The effect of additive noise on dynamical hysteresis

We investigate the properties of hysteresis cycles produced by a one-dimensional, periodically forced Langevin equation. We show that depending on amplitude and frequency of the forcing and on noise intensity, there are three qualitatively different types of hysteresis cycles. Below a critical noise intensity, the random area enclosed by hysteresis cycles is concentrated near the deterministic area, which is different for small and large driving amplitude. Above this threshold, the area of typical hysteresis cycles depends, to leading order, only on the noise intensity. In all three regimes, we derive mathematically rigorous estimates for expectation, variance, and the probability of deviations of the hysteresis area from its typical value.Comment: 30 pages, 5 figure

Computation of saddle type slow manifolds using iterative methods

This paper presents an alternative approach for the computation of trajectory segments on slow manifolds of saddle type. This approach is based on iterative methods rather than collocation-type methods. Compared to collocation methods, that require mesh refinements to ensure uniform convergence with respect to $\epsilon$, appropriate estimates are directly attainable using the method of this paper. The method is applied to several examples including: A model for a pair of neurons coupled by reciprocal inhibition with two slow and two fast variables and to the computation of homoclinic connections in the FitzHugh-Nagumo system.Comment: To appear in SIAM Journal of Applied Dynamical System

Robust Mixing for Ab-Initio Quantum Mechanical Calculations

We study the general problem of mixing for ab-initio quantum-mechanical problems. Guided by general mathematical principles and the underlying physics, we propose a multisecant form of Broydens second method for solving the self-consistent field equations of Kohn-Sham density functional theory. The algorithm is robust, requires relatively little finetuning and appears to outperform the current state of the art, converging for cases that defeat many other methods. We compare our technique to the conventional methods for problems ranging from simple to nearly pathological.Comment: 32 Pages, 4 figure

Exceptional Laguerre and Jacobi polynomials and the corresponding potentials through Darboux-Crum Transformations

Simple derivation is presented of the four families of infinitely many shape invariant Hamiltonians corresponding to the exceptional Laguerre and Jacobi polynomials. Darboux-Crum transformations are applied to connect the well-known shape invariant Hamiltonians of the radial oscillator and the Darboux-P\"oschl-Teller potential to the shape invariant potentials of Odake-Sasaki. Dutta and Roy derived the two lowest members of the exceptional Laguerre polynomials by this method. The method is expanded to its full generality and many other ramifications, including the aspects of generalised Bochner problem and the bispectral property of the exceptional orthogonal polynomials, are discussed.Comment: LaTeX2e with amsmath, amssymb, amscd 26 pages, no figure

Beyond the Fokker-Planck equation: Pathwise control of noisy bistable systems

We introduce a new method, allowing to describe slowly time-dependent Langevin equations through the behaviour of individual paths. This approach yields considerably more information than the computation of the probability density. The main idea is to show that for sufficiently small noise intensity and slow time dependence, the vast majority of paths remain in small space-time sets, typically in the neighbourhood of potential wells. The size of these sets often has a power-law dependence on the small parameters, with universal exponents. The overall probability of exceptional paths is exponentially small, with an exponent also showing power-law behaviour. The results cover time spans up to the maximal Kramers time of the system. We apply our method to three phenomena characteristic for bistable systems: stochastic resonance, dynamical hysteresis and bifurcation delay, where it yields precise bounds on transition probabilities, and the distribution of hysteresis areas and first-exit times. We also discuss the effect of coloured noise.Comment: 37 pages, 11 figure

КЛИНИЧЕСКИЕ И ЭПИДЕМИОЛОГИЧЕСКИЕ ЗАКОНОМЕРНОСТИ ВЕТРЯНОЙ ОСПЫ У ВЗРОСЛЫХ В ПРИМОРСКОМ КРАЕ

Objective: to establish clinical and epidemiological patterns of chickenpox in adults in the Primorsky Territory to improve treatment and prevention work. Materials and methods: The study used statistics on the incidence of chicken pox in the Primorsky Territory from 2009 to 2017 and demographic data on the population for the specified period. A clinical characteristic of 102 cases of varicella in patients hospitalized in the infectious disease ward of Primorsky Regional Clinical Hospital No. 2 (Vladivostok) in 2015-2017 is given. Results: From 2009 to 2017, the incidence of chickenpox in the Primorsky Territory was consistently high, with an average of 636.2 per 100,000 population. Analysis of the structure of patients showed that the maximum proportion of cases of chicken pox occur in organized children 3-6 years old (55.9%), schoolchildren (19.1%) and children 1–2 years old (11.6%). The proportion of 15–17 year olds was 3.0%. Adults, from 18 years old – 4.9%. The proportion of unorganized children 3–6 years old (2.9%) and children in the first year of life (2.6%) is not high. High risk of infection was detected in Vladivostok (57.8 per 100.000), Yakovlevsky (52.1% ooo), Olginsky (45.9% ooo), Lazovsky (42.8% ooo) and Shkotovsky (40, 2% ooo) areas. In the annual dynamics, a pronounced spring-summer seasonal rise was formed. The clinical course of varicella in the group of adults was characterized by an average duration of the initial period of the disease of 1.4 ± 0.7 days. For clinical diagnosis, an important time was the appearance of the rash, which varied within four days. The phenomenon of “spillage” was noted in 97% of patients. The rash was polymorphic, with the formation of crusts. The period of height was characterized by the presence of catarrhal manifestations. The granularity of the posterior pharyngeal wall was observed in all patients without exception. In 84.3% of patients there was a bright hyperemia of the pharynx. In 14.7% of cases, enantema was detected on the soft palate, palatine arches, and posterior pharyngeal wall. Dry cough was detected in 28 of 103 patients, which probably increased their epidemiological significance. Findings: The clinical picture of varicella was typical and was represented by a combination of intoxication, exanthema, and catarrhal syndrome. The epidemic process of varicella in adults in Primorsky Krai was characterized by relative autonomy of manifestations in adults, children up to one year old, and unorganized children. When implementing chickenpox epidemiological surveillance, priority should be given to interventions in epidemic foci.Цель: установление клинических и эпидемиологических закономерностей ветряной оспы у взрослых в Приморском крае для совершенствования лечебной и профилактической работы. Материалы и методы: в исследовании использованы статистические данные по заболеваемости ветряной оспой в Приморском крае с 2009 по 2017 г. и демографические данные по численности населения за указанный период. Дана клиническая характеристика 102 случаев ветряной оспы у пациентов, госпитализированных в инфекционное отделение Приморской краевой клинической больницы №2 (г. Владивосток) в 2015–2017 гг. Результаты: с 2009 по 2017 г. заболеваемость ветряной оспой в Приморском крае была стабильно высокой, со средним уровнем 636,2 на 100 000 населения. Анализ структуры больных показал, что максимальная доля случаев ветряной оспы приходится на организованных детей 3–6 лет (55,9%), школьников (19,1%) и детей 1–2 лет (11,6%). Доля лиц 15–17 лет составила 3,0%; взрослых (с 18 лет) – 4,9%. Невысока доля неорганизованных детей 3–6 лет (2,9%) и детей первого года жизни (2,6%). К территориям риска по заболеваемости ветряной оспой взрослых были отнесены Владивосток (57,8 на 100,000), Яковлевский (52,1%ooo), Ольгинский (45,9%ooo), Лазовский (42,8%ooo) и Шкотовский (40,2%ooo) районы. В годовой динамике формировался ярко выраженный весенне-летний сезонный подъем. Клиническое течение ветряной оспы в группе взрослых лиц характеризовалось средней продолжительностью начального периода болезни 1,4±0,7 дня. Для клинической диагностики важным являлось время появления сыпи, варьировавшее в пределах 4 суток. Феномен «подсыпания» был отмечен у 97% больных. Сыпь была полиморфной, с образованием корочек. Период разгара характеризовался наличием катаральных проявлений. Зернистость задней стенки глотки наблюдалась у всех без исключения пациентов. У 84,3% больных отмечалась яркая гиперемия зева. В 14,7% случаев выявлена энантема на мягком небе, небных дужках, задней стенке глотки. Сухой кашель выявлен у 28 из 102 больных, что, вероятно, увеличивало их эпидемиологическую значимость.Выводы: клиническая картина ветряной оспы была типичной и представлена сочетанием интоксикационного, экзантемного и катарального синдрома. Эпидемический процесс ветряной оспы у взрослых в Приморском крае характеризовался относительной автономностью проявлений у взрослых, детей до 1 года и неорганизованных детей. При осуществлении эпидемического надзора за ветряной оспой следует отдавать приоритет мероприятиям в эпидемических очагах