Este trabalho apresenta o projeto de um bloco LNA-Misturador dentro de um mesmo circuito integrado para aplicações em um receptor Bluetooth 2;45GHz. Uma estratégia de projeto bem clara, concisa e com uma boa base física e matemática foi desenvolvida para auxiliar o processo de projeto de um bloco LNA-Misturador, composto por um LNA cascode em cascata com um misturador de chaveamento de corrente com entradas simples e degeneração indutiva nas fontes dos estágios de transcondutância. Esta estratégia foi adaptada de trabalhos apresentados na literatura. A estratégia de projeto proposta considera o compromisso entre ruído, linearidade, ganho, dissipação de potência, casamento de impedâncias e isolamento de portas, usando as dimensões dos dispositivos e condições de polarização como variáveis de projeto. Com base nesta estratégia se obteve um bloco LNA-Misturador que atinge algumas especificações propostas. Um bloco LNA-Misturador foi projetado e fabricado em uma tecnologia CMOS 0;35µm para validar a estratégia de projeto proposta. Além disso, para atingir os objetivos, durante o desenvolvimento deste trabalho foi dada atenção especial no projeto dos indutores. Foi projetado, fabricado e medido um chip de teste. Para tal fim foram aplicadas técnicas e estruturas de de-embedding nas medidas para conseguir resultados mais confiáveis. Os resultados experimentais obtidos para os indutores e os resultados preliminares do bloco LNA-Misturador s~ao satisfatórios de acordo com as especificações e os esperados das simulações. No entanto, os indutores integrados degradam significativamente o desempenho do bloco LNA-Misturador. Se forem usados processos de fabricação nos quais os indutores apresentem melhor desempenho, os resultados do bloco LNA-Misturador aplicando a estratégia de projeto desenvolvida neste trabalho podem ser melhorados. Finalmente, é importante ressaltar que a estratégia de projeto proposta neste trabalho já está sendo usada e adaptada em outros projetos com o propósito de melhorar os resultados obtidos, e conseguir auxiliar o processo de projeto deste tipo de blocos. / This work presents a fully integrated LNA-Mixer design for a Bluetooth receiver application at 2:45GHz. A concise design strategy with good physics and mathematics basis was developed to assist the design process of a LNA-Mixer block, formed by a cascode LNA in cascade to a single balanced current commutation Mixer with inductive degeneration. This strategy was adapted from literature and considers the trade-offs between noise, linearity, gain, power dissipation, impedance matching and ports isolation, using the device dimensions and bias conditions as design variables. Based on this strategy, the proposed LNA-Mixer design specifications were achieved. To validate the proposed design strategy, the LNA-Mixer were fabricated in a 0:35µm CMOS process. Furthermore, to achieve the specifications, during the development of this work a special attention to the RF CMOS inductors was given. A test chip was designed, fabricated and measured applying de-embedding structures to obtain more reliable results. The experimental results obtained for the inductors and the preliminary results for the LNA-Mixer are satisfactory compared to the specifications and as expected from simulations. However, the integrated inductors degrade the performance of the block significantly and if a manufacturing process in which the inductor has better performance is used, the resulting LNA-Mixer design applying the strategy developed in this work can be improved. Finally, it is important to highlight that the design strategy proposed in this work is already being used and adapted in other designs in order to improve the results, and to assist the design process of such blocks.

Knowledge bases are employed in a variety of applications, from natural language processing to semantic web search; alas, in practice, their usefulness is hurt by their incompleteness. To address this issue, several techniques aim at performing knowledge base completion, of which embedding models are efficient, attain state-of-the-art accuracy, and eliminate the need for feature engineering. However, embedding models predictions are notoriously hard to interpret. In this work, we propose model-agnostic methods that allow one to interpret embedding models by extracting weighted Horn rules from them. More specifically, we show how the so-called \"pedagogical techniques\", from the literature on neural networks, can be adapted to take into account the large-scale relational aspects of knowledge bases, and show experimentally their strengths and weaknesses. / Bases de conhecimento apresentam diversas aplicações, desde processamento de linguagem natural a pesquisa semântica da web; contudo, na prática, sua utilidade é prejudicada por não serem totalmente completas. Para solucionar esse problema, diversas técnicas focam em completar bases de conhecimento, das quais modelos de embedding são eficientes, atingem estado da arte em acurácia, e eliminam a necessidade de fazer-se engenharia de características dos dados de entrada. Entretanto, as predições dos modelos de embedding são notoriamente difíceis de serem interpretadas. Neste trabalho, propomos métodos agnósticos a modelo que permitem interpretar modelos de embedding através da extração de regras Horn ponderadas por pesos dos mesmos. Mais espeficicamente, mostramos como os chamados \"métodos pedagógicos\", da literatura de redes neurais, podem ser adaptados para lidar com os aspectos relacionais e de larga escala de bases de conhecimento, e mostramos experimentalmente seus pontos fortes e fracos.

Knowledge bases are employed in a variety of applications, from natural language processing to semantic web search; alas, in practice, their usefulness is hurt by their incompleteness. To address this issue, several techniques aim at performing knowledge base completion, of which embedding models are efficient, attain state-of-the-art accuracy, and eliminate the need for feature engineering. However, embedding models predictions are notoriously hard to interpret. In this work, we propose model-agnostic methods that allow one to interpret embedding models by extracting weighted Horn rules from them. More specifically, we show how the so-called \"pedagogical techniques\", from the literature on neural networks, can be adapted to take into account the large-scale relational aspects of knowledge bases, and show experimentally their strengths and weaknesses. / Bases de conhecimento apresentam diversas aplicações, desde processamento de linguagem natural a pesquisa semântica da web; contudo, na prática, sua utilidade é prejudicada por não serem totalmente completas. Para solucionar esse problema, diversas técnicas focam em completar bases de conhecimento, das quais modelos de embedding são eficientes, atingem estado da arte em acurácia, e eliminam a necessidade de fazer-se engenharia de características dos dados de entrada. Entretanto, as predições dos modelos de embedding são notoriamente difíceis de serem interpretadas. Neste trabalho, propomos métodos agnósticos a modelo que permitem interpretar modelos de embedding através da extração de regras Horn ponderadas por pesos dos mesmos. Mais espeficicamente, mostramos como os chamados \"métodos pedagógicos\", da literatura de redes neurais, podem ser adaptados para lidar com os aspectos relacionais e de larga escala de bases de conhecimento, e mostramos experimentalmente seus pontos fortes e fracos.

O atual sucesso da Internet vem inibindo a disseminação de novas arquiteturas e protocolos de rede. Especificamente, qualquer modificação no núcleo da rede requer comum acordo entre diversas partes. Face a isso, a Virtualização de Redes vem sendo proposta como um atributo diversificador para a Internet. Tal paradigma promove o desenvolvimento de novas arquiteturas e protocolos por meio da criação de múltiplas redes virtuais sobrepostas em um mesmo substrato físico. Adicionalmente, aplicações executando sobre uma mesma rede física podem ser isoladas mutuamente, propiciando a independência funcional entre as mesmas. Uma de suas mais promissoras vantagens é a capacidade de limitar o escopo de ataques, através da organização de uma infraestrutura em múltiplas redes virtuais, isolando o tráfego das mesmas e impedindo interferências. Contudo, roteadores e enlaces virtuais permanecem vulneráveis a ataques e falhas na rede física subjacente. Particularmente, caso determinado enlace do substrato seja comprometido, todos os enlaces virtuais sobrepostos (ou seja, alocados neste) serão afetados. Para lidar com esse problema, a literatura propõe dois tipos de estratégias: as que reservam recursos adicionais do substrato como sobressalentes, protegendo contra disrupções; e as que utilizam migração em tempo real para realocar recursos virtuais comprometidos. Ambas estratégias acarretam compromissos: o uso de recursos sobressalentes tende a tornar-se custoso ao provedor de infraestrutura, enquanto a migração de recursos demanda um período de convergência e pode deixar as redes virtuais inoperantes durante o mesmo. Esta dissertação apresenta ORE (Opportunistic Resilience Embedding – Mapeamento com Resiliência Oportunística), uma nova abordagem de mapeamento de redes para proteger enlaces virtuais contra disrupções no substrato físico. ORE é composto por duas estratégias: uma proativa, na qual enlaces virtuais são alocados em múltiplos caminhos para mitigar o impacto de uma disrupção; e uma reativa, a qual tenta recuperar, parcial ou integralmente, a capacidade perdida nos enlaces virtuais afetados. Ambas são modeladas como problemas de otimização. Ademais, como o mapeamento de redes virtuais é NP-Difícil, ORE faz uso de uma meta-heurística baseada em Simulated Annealing para resolver o problema de forma eficiente. Resultados numéricos mostram que ORE pode prover resiliência a disrupções por um custo mais baixo. / Recently, the Internet’s success has prevented the dissemination of novel networking architectures and protocols. Specifically, any modification to the core of the network requires agreement among many different parties. To address this situation, Network Virtualization has been proposed as a diversifying attribute for the Internet. This paradigm promotes the development of new architectures and protocols by enabling the creation of multiple virtual networks on top of a same physical substrate. In addition, applications running over the same physical network can be isolated from each other, thus allowing them to coexist independently. One of the main advantages of this paradigm is the use of isolation to limit the scope of attacks. This can be achieved by creating different, isolated virtual networks for each task, so traffic from one virtual network does not interfere with the others. However, routers and links are still vulnerable to attacks and failures on the underlying network. Particularly, should a physical link be compromised, all embedded virtual links will be affected. Previous work tackled this problem with two main strategies: using backup resources to protect against disruptions; or live migration to relocate a compromised virtual resource. Both strategies have drawbacks: backup resources tend to be expensive for the infrastructure provider, while live migration may leave virtual networks inoperable during the recovery period. This dissertation presents ORE (Opportunistic Resilience Embedding), a novel embedding approach for protecting virtual links against substrate network disruptions. ORE’s design is two-folded: while a proactive strategy embeds virtual links into multiple substrate paths in order to mitigate the initial impact of a disruption, a reactive one attempts to recover any capacity affected by an underlying disruption. Both strategies are modeled as optimization problems. Additionally, since the embedding problem is NP-Hard, ORE uses a Simulated Annealing-based meta-heuristic to solve it efficiently. Numerical results show that ORE can provide resilience to disruptions at a lower cost.

Estudamos inicialmente resultados de classificação de difeomorfismos de produtos de esferas de mesma dimensão. Tratado isto, estudamos os mergulhos suaves de produtos de três esferas, sendo a primeira de dimensão um e as demais de dimensão maior ou igual a um, com a dimensão da última maior ou igual a da segunda, em uma esfera em codimensão um, e buscamos a total caracterização do fecho das duas componentes conexas do complementar de tais mergulhos. Tratamos com enfoque especial os mergulhos do produto de três esferas de dimensão um na esfera de dimensão quatro, e, finalmente, estudamos problemas de classificação de mergulhos PL localmente não-enodados de somas conexas de toros em codimensão um. / We study initially results of classification of difeomorfisms of Cartesian products of spheres of same dimension. Treated this, we study the smooth embeddings of cartesian products of three spheres, being the first one of dimension one and excessively of bigger or equal dimension to one, with the dimension of the last equal greater or of second, in a sphere in codimension one, and search the total characterization of the latch of the two connected components of complementing of such embeddings. We deal with special approach the embeddings of the product to three spheres to dimension one in the sphere dimension four, and, finally, we study problems of classification of PL locally unknotted embeddings of connected sums of torus on codimension one.

La virtualisation de réseau permet de créer des réseaux logiques, dits virtuels sur un réseau physique partagé dit substrat. Pour ce faire, le problème d’allocation des ressources aux réseaux virtuels doit être résolu efficacement. Appelé VNE (Virtual Network Embedding), ce problème consiste à faire correspondre à chaque nœud virtuel un nœud substrat d’un côté, et de l’autre, à tout lien virtuel un ou plusieurs chemins substrat, de manière à optimiser un objectif tout en satisfaisant un ensemble de contraintes. Les ressources de calcul des nœuds et les ressources de bande passante des liens sont souvent optimisées dans un seul réseau substrat. Dans le contexte multi-domaine où la connaissance de l’information de routage est incomplète, l’optimisation des ressources de nœuds et de liens est difficile et souvent impossible à atteindre. Par ailleurs, pour assurer la continuité de service même après une panne, le VNE doit être réalisé de manière à faire face aux pannes. Dans cette thèse, nous étudions le problème d’allocation de ressources (VNE) sous diverses exigences. Pour offrir la virtualisation dans le contexte de réseau substrat multi-domaines, nous proposons une méthode de mappage conjoint des liens inter-domaines et intra-domaines. Avec une information réduite et limitées annoncées par les domaines, notre méthode est capable de mapper simultanément les liens intra-domaines et les liens inter-domaines afin d’optimiser les ressources. De plus, pour améliorer la robustesse des réseaux virtuels, nous proposons un algorithme d’évitement des pannes qui minimise la probabilité de panne des réseaux virtuels. Des solutions exactes et heuristiques sont proposées et détaillées pour des liens à bande passante infinie ou limitée. En outre, nous combinons l’algorithme d’évitement des pannes avec la protection pour proposer un VNE robuste et résistant aux pannes. Avec cette nouvelle approche, les liens protégeables puis les liens les moins vulnérables sont prioritairement sélectionnés pour le mappage des liens. Pour déterminer les liens protégeables, nous proposons une heuristique qui utilise l’algorithme du maxflow afin de vérifier etdedéterminerlesliensprotégeablesàl’étapedumappagedesliensprimaires. Encasd’insuffisance de ressources pour protéger tous les liens primaires, notre approche sélectionne les liens réduisant la probabilité de panne. / Network virtualization allows to create logical or virtual networks on top of a shared physical or substrate network. The resource allocation problem is an important issue in network virtualization. It corresponds to a well known problem called virtual network embedding (VNE). VNE consists in mapping each virtual node to one substrate node and each virtual link to one or several substrate paths in a way that the objective is optimized and the constraints verified. The objective often corresponds to the optimization of the node computational resources and link bandwidth whereas the constraints generally include geographic location of nodes, CPU, bandwidth, etc. In the multi-domain context where the knowledge of routing information is incomplete, the optimization of node and link resources are difficult and often impossible to achieve. Moreover, to ensure service continuity even upon failure, VNE should cope with failures by selecting the best and resilient mappings. Inthisthesis,westudytheVNEresourceallocationproblemunderdifferentrequirements. To embed a virtual network on multi-domain substrate network, we propose a joint peering and intra domain link mapping method. With reduced and limited information disclosed by the domains, our downsizing algorithm maps the intra domain and peering links in the same stage so that the resource utilization is optimized. To enhance the reliability of virtual networks, we propose a failure avoidance approach that minimizes the failure probability of virtual networks. Exact and heuristic solutions are proposed and detailed for the infinite and limited bandwidth link models. Moreover, we combine the failure avoidance with the failure protection in our novel protection-level-aware survivable VNE in order to improve the reliability. With this last approach, the protectable then the less vulnerable links are first selected for link mapping. To determine the protectable links, we propose a maxflow based heuristic that checks for the existence of backup paths during the primary mapping stage. In case of insufficient backup resources, the failure probability is reduced.

In the development of a nonlinear transistor model, several measurements are used to extract equivalent circuit parameters. The current-voltage (IV) characteristic of a transistor is one of the measurement data sets that allows the nonlinear model parameters to be extracted. The accuracy of the IV measurement greatly influences the accuracy of the large-signal model. Numerous works have reported the inadequacy of traditional static DC IV measurements to accurately predict radio-frequency (RF) behavior for many devices. This inaccuracy results from slow processes in the device that do not have time to completely respond to the quick changes in terminal conditions when the device is operating at high frequencies; however, these slow processes respond fully to reach a new steady-state condition in the DC sweep measurement. The two dominant processes are self-heating of the device and changes in trap occupancy. One method of allowing the thermal and trap conditions to remain in a state comparable to that of RF operation is to perform pulsed IV measurements to obtain the IV curves. In addition, thermal correction can be used to adjust the IV curves to compensate for self-heating in the case that the predominant effect in the device is thermal. To gain a better understanding of pulsed IV measurement techniques, measurement waveforms of a commercially available pulsed IV analyzer are examined in the time domain. In addition, the use of bias tees with pulsed IV measurement is explored; such a setup may be desired to maintain stability or to enable simultaneous pulsed S-parameter and pulsed IV measurement. In measurements with bias tees, the pulse length setting must be long enough to allow the voltage across the inductor to change before the measurement is made. In many circumstances, it is beneficial to compare different sets of IV curves for a device. The comparison of pulsed and static IV measurements, measured and modeled IV measurements, as well as two measurements with identical settings on the same instrument (to ascertain instrument repeatability) can be performed using the proposed normalized difference unit (NDU). This unit provides a comparison that equally weights the two sets of data to be compared. Due to the normalization factor used, the value of the NDU is independent of the size of the device for which the IV curves are compared. The variety of comparisons for which this unit can be used and its ability to present differences quantitatively allow it to be used as a robust metric for comparing IV curves. Examples of the use of the NDU shown include determination of measurement repeatability, comparison of pulsed and static IV data, and a comparison of model fits. The NDU can also be used to isolate thermal and trapping processes and to give the maximum pulse length that can be used for pulsed IV measurement without contamination by each of these processes. Plotting the NDU comparing static and pulsed IV data versus pulse length shows this maximum pulse length that can be used for each effect, while a plot of the NDU comparing pulsed IV data for two quiescent bias points of equal power dissipation reveals only differences due to trapping effects. In this way, trapping effects can be distinguished from thermal effects. Electrothermal modeling has arisen as a method of correcting for self-heating processes in a device with predominantly thermal effects. A parallel RC circuit is used to model channel temperature as a function of ambient temperature and power dissipated in the channel or junction. A technique is proposed for thermal resistance measurement and compared with a technique found in the literature. It is demonstrated that the thermal time constant can be measured from a plot of the NDU versus pulse length, and the thermal capacitance is then obtained using the thermal resistance and time constant. Finally, the results obtained through the thermal resistance measurement procedures are used to thermally correct static IV curves. Because trapping effects are negligible, it is shown that IV curves corresponding to different quiescent bias points for a Si LDMOSFET can be synthesized from three sets of static IV data taken at different ambient temperatures. The results obtained from this correction process for two quiescent bias points are compared to the pulsed IV results for these quiescent bias points and found to be quite accurate. Use of the methods presented in this work for obtaining more accurate transistor IV data data should assist in allowing more accurate nonlinear models to be obtained.

A new approach to image-space hardware-based rendering of Constructive Solid Geometry (CSG) models is presented. The work is motivated by the evolving functionality and performance of computer graphics hardware. This work is also motivated by a specific industrial application --- interactive verification of five axis grinding machine tool programs. The goal is to minimise the amount of time required to render each frame in an animation or interactive application involving boolean combinations of three dimensional shapes. The Sequenced Convex Subtraction (SCS) algorithm utilises sequenced subtraction of convex objects for the purpose of interactive CSG rendering. Concave shapes must be decomposed into convex shapes for the purpose of rendering. The length of Permutation Embedding Sequences (PESs) used as subtraction sequences are shown to have a quadratic lower bound. In many situations shorter sequences can be used, in the best case linear. Approaches to s ubtraction sequence encoding are presented including the use of object-space overlap information. The implementation of the algorithm is experimentally shown to perform better on modern commodity graphics hardware than previously reported methods. This work also examines performance aspects of the SCS algorithm itself. Overall performance depends on hardware characteristics, the number and spatial arrangement of primitives, and the structure and boolean operators of the CSG tree.

<p>This thesis treats inequalities of Carlson type, i.e. inequalities of the form</p><p><mml:math><mml:semantics><mml:mrow><mml:mrow><mml:msub><mml:mi>∥f∥</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo mml:stretchy="false">≤</mml:mo><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:munderover><mml:mo mml:stretchy="false">∏</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo mml:stretchy="false">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>m</mml:mi></mml:munderover><mml:msubsup><mml:mi>∥f∥</mml:mi><mml:msub><mml:mi>A</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:msub><mml:mo mml:stretchy="false">θ</mml:mo><mml:mi>i</mml:mi></mml:msub></mml:msubsup></mml:mrow></mml:mrow></mml:semantics></mml:math></p><p>where <mml:math><mml:semantics><mml:mrow><mml:mrow><mml:munderover><mml:mo mml:stretchy="false">∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo mml:stretchy="false">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>m</mml:mi></mml:munderover><mml:msub><mml:mo mml:stretchy="false">θ</mml:mo><mml:mn>i </mml:mn></mml:msub></mml:mrow><mml:mo mml:stretchy="false">=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math> and <i>K</i> is some constant, independent of the function <i>f</i>. <i>X</i> and <mml:math><mml:semantics><mml:msub><mml:mi>A</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:semantics></mml:math> are normed spaces, embedded in some Hausdorff topological vector space. In most cases, we have <mml:math><mml:semantics><mml:mrow><mml:mi>m</mml:mi><mml:mo mml:stretchy="false">=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:semantics></mml:math>, and the spaces involved are weighted Lebesgue spaces on some measure space. For example, the inequality</p><p><mml:math><mml:semantics><mml:mrow><mml:mrow><mml:munderover><mml:mo mml:stretchy="false">∫</mml:mo><mml:mn>0</mml:mn><mml:mo mml:stretchy="false">∞</mml:mo></mml:munderover><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo mml:stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo mml:stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi mml:fontstyle="italic">dx</mml:mi><mml:mo mml:stretchy="false">≤</mml:mo><mml:msqrt><mml:mo mml:stretchy="false">π</mml:mo></mml:msqrt></mml:mrow><mml:msup><mml:mfenced mml:open="(" mml:close=")"><mml:mrow><mml:mrow><mml:munderover><mml:mo mml:stretchy="false">∫</mml:mo><mml:mn>0</mml:mn><mml:mo mml:stretchy="false">∞</mml:mo></mml:munderover><mml:msup><mml:mi>f</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:mo mml:stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo mml:stretchy="false">)</mml:mo></mml:mrow><mml:mi mml:fontstyle="italic">dx</mml:mi></mml:mrow></mml:mfenced><mml:mrow><mml:mn>1</mml:mn><mml:mo mml:stretchy="false">/</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mfenced mml:open="(" mml:close=")"><mml:mrow><mml:mrow><mml:munderover><mml:mo mml:stretchy="false">∫</mml:mo><mml:mn>0</mml:mn><mml:mo mml:stretchy="false">∞</mml:mo></mml:munderover><mml:msup><mml:mi>x</mml:mi><mml:mn>2 </mml:mn></mml:msup></mml:mrow><mml:msup><mml:mi>f</mml:mi><mml:mn>2 </mml:mn></mml:msup><mml:mrow><mml:mo mml:stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo mml:stretchy="false">)</mml:mo></mml:mrow><mml:mi mml:fontstyle="italic">dx</mml:mi></mml:mrow></mml:mfenced><mml:mrow><mml:mn>1</mml:mn><mml:mo mml:stretchy="false">/</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:semantics></mml:math></p><p>first proved by F. Carlson, is the above inequality with <mml:math><mml:semantics><mml:mrow><mml:mi>m</mml:mi><mml:mo mml:stretchy="false">=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:semantics></mml:math>, <mml:math><mml:semantics><mml:mrow><mml:mrow><mml:msub><mml:mo mml:stretchy="false">θ</mml:mo><mml:mn>1 </mml:mn></mml:msub><mml:mo mml:stretchy="false">=</mml:mo><mml:msub><mml:mo mml:stretchy="false">θ</mml:mo><mml:mn>2 </mml:mn></mml:msub></mml:mrow><mml:mo mml:stretchy="false">=</mml:mo><mml:mfrac><mml:mn>1 </mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:semantics></mml:math>, <mml:math><mml:semantics><mml:mrow><mml:mrow><mml:mi>X</mml:mi><mml:mo mml:stretchy="false">=</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mo mml:stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mo mml:stretchy="false">ℝ</mml:mo><mml:mrow><mml:mo mml:stretchy="false">+</mml:mo><mml:mn>, </mml:mn></mml:mrow></mml:msub><mml:mi mml:fontstyle="italic">dx</mml:mi></mml:mrow><mml:mo mml:stretchy="false">)</mml:mo></mml:mrow><mml:mn>, </mml:mn><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mn>1 </mml:mn></mml:msub><mml:mo mml:stretchy="false">=</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>2 </mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mo mml:stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mo mml:stretchy="false">ℝ</mml:mo><mml:mrow><mml:mo mml:stretchy="false">+</mml:mo><mml:mn>, </mml:mn></mml:mrow></mml:msub><mml:mi mml:fontstyle="italic">dx</mml:mi></mml:mrow><mml:mo mml:stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math> and <mml:math><mml:semantics><mml:mrow><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mn>2 </mml:mn></mml:msub><mml:mo mml:stretchy="false">=</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mn>2 </mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mo mml:stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mo mml:stretchy="false">ℝ</mml:mo><mml:mrow><mml:mo mml:stretchy="false">+</mml:mo><mml:mn>, </mml:mn></mml:mrow></mml:msub><mml:msup><mml:mi>x</mml:mi><mml:mn>2 </mml:mn></mml:msup><mml:mi mml:fontstyle="italic">dx</mml:mi></mml:mrow><mml:mo mml:stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math>. In different situations, suffcient, and sometimes necessary, conditions are given on the weights in order for a Carlson type inequality to hold for some constant <i>K</i>. Carlson type inequalities have applications to e.g. moment problems, Fourier analysis, optimal sampling, and interpolation theory.</p>

This thesis treats inequalities of Carlson type, i.e. inequalities of the form &lt;mml:math&gt;&lt;mml:semantics&gt;&lt;mml:mrow&gt;&lt;mml:mrow&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;∥f∥&lt;/mml:mi&gt;&lt;mml:mi&gt;x&lt;/mml:mi&gt;&lt;/mml:msub&gt;&lt;mml:mo mml:stretchy="false"&gt;≤&lt;/mml:mo&gt;&lt;mml:mi&gt;K&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;mml:mrow&gt;&lt;mml:munderover&gt;&lt;mml:mo mml:stretchy="false"&gt;∏&lt;/mml:mo&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;i&lt;/mml:mi&gt;&lt;mml:mo mml:stretchy="false"&gt;=&lt;/mml:mo&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;/mml:mrow&gt;&lt;mml:mi&gt;m&lt;/mml:mi&gt;&lt;/mml:munderover&gt;&lt;mml:msubsup&gt;&lt;mml:mi&gt;∥f∥&lt;/mml:mi&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;A&lt;/mml:mi&gt;&lt;mml:mi&gt;i&lt;/mml:mi&gt;&lt;/mml:msub&gt;&lt;mml:msub&gt;&lt;mml:mo mml:stretchy="false"&gt;θ&lt;/mml:mo&gt;&lt;mml:mi&gt;i&lt;/mml:mi&gt;&lt;/mml:msub&gt;&lt;/mml:msubsup&gt;&lt;/mml:mrow&gt;&lt;/mml:mrow&gt;&lt;/mml:semantics&gt;&lt;/mml:math&gt; where &lt;mml:math&gt;&lt;mml:semantics&gt;&lt;mml:mrow&gt;&lt;mml:mrow&gt;&lt;mml:munderover&gt;&lt;mml:mo mml:stretchy="false"&gt;∑&lt;/mml:mo&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;i&lt;/mml:mi&gt;&lt;mml:mo mml:stretchy="false"&gt;=&lt;/mml:mo&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;/mml:mrow&gt;&lt;mml:mi&gt;m&lt;/mml:mi&gt;&lt;/mml:munderover&gt;&lt;mml:msub&gt;&lt;mml:mo mml:stretchy="false"&gt;θ&lt;/mml:mo&gt;&lt;mml:mn&gt;i &lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;/mml:mrow&gt;&lt;mml:mo mml:stretchy="false"&gt;=&lt;/mml:mo&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;/mml:mrow&gt;&lt;/mml:semantics&gt;&lt;/mml:math&gt; and K is some constant, independent of the function f. X and &lt;mml:math&gt;&lt;mml:semantics&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;A&lt;/mml:mi&gt;&lt;mml:mi&gt;i&lt;/mml:mi&gt;&lt;/mml:msub&gt;&lt;/mml:semantics&gt;&lt;/mml:math&gt; are normed spaces, embedded in some Hausdorff topological vector space. In most cases, we have &lt;mml:math&gt;&lt;mml:semantics&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;m&lt;/mml:mi&gt;&lt;mml:mo mml:stretchy="false"&gt;=&lt;/mml:mo&gt;&lt;mml:mn&gt;2&lt;/mml:mn&gt;&lt;/mml:mrow&gt;&lt;/mml:semantics&gt;&lt;/mml:math&gt;, and the spaces involved are weighted Lebesgue spaces on some measure space. For example, the inequality &lt;mml:math&gt;&lt;mml:semantics&gt;&lt;mml:mrow&gt;&lt;mml:mrow&gt;&lt;mml:munderover&gt;&lt;mml:mo mml:stretchy="false"&gt;∫&lt;/mml:mo&gt;&lt;mml:mn&gt;0&lt;/mml:mn&gt;&lt;mml:mo mml:stretchy="false"&gt;∞&lt;/mml:mo&gt;&lt;/mml:munderover&gt;&lt;mml:mi&gt;f&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;mml:mrow&gt;&lt;mml:mo mml:stretchy="false"&gt;(&lt;/mml:mo&gt;&lt;mml:mi&gt;x&lt;/mml:mi&gt;&lt;mml:mo mml:stretchy="false"&gt;)&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;mml:mrow&gt;&lt;mml:mi mml:fontstyle="italic"&gt;dx&lt;/mml:mi&gt;&lt;mml:mo mml:stretchy="false"&gt;≤&lt;/mml:mo&gt;&lt;mml:msqrt&gt;&lt;mml:mo mml:stretchy="false"&gt;π&lt;/mml:mo&gt;&lt;/mml:msqrt&gt;&lt;/mml:mrow&gt;&lt;mml:msup&gt;&lt;mml:mfenced mml:open="(" mml:close=")"&gt;&lt;mml:mrow&gt;&lt;mml:mrow&gt;&lt;mml:munderover&gt;&lt;mml:mo mml:stretchy="false"&gt;∫&lt;/mml:mo&gt;&lt;mml:mn&gt;0&lt;/mml:mn&gt;&lt;mml:mo mml:stretchy="false"&gt;∞&lt;/mml:mo&gt;&lt;/mml:munderover&gt;&lt;mml:msup&gt;&lt;mml:mi&gt;f&lt;/mml:mi&gt;&lt;mml:mn&gt;2&lt;/mml:mn&gt;&lt;/mml:msup&gt;&lt;/mml:mrow&gt;&lt;mml:mrow&gt;&lt;mml:mo mml:stretchy="false"&gt;(&lt;/mml:mo&gt;&lt;mml:mi&gt;x&lt;/mml:mi&gt;&lt;mml:mo mml:stretchy="false"&gt;)&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;mml:mi mml:fontstyle="italic"&gt;dx&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;/mml:mfenced&gt;&lt;mml:mrow&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;mml:mo mml:stretchy="false"&gt;/&lt;/mml:mo&gt;&lt;mml:mn&gt;4&lt;/mml:mn&gt;&lt;/mml:mrow&gt;&lt;/mml:msup&gt;&lt;mml:msup&gt;&lt;mml:mfenced mml:open="(" mml:close=")"&gt;&lt;mml:mrow&gt;&lt;mml:mrow&gt;&lt;mml:munderover&gt;&lt;mml:mo mml:stretchy="false"&gt;∫&lt;/mml:mo&gt;&lt;mml:mn&gt;0&lt;/mml:mn&gt;&lt;mml:mo mml:stretchy="false"&gt;∞&lt;/mml:mo&gt;&lt;/mml:munderover&gt;&lt;mml:msup&gt;&lt;mml:mi&gt;x&lt;/mml:mi&gt;&lt;mml:mn&gt;2 &lt;/mml:mn&gt;&lt;/mml:msup&gt;&lt;/mml:mrow&gt;&lt;mml:msup&gt;&lt;mml:mi&gt;f&lt;/mml:mi&gt;&lt;mml:mn&gt;2 &lt;/mml:mn&gt;&lt;/mml:msup&gt;&lt;mml:mrow&gt;&lt;mml:mo mml:stretchy="false"&gt;(&lt;/mml:mo&gt;&lt;mml:mi&gt;x&lt;/mml:mi&gt;&lt;mml:mo mml:stretchy="false"&gt;)&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;mml:mi mml:fontstyle="italic"&gt;dx&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;/mml:mfenced&gt;&lt;mml:mrow&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;mml:mo mml:stretchy="false"&gt;/&lt;/mml:mo&gt;&lt;mml:mn&gt;4&lt;/mml:mn&gt;&lt;/mml:mrow&gt;&lt;/mml:msup&gt;&lt;/mml:mrow&gt;&lt;/mml:semantics&gt;&lt;/mml:math&gt; first proved by F. Carlson, is the above inequality with &lt;mml:math&gt;&lt;mml:semantics&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;m&lt;/mml:mi&gt;&lt;mml:mo mml:stretchy="false"&gt;=&lt;/mml:mo&gt;&lt;mml:mn&gt;2&lt;/mml:mn&gt;&lt;/mml:mrow&gt;&lt;/mml:semantics&gt;&lt;/mml:math&gt;, &lt;mml:math&gt;&lt;mml:semantics&gt;&lt;mml:mrow&gt;&lt;mml:mrow&gt;&lt;mml:msub&gt;&lt;mml:mo mml:stretchy="false"&gt;θ&lt;/mml:mo&gt;&lt;mml:mn&gt;1 &lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;mml:mo mml:stretchy="false"&gt;=&lt;/mml:mo&gt;&lt;mml:msub&gt;&lt;mml:mo mml:stretchy="false"&gt;θ&lt;/mml:mo&gt;&lt;mml:mn&gt;2 &lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;/mml:mrow&gt;&lt;mml:mo mml:stretchy="false"&gt;=&lt;/mml:mo&gt;&lt;mml:mfrac&gt;&lt;mml:mn&gt;1 &lt;/mml:mn&gt;&lt;mml:mn&gt;2&lt;/mml:mn&gt;&lt;/mml:mfrac&gt;&lt;/mml:mrow&gt;&lt;/mml:semantics&gt;&lt;/mml:math&gt;, &lt;mml:math&gt;&lt;mml:semantics&gt;&lt;mml:mrow&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;X&lt;/mml:mi&gt;&lt;mml:mo mml:stretchy="false"&gt;=&lt;/mml:mo&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;L&lt;/mml:mi&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;/mml:mrow&gt;&lt;mml:mrow&gt;&lt;mml:mo mml:stretchy="false"&gt;(&lt;/mml:mo&gt;&lt;mml:mrow&gt;&lt;mml:msub&gt;&lt;mml:mo mml:stretchy="false"&gt;ℝ&lt;/mml:mo&gt;&lt;mml:mrow&gt;&lt;mml:mo mml:stretchy="false"&gt;+&lt;/mml:mo&gt;&lt;mml:mn&gt;, &lt;/mml:mn&gt;&lt;/mml:mrow&gt;&lt;/mml:msub&gt;&lt;mml:mi mml:fontstyle="italic"&gt;dx&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;mml:mo mml:stretchy="false"&gt;)&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;mml:mn&gt;, &lt;/mml:mn&gt;&lt;mml:mrow&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;A&lt;/mml:mi&gt;&lt;mml:mn&gt;1 &lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;mml:mo mml:stretchy="false"&gt;=&lt;/mml:mo&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;L&lt;/mml:mi&gt;&lt;mml:mn&gt;2 &lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;/mml:mrow&gt;&lt;mml:mrow&gt;&lt;mml:mo mml:stretchy="false"&gt;(&lt;/mml:mo&gt;&lt;mml:mrow&gt;&lt;mml:msub&gt;&lt;mml:mo mml:stretchy="false"&gt;ℝ&lt;/mml:mo&gt;&lt;mml:mrow&gt;&lt;mml:mo mml:stretchy="false"&gt;+&lt;/mml:mo&gt;&lt;mml:mn&gt;, &lt;/mml:mn&gt;&lt;/mml:mrow&gt;&lt;/mml:msub&gt;&lt;mml:mi mml:fontstyle="italic"&gt;dx&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;mml:mo mml:stretchy="false"&gt;)&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;/mml:mrow&gt;&lt;/mml:semantics&gt;&lt;/mml:math&gt; and &lt;mml:math&gt;&lt;mml:semantics&gt;&lt;mml:mrow&gt;&lt;mml:mrow&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;A&lt;/mml:mi&gt;&lt;mml:mn&gt;2 &lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;mml:mo mml:stretchy="false"&gt;=&lt;/mml:mo&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;L&lt;/mml:mi&gt;&lt;mml:mn&gt;2 &lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;/mml:mrow&gt;&lt;mml:mrow&gt;&lt;mml:mo mml:stretchy="false"&gt;(&lt;/mml:mo&gt;&lt;mml:mrow&gt;&lt;mml:msub&gt;&lt;mml:mo mml:stretchy="false"&gt;ℝ&lt;/mml:mo&gt;&lt;mml:mrow&gt;&lt;mml:mo mml:stretchy="false"&gt;+&lt;/mml:mo&gt;&lt;mml:mn&gt;, &lt;/mml:mn&gt;&lt;/mml:mrow&gt;&lt;/mml:msub&gt;&lt;mml:msup&gt;&lt;mml:mi&gt;x&lt;/mml:mi&gt;&lt;mml:mn&gt;2 &lt;/mml:mn&gt;&lt;/mml:msup&gt;&lt;mml:mi mml:fontstyle="italic"&gt;dx&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;mml:mo mml:stretchy="false"&gt;)&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;/mml:mrow&gt;&lt;/mml:semantics&gt;&lt;/mml:math&gt;. In different situations, suffcient, and sometimes necessary, conditions are given on the weights in order for a Carlson type inequality to hold for some constant K. Carlson type inequalities have applications to e.g. moment problems, Fourier analysis, optimal sampling, and interpolation theory.