This thesis is primarily concerned with properties of nilpotence and algebraicity in algebras over fields. We study properties of certain non-commutative polynomials, which are called the order-symmetric polynomials. We give an alternative proof to Amitsur's theorem that algebraic algebras over uncountable fields have locally bounded degree. We also prove that the associated graded algebra of a filtered algebraic algebra over an uncountable field is nil

Stress is among the strongest signals promoting neuroplasticity: Stress signals, indicating real or perceived danger, lead to alterations of neuronal function and often structure, designed to adapt to the changed conditions and promote survival. Corticotropin releasing factor (CRF) is expressed and released in several types of neuronal populations that are involved in cognition, emotion and the regulation of autonomic and endocrine function. CRF expressing neurons undergo functional and structural plasticity during stress and, in addition, the peptide acts via specific receptors to promote plasticity of target neurons.

Novel chiral iron phosphate complexes were prepared as catalysts for asymmetric oxidative coupling reactions. These catalysts were applied for the synthesis of enantio-enriched C₁- and C₂-symmetric BINOLs, in which the 3 and 3' positions are available for chemical modifications. It was proposed that the reaction takes place via an oxidative radical-anion coupling mechanism. A destructive BINOL racemization that competes with the enantioselective oxidative coupling of 2-naphthols was revealed, thereby offering new insights into this highly important reaction.

We prove a Plancherel theorem for a nonlinear Fourier transform in two dimensions arising in the Inverse Scattering method for the defocusing Davey-Stewartson II equation. We then use it to prove global well-posedness and scattering in $L^2$ for defocusing DSII. This Plancherel theorem also implies global uniqueness in the inverse boundary value problem of Calder on in dimension $2$, for conductivities $\sigma>0$ with $\log \sigma \in \dot H^1$. The proof of the nonlinear Plancherel theorem includes new estimates on classical fractional integrals, as well as a new result on $L^2$-boundedness of pseudo-differential operators with non-smooth symbols, valid in all dimensions.

Vulnerability to emotional disorders including depression derives from interactions between genes and environment, especially during sensitive developmental periods. Across evolution, maternal care is a key source of environmental sensory signals to the developing brain, and a vast body of work has linked quantitative and qualitative aspects of maternal care to emotional outcome in children and animals. However, the fundamental properties of maternal signals, that promote advantageous vs pathological outcomes in the offspring, are unknown and have been a topic of intense study. We studied emotional outcomes of adolescent rats reared under routine or impoverished environments, and used mathematical approaches to analyze the nurturing behaviors of the dams. Unexpectedly, whereas the quantity and typical qualities of maternal care behaviors were indistinguishable in the two environments, their patterns and rhythms differed drastically and influenced emotional outcomes. Specifically, unpredictable, fragmented maternal care patterns translated into high-entropy rates of sensory signals to the offspring in the impoverished cages. During adolescence, these offspring had significant reductions in sucrose preference and in peer-play, two independent measures of the ability to experience pleasure. This adolescent anhedonia, often a harbinger of later depression, was not accompanied by measures of anxiety or helplessness. Dopaminergic pleasure circuits underlying anhedonia are engaged by predictable sequences of events, and predictable sensory signals during neonatal periods may be critical for their maturation. Conversely, unpredictability maternal-derived signals may disrupt these developmental processes, provoking anhedonia. In sum, high-entropy and fragmented patterns of maternal-derived sensory input to the developing brain predicts, and might promote, the development of anhedonia in rodents, with potential clinical implications.

We introduce a Bayesian semi-supervised method for estimating cell counts from DNA methylation by leveraging an easily obtainable prior knowledge on the cell-type composition distribution of the studied tissue. We show mathematically and empirically that alternative methods which attempt to infer cell counts without methylation reference only capture linear combinations of cell counts rather than provide one component per cell type. Our approach allows the construction of components such that each component corresponds to a single cell type, and provides a new opportunity to investigate cell compositions in genomic studies of tissues for which it was not possible before.

The complex effects of stress on learning and memory are mediated, in part, by stress-induced changes in the composition and structure of excitatory synapses. In the hippocampus, the effects of stress involve several factors including glucocorticoids and the stress-released neuropeptide corticotropin-releasing hormone (CRH), which influence the integrity of dendritic spines and the structure and function of the excitatory synapses they carry. CRH, at nanomolar, presumed-stress levels, rapidly abolishes short-term synaptic plasticity and destroys dendritic spines, yet the mechanisms for these effects are not fully understood. Here we tested the hypothesis that glutamate receptor-mediated processes, which shape synaptic structure and function, are engaged by CRH and contribute to spine destabilization. In cultured rat hippocampal neurons, CRH application reduced dendritic spine density in a time- and dose-dependent manner, and this action depended on the CRH receptor type 1. CRH-mediated spine loss required network activity and the activation of NMDA, but not of AMPA receptors; indeed GluR1-containing dendritic spines were resistant to CRH. Downstream of NMDA receptors, the calcium-dependent enzyme, calpain, was recruited, resulting in the breakdown of spine actin-interacting proteins including spectrin. Pharmacological approaches demonstrated that calpain recruitment contributed critically to CRH-induced spine loss. In conclusion, the stress hormone CRH co-opts mechanisms that contribute to the plasticity and integrity of excitatory synapses, leading to selective loss of dendritic spines. This spine loss might function as an adaptive mechanism preventing the consequences of adverse memories associated with severe stress.

High costs and technical limitations of cell sorting and single-cell techniques currently restrict the collection of large-scale, cell-type-specific DNA methylation data. This, in turn, impedes our ability to tackle key biological questions that pertain to variation within a population, such as identification of disease-associated genes at a cell-type-specific resolution. Here, we show mathematically and empirically that cell-type-specific methylation levels of an individual can be learned from its tissue-level bulk data, conceptually emulating the case where the individual has been profiled with a single-cell resolution and then signals were aggregated in each cell population separately. Provided with this unprecedented way to perform powerful large-scale epigenetic studies with cell-type-specific resolution, we revisit previous studies with tissue-level bulk methylation and reveal novel associations with leukocyte composition in blood and with rheumatoid arthritis. For the latter, we further show consistency with validation data collected from sorted leukocyte sub-types.

Epigenetic changes are crucial for the generation of immunological memory. Failure to generate or maintain these changes will result in poor memory responses. Similarly, augmenting or stabilizing the correct epigenetic states offers a potential method of enhancing memory. Yet the transcription factors that regulate these processes are poorly defined. We find that the transcription factor Oct1 and its cofactor OCA-B are selectively required for the in vivo generation of CD4(+) memory T cells. More importantly, the memory cells that are formed do not respond properly to antigen reencounter. In vitro, both proteins are required to maintain a poised state at the Il2 target locus in resting but previously stimulated CD4(+) T cells. OCA-B is also required for the robust reexpression of multiple other genes including Ifng. ChIPseq identifies ∼50 differentially expressed direct Oct1 and OCA-B targets. We identify an underlying mechanism involving OCA-B recruitment of the histone lysine demethylase Jmjd1a to targets such as Il2, Ifng, and Zbtb32. The findings pinpoint Oct1 and OCA-B as central mediators of CD4(+) T cell memory.