We study log varieties, over a field of characteristic zero, which are generically logarithmically smooth and fs in the Kummer normally log étale topology. As an application, we prove an analog of Abramovich-Temkin-Wlodarczyk’s log resolution of singularities of fs log schemes in the Kummer fs setting.<br>

No início do século XVII, o escocês John Napier revolucionou os métodos de cálculo da época com a invenção dos logaritmos. O logaritmo de Napier não era exatamente o que usamos hoje. Naquela época, o trabalho de multiplicação, divisão, cálculo de potências e extração de raízes eram trabalhosos e feitos a partir de senos. Surgiram as primeiras tábuas de logaritmos, inventadas independentemente por John Napier (1550-1617) e Jost Bürgi (1552-1632). Pouco depois, Henry Briggs (1561-1631) aperfeiçoou essas tábuas, apresentando os logaritmos decimais. A contribuição fundamental dos logaritmos é a de facilitar os cálculos através da transformação de operações de multiplicação em adição e de operações de divisão em subtração. Essas transformações foram de grande importância nos cálculos trabalhosos que estavam envolvidos em Astronomia e Navegação. Em 1632, um matemático inglês chamado William Oughtred inventou a régua de cálculo, com base na \"Tábua de Napier\". Esse foi um grande passo em direção à calculadora e à construção dos computadores. Nesse trabalho propomos a utilização da régua de cálculo no ensino das propriedades dos logaritmos. Para tanto, foram estudados tópicos como a história dos logaritmos, a função logaritmo, a caracterização das funções logarítmicas, a associação de logaritmos a progressões aritméticas e geométricas e o uso de uma régua de cálculo / In the early seventeenth century, the Scotsman John Napier revolutionized the calculation methods of that time with the invention of logarithms. The Napier logarithm was not exactly the same as we use now. At that time, the multiplication, division, exponents calculation and extracting roots were demanded extensive labor. John Napier (1550-1617) and Jost Bürgi (1552-1632) invented independently the first logarithm tables. Shortly after, Henry Briggs (1561-1631) improved these boards, presenting the decimal logarithms. The main contribution of logarithms is to make calculations easier by transforming multiplication operations into addition ones and division operations into subtraction ones. These changes have been of great importance in laborious calculations that involved Astronomy and Navigation. In 1632, an English mathematician called William Oughtred invented the slide ruler, based on the \"Napier board\". This was a big step towards the invention of the calculator and the computer. In this work we propose the use of the slide ruler in teaching the properties of logarithms. Thus, topics such as the history of logarithms, the logarithm function, the characterization of logarithmic functions, the association of the logarithms with arithmetical and geometrical progressions, and the use of a slide ruler were studied

Thesis (M.A.) in Mathematics--University of Maine, 2001. / Includes vita. Includes bibliographical references (leaf 45).

Thesis (Ph. D.)--Oregon State University, 2008. / Printout. Includes bibliographical references (leaves 165-173). Also available on the World Wide Web.

abstract: There have been a number of studies that have examined students’ difficulties in understanding the idea of logarithm and the effectiveness of non-traditional interventions. However, there have been few studies that have examined the understandings students develop and need to develop when completing conceptually oriented logarithmic lessons. In this document, I present the three papers of my dissertation study. The first paper examines two students’ development of concepts foundational to the idea of logarithm. This paper discusses two essential understandings that were revealed to be problematic and essential for students’ development of productive meanings for exponents, logarithms and logarithmic properties. The findings of this study informed my later work to support students in understanding logarithms, their properties and logarithmic functions. The second paper examines two students’ development of the idea of logarithm. This paper describes the reasoning abilities two students exhibited as they engaged with tasks designed to foster their construction of more productive meanings for the idea of logarithm. The findings of this study provide novel insights for supporting students in understanding the idea of logarithm meaningfully. Finally, the third paper begins with an examination of the historical development of the idea of logarithm. I then leveraged the insights of this literature review and the first two papers to perform a conceptual analysis of what is involved in learning and understanding the idea of logarithm. The literature review and conceptual analysis contributes novel and useful information for curriculum developers, instructors, and other researchers studying student learning of this idea. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2018

No início do século XVII, o escocês John Napier revolucionou os métodos de cálculo da época com a invenção dos logaritmos. O logaritmo de Napier não era exatamente o que usamos hoje. Naquela época, o trabalho de multiplicação, divisão, cálculo de potências e extração de raízes eram trabalhosos e feitos a partir de senos. Surgiram as primeiras tábuas de logaritmos, inventadas independentemente por John Napier (1550-1617) e Jost Bürgi (1552-1632). Pouco depois, Henry Briggs (1561-1631) aperfeiçoou essas tábuas, apresentando os logaritmos decimais. A contribuição fundamental dos logaritmos é a de facilitar os cálculos através da transformação de operações de multiplicação em adição e de operações de divisão em subtração. Essas transformações foram de grande importância nos cálculos trabalhosos que estavam envolvidos em Astronomia e Navegação. Em 1632, um matemático inglês chamado William Oughtred inventou a régua de cálculo, com base na \"Tábua de Napier\". Esse foi um grande passo em direção à calculadora e à construção dos computadores. Nesse trabalho propomos a utilização da régua de cálculo no ensino das propriedades dos logaritmos. Para tanto, foram estudados tópicos como a história dos logaritmos, a função logaritmo, a caracterização das funções logarítmicas, a associação de logaritmos a progressões aritméticas e geométricas e o uso de uma régua de cálculo / In the early seventeenth century, the Scotsman John Napier revolutionized the calculation methods of that time with the invention of logarithms. The Napier logarithm was not exactly the same as we use now. At that time, the multiplication, division, exponents calculation and extracting roots were demanded extensive labor. John Napier (1550-1617) and Jost Bürgi (1552-1632) invented independently the first logarithm tables. Shortly after, Henry Briggs (1561-1631) improved these boards, presenting the decimal logarithms. The main contribution of logarithms is to make calculations easier by transforming multiplication operations into addition ones and division operations into subtraction ones. These changes have been of great importance in laborious calculations that involved Astronomy and Navigation. In 1632, an English mathematician called William Oughtred invented the slide ruler, based on the \"Napier board\". This was a big step towards the invention of the calculator and the computer. In this work we propose the use of the slide ruler in teaching the properties of logarithms. Thus, topics such as the history of logarithms, the logarithm function, the characterization of logarithmic functions, the association of the logarithms with arithmetical and geometrical progressions, and the use of a slide ruler were studied

We show that replicator equations follow naturally from the exponential affine structure of the simplex known from information geometry. It is then natural to call replicator equations linear if their fitness function is affine. For such linear replicator equations an explicit solution can be found. The approach is also demonstrated for the example of Eigen’s hypercycle, where some new analytic results are obtained using the explicit solution.

This thesis presents the rate distortion analysis of conditional motion estimation, a process in which motion computation is restricted to only active pixels in the video. We model active pixels as independent and identically distributed Gaussian process and inactive pixels as Gaussian-Markov process and derive the rate distortion function based on conditional motion estimation. Rate-Distortion curves for the conditional motion estimation scheme are also presented. In addition this thesis also presents the hardware implementation of a block based motion estimation algorithm. Block matching algorithms are difficult to implement on FPGA chip due to its complexity. We implement 2D-Logarithmic search algorithm to estimate the motion vectors for the image. The matching criterion used in the algorithm is Sum of Absolute Differences (SAD). VHDL code for the motion estimation algorithm is verified using ISim and is implemented using Xilinx ISE Design tool. Synthesis results for the algorithm are also presented.

We consider the asymptotic minimum rate under the logarithmic loss distortion constraint. More specifically, we find the asymptotic minimum rate expression when given distortions get close to 0. The problem under consideration is separate encoding and joint decoding of correlated two information sources, subject to a logarithmic loss distortion constraint. We introduce a test channel, whose transition probability (conditional probability mass function) captures the encoding and decoding process. Firstly, we find the expression for the special case of doubly symmetric binary sources with binary-output test channels. Then the result is extended to the case where the test channels are arbitrary. When given distortions get close to 0, the asymptotic rate coincides with that for the aforementioned special case. Finally, we consider the general case and show that the key findings for the special case continue to hold. / Thesis / Master of Applied Science (MASc)

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