This thesis is of the three article format. All three articles contribute to the literature on optimal fiscal policy with exogeneous government expenditures and distortionary taxation following Lucas and Stokey (1983) and Aiyagari et al. (2002) (AMSS). The first article extends the framework of AMSS by modelling agents ex ante heterogeneous in deterministic labour productivity trends in an infinite-horizon production economy with incomplete markets. The government does not use transfers. When the productivities of different agents grow at different rates, there is a conflict over the timing of tax collection. This is explored in a two-period model. The infinite-horizon model with two agents (‘lowskilled’ and ‘high-skilled’) is used to quantitatively analyse the impact of productivity trends observed in recent decades on the optimal policy. The impact is significant. The model can contribute to explaining the increase in government debt in many advanced economies in recent decades. The optimal policy strongly depends on Pareto weights but welfare of the agents does not. Political economics implications are discussed. The second article analyses the impact of heterogeneous productivity trends on the optimal policy when the social planner can use transfers. There is now conflict over the timing and the level of taxation, and it is explored in a two-period model. The optimal policy is studied in the same environment as in the first article. For most Pareto weights, the change in the tax rate is less pronounced than in the model without transfers, but still greater than the expected change due to shocks. The optimal policy and the welfare of the agents strongly depend on Pareto weights. Policy implications are discussed. The optimal policy in the horizon of decades is significantly affected by even a modest heterogeneity in the growth rates of the agents. Solution methods common to all three articles are discussed. In the third article the closed economy model of AMSS is extended into an open economy setting with two countries. The government of each country finances its exogeneous stochastic expenditures by distortionary labour taxation, and issues one-period bonds. The Ramsey planner chooses policy for both countries, and a no-arbitrage condition on the return of bonds of the two countries restricts her choices. The optimal policy is quantitatively studied in a calibrated model with ex-ante identical countries and equal Pareto weights, and three settings are compared in terms of policy and welfare: autarky (closed economy), partial union (international borrowing allowed), and full union (transfers between governments allowed).

abstract: Since the seminal work of Tur ́an, the forbidden subgraph problem has been among the central questions in extremal graph theory. Let ex(n;F) be the smallest number m such that any graph on n vertices with m edges contains F as a subgraph. Then the forbidden subgraph problem asks to find ex(n; F ) for various graphs F . The question can be further generalized by asking for the extreme values of other graph parameters like minimum degree, maximum degree, or connectivity. We call this type of question a Tura ́n-type problem. In this thesis, we will study Tura ́n-type problems and their variants for graphs and hypergraphs. Chapter 2 contains a Tura ́n-type problem for cycles in dense graphs. The main result in this chapter gives a tight bound for the minimum degree of a graph which guarantees existence of disjoint cycles in the case of dense graphs. This, in particular, answers in the affirmative a question of Faudree, Gould, Jacobson and Magnant in the case of dense graphs. In Chapter 3, similar problems for trees are investigated. Recently, Faudree, Gould, Jacobson and West studied the minimum degree conditions for the existence of certain spanning caterpillars. They proved certain bounds that guarantee existence of spanning caterpillars. The main result in Chapter 3 significantly improves their result and answers one of their questions by proving a tight minimum degree bound for the existence of such structures. Chapter 4 includes another Tur ́an-type problem for loose paths of length three in a 3-graph. As a corollary, an upper bound for the multi-color Ramsey number for the loose path of length three in a 3-graph is achieved. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2018

Ingeniero Civil Matemático / Un coloreo de aristas de un grafo se llama γ-promedio si es que el número promedio de colores incidentes a cada vértice es a lo más γ. Dados n, m enteros positivos y γ un real positivo, el número de Turán promedio coloreado T(n, K_m, γ-promedio) corresponde a la máxima cantidad de aristas que puede tener un grafo de n vértices de manera que exista un coloreo γ-promedio que no contenga ninguna copia monocromática de K_m. Esta noción fue introducida por Caro, quien observa que la expansión (blow-up) de un grafo completo γ-promedio coloreado sin copias monocromáticas de K_m también es γ-promedio coloreado y tampoco posee copias monocromáticas de K_m. Con ello, Caro se pregunta de si el máximo de aristas buscado se alcanza en la expansión de un grafo completo γ-promedio coloreado que maximice el número de vértices bajo la condición de no contener una copia monocromática de K_m (un coloreo extremal para el número de Ramsey γ-promedio). Yuster probó que la respuesta es afirmativa para el caso m=3 y γ=2, y además conjeturó que la respuesta es siempre afirmativa para todos los γ = ℓ ∈ N. En la presente memoria se demuestra esta conjetura de Yuster cuando m >= ℓ(ℓ+1)+1. Por otro lado, se demuestra también que la respuesta a la pregunta de Caro es negativa para un conjunto no numerable de valores de γ ∉ N.

The history of the Revolutionary War Fort Watauga located in present day Elizabethton, TN has yet to be completed. The critique of several Tennessee historians including J. G. M. Ramsey, Lyman C. Draper, Samuel Cole Williams, and others uncovered discrepancies and errors in Fort Watauga’s written history. These problems forced a rethinking of the fort’s actual location, which raised questions as to the reliability of the archaeological excavation in 1974 as well as to the historians who wrote about the fort during the last 200+ years. Evidence found pointed to new accounts of the fort’s history never before published as well as possible alternate locations of the original fort. An updated version of Fort Watauga’s history was included to serve as a new starting point for future historians and students interested in the Fort Watauga’s history. An updated version of Fort Watauga’s history was included to serve as a new starting point for future historians and students interested in Fort Watauga’s history.

The core idea of Ramsey theory is that complete disorder is impossible. Given a large structure, no matter how complex it is, we can always find a smaller substructure that has some sort of order. One view of this problem is in edge-colorings of complete graphs. For any graphs G, H1, ..., Hk, we write G → (H1, ..., Hk), or G → (H)k when H1 = ··· = Hk = H, if every k-edge-coloring of G contains a monochromatic Hi in color i for some i ∈ {1,...,k}. The Ramsey number rk(H1, ..., Hk) is the minimum integer n such that Kn → (H1, ..., Hk), where Kn is the complete graph on n vertices. Computing rk(H1, ..., Hk) is a notoriously difficult problem in combinatorics. A weakening of this problem is to restrict ourselves to Gallai colorings, that is, edge-colorings with no rainbow triangles. From this we define the Gallai-Ramsey number grk(K3,G) as the minimum integer n such that either Kn contains a rainbow triangle, or Kn → (G)k . In this thesis, we determine the Gallai-Ramsey numbers for C7 with multiple colors. We believe the method we developed can be applied to find grk(K3, C2n+1) for any integer n ≥ 2, where C2n+1 denotes a cycle on 2n + 1 vertices.

The core idea of Ramsey theory is that complete disorder is impossible. Given a large structure, no matter how complex it is, we can always find a smaller substructure that has some sort of order. For a graph H, the k-color Ramsey number r(H; k) of H is the smallest integer n such that every k-edge-coloring of Kn contains a monochromatic copy of H. Despite active research for decades, very little is known about Ramsey numbers of graphs. This is especially true for r(H; k) when k is at least 3, also known as the multicolor Ramsey number of H. Let Sn denote the star on n+1 vertices, the graph with one vertex of degree n (the center of Sn) and n vertices of degree 1. The double star S(n,m) is the graph consisting of the disjoint union of Sn and Sm together with an edge joining their centers. In this thesis, we study the multicolor Ramsey number of double stars. We obtain upper and lower bounds for r(S(n,m); k) when k is at least 3 and prove that r(S(n,m); k) = nk + m + 2 for k odd and n sufficiently large. We also investigate a new variant of the Ramsey number known as the list Ramsey number. Let L be an assignment of k-element subsets of the positive integers to the edges of Kn. A k-edge-coloring c of Kn is an L-coloring if c(e) belongs to L(e) for each edge e of Kn. The list Ramsey number rl(H; k) of H is the smallest integer n such that there is some L for which every L-coloring of Kn contains a monochromatic copy of H. In this thesis, we study rl(S(1,1); p) and rl(Sn; p), where p is an odd prime number.

No description available.

Apresentamos os aspectos mais relevantes para se realizar um chafariz a átomos frios de 133Cs, para operá-lo como um padrão primário de tempo e freqüência para realizar a definição do segundo. O objetivo principal desse trabalho foi a otimização do sistema experimental do chafariz de átomos em duas partes críticas: a região de interação e o sistema de detecção. Além disso, otimizamos as diversas fases da manipulação óptica que fazem parte do ciclo de operação de um chafariz com o intuito de aumentar a relação sinal ruído na região de detecção. Quando os átomos são lançados com uma velocidade de 3,39ms-1, atingem o ápice a 60cm acima da região de captura e passam 360ms na região de vôo livre. A temperatura dos átomos na região de detecção não ultrapassa 15,5K e a diferença de população observada através das Franjas de Ramsey, que são a assinatura característica de um padrão de freqüência atômico, tem a largura a meia altura de 1,4Hz. A largura de linha e a relação sinal ruído da franja implicam em uma estabilidade a curto prazo de 5,18×1012. Alguns causadores de deslocamentos de freqüência como a radiação de corpo negro, o efeito Doppler de segunda ordem, o efeito gravitacional e o efeito Zeeman de segunda ordem foram avaliados. / We present some relevant aspects for the realization of a 133Cs fountain, intended to operate as a primary frequency standard to realize the definition of the second. The main goal of this work is the optimization of the experimental setup of the atomic fountain in its most critical parts: the interaction region and the detection system. Furthermore, the several phases of the optical manipulation which concern the operation cycle of an atomic fountain were also optimized. These procedures allowed us to increase the signal to noise ratio in the detection signal. When the launching velocity is about 3.39 m/s, the atoms reach the apogee about 60 cm above the capture region and spend 360 ms in the free flight zone. The atomic ensemble temperature in the detection region is about 15.5K and the population difference between the two fundamental level is measured through the Ramsey fringe with a linewidth of 1.4 Hz. This linewidth, allied to the obtained S/N resulted in the measured short-term stability of 5.18×1012. Some frequency shift as the black body radiation shift, gravitational shift, second order Doppler shift and second order Zeeman shift were measured.

Apresentamos os aspectos mais relevantes para a construção e caracterização de um padrão primário de tempo e frequência compacto. Nesse sistema a amostra de átomos de referência é preparada a partir de uma armadilha magneto-óptica. Após a obtenção de uma amostra significativa de átomos os feixes laser são desligados e, durante a expansão livre, a nuvem atômica é submetida a uma sequência de pulsos de micro-ondas que caracterizam o método de Ramsey. Os pulsos de micro-ondas são aplicados em uma cavidade de micro-ondas, esculpida dentro da câmara de vácuo. Todo o ciclo de funcionamento é implementado dentro dessa cavidade e por esse motivo o padrão é considerado compacto. Quando os átomos são interrogados com pulsos de τ = 1 ms separados por Τ = 8 ms obtemos uma largura a meia altura da franja central de 52 Hz e uma estabilidade a curto prazo de 10-13. Alguns causadores de deslocamentos de frequência foram avaliados. / We present some relevant aspects for the construction and characterization of a compact primary frequency standard. In this system, the atomic ensemble is prepared with a magnetooptical trap. Once a significant number of cold atoms is obtained, the laser beams are turned off and, during the free expansion, the cloud is submitted to a sequence of two microwave pulses, characterizing the well-known Ramsey method. The microwave pulses are applied in a microwave cavity that was sculpted inside the vacuum chamber. All the working cycle takes place into this cavity. When the atoms are interrogated by two pulses of τ = 1 ms separated by Τ = 8 ms, the linewidth of the central fringe is 52 Hz and a short-term stability 10-13 is observed. Some frequency shifts were evaluated.

Esta dissertação descreve os recentes desenvolvimentos do Chafariz Atômico localizado no Instituto de Física de São Carlos. Ele consiste de um aparato experimental que provê uma referência de freqüência (e tempo) de altíssima precisão. Para conseguir tal qualidade metrológica, esse sistema trava a freqüência de um oscilador eletrônico, baseado em um cristal de quartzo, na freqüência relativa a uma transição atômica, de uma amostra de átomos resfriados. O átomo utilizado é o 133Cs e a transição utilizada corresponde aos dois níveis hiperfínos do seu estado fundamental. O ciclo de funcionamento é composto por uma etapa de aprisionamento a laser dos átomos e bombeamento óptico, para que todos os átomos se encontrem num mesmo e determinado nível de energia. Esses átomos são então lançados opticamente contra a gravidade através de uma cavidade de microondas. Em trajetória balística e livre de interferências externas, o conjunto de átomos sofre uma possível mudança de estado, dependendo das características de potência e freqüência do sinal de microondas injetado na cavidade. Esta probabilidade de transição é o sinal de erro utilizado para travar em malha fechada o gerador de microondas que alimenta a cavidade. Os melhores resultados obtidos neste experimento foram uma estabilidade de 5x10-12τ-1/2 resultante de um sinal com 3Hz de largura a meia altura (FWHM) da franja central. Fizemos também uma avaliação preliminar dos principais deslocamentos de freqüência e uma análise de interação espacial dos átomos com o campo de microondas. / This paper describes the recent developments of the Atomic Fountain located at the São Carlos Physics Institute. It provides a very high resolution frequency (and time) reference. This is achieved locking an electronic oscillator, based on a quartz crystal, to an atomic resonance of a cold atomic sample. Our laboratory uses the 133Cs atom, using as the referenced transition that corresponds to the two hyperfine energy levels of the ground state. The operating cycle comprises a stage of laser trapping atoms or optical pumping, so that all atoms are within the same atomic state. These atoms are then launched optically against gravity through a microwave cavity. In ballistic trajectory and free from external interference, the set of atoms undergoes a change of state, depending on the power and frequency of the microwave signal injected into the cavity. This transition probability is the error signal used to lock the microwave generator supplying the cavity in a closed loop. The best result obtained in this experiment was a stability of 5x10-12τ-1/2 resulting from signal with a 3 Hz half width (FWHM) of the central fringe. We also provide a preliminary assessment of the main frequency shifts and an analysis of spatial interaction of atoms with the microwave field.