies their instability. Moreover, we demonstrate that group words, and in particular quantifierfree definable sets, define stable sets in bounded balls of free products of groups using a finite version of Ramsey’s theorem. Finally, we introduce certain groups constructed as special towers of free products and HNN-extensions.

In this dissertation, we are interested in studying several parameters of graphs and understanding their extreme values. We begin in Chapter~$2$ with a question on edge colouring. When can a partial proper edge colouring of a graph of maximum degree $\Delta$ be extended to a proper colouring of the entire graph using an `optimal' set of colours? Albertson and Moore conjectured this is always possible provided no two precoloured edges are within distance $2$. The main result of Chapter~$2$ comes close to proving this conjecture. Moreover, in Chapter~$3$, we completely answer the previous question for the class of planar graphs. Next, in Chapter~$4$, we investigate some Ramsey theoretical problems. We determine exactly what minimum degree a graph $G$ must have to guarantee that, for any two-colouring of $E(G)$, we can partition $V(G)$ into two parts where each part induces a connected monochromatic subgraph. This completely resolves a conjecture of Bal and Debiasio. We also prove a `covering' version of this result. Finally, we study another variant of these problems which deals with coverings of a graph by monochromatic components of distinct colours. The following saturation problem proposed by Barrus, Ferrara, Vandenbussche, and Wenger is considered in Chapter~$5$. Given a graph $H$ and a set of colours $\{1,2,\ldots,t\}$ (for some integer $t\geq |E(H)|$), we define $sat_{t}(n, R(H))$ to be the minimum number of $t$-coloured edges in a graph on $n$ vertices which does not contain a rainbow copy of $H$ but the addition of any non-edge in any colour from $\{1,2,\ldots,t\}$ creates such a copy. We prove several results concerning these extremal numbers. In particular, we determine the correct order of $sat_{t}(n, R(H))$, as a function of $n$, for every connected graph $H$ of minimum degree greater than $1$ and for every integer $t\geq e(H)$. In Chapter~$6$, we consider the following question: under what conditions does a Hamiltonian graph on $n$ vertices possess a second cycle of length at least $n-o(n)$? We prove that the `weak' assumption of a minimum degree greater or equal to $3$ guarantees the existence of such a long cycle. We solve two problems related to majority colouring in Chapter~$7$. This topic was recently studied by Kreutzer, Oum, Seymour, van der Zypen and Wood. They raised the problem of determining, for a natural number $k$, the smallest positive integer $m = m(k)$ such that every digraph can be coloured with $m$ colours, where each vertex has the same colour as at most a proportion of $\frac{1}{k}$ of its out-neighbours. Our main theorem states that $m(k) \in \{2k-1, 2k\}$. We study the following problem, raised by Caro and Yuster, in Chapter~$8$. Does every graph $G$ contain a `large' induced subgraph $H$ which has $k$ vertices of degree exactly $\Delta(H)$? We answer in the affirmative an approximate version of this question. Indeed, we prove that, for every $k$, there exists $g(k)$ such that any $n$ vertex graph $G$ with maximum degree $\Delta$ contains an induced subgraph $H$ with at least $n-g(k)\sqrt{\Delta}$ vertices such that $V(H)$ contains at least $k$ vertices of the same degree $d \ge \Delta(H)-g(k)$. This result is sharp up to the order of $g(k)$. %Subsequently, we investigate a concept called $\textit{path-pairability}$. A graph is said to be path-pairable if for any pairing of its vertices there exist a collection of edge-disjoint paths routing the the vertices of each pair. A question we are concerned here asks whether every planar path pairable graph on $n$ vertices must possess a vertex of degree linear in $n$. Indeed, we answer this question in the affirmative. We also sketch a proof resolving an analogous question for graphs embeddable on surfaces of bounded genus. Finally, in Chapter~$9$, we move on to examine $k$-linked tournaments. A tournament $T$ is said to be $k$-linked if for any two disjoint sets of vertices $\{x_1,\ldots ,x_k\}$ and $\{y_1,\dots,y_k\}$ there are directed vertex disjoint paths $P_1,\dots, P_k$ such that $P_i$ joins $x_i$ to $y_i$ for $i = 1,\ldots, k$. We prove that any $4k$ strongly-connected tournament with sufficiently large minimum out-degree is $k$-linked. This result comes close to proving a conjecture of Pokrovskiy.

Li, Nan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 88-90). / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Overview --- p.1 / Chapter 1.2 --- Literature on Illegal Immigration --- p.2 / Chapter 1.3 --- Literature on Income Distribution --- p.4 / Chapter 1.4 --- Outline and Contribution --- p.5 / Chapter 2 --- The Solow Model of Illegal Immigration --- p.7 / Chapter 2.1 --- The Basic Model --- p.7 / Chapter 2.2 --- Equilibrium and Transitional Dynamics --- p.10 / Chapter 2.2.1 --- Aggregate level --- p.10 / Chapter 2.2.2 --- Individual level --- p.12 / Chapter 3 --- Sensitivity Analysis in the Solow Model --- p.16 / Chapter 3.1 --- Benchmark Example --- p.16 / Chapter 3.2 --- Input Shares --- p.18 / Chapter 3.3 --- Share of Unskilled Labor 0 --- p.20 / Chapter 3.4 --- Different Initial Value of Capital --- p.21 / Chapter 4 --- The Ramsey Model of Illegal Immigration --- p.23 / Chapter 4.1 --- The Basic Model --- p.24 / Chapter 4.2 --- Transitional Dynamics and Equilibrium --- p.26 / Chapter 4.2.1 --- Transitional Dynamics --- p.26 / Chapter 4.2.2 --- Steady State --- p.27 / Chapter 5 --- Sensitivity Analysis in the Ramsey Model --- p.28 / Chapter 5.1 --- Benchmark Example --- p.29 / Chapter 5.2 --- Input Share --- p.30 / Chapter 5.3 --- Elasticity of Substitution 0 --- p.31 / Chapter 5.4 --- Penalty Ratio r --- p.32 / Chapter 5.5 --- Different Initial Value of Capital --- p.33 / Chapter 5.5.1 --- Case 1 k10 < k20 --- p.34 / Chapter 5.5.2 --- Case 2 k10 > k20 --- p.35 / Chapter 6 --- Comparison and Conclusion --- p.36

In the first chapter, I consider the institutional structures as well as the doctrines typically encountered in the surface water sector. To investigate the sources and methods of government support in the water sector, I categorize different sorts of government support according to the location of water along the water cycle. I conclude the section with examples of observed water markets. In the second chapter, I consider the problem of water usage, developing a model to analyze the optimal pricing of water within a second-best economy. As a water supplier, the local government may price discriminate across consumers and farmers. I introduce the second-best pricing scheme, derive conditions for the marginal-cost pricing and inverse-elasticity rules, and analyze when the government optimally deviates from these two pricing schemes. In the third chapter, I provide an analysis of the data I collected from Turkey. First, I examine the data on reservoir flows, including service share and fixed costs of the reservoirs. Then, I provide details about the relationship between the quantity and price of irrigation and of tap water. Finally, in the fourth chapter, I apply the theoretical framework to the data from Turkey. In Turkey, the current water-pricing policy is dictated by the sole objective of breaking-even in each period. This results in large withdrawals, which is not sustainable in the long-run, hence not optimal. I analyze the dynamic optimal water resource management problem of a benevolent government. I compare the implications of the current and the optimal pricing policies.

<p>This dissertation computes the optimal monetary and fiscal policy for small open and emerging economies in an estimated medium-scale model. The model departs from the conventional approach as it encompasses all the major nominal and real rigidities normally found in the literature in a single framework. After estimating the model using Bayesian techniques for one small open economy and one emerging economy, the Ramsey solution for the optimal monetary and fiscal policy is computed. Results show that foreign shocks have a strong influence in the dynamics of emerging economies, when compared to the designed optimal policy for a developed small open economy. For both economies, inflation is low, but very volatile, while taxes follow the traditional results in the literature with high taxes over labor income and low taxes for capital income.</p> / Dissertation

En 2003, Kechris, Pestov et Todorcevic démontrèrent que la structure de certains espaces métriques - dits ultrahomogènes - est intimement liée au comportement combinatoire de la classe de leurs sous-espaces métriques finis. La présente thèse a pour but d'explorer les différents aspects de cette connexion. Dans la première partie, la notion d'ultrahomogénéité métrique et les espaces ultrahomogènes complets séparables les plus remarquables, à savoir la sphère unité S_H de l'espace de Hilbert, l'espace de Baire et la sphère d'Urysohn S_U (à isométrie près, le seul espace complet séparable ultrahomogène et universel pour la classe des espaces métriques séparables de diamètre inférieur à 1) sont présentés. Dans la seconde partie, la notion de classe de Ramsey d'espaces métriques finis ordonnés est introduite et mise en lien avec les propriétés dynamiques des groupes d'isométries des espaces ultrahomogènes. Une importance particulière est attachée au théorème de Nesetril et à sa conséquence (originalement due à Pestov) selon laquelle toute action continue du groupe des autoisométries de S_U sur un compact admet un point fixe. Des résultats analogues sont ensuite obtenus dans d'autres cas, en particulier les espaces ultramétriques et l'espace de Baire. La troisième partie est quant à elle axée sur la notion de stabilité par oscillations. Pour la sphere de l'espace de Hilbert, la stabilité par oscillations n'est pas satisfaite ; il sagit d'un résultat essentiel en analyse fonctionnelle dû à Odell et Schlumprecht et équivalent à l'existence d'une application uniformément continue f de S_H dans [0,1] qui ne stabilise (ne devient presque constante) sur aucune copie isométrique de S_H dans S_H. En revanche, pour la majorité des autres espaces séparables ultrahomogènes, rien ne permet de démontrer ou de réfuter la stabilité par oscillations. C'est à ce problème qu'est consacré l'essentiel de la dernière partie. Cela conduit à la caractérisation complète des espaces ultramétriques séparables ultrahomogènes stables par oscillations et à une solution partielle dans le cas de la sphère d'Urysohn S_U.

In this thesis, we present a new viewpoint of the universal minimal flow in the language of near ultrafilters. We apply this viewpoint to generalize results of Kechris, Pestov and Todorcevic about a connection between groups of automorphisms of structures and structural Ramsey theory from countable to uncountable structures. This allows us to provide new examples of explicit descriptions of universal minimal flows as well as of extremely amenable groups. We identify new classes of finite structures satisfying the Ramsey property and apply the result to the computation of the universal minimal flow of the group of automorphisms of $\P(\omega_1)/\fin$ as well as of certain closed subgroups of groups of homeomorphisms of Cantor cubes. We furthermore apply our theory to groups of isometries of metric spaces and the problem of unique amenability of topological groups. The theory combines tools from set theory, model theory, Ramsey theory, topological dynamics and ergodic theory, and homogeneous structures.