We consider various infinite series and examine their arithmetic nature. Series of interest are of the form $$\sum_{n =0}^{\infty} \frac{f(n)A(n)}{B(n)}, \;\;\;\; \sum_{n \in \mathbb{Z}}\frac{f(n)A(n)}{B(n)}, \;\;\;\; \sum_{n=0}^{\infty} \frac{z^n A(n)}{B(n)}$$ where $f$ is algebraic valued periodic function, $A(x), B(x) \in \overline{\mathbb{Q}}[x]$ and $z$ is an algebraic number with $|z| \leq 1$. We also examine multivariable extensions $$\sum_{n_1, \ldots, n_k = 0}^{\infty} \frac{f(n_1, \ldots, n_k)A_1(n_1) \cdots A_k(n_k)}{B_1(n_1) \cdots B_k(n_k)}$$ and $$\sum_{n_1, \ldots, n_k \in \mathbb{Z}} \frac{f(n_1, \ldots, n_k)A_1(n_1) \cdots A_k(n_k)}{B_1(n_1) \cdots B_k(n_k)}.$$ These series are all very natural things to write down and we would like to understand them better. We calculate closed forms using various techniques. For example, we use relations between Hurwitz zeta functions, digamma functions, polygamma functions, Fourier analysis, discrete Fourier transforms, among other objects and techniques. Once closed forms are found, we make use of some of the well-known transcendental number theory including the theorem of Baker regarding linear forms in logarithms of algebraic numbers to determine their arithmetic nature. In one particular setting, we extend the work of Bundschuh \cite{bundschuh} by proving the following series are all transcendental for positive $c \in \mathbb{Q} \setminus \mathbb{Z}$ and $k$ a positive integer: $$\sum_{n \in \mathbb{Z}} \frac{1}{(n^2 + c)^k}, \;\;\; \sum_{n \in \mathbb{Z}} \frac{1}{(n^4 - c^4)^{2k}}, \;\;\; \sum_{n \in \mathbb{Z}} \frac{1}{(n^6 - c^6)^{2k}}, \;\;\; \sum_{n \in \mathbb{Z}} \frac{1}{(n^3 \pm c^3)^{2k}}$$ $$\sum_{n \in \mathbb{Z}} \frac{1}{n^3 \pm c^3}, \;\;\; \sum_{|n| \geq 2} \frac{1}{n^3 -1}, \;\;\; \sum_{|n| \geq 2} \frac{1}{n^4 -1}, \;\;\; \sum_{|n| \geq 2} \frac{1}{n^6 -1}.$$ Bundschuh conjectured that the last three series are transcendental, but we offer the first unconditional proofs of transcendence. We also show some conditional results under the assumption of some well-known conjectures. In particular, for $A_i(x), B_i(x) \in \overline{\mathbb{Q}}[x]$ with each $B_i(x)$ has only simple rational roots, if Schanuel's conjecture is true, the series (avoiding roots of the denominator) $$\mathop{{\sum}}_{n_1, \ldots, n_k =0}^{\infty} \frac{f(n_1, \ldots, n_k)A_1(n_1) \cdots A_k(n_k)}{B_1(n_1) \cdots B_k(n_k)}$$ is either an effectively computable algebraic number or transcendental. We also show that Schanuel's conjecture implies that the series $$\sum_{n \in \mathbb{Z}} \frac{A(n)}{B(n)}$$ is either zero or transcendental, when $B(x)$ has non-integral roots. We develop a general theory, analyzing various infinite series throughout. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2009-04-23 10:45:19.735

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We apply anisotropic Banach space methods together with operator renewal theory to obtain polynomial rates of decay of correlations for a class of generalized baker's transformations. The polynomial rates were proved for a smaller class of observables in a 2013 paper of Bose and Murray by fundamentally different methods. Our approach provides a direct analysis of the Frobenius-Perron operator associated to a generalized baker's transformation in contrast to the paper of Bose and Murray where decay rates are obtained for a factor map and lifted to the full map. / Graduate

Orientador: Paulo Jose Samenho Moran / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Quimica / Made available in DSpace on 2018-08-15T03:11:12Z (GMT). No. of bitstreams: 1 Martins_RodrigodosSantos_M.pdf: 9288307 bytes, checksum: f6f89494491cb3965ccd1e83f1ff7920 (MD5) Previous issue date: 2009 / Resumo: Em estudos recentes, o racemato da 2-(4-clorofeniltio)-1-(3,4-dimetoxifenil)-1- propanona (15b), molécula análoga sintética das neolignanas isoladas da Virola surinamensis, apresentaram atividade anti-leismania. Neste trabalho com o objetivo de sintetizar 15b opticamente pura, obtivemos (2S)-2-hidroxi-1-(3,4-dimetoxifenil)-1- propanona (2c) através da biorredução da 1-(3,4-dimetoxifenil)-1,2-propanodiona (1c) mediada pelo fermento de pão (Saccharomyces cerevisiae) com 60% de rendimento e 93% de e.e. Consecutivamente testou-se uma série de protocolos para a realização de uma SN2 entre 2c e o p-clorotiofenol para a síntese de 15b enantiomericamente pura. A transformação do grupo hidroxila de (+)-(2S)-2c em um éster sulfônico seguida da substituição via catálise por transferência de fase (CTF) provou ser a melhor estratégia na obtenção de 15b. Ao fim do trabalho se sintetizou ambos os enantiômeros de 15b, a (-)-15b com 10% de rendimento e 95% de e.e., a partir do éster sulfônico de 2c, e a (+)-15b com 90% de rendimento e 60% de e.e. partindo-se da (-)-2-cloro-1-(3,4-dimetoxifenil)-1- propanona, um co-produto da síntese do éster sulfônico de 2c. Conjuntamente, estudou-se uma metodologia alternativa na obtenção de 2-hidroxi-1-aril-1-propanona (2) através da resolução cinética de fenilpropanodiois, realizada por meio de uma oxidação biocatalítica mediada pelo fermento de pão. Após a otimização de alguns parâmetros reacionais, partindo-se do (±)-anti-1-fenil-1,2-propanodiol (4a), se obteve a (2S)-2-hidroxi-1-fenil-1-propanona (2a) com 25% de rendimento e 83% de e.e. e a (1S,2R)- 4a com 30% de rendimento e 68% de e.e. depois de 7 dias de reação. O mesmo procedimento foi realizado com outros microrganismos, com destaque para o resultado da bioxidação mediada pelo Geotrichum candidum, onde se obteve somente a (1S,2R)-4a com 30% de rendimento e 99% de e.e. após 5 dias de reação. / Abstract: Recently, the racemate 2-(4-chlorophenylthio)-1-(3,4-dimethoxyphenyl)-1- propanone (15b) that is an analog synthetic of neolignans, isolated from Virola surinamensis shows anti-leismania activity. In this work, we performed the bioreduction of 1-(3,4-dimethoxy)1,2-propanodione (1c) mediated by baker yeast (Saccharomyces cerevisiae) affording (2S)-2-hydroxy-1-(3,4-dimethoxyphenyl)-1-propanone (2c) in 60% yield and 93% e.e. which is an intermediate for 15b. Consecutively, protocols were applied to perform SN2 reaction of 2c and p-chlorotiophenol in order to synthesize optically pure 15b. The reaction of sulfonic ester of (+)-(2S)-2c with p-chlorotiophenol catalyzed by phase transfer catalysis (PTC) shows to be a better strategy to obtain 15b. Both enantiomers of 15b were synthesized, (-)-15b was obtained in 10% yield and >95% e.e. from sulfonic ester from 2c, and (+)-15b was obtained in 90% yield and 60% e.e. from (-)-2-chloro-1- (3,4-dimethoxyphenyl)-1-propanone that was a by-product formed in the synthesis of sulfonic ester from 2c. In addition, an alternative methodology for obtaining 2-hydroxy-1-aryl-1-propanone (2) was applied through kinetic resolution of 1-phenylpropane-1,2-propanediol by biocatalytic oxidation mediated by bakers yeast. After optimization of this process, the (2S)-2-hydroxy-1-phenyl-1-propanone (2a) and (1S,2R)-4a were obtained in 25% yield and 83% e.e. and 30% yield and 68% e.e. respectively from (±)-anti-1-phenyl-1,2-propanediol (anti-4a) after 7 days. The same protocol was applied with other microorganisms as Geotrichum candidum that produced (1S,2R)-4a in 30% yield and 99% e.e. after 5 days. / Mestrado / Quimica Organica / Mestre em Química

Chapter one of my thesis has been published in the May 2008 issue of the journal Evolution under the title “A TALE OF TWO CONTINENTS: BAKER’S RULE AND THE MAINTENANCE OF SELF-INCOMPATIBILITY IN LYCIUM (SOLANACEAE).” This chapter was co-authored by Dr. Jill S. Miller and Dr. Rachel Levin.

In the large-scale production of <i>Saccharomyces cerevisiae</i> (baker's yeast), oxygen transfer, which is one of the major limiting factors, is improved by using high agitation rates. However, high agitation rates subject the microorganisms to high shear stress and caused high power consumption. A microbubble dispersion (MBD) method was investigated to improve oxygen transfer at low agitation rates and thus reduce power consumption and shear stress on the microorganisms. The experiments were conducted at the 1-liter level and subsequently scaled-up to 50-liters using a constant volumetric oxygen transfer coefficient (<i>k_La</i>) method for scaling. In comparison to a conventional air-sparged fermentation, the MBD method considerably improved the cell mass yield, growth rate and power consumption in the 50-liter fermentor. Cell mass production in the MBD system at agitation rate of 150 rpm was about the same as those obtained for a conventional air-sparged system agitatid at 500 rpm. Power consumption in the conventional air-sparged system was three-fold that required for the same biomass yield in the MBD system. However, at the 1-liter scale, the MBD system did not show any significant advantage over the air-sparged system because of the high power consumption. / Master of Science

A primeira parte desta tese consiste na preparação de compostos aromáticos contendo átomos de selênio e de telúrio. Quatro metodologias foram utilizadas para essa finalidade, a saber: orto-metalação de compostos contendo oxigênio, troca metal-halogênio em haletos aromáticos substituídos, reação de substituição eletrofílica aromática entre tetracloreto de telúrio e compostos aromáticos ativados e reação de sais de diazônio com disselenetos e diteluretos orgânicos. Alguns dos teluretos preparados foram usados em reações de acoplamento com alcinos terminais catalisada por Paládio, em reações de troca telúrio-lítio e em reações de oxidação de Te (II) a Te(IV). A segunda parte da tese consiste no uso de calcogenetos de arila em reações biocatalisadas. Inicialmente foram estudadas biotransformações em substratos não contendo átomo de selênio e de telúrio. Reações de redução de carbonila e de desracemização de álcoois foram observadas por ação de fungos e de raízes de plantas. Meta e para organosseleno acetofenonas foram reduzidas aos organosseleno feniletanóis correspondentes por ação de fermento de pão, células íntegras de fungos e por Daucus carota com conversões e excessos enantioméricos que chegaram a >99%. Orto organosseleno e metiltio feniletanóis foram resolvidos em seus isômeros R e S por ação de lipase imobilizada (Novozyme 435) em presença de acetato de vinila com excessos enantioméricos acima de 99% . / The first part of this thesis shows the preparation of organic compounds containing selenium or tellurium. For this purpose, four methodologies were applied: ortho-metallation of 1-phenylethanol, metal-halogen exchange involving aromatic halides, electrophilic aromatic substitution using tellurium tetrachloride and reactions with aryldiazonium salts and diselenides or ditellurides. Next, the aryl tellurides were applied in Palladium catalyzed coupling reactions with terminal alkynes, tellurium-lithium exchange reactions and oxidation of Te(II) to Te(IV). The second part of this thesis consists in the inverstigation of biocatalyzed reactions. Biotransformations of substrates without Se or Te atoms were initially investigated. Carbonyl reduction reactions and deracemization of secondary alcohols were observed by means of whole fungal cells and plants. Meta and para organoseleno acetophenones were then reduced with baker´s yeast, whole fungal cells and Daucus carota, yielding the corresponding organoseleno phenylethanols optically pure with enantiomeric excess up to >99%. Ortho organoseleno e methylthio phenylethanols were resolved in both enantiomeric forms by reacting them with immobilized lipase (Novozyme 435) and vinyl acetate in hexane. High values of enantiomeric excess (>99%) were obtained.

Resumo: No presente trabalho, a atividade da enzima glicerol quinase (GK; EC 2.7.1.30; ATP: glicerol 3-fosfotransferase), proveniente de extratos de levedura seca de panificação, foi otimizada. A melhor preparação enzimática da GK foi obtida por rompimento celular com esferas de vidro, durante sete minutos, com lise de 54,2% das células. O extrato celular foi parcialmente purificado com 1% de sulfato de estreptomicina, antes da precipitação com igual volume de solução a 30% (m/v) de polietilenoglicol 3350, e posteriormente dialisado. A atividade máxima da GK foi obtida em pH 10,0, a 60ºC e 50mM de substrato, por metodologia clássica. A enzima apresentou alta estabilidade térmica ― a atividade foi completamente mantida até 50ºC, durante uma hora ― e em pH entre 6,0 e 8,0. Além disso, manteve-se estável, por quatro meses, a 4°C, na presença de azida de sódio 0,05% e cloreto de cobalto 10mM, e, por até oito meses, com o extrato liofilizado. Calculados pelos métodos de Lineweaver-Burk, Hanes-Woolf e Eadie-Hofstee, o valor da constante de Michaelis (Km) da enzima variou entre 1,99mM e 3,11mM, e a Vmax, entre 1,14U/mL e 1,19U/mL. Utilizou-se a metodologia de superfície de resposta (MSR) para melhor definição dos parâmetros da reação enzimática, observando-se valores ótimos de atividades a temperaturas entre 52ºC e 56ºC, pH entre 10,2 e 10,5 e concentração de substrato de 150mM a 170mM. A MSR mostrou-se adequada para modelar a reação e maximizar a atividade da glicerol quinase. Este método, de baixo custo, dosa a glicerol quinase em uma seqüência de reações, sendo de grande importância para diversas indústrias, como a de alimentos, açúcar e álcool. / Abstract: In the present study, the activity of the enzyme glycerol kinase (GK; EC 2.7.1.30; ATP: glycerol 3-phosphotransferase) from dry baker's yeast, was optimized. The best enzymatic preparation of GK was obtained by cell disruption with glass beads, for seven minutes, with 54.2% of lysed cells. Cell extract was partially purified with 1% of streptomycin sulphate, before the precipitation with equal volume of a 30% solution (m/v) of polyethylene glycol 3350, and then it was dialyzed. The maximum activity of GK was obtained with pH 10.0, at 60ºC and 50mM of substrate, by the classic methodology. The enzyme presented high thermal stability ― the activity was completely maintained up to 50ºC, during one hour ― and at pH between 6.0 and 8.0. Besides, it was stable, for four months, at 4°C, in the presence of sodium azide 0.05% and cobalt chloride 10mM, and, for up to eight months, with the lyophilized extract. The value of the Michaelis constant (Km) of the enzyme was calculated by the methods of Lineweaver-Burk, Hanes-Woolf and Eadie-Hofstee,and it varied between 1.99mM and 3.11mM, and Vmax, between 1.14U/mL and 1.19U/mL. Response surface methodology (RSM) was used for better definition of the parameters of the enzymatic reaction, being observed higher activity values at temperatures between 52ºC and 56ºC, pH between 10.2 and 10.5 and substrate concentration from 150mM to 170mM. RSM showed to be an adequate approach for modeling the reaction and maximizing the glycerol kinase activity. This low cost method doses glycerol kinase in a sequence of reactions, being of great importance for many industries, like food, sugar and alcohol. / Orientador: Maristela de Freitas Sanches Peres / Coorientador: Edwil Aparecida de Lucca Gattás / Banca: Luis Henrique Souza Guimarães / Banca: Rubens Monti / Mestre

A primeira parte desta tese consiste na preparação de compostos aromáticos contendo átomos de selênio e de telúrio. Quatro metodologias foram utilizadas para essa finalidade, a saber: orto-metalação de compostos contendo oxigênio, troca metal-halogênio em haletos aromáticos substituídos, reação de substituição eletrofílica aromática entre tetracloreto de telúrio e compostos aromáticos ativados e reação de sais de diazônio com disselenetos e diteluretos orgânicos. Alguns dos teluretos preparados foram usados em reações de acoplamento com alcinos terminais catalisada por Paládio, em reações de troca telúrio-lítio e em reações de oxidação de Te (II) a Te(IV). A segunda parte da tese consiste no uso de calcogenetos de arila em reações biocatalisadas. Inicialmente foram estudadas biotransformações em substratos não contendo átomo de selênio e de telúrio. Reações de redução de carbonila e de desracemização de álcoois foram observadas por ação de fungos e de raízes de plantas. Meta e para organosseleno acetofenonas foram reduzidas aos organosseleno feniletanóis correspondentes por ação de fermento de pão, células íntegras de fungos e por Daucus carota com conversões e excessos enantioméricos que chegaram a >99%. Orto organosseleno e metiltio feniletanóis foram resolvidos em seus isômeros R e S por ação de lipase imobilizada (Novozyme 435) em presença de acetato de vinila com excessos enantioméricos acima de 99% . / The first part of this thesis shows the preparation of organic compounds containing selenium or tellurium. For this purpose, four methodologies were applied: ortho-metallation of 1-phenylethanol, metal-halogen exchange involving aromatic halides, electrophilic aromatic substitution using tellurium tetrachloride and reactions with aryldiazonium salts and diselenides or ditellurides. Next, the aryl tellurides were applied in Palladium catalyzed coupling reactions with terminal alkynes, tellurium-lithium exchange reactions and oxidation of Te(II) to Te(IV). The second part of this thesis consists in the inverstigation of biocatalyzed reactions. Biotransformations of substrates without Se or Te atoms were initially investigated. Carbonyl reduction reactions and deracemization of secondary alcohols were observed by means of whole fungal cells and plants. Meta and para organoseleno acetophenones were then reduced with baker´s yeast, whole fungal cells and Daucus carota, yielding the corresponding organoseleno phenylethanols optically pure with enantiomeric excess up to >99%. Ortho organoseleno e methylthio phenylethanols were resolved in both enantiomeric forms by reacting them with immobilized lipase (Novozyme 435) and vinyl acetate in hexane. High values of enantiomeric excess (>99%) were obtained.

Thesis (M.S.)--West Virginia University, 2003. / Title from document title page. Document formatted into pages; contains ix, 99 p. : ill. (some col.), maps (some col.). Includes abstract. Includes bibliographical references (p. 69-71).