Thesis advisor: Solomon Friedberg / We study the generalized theta lifting between the double covers of split special orthogonal groups, which uses the non-minimal theta representations constructed by Bump, Friedberg and Ginzburg. We focus on the theta liftings of non-generic representations and make a conjecture that gives an upper bound of the first non-zero occurrence of the liftings, depending only on the unipotent orbit. We prove both global and local results that support the conjecture. / Thesis (PhD) — Boston College, 2021. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Indiana University-Purdue University Indianapolis (IUPUI) / Early life trauma is a risk factor for a number of neuropsychiatric disorders, including schizophrenia (SZ) and depression. Animal models have played a critical role in understanding how early-life trauma may evoke changes in behavior and biomarkers of altered brain function that resemble these neuropsychiatric disorders. However, since SZ is a complex condition with multifactorial etiology, it is difficult to model the breadth of this condition in a single animal model. Considering this, it is necessary to develop rodent models with clearly defined subsets of pathologies observed in the human condition and their developmental trajectory. Episodic memory is among the cognitive deficits observed in SZ. Theta (6-10 Hz), low gamma (30-50 Hz), and high gamma (50-100 Hz) frequencies in the hippocampus (HC) are critical for encoding and retrieval of memory. Also, theta-gamma comodulation, defined as correlated fluctuations in power between these frequencies, may provide a mechanism for coding episodic sequences by coordinating neuronal activity at timescales required for memory encoding and retrieval. Given that patients with SZ have impaired recognition memory, the overall objectives of these experiments were to assess local field potential (LFP) recordings in the theta and gamma range from the dorsal HC during a recognition memory task in an animal model that exhibits a subclass of symptoms that resemble SZ. In Aim 1, LFPs were recorded from the HC to assess theta and gamma power to determine whether rats that were maternally deprived (MD) for 24-hrs on postnatal day (PND 9), had altered theta and high/low gamma power compared to sham rats during novel object recognition (NOR). Brain activity was recorded while animals underwent NOR on PND 70, 74, and 78. In Aim 2, the effects of theta-low gamma comodulation and theta-high gamma comodulation in the HC were assessed during NOR between sham and MD animals. Furthermore, measures of maternal care were taken to assess if high or low licking/grooming behaviors influenced recognition memory. It was hypothesized that MD animals would have impaired recognition memory and lower theta and low/high gamma power during interaction with both objects compared to sham animals. Furthermore, it was hypothesized that sham animals would have higher theta-gamma comodulation during novel object exploration compared to the familiar object, which would be higher than the MD group. Measures of weight, locomotor activity, and thigmotaxis were also assessed. MD animals were impaired on the NOR task and had no change in theta or low/high gamma power or theta-gamma comodulation when interacting with the novel or familiar object during trials where they performed unsuccessfully or successfully. However, higher theta and gamma power and theta-gamma comodulation was observed in sham animals depending on the object they were exploring or whether it was a successful or unsuccessful trial. These data indicate altered functioning of the HC following MD and a dissociation between brain activity and behavior in this group, providing support that early life trauma can induce cognitive and physiological impairments that are long-lasting. In conclusion, these data identify a model of early life stress with a translational potential, given that there are points of contact between human studies and the MD model. Furthermore, these data provide a set of tools that could be used to further explore how these altered neural mechanisms may influence cognition and behavior.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18766号 / 理博第4024号 / 新制||理||1580(附属図書館) / 31717 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 伊藤 哲史, 教授 上田 哲生, 教授 雪江 明彦 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

The hippocampal CA1 is crucial for myriad types of learning and memory. It is theorized to provide a spatiotemporal framework for the encoding of relevant information during learning, allowing an individual to create a cognitive map of its environment and experiences. To probe CA1 network dynamics that underlie such complex cognitive function, in this work we used recently developed cellular optical imaging techniques that provide high spatial and temporal resolutions. Genetically-encoded calcium indicators offer the ability to record intracellular calcium dynamics, a proxy of neural activity, from hundreds of cells in behaving animals with single cell resolution in genetically-defined cell types. In complement, recently developed genetically-encoded voltage indicators have enabled direct recording of transmembrane voltage of individual genetically-defined cells in behaving animals. The work presented here uses the genetically-encoded calcium indicator GCaMP6f and the genetically-encoded voltage indicator SomArchon to interrogate the activities of individual hippocampal CA1 neurons and their relationship to the dynamics of the broader network during behavior. First, we provide the first in vivo, real-time evidence that two unique populations of CA1 cells encode trace conditioning and extinction learning, two distinct phases of hippocampal-dependent learning. The population of cells responsible for the representation of extinction learning emerges within one session of extinction training. Second, we perform calcium imaging in a mouse model containing a total knockout of NEXMIF, a gene causative of autism spectrum disorder. We reveal that loss of NEXMIF causes over-synchronization of the CA1 circuit, particularly during locomotion, impairing the information encoding capacity of the network. Finally, we conduct voltage imaging of CA1 pyramidal cells and parvalbumin (PV)-positive interneurons, with simultaneous recording of local field potential (LFP), to characterize how cellular-level membrane dynamics and spiking relate to network-level LFP. We demonstrate that in PV neurons, membrane potential oscillations in the theta frequency range show consistent synchrony with LFP theta oscillations and organize spike timing of the PV population relative to LFP theta, indicating that PV interneurons orchestrate theta rhythmicity in the CA1 network. In summary, this dissertation utilizes genetically-encoded optical reporters of neural activity, providing critical insights into the function of the CA1 as a flexible, diverse network of individual neurons.

No description available.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第22968号 / 理博第4645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)准教授 市野 篤史, 教授 池田 保, 教授 加藤 周 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

We investigate the ultraconnectivity condition on graphs, and provide further connections between critical and ultraconnected graphs in the positive definite partial matrix completion problem. We completely characterize when the join of graphs is ultraconnected, and prove that ultraconnectivity is preserved by Cartesian products. We completely characterize when adding a vertex to an ultraconnected graph preserves ultraconnectivity. We also derive bounds on the number of vertices which guarantee ultraconnectivity of certain classes of regular graphs. We give results from our exhaustive enumeration of ultraconnected graphs up to 11 vertices. Using techniques involving the Lovász theta parameter for graphs, we prove certain classes of graphs are critical (and hence ultraconnected) in the positive definite partial matrix completion problem.

This paper includes some basic ideas about the computation of a function theta(G), the theta number of a graph G, which is known as the Lovasz number of G. theta(G^c) lies between two hard-to-compute graph numbers omega(G), the size of the largest lique in a graph G, and chi(G), the minimum number of colors need to properly color the vertices of G. Lovasz and Grotschel called this the "Sandwich Theorem". Donald E. Knuth gives four additional definitions of theta, theta_1, theta_2, theta_3, theta_4 and proves that they are all equal. First I am going to describe the proof of the equality of theta, theta_1 and theta_2 and then I will show the calculation of the theta function for some specific graphs: K_n, graphs related to K_n, and C_n. This will help us understand the theta function, an important function for graph theory. Some of the results are calculated in different ways. This will benefit students who have a basic knowledge of graph theory and want to learn more about the theta function.

<p>The main focus of this dissertation is to find and explain the periodic points of certain algebraic functions that are related to some modular functions, which themselves can be represented by continued fractions. Some of these continued fractions are first explored by Srinivasa Ramanujan in early 20th century. Later on, much work has been done in terms of studying the continued fractions, and proving several relations, identities, and giving different representations for them.</p> <p><br></p> <p>The layout of this report is as follows. Chapter 1 has all the basic background knowledge and ingredients about algebraic number theory, class field theory, Ramanujan’s theta functions, etc. In Chapter 2, we look at the Ramanujan-Göllnitz-Gordon continued fraction that we call v(τ) and evaluate it at certain arguments in the field K = Q(√−d), with −d ≡ 1 (mod 8), in which the ideal (2) = ℘₂℘′₂  is a product of two prime ideals. We prove several identities related to itself and with other modular functions. Some of these are new, while some of them are known but with different proofs. These values of v(τ) are shown to generate the inertia field of ℘₂ or ℘′₂ in an extended ring class field over the field K. The conjugates over Q of these same values, together with 0, −1 ± √2, are shown to form the exact set of periodic points of a fixed algebraic function ˆF(x), independent of d. These are analogues of similar results for the Rogers-Ramanujan continued fraction. See [1] and [2]. This joint work with my advisor Dr. Morton, is submitted for publication to the New York Journal.</p> <p><br> In Chapters 3 and 4, we take a similar approach in studying two more continued fractions c(τ) and u(τ), the first of which is more commonly known as the Ramanujan’s cubic continued fraction. We show what fields a value of this continued fraction generates over Q, and we describe how the periodic points for described functions arise as values of these continued fractions. Then in the last chapter, we summarise all these results, give some possible directions for future research as well as mentioning some conjectures.</p>

Quantum systems in the d-dimensional Hilbert space are considered. The mutually unbiased bases is a deep problem in this area. The problem of finding all mutually unbiased bases for higher (non-prime) dimension is still open. We derive an alternate approach to mutually unbiased bases by studying a weaker concept which we call weak mutually unbiased bases. We then compare three rather different structures. The first is weak mutually unbiased bases, for which the absolute value of the overlap of any two vectors in two different bases is 1/√k (where k∣d) or 0. The second is maximal lines through the origin in the Z(d) × Z(d) phase space. The third is an analytic representation in the complex plane based on Theta functions, and their zeros. The analytic representation of the weak mutually unbiased bases is defined with the zeros examined. It is shown that there is a correspondence (triality) that links strongly these three apparently different structures. We give an explicit breakdown of this triality.