A section of the total tangent space of a scheme X of finite type over a field k, i.e. a vector field on X, corresponds to an X-valued 1-jet on X. In the language of jets the notion of a vector field becomes functorial, and the total tangent space constitutes one of an infinite family of jet schemes Jm(X) for m ≥ 0. We prove that there exist families of “logarithmic” jet schemes JDm(X) for m ≥ 0, in the category of k-schemes of finite type, associated to any given X and its family of divisors D = (D₁, . . . ,Dr). The sections of JD₁(X) correspond to so-called vector fields on X with logarithmic poles along the family of divisors D = (D₁, . . . ,Dr). To prove this, we first introduce the categories of pairs (X,D) where D is as mentioned, an r-tuple of (effective Cartier) divisors on the scheme X. The categories of pairs provide a convenient framework for working with only those jets that pull back families of divisors.

In this thesis a single stage detector logarithmic video amplifier is designed with a dynamic range of 40dB in 2-6GHz frequency band. Since the detector logarithmic video amplifier (DLVA) is used to convert the power of the RF signals to video voltages in logarithmic scale, it can be regarded as a logarithmic converter instead of logarithmic amplifier. The design is composed of two main parts: The Schottky diode detector rectifies the incoming RF signal and produces a video voltage and the logarithmic amplifier transforms the scale of the video voltage from linear scale to logarithmic scale in order to observe the RF signals with a wide amplitude range. The approximation of the logarithmic function is obtained by the summation of the output currents of the differential amplifiers operating as logarithmic stages. Offset voltage of the DLVA is minimized in order to obtain maximum sensitivity / this makes the detection of RF signals with low power possible. The study is composed of mainly three parts: First, brief information about logarithmic amplification techniques is given and the circuit architecture is developed for logarithmic amplification and video detection, second these circuits are simulated and finally the design is implemented and tested.

Στα ασύρµατα δίκτυα η τεχνολογία MIMO-OFDM έχει ευρέως υιοθετηθεί µε στόχο την αύξηση του ρυθµού δεδοµένων σε υπηρεσίες υψηλής ποιότητας. Στο δέκτη ενός συστήµατος MIMO-OFDM ο υπολογισµός του µητρώου συντελεστών που χρειάζεται ο εξισωτής (equalizer) είναι ένα κρίσιµο σηµείο µε υψηλή υπολογιστική πολυπλοκότητα. Η καθυστέρηση που απαιτείται για την εκτέλεση του συγκεκριµένου υπολογισµού επηρεάζει άµεσα το ρυθµό περάτωσης (throughput) και την ποιότητα υπηρεσίας (QoS). Σε συστήµατα wi-fi (για παράδειγµα στο IEEE 802.11ac) στην αρχή κάθε πακέτου µεταδίδονται προσυµφωνηµένα σύµβολα, ώστε να εκπαιδεύσουν τον εξισωτή. Η συµβατική µέθοδος περιµένει να έρθουν όλα τα σύµβολα και στη συνέχεια υπολογίζει τοµητρώο του εξισωτή. Μια πιο πρόσφατη µέθοδος που αποσκοπεί στη µείωση της παραπάνω καθυστέρησης είναι ο προοδευτικός υπολογισµός του µητρώου του εξισωτή. Στη µέθοδο αυτή, γίνονται υπολογισµοί ανά σύµβολο εκπαίδευσης χωρίς να επηρεάζεται το τελικό αποτέλεσµα. Στα πλαίσια της διπλωµατικής υλοποιήθηκαν αρχιτεκτονικές υλικού για εξισωτές που αξιοποιούν την ανωτέρω µέθοδο και απεικονίστηκαν σε αναπτυξιακό σύστηµα µε FPGA. Επίσης, διερευνήθηκε ο ρόλος της αναπαράστασης των δεδοµένων στην πολυπλοκότητα του συστήµατος και βελτιστοποιήθηκαν οι σχετικές σχεδιαστικές παράµετροι, µε έµφαση στη χρήση της λογαριθµικής αριθµητικής. / In wireless networks, the MIMO-OFDM technology has been widely adopted in order to increase the data rate at high quality of service (QoS). In the receiver, the MIMO equalizer matrix calculation is an important part with high computational complexity. The delay of the matrix calculation affects directly the system throughput and QoS. In wi-fi systems (e.g. IEEE 802.11ac), training symbols are transmitted at the beginning of every frame, in order to train the equalizer. The conventional method waits all the training symbols to arrive before starting the calculations. A recent method proposes the progressive calculation of the equalizer matrix in order to decrease the processing delay. In this method, the equalizer matrix calculation starts upon receiving the first training symbol. The object of this thesis is the development of hardware architectures for equalizers that follow the progressive method and their implementation on FPGA. Furthermore, we analyzed the impact of the data representation on the system complexity and optimized the respective design parameters, using logarithmic number system.

Αρχικά εισάγονται οι ανισότητες Sobolev για τις οποίες ο ρόλος της διάστασης είναι θεμελιώδης και στην συνέχεια δείχνουμε τον τρόπο με τιν οποίο αυτές εξασθενίζουν. Ακόμα θα δούμε πως με τη βοήθεια των ανισοτήτων εντροπίας ενέργειας γίνεται κατανοητός ο απειροδιάστατος χαρακτήρας της λογαριθμικής ανισότητας Sobolev. / Firstly we introduce the Sobolev inequalities that the role of the dimension is vital. Furthermore, we will show the way that they have been weakened. We see also with the help of the inequalities of Entropy-Energy how we can clarify the infinite dimensional character of the logarithmic Sobolev inequalities.

A section of the total tangent space of a scheme X of finite type over a field k, i.e. a vector field on X, corresponds to an X-valued 1-jet on X. In the language of jets the notion of a vector field becomes functorial, and the total tangent space constitutes one of an infinite family of jet schemes Jm(X) for m ≥ 0. We prove that there exist families of “logarithmic” jet schemes JDm(X) for m ≥ 0, in the category of k-schemes of finite type, associated to any given X and its family of divisors D = (D₁, . . . ,Dr). The sections of JD₁(X) correspond to so-called vector fields on X with logarithmic poles along the family of divisors D = (D₁, . . . ,Dr). To prove this, we first introduce the categories of pairs (X,D) where D is as mentioned, an r-tuple of (effective Cartier) divisors on the scheme X. The categories of pairs provide a convenient framework for working with only those jets that pull back families of divisors.

Molecular simulation is an essential tool in understanding complex chemical and biochemical processes as real-life experiments prove increasingly costly or infeasible in practice . This thesis is devoted to methodological aspects of molecular simulation, with a particular focus on computing transition paths and their associated free energy profiles. The first part is dedicated to computational methods for reaction path and transition state searches on a potential energy surface. In Chapter 3 we propose an improvement to a widely-used transition state search method, the Activation Relaxation Technique (ART). We also present a local convergence study of a prototypical algorithm. The second part is dedicated to free energy computations. We focus in particular on an adaptive importance sampling technique, the Adaptive Biasing Force (ABF) method. The first contribution to this field, presented in Chapter 5, consists in showing the applicability to a large molecular system of a new parallel implementation, named multiple-walker ABF (MW-ABF). Numerical experiments demonstrated the robustness of MW-ABF against artefacts arising due to poorly chosen or oversimplified reaction coordinates. These numerical findings inspired a new study of the longtime convergence of the ABF method, as presented in Chapter 6. By studying a slightly modified model, we back our numerical results by showing a faster theoretical rate of convergence of ABF than was previously shown

Orientador: Dimitar Kolev Dimitrov / Banca: Valdir Antônio Menegatto / Banca: Ali Messaoudi / Resumo: Estudamos alguns tópicos da Teoria do Potencial Logarítmico. Enfatizamos o problema de caracterizar a medida do equilíbrio. Provamos um resultado sobre a assintótica da medida contadora de zeros, associada com uma classe de polinômios. / Abstract: We study some basic topics of The Theory of the Logarithmic potential. We emphasize on the problem by characterizing the equilibrium measure. A result on the asymptotics of the zero counting measure associated with a class of polynomials is proved. / Mestre

[pt] O estudo teve como objetivo abordar o ensino das Funções Exponenciais e Logarítmicas quanto ao aspecto teórico para o fortalecimento dos conceitos. Para alcançar este objetivo, este trabalho foi feito baseado em pesquisas bibliográficas em livros, artigos, dissertações entre outras fontes. Antes de abordarmos o assunto de exponenciais explanamos sobre o conceito de funções e citamos a importância das tabelas. Abordamos o assunto de exponenciais citando algumas de suas aplicações e algumas demonstrações. Mostramos também o processo de construção da tabela de logaritmos decimais. Para fortalecer o conhecimento sobre o assunto faremos uso de calculadoras e planilhas eletrônicas em algumas atividades propostas para mostrar propriedades das funções exponenciais e logarítmicas e a importante constante matemática e, que aparece naturalmente em fenômenos da Natureza. Esperamos contribuir de forma positiva para o interesse e aprimoramento de professores no assunto explanado e, principalmente, para a motivação dos alunos em estudar e compreender melhor a importância dos logaritmos. / [en] This study aims to approach the teaching of exponential and logarithmic functions regarding the theoretical aspects to the enhancement of concepts. In order to achieve this goal, this study was based on bibliographic research, articles, and dissertations, among other sources. Before we approached the issue on exponentials, we went over the concept of functions and highlited the importance of tables. We also approached the issue of exponentials citing from some of its applications and demonstrations. We brought up, as well, the process of setting up the decimal logarithmic table. In order to enhance the knowledge on the subject, we will make use of calculators and tables so as to reinforce the knowledge over the subject in some activities proposed to demonstrate the properties of the exponential and logarithmic functions as well as the important mathematical constant e which occurs naturally in natural phenomena. We hope to contribute positively to the bringing out of interest to teachers and likewise enhancement to their knowledge on the subject studied and, specially, motivation to students generating a better understanding over the importance of logarithms.

Submitted by MARCIA EIKO MOTOKI null (marcikom@yahoo.com.br) on 2016-02-18T16:26:31Z No. of bitstreams: 1 V_Final_Dissertação_PROFMAT.pdf: 2562973 bytes, checksum: 76528c6285a91b9e64a1202f9298773f (MD5) / Approved for entry into archive by Ana Paula Grisoto (grisotoana@reitoria.unesp.br) on 2016-02-19T19:18:27Z (GMT) No. of bitstreams: 1 motoki_me_me_prud.pdf: 2562973 bytes, checksum: 76528c6285a91b9e64a1202f9298773f (MD5) / Made available in DSpace on 2016-02-19T19:18:27Z (GMT). No. of bitstreams: 1 motoki_me_me_prud.pdf: 2562973 bytes, checksum: 76528c6285a91b9e64a1202f9298773f (MD5) Previous issue date: 2016-01-22 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Os logaritmos foram criados, na primeira metade do século XVII, para facilitar os cálculos matemáticos tornando-se um instrumento de cálculo eficiente, pois tem como propriedade fundamental transformar produtos em soma. Atualmente, mesmo com o uso de modernas máquinas de calcular ao alcance de todos, sua importância não é menor do que foi no passado, pois está relacionada a vários fenômenos naturais. Apesar das diversas aplicações, além da Matemática Financeira (juros simples e contínuos) há também aplicações na Física, Química, Biologia, Geografia e Música. Na minha experiência profissional, resolver problemas que inclua os logaritmos é considerado complicado por muitos estudantes, que não compreendem seu conceito e nem a sua utilidade. O presente trabalho trata de uma abordagem para solucionar situações-problemas envolvendo os logaritmos, através de um esquema de resolução que explora os detalhes do enunciado, organizando os dados relevantes, a transposição para a linguagem matemática e desenvolvimento dos cálculos até a resposta final. Para atingir tal objetivo, é feito uma revisão do surgimento dos logaritmos, do conceito de potenciação, função exponencial, logaritmo como área e função logarítmica. Além disso, é apresentado o logaritmo aplicado em diversas áreas. / Logarithms were created in the first half of the seventeenth century, to facilitate the mathematics becoming an efficient calculation tool, it has become a fundamental property products sum. Currently, even with the use of modern calculators available to all, its importance is no less than in the past, because it is related to various natural phenomena. Despite several applications in addition to the Financial Mathematics (simple and continuous interest) there are also applications in physics, chemistry, biology, geography and music. In my professional experience, solve problems involving logarithms is considered complicated by many students who do not understand its concept nor its usefulness. This work is an approach to solving problem situations involving logarithms, through a resolution scheme that explores the details of the statement by organizing relevant data, transposed into mathematical language and development of calculations to the final answer. To achieve this goal, it is made a revision of emergence of logarithms, the concept of empowerment, exponential function, logarithm as area and logarithmic function. Moreover, the logarithm is shown applied in several areas.

CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / Neste trabalho vamos fazer uma abordagem elementar sobre a funÃÃo exponencial visando entender o significado de potÃncias com expoente natural, inteiro, racional e irracional bem como as propriedades que fazem dela uma das funÃÃes mais importantes da MatemÃtica. Paralelamente, serà feita outra abordagem mostrando essas propriedades usando uma ferramenta poderosa da MatemÃtica: o cÃlculo diferencial e integral. TambÃm vamos tratar da funÃÃo logarÃtmica por ela ser a inversa da funÃÃo exponencial e por ser tÃo importante quanto esta. / In this work we make an elementary approach to the exponential function in order to understand the meaning of powers with natural exponent, integer, rational and irrational as well as the properties that make it one of the most important functions of mathematics. In parallel, another approach will be showing these properties using a powerful tool of mathematics: differential and integral calculus. We will also discuss the logarithmic function because it is the inverse of the exponential function and being as important as this.