Neste trabalho, estudamos duas identidades teta uma devida a Ramanujan e outra a Andrews. Provamos essas identidades de forma analítica e as interpretamos através de argumentos combinat orios como funções geradoras para o número de parti ções contadas considerando pesos para elas. A partir das demonstrações anal tics deduzimos tamb em algumas identidades envolvendo funções partições. / In this work, we study two theta identities one of them due to Ramanujan and the other due to Andrews. We prove them in an analytical way and we interpret them using combinatorial arguments as generating functions for partitions counted attributing weights to them. From the analytical proofs we also deduce some identities involving partition functions.

Neste trabalho, estudamos duas identidades teta uma devida a Ramanujan e outra a Andrews. Provamos essas identidades de forma analítica e as interpretamos através de argumentos combinat orios como funções geradoras para o número de parti ções contadas considerando pesos para elas. A partir das demonstrações anal tics deduzimos tamb em algumas identidades envolvendo funções partições. / In this work, we study two theta identities one of them due to Ramanujan and the other due to Andrews. We prove them in an analytical way and we interpret them using combinatorial arguments as generating functions for partitions counted attributing weights to them. From the analytical proofs we also deduce some identities involving partition functions.

Indiana University-Purdue University Indianapolis (IUPUI) / The development of alcohol use disorder (AUD) is believed to involve functional adaptations in corticostriatal projections which regulate the reinforcing properties of ethanol (EtOH). To further our understanding of how repeated EtOH consumption impacts the corticostriatal circuit, extracellular electrophysiological recordings (local field potentials; LFPs) were gathered from the nucleus accumbens and prefrontal cortex of female and male C57BL/6J mice voluntarily consuming EtOH or water using ‘drinking-in-the-dark’ (DID) procedures. Mice were given 15 consecutive days of two-hours of access to EtOH (20% v/v), three hours into the dark cycle while LFPs were recorded. To determine the impact of repeated EtOH consumption on neural activity between these brain regions, theta phase-locking value (PLV, a measure of synchrony) was calculated. Specifically, theta PLV was calculated during active drinking periods (bouts) and average PLV during the first bout was compared to the last bout to determine within session changes in synchrony. Results indicated significantly lower PLV during the last bout than the first bout. Additionally, longer bouts predicted lower PLV during the last bout, but not the first bout when mice were consuming EtOH. These results may suggest that alcohol intoxication decreases corticostriatal synchrony over a drinking period. Results considering changes in theta power spectral density (PSD) indicated an increase in PSD when mice were given access to water during the typical EtOH access time following the 15-day EtOH drinking history. This effect was not seen when mice were drinking water prior to EtOH access and may be indicative of a successive negative contrast effect. This work identifies unique functional characteristics of corticostriatal communication associated with binge-like EtOH intake and sets the stage for identifying the biological mechanisms subserving them.

A class of hippocampal interneurons in CA1, bordering the lacunosum-moleculare and radiatum hippocampal layers (the LM/R cell), has been shown to exhibit membrane potential oscillations (MPOs) subthreshold to action potential generation. MPOs occur at theta frequency (4-12 Hz) and are of interest because of their putative role in promoting network level theta activity. MPOs arise without synaptic input suggesting that they originate through interactions in the cell’s repertoire of currents. To investigate the generation of MPOs we develop a single compartment model of the cell based on the physiological characterization of its currents. The model includes both deterministic current models and white noise. Our analysis allows for a complete characterization of the cell’s dynamics over the subthreshold range and shows that MPOs arise through the interaction between current dynamics and system noise. We find that MPOs show a particular dependency on the A-type potassium and persistent sodium current magnitudes.

A class of hippocampal interneurons in CA1, bordering the lacunosum-moleculare and radiatum hippocampal layers (the LM/R cell), has been shown to exhibit membrane potential oscillations (MPOs) subthreshold to action potential generation. MPOs occur at theta frequency (4-12 Hz) and are of interest because of their putative role in promoting network level theta activity. MPOs arise without synaptic input suggesting that they originate through interactions in the cell’s repertoire of currents. To investigate the generation of MPOs we develop a single compartment model of the cell based on the physiological characterization of its currents. The model includes both deterministic current models and white noise. Our analysis allows for a complete characterization of the cell’s dynamics over the subthreshold range and shows that MPOs arise through the interaction between current dynamics and system noise. We find that MPOs show a particular dependency on the A-type potassium and persistent sodium current magnitudes.

The study of periods of automorphic forms using the theta correspondence and the Weil representation was initiated by Waldspurger and his work relating Fourier coefficients of modular forms of half-integral weight, periods over tori of modular forms of integral weight and special values of L-functions attached to these modular forms. In this thesis, we show that there are general relations among periods of automorphic forms on groups related by the theta correspondence. For example, if G is a symplectic group and H is an orthogonal group over a number field k, these relations are identities equating Fourier coefficients of cuspidal automorphic forms on G (relative to the Siegel parabolic subgroup) and periods of cuspidal automorphic forms on H over orthogonal subgroups. These identities are quite formal and follow from the basic properties of theta functions and the Weil representation; further study is required to show how they compare to the results of Waldspurger. The second part of this thesis shows that, under some restrictions, the identities alluded to above are the result of a comparison of nonstandard relative traces formulas. In this comparison, the relative trace formula for H is standard however the relative trace formula for G is novel in that it involves the trace of an operator built from theta functions. The final part of this thesis explores some preliminary results on local height pairings of special cycles on the p-adic upper half plane following the work of Kudla and Rapoport. These calculations should appear as the local factors of arithmetic orbital integrals in an arithmetic relative trace formula built from arithmetic theta functions as in the work of Kudla, Rapoport and Yang. Further study is required to use this approach to relate Fourier coefficients of modular forms of half-integral weight and arithmetic degrees of cycles on Shimura curves (which are the analogues in the arithmetic situation of the periods of automorphic forms over orthogonal subgroups).

The study of periods of automorphic forms using the theta correspondence and the Weil representation was initiated by Waldspurger and his work relating Fourier coefficients of modular forms of half-integral weight, periods over tori of modular forms of integral weight and special values of L-functions attached to these modular forms. In this thesis, we show that there are general relations among periods of automorphic forms on groups related by the theta correspondence. For example, if G is a symplectic group and H is an orthogonal group over a number field k, these relations are identities equating Fourier coefficients of cuspidal automorphic forms on G (relative to the Siegel parabolic subgroup) and periods of cuspidal automorphic forms on H over orthogonal subgroups. These identities are quite formal and follow from the basic properties of theta functions and the Weil representation; further study is required to show how they compare to the results of Waldspurger. The second part of this thesis shows that, under some restrictions, the identities alluded to above are the result of a comparison of nonstandard relative traces formulas. In this comparison, the relative trace formula for H is standard however the relative trace formula for G is novel in that it involves the trace of an operator built from theta functions. The final part of this thesis explores some preliminary results on local height pairings of special cycles on the p-adic upper half plane following the work of Kudla and Rapoport. These calculations should appear as the local factors of arithmetic orbital integrals in an arithmetic relative trace formula built from arithmetic theta functions as in the work of Kudla, Rapoport and Yang. Further study is required to use this approach to relate Fourier coefficients of modular forms of half-integral weight and arithmetic degrees of cycles on Shimura curves (which are the analogues in the arithmetic situation of the periods of automorphic forms over orthogonal subgroups).

Thesis (M.S.)--Virginia Polytechnic Institute and State University, 1993. / Vita. Abstract. Includes bibliographical references (leaves 89-106). Also available via the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 114-118). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 114-118). Also available on the Internet.