O objetivo principal deste trabalho e descrever os números inteiros que podem ser representados nas formas 9x2+16y2+36z2+16yz+4xz+8xy e 9x2+17y2+ 32z2 - 8yz + 8xz + 6xy. Para isso, utilizamos uma série de resultados envolvendo funções theta, como a identidade do produto triplo de Jacobi e equações modulares. / The main goal of this work is to describe the integers which can be written in the forms 9x2 + 16y2 + 36z2 + 16yz + 4xz + 8xy and 9x2 + 17y2 + 32z2 - 8yz + 8xz + 6xy. To do so, we use a series of results concerning theta functions, such as the Jacobi triple product identity and modular equations.

Neste trabalho, com base em representações por matrizes de duas linhas para alguns tipos de partição (algumas já conhecidas e outras novas), identificamos propriedades sugeridas por classificá-las de acordo com a soma dos elementos de sua segunda linha. Esta soma sempre fornece alguma propriedade da partição relacionada. Se considerarmos versões sem sinal de algumas funções Mock Theta, seu termo geral pode ser interpretado como função geradora para algum tipo de partição com restrições. Para retornar aos coeficientes originais, é possível definir um peso para cada matriz e depois somá-las para contá-los. Uma representação análoga para essas partições nos permite observar propriedades sobre elas, novamente por meio de uma classificação referente à soma dos seu elementos da segunda linha. Esta seriação é feita por meio de tabelas criadas pelo software matemático Maple, as quais nos sugerem padrões e identidades relacionadas com outros tipos de partições conhecidas e, muitas vezes, encontrando uma fórmula fechada para contá-las. Tendo as conjecturas obtidas, elas são provadas por meio de bijeções entre conjuntos ou por contagem. / In this work, based on representations by matrices of two lines for some kind of partition (some already known and other new ones), we identify properties suggested by classifying them according to the sum of its second line. This sum always provides some properties of the related partition. If we consider unsigned versions of some Mock Theta Functions, its general term can be interpreted as generating function for some kind of partition with restrictions. To come back to the original coefficients, you can set a weight for each array and so add them to evaluate the coefficients. An analogous representation for partitions allows us to observe properties, again by classificating them according to the sum of its elements on the second row. This classification is made by means of tables created by mathematical software Maple, which suggest patterns, identities related to other known types of partitions and often, finding a closed formula to count them. Having established conjectured identities, all are proved by bijections between sets or counting methods.

O objetivo principal deste trabalho e descrever os números inteiros que podem ser representados nas formas 9x2+16y2+36z2+16yz+4xz+8xy e 9x2+17y2+ 32z2 - 8yz + 8xz + 6xy. Para isso, utilizamos uma série de resultados envolvendo funções theta, como a identidade do produto triplo de Jacobi e equações modulares. / The main goal of this work is to describe the integers which can be written in the forms 9x2 + 16y2 + 36z2 + 16yz + 4xz + 8xy and 9x2 + 17y2 + 32z2 - 8yz + 8xz + 6xy. To do so, we use a series of results concerning theta functions, such as the Jacobi triple product identity and modular equations.

Arbetet undersöker prestanda och estetik hos A*PS och Theta* inom olika miljötyper, där miljötyper är ett generaliserat begrepp för spelmiljöer och innefattar en struktur samt ett antal blockerande objekt. Båda teknikerna är varianter av A*-algoritmen och används för att åstadkomma estetisk vägplanering. Vägplanering innebär att finna en kortaste väg från en punkt A till en punkt B i en graf. En populär grafrepresentation i spel är rutnät med blockerande och icke-blockerande noder. Ett problem med rutnät är att algoritmerna tenderar att planera sina vägar längst med kanterna och därav resultera i sicksackformade vägar. A*PS och Theta* löser detta genom att utföra estetiska optimeringar som tillåter vägplaneringen att göras längst med grafens kanter, men samtidigt inte låta vägarna begränsas till att direkt följa dessa. Med experiment visar arbetet att Theta* i majoriteten av fallen hittar kortare och mer realistiska vägar. Vidare föreslås framtida arbeten inom området.

I obtained 182 session graphs from 10 client records from a university-based neurotherapy clinic and from a private practitioner. These graphs were used to examine the relationship of therapeutic crossover activity (defined as at least 3 minutes in duration and at least 1μv in amplitude) with and without predetermined amplitude thresholds of beta (15-20Hz) to client reports of imagery and to treatment outcomes. Crosstab analysis revealed that significantly more reports of imagery were observed in the therapeutic crossover with beta condition and that higher amplitudes of slower brainwave activity correlated with progression to deeper states of consciousness. Multi-level modeling revealed a significant interaction between therapeutic crossover activity, higher beta frequency amplitude, and reported salient imagery. Due to small sample size, significance testing was not deemed appropriate. However, observation in change of pre-post scores suggested that individuals who experienced more therapeutic crossover with sufficient beta amplitude conditions had greater improvements on post-test measures (BAI, BDI, BHS, PSQI and MMPI) than those with no or few crossovers. Higher amplitudes of slower brainwave activity correlated with progression to deeper states of consciousness, with delta amplitude positively correlating with transpersonal states. Reports of imagery and/or biographical memory are much more likely to occur during theta-alpha crossover activity characterized by 3 minutes or more in duration, one microvolt or more in amplitude, and 3.75μv amplitude or more of beta. This defined therapeutic crossover condition does appear to facilitate recall of imagery and memories during alpha-theta neurofeedback and was related to better treatment outcomes.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第22971号 / 理博第4648号 / 新制||理||1668(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 入谷 寛, 教授 吉川 謙一, 教授 加藤 毅 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Theta-burst stimulation (TBS) is a form of repetitive transcranial magnetic stimulation (TMS) developed to induce neuroplasticity. TBS usually consists of 50 Hz bursts at 5 Hz intervals. When applied intermittently, it can lead to facilitation of motor evoked potentials (MEPs), although these effects can be variable between individuals. Here, we aimed to determine whether a version of intermittent TBS (iTBS) consisting of 30 Hz bursts at 6 Hz intervals would produce less variable modulation. Nineteen healthy adults underwent single-pulse TMS to assess corticomotor excitability at baseline as reflected in MEP amplitude. 30 Hz iTBS was then administered and MEP amplitude was reassessed at 5-, 20- and 45-mins after the iTBS protocol. Compared to baseline, MEPs were significantly facilitated up to 45-min post-iTBS and most participants exhibited the expected facilitation. These observations suggest that 30 Hz/6 Hz iTBS may provide a sound alternative to induce consistent neuromodulatory effects over the commonly used 50 Hz/5 Hz protocol.

Working memory (WM) is a crucial component of cognitive function that affects learning, reasoning, and problem solving, all of which are important for daily functioning. Therefore, addressing factors that can impact working memory, such as stress, are incredibly relevant to understanding WM efficiency. WM is an important component of higher order cognitive function and high WM capacity has been shown to be important for academic and occupational performance. Thus, understanding the relationship between stress-related factors and WM could aid in identifying strategies to mitigate the deleterious effects of stress on working memory. Although some previous research has indicated a negative impact of acute stress on WM, other research has indicated no impact or even a positive impact of stress on WM. As the relationship between acute stress and WM is mixed, examining other stress-related factors may provide further insight into the relationship. The current study examines how adverse childhood experiences (ACEs) and acute stress influence WM, and how frontal theta and alpha activity are affected by WM task demands. Participants completed a working memory task while their EEG was recorded. Participants then completed the PANAS to assess their current emotional state. Following the PANAS, participants viewed a stressful or neutral video as an acute stress induction, followed by a second PANAS to ensure effectiveness of stress induction. Participants then completed the WM task a second time. Finally, the participants completed the ACEs questionnaire. Bayesian linear mixed effects models were used to examine the relationships between ACEs, acute stress, WM, and frontal theta and alpha frequencies. Findings suggest there is not enough evidence to support a relationship between acute stress, ACEs, WM, and WM-related theta and alpha. While the current study did not reveal a relationship, future research should explore how acute stress and exposure to specific stressors during childhood could explain individual differences in WM.

Equations of the form 𝑎𝑥³ + 𝑏𝑦³ = 1, where the constants 𝑎 and 𝑏 are integers of some number field such that 𝑎𝑥³ + 𝑏𝑦³ is irreducible, are a particularly significant class of cubic Thue equations that notably includes the cubic Pell equation. For a positive cubefree rational integer 𝑑, we consider the family of equations of the form 𝑚𝑥³ − 𝑑𝑛𝑦³ = 1 where 𝑚 and 𝑛 are squarefree. We define an 𝐿-function associated to 𝑑 whose nonvanishing coefficients correspond to the nontrivial solutions of those equations. That definition uses expressions related to the cubic theta function Q (􏰇√ 􏰈-), and we study that 𝐿-function’s analytic properties by using a method generalizing the approach used by Takhtajan and Vinogradov to derive a trace formula using the quadratic theta function for Q. We construct its meromorphic continuation and determine the locations and orders of its poles. Specifically, the poles occur at the eigenvalues of the Laplacian for the Maass forms 𝑢_𝑗 , 𝑗 = 1, 2, 3, · · · in the discrete spectrum, with a double pole at 𝑠 = ½ and possible simple poles at 𝑠=𝑠_𝑗,1−𝑠_𝑗,where𝜆𝑗 =2𝑠_𝑗(2−2𝑠_𝑗)istheLaplaceeigenvalueof𝑢𝑗 and𝜆𝑗 ≠1.

Quantum systems in a d-dimensional Hilbert space are considered, where the phase spase is Z(d) x Z(d). An analytic representation in a cell S in the complex plane using Theta functions, is defined. The analytic functions have exactly d zeros in a cell S. The reproducing kernel plays a central role in this formalism. Wigner and Weyl functions are also studied. Quantum systems with positions in a circle S and momenta in Z are also studied. An analytic representation in a strip A in the complex plane is also defined. Coherent states on a circle are studied. The reproducing kernel is given. Wigner and Weyl functions are considered.