Laplacian Growth, Elliptic Growth, and Singularities of the Schwarz Potential

The Schwarz function has played an elegant role in understanding and in generating new examples of exact solutions to the Laplacian growth (or "Hele- Shaw") problem in the plane. The guiding principle in this connection is the fact that "non-physical" singularities in the "oil domain" of the Schwarz function are stationary, and the "physical" singularities obey simple dynamics. We give an elementary proof that the same holds in any number of dimensions for the Schwarz potential, introduced by D. Khavinson and H. S. Shapiro [17] (1989). A generalization is also given for the so-called "elliptic growth" problem by defining a generalized Schwarz potential. New exact solutions are constructed, and we solve inverse problems of describing the driving singularities of a given flow. We demonstrate, by example, how \mathbb{C}^n - techniques can be used to locate the singularity set of the Schwarz potential. One of our methods is to prolong available local extension theorems by constructing "globalizing families". We make three conjectures in potential theory relating to our investigation

Peptide KED: molecular-genetic aspects of neurogenesis regulation in Alzheimer’s disease

Neuroprotective peptides are promising candidate molecules for the treatment of Alzheimer’s disease (AD). Oral application of KED (Lys-Glu-Asp) improved memory and attention in elderly individuals with functional CNS disorders. Peptide KED also restores synaptic plasticity in in vitro model of AD. This review is focused on the analysis of the influence of KED peptide on the expression of genes and synthesis of proteins regulating apoptosis, aging, neurogenesis, and involved in AD pathogenesi

Peptides: Prospects for use in the treatment of COVID-19

There is a vast practice of using antimalarial drugs, RAS inhibitors, serine protease inhibitors, inhibitors of the RNA-dependent RNA polymerase of the virus and immunosuppressants for the treatment of the severe form of COVID-19, which often occurs in patients with chronic diseases and older persons. Currently, the clinical efficacy of these drugs for COVID-19 has not been proven yet. Side effects of antimalarial drugs can worsen the condition of patients and increase the likelihood of death. Peptides, given their physiological mechanism of action, have virtually no side effect

Патогенез, диагностика и эффективное лечение пептидами эпифиза аспириновой бронхиальной астмы

Characteristic features of history and clinical course of the disease, melatonin and nitric oxide production and thrombocyte functional activity were searched in 133 patients with aspirin-induced (Al) asthma. Control groups included 143 patients with non-AI asthma and 54 healthy volunteers. The Al asthma was found to be a specific pathology based on a reduction in the melatonin synthesis by the diffuse neuroendocrine (APUD) system cells. This fact causes simultaneous failure and imbalance of the nervous, endocrine, immune and other systems, functional disorders of the thrombocytes, the vascular system and the homeostasis in general and is accompanied by the asthmatic syndrome development. An increased sensitivity of the thrombocyte receptors and their perverted response on the melatonin and its metabolite having the chemical structure similar to that of the acetylsalicylic acid can be a background of intolerability to aspirin in Al asthma patients and to provide a progress of the disease. Epithalamine and epiphamine were shown to be pathogenic drugs for Al asthma.Изучены особенности анамнеза, клинического течения заболевания, продукция мелатонина, окиси азота и функциональная активность тромбоцитов у 133 больных аспириновой бронхиальной астмой (АсБА). Контрольные группы составили 143 больных бронхиальной астмой без непереносимости аспирина (НБА) и 54 здоровых добровольца. Установлено, что аспириновая астма — особое заболевание, в основе которого лежит снижение продукции мелатонина клетками диффузной нейроэндокринной системы (АПУД-системы), что ведет к одновременно возникающей недостаточности и дисбалансу функций нервной, эндокринной, иммунной и других систем организма, к нарушениям функционального состояния тромбоцитов, сосудистой системы, гомеостаза в целом и сопровождается развитием астматического синдрома. Повышение чувствительности рецепторного аппарата тромбоцитов и извращенная реакция не только на мелатонин, но и на его метаболит, сходный по химической структуре с ацетилсалициловой кислотой (АСК), может служить предпосылкой к появлению симптомов непереносимости аспирина у больных АсБА и способствовать прогрессированию заболевания. Показано, что эпиталамин и эпифамин являю тся препаратами стратегической патогенетической терапии АсБА

Mathematics of Gravitational Lensing: Multiple Imaging and Magnification

The mathematical theory of gravitational lensing has revealed many generic and global properties. Beginning with multiple imaging, we review Morse-theoretic image counting formulas and lower bound results, and complex-algebraic upper bounds in the case of single and multiple lens planes. We discuss recent advances in the mathematics of stochastic lensing, discussing a general formula for the global expected number of minimum lensed images as well as asymptotic formulas for the probability densities of the microlensing random time delay functions, random lensing maps, and random shear, and an asymptotic expression for the global expected number of micro-minima. Multiple imaging in optical geometry and a spacetime setting are treated. We review global magnification relation results for model-dependent scenarios and cover recent developments on universal local magnification relations for higher order caustics.Comment: 25 pages, 4 figures. Invited review submitted for special issue of General Relativity and Gravitatio

Spectral bounds for the Neumann-Poincaré operator on planar domains with corners

The boundary double layer potential, or the Neumann-Poincaré operator, is studied on the Sobolev space of order 1/2 along the boundary, coinciding with the space of charges giving rise to double layer potentials with finite energy in the whole space. Poincaré’s program of studying the spectrum of the boundary double layer potential is developed in complete generality on closed Lipschitz hypersurfaces in euclidean space. Furthermore, the Neumann-Poincaré operator is realized as a singular integral transform bearing similarities to the Beurling-Ahlfors transform in 2 dimensions. As an application, in the case of planar curves with corners, bounds for the spectrum of the Neumann-Poincaré operator are derived from recent results in quasi-conformal mapping theory