Hypercomplex quantum mechanics

The fundamental axioms of the quantum theory do not explicitly identify the algebraic structure of the linear space for which orthogonal subspaces correspond to the propositions (equivalence classes of physical questions). The projective geometry of the weakly modular orthocomplemented lattice of propositions may be imbedded in a complex Hilbert space; this is the structure which has traditionally been used. This paper reviews some work which has been devoted to generalizing the target space of this imbedding to Hilbert modules of a more general type. In particular, detailed discussion is given of the simplest generalization of the complex Hilbert space, that of the quaternion Hilbert module.Comment: Plain Tex, 11 page

Development of a Molecular Signature to Monitor Pharmacodynamic Responses Mediated by In Vivo Administration of Glucocorticoids

© 2018 American College of Rheumatology. Objective: To develop an objective, readily measurable pharmacodynamic biomarker of glucocorticoid (GC) activity. Methods: Genes modulated by prednisolone were identified from in vitro studies using peripheral blood mononuclear cells from normal healthy volunteers. Using the criteria of a \u3e2-fold change relative to vehicle controls and an adjusted P value cutoff of less than 0.05, 64 up-regulated and 18 down-regulated genes were identified. A composite score of the up-regulated genes was generated using a single-sample gene set enrichment analysis algorithm. Results: GC gene signature expression was significantly elevated in peripheral blood leukocytes from normal healthy volunteers following oral administration of prednisolone. Expression of the signature increased in a dose-dependent manner, peaked at 4 hours postadministration, and returned to baseline levels by 48 hours after dosing. Lower expression was detected in normal healthy volunteers who received a partial GC receptor agonist, which is consistent with the reduced transactivation potential of this compound. In cohorts of patients with systemic lupus erythematosus and patients with rheumatoid arthritis, expression of the GC signature was negatively correlated with the percentages of peripheral blood lymphocytes and positively correlated with peripheral blood neutrophil counts, which is consistent with the known biology of the GC receptor. Expression of the signature largely agreed with reported GC use in these populations, although there was significant interpatient variability within the dose cohorts. Conclusion: The GC gene signature identified in this study represents a pharmacodynamic marker of GC exposure

On Measuring Non-Recursive Trade-Offs

We investigate the phenomenon of non-recursive trade-offs between descriptional systems in an abstract fashion. We aim at categorizing non-recursive trade-offs by bounds on their growth rate, and show how to deduce such bounds in general. We also identify criteria which, in the spirit of abstract language theory, allow us to deduce non-recursive tradeoffs from effective closure properties of language families on the one hand, and differences in the decidability status of basic decision problems on the other. We develop a qualitative classification of non-recursive trade-offs in order to obtain a better understanding of this very fundamental behaviour of descriptional systems

Small ball probability, Inverse theorems, and applications

Let $\xi$ be a real random variable with mean zero and variance one and $A={a_1,...,a_n}$ be a multi-set in $\R^d$. The random sum $$S_A := a_1 \xi_1 + ... + a_n \xi_n$$ where $\xi_i$ are iid copies of $\xi$ is of fundamental importance in probability and its applications. We discuss the small ball problem, the aim of which is to estimate the maximum probability that $S_A$ belongs to a ball with given small radius, following the discovery made by Littlewood-Offord and Erdos almost 70 years ago. We will mainly focus on recent developments that characterize the structure of those sets $A$ where the small ball probability is relatively large. Applications of these results include full solutions or significant progresses of many open problems in different areas.Comment: 47 page

Exploring new physics frontiers through numerical relativity

The demand to obtain answers to highly complex problems within strong-field gravity has been met with significant progress in the numerical solution of Einstein's equations - along with some spectacular results - in various setups. We review techniques for solving Einstein's equations in generic spacetimes, focusing on fully nonlinear evolutions but also on how to benchmark those results with perturbative approaches. The results address problems in high-energy physics, holography, mathematical physics, fundamental physics, astrophysics and cosmology