Asymptotically accurate error estimates of exponential convergence for the trapezoidal rule

In many applied problems, efficient calculation of quadratures with high accuracy is required. The examples are: calculation of special functions of mathematical physics, calculation of Fourier coefficients of a given function, Fourier and Laplace transformations, numerical solution of integral equations, solution of boundary value problems for partial differential equations in integral form, etc. For grid calculation of quadratures, the trapezoidal, the mean and the Simpson methods are usually used. Commonly, the error of these methods depends quadratically on the grid step, and a large number of steps are required to obtain good accuracy. However, there are some cases when the error of the trapezoidal method depends on the step value not quadratically, but exponentially. Such cases are integral of a periodic function over the full period and the integral over the entire real axis of a function that decreases rapidly enough at infinity. If the integrand has poles of the first order on the complex plane, then the Trefethen-Weidemann majorant accuracy estimates are valid for such quadratures. In the present paper, new error estimates of exponentially converging quadratures from periodic functions over the full period are constructed. The integrand function can have an arbitrary number of poles of an integer order on the complex plane. If the grid is sufficiently detailed, i.e., it resolves the profile of the integrand function, then the proposed estimates are not majorant, but asymptotically sharp. Extrapolating, i.e., excluding this error from the numerical quadrature, it is possible to calculate the integrals of these classes with the accuracy of rounding errors already on extremely coarse grids containing only ~ 10 steps.Во многих прикладных задачах требуется экономичное вычисление квадратур с высокой точностью. Примерами являются: вычисление специальных функций математической физики, расчёт коэффициентов Фурье заданной функции, преобразования Фурье и Лапласа, численное решение интегральных уравнений, решение краевых задач для уравнений в частных производных в интегральной форме и т.д. Для сеточного вычисления квадратур обычно используют методы трапеций, средних и Симпсона. Обычно погрешность этих методов зависит от шага степенным образом, и для получения хорошей точности требуется большое число шагов. Однако существует ряд случаев, когда погрешность метода трапеций зависит от величины шага не квадратично, а экспоненциально. Такими случаями являются интеграл от периодической функции по полному периоду и интеграл по всей числовой прямой от функции, достаточно быстро убывающей на бесконечности. Если подынтегральная функция имеет полюса первого порядка в комплексной плоскости, то для таких квадратур справедливы мажорантные оценки точности Трефетена и Вайдемана. В работе построены новые оценки погрешности экспоненциально сходящихся квадратур от периодических функций по полному периоду. Подынтегральная функция может иметь произвольное число полюсов целого порядка на комплексной плоскости. Если сетка достаточно подробная (разрешает профиль подынтегральной функции), то предлагаемые оценки являются не мажорантными, а асимптотически точными. Экстраполируя, то есть исключая эту погрешность из численной квадратуры, можно вычислять интегралы указанных классов с точностью ошибок округления уже на чрезвычайно грубых сетках, содержащих всего ~ 10 шагов

Shifted Sobol points and multigrid Monte Carlo simulation

Multidimensional integrals arise in many problems of physics. For example, moments of the distribution function in the problems of transport of various particles (photons, neutrons, etc.) are 6-dimensional integrals. When calculating the coefficients of electrical conductivity and thermal conductivity, scattering integrals arise, the dimension of which is equal to 12. There are also problems with a significantly large number of variables. The Monte Carlo method is the most effective method for calculating integrals of such a high multiplicity. However, the efficiency of this method strongly depends on the choice of a sequence that simulates a set of random numbers. A large number of pseudo-random number generators are described in the literature. Their quality is checked using a battery of formal tests. However, the simplest visual analysis shows that passing such tests does not guarantee good uniformity of these sequences. The magic Sobol points are the most effective for calculating multidimensional integrals. In this paper, an improvement of these sequences is proposed: the shifted magic Sobol points that provide better uniformity of points distribution in a multidimensional cube. This significantly increases the cubature accuracy. A significant difficulty of the Monte Carlo method is a posteriori confirmation of the actual accuracy. In this paper, we propose a multigrid algorithm that allows one to find the grid value of the integral simultaneously with a statistically reliable accuracy estimate. Previously, such estimates were unknown. Calculations of representative test integrals with a high actual dimension up to 16 are carried out. The multidimensional Weierstrass function, which has no derivative at any point, is chosen as the integrand function. These calculations convincingly show the advantages of the proposed methods.Многомерные интегралы возникают во многих задачах физики. Например, моменты функции распределения в задачах переноса различных частиц (фотонов, нейтронов и др.) являются 6-мерными интегралами. При расчёте коэффициентов электропроводности и теплопроводности возникают интегралы рассеяния, размерность которых равна 12. Возникают задачи и с существенно большим числом переменных. Для вычисления интегралов столь высокой кратности наиболее эффективен метод Монте-Карло. Однако работоспособность этого метода сильно зависит от выбора последовательности, имитирующей набор случайных чисел. В литературе описано большое количество генераторов псевдослучайных чисел. Их качество проверяется с помощью батарей формальных тестов. Однако простейший визуальный анализ показывает, что прохождение таких тестов не гарантирует хорошей равномерности этих последовательностей. Для вычисления многомерных интегралов наиболее эффективны магические точки Соболя. В данной работе предложено усовершенствование этих последовательностей - смещённые магические точки Соболя, обеспечивающие большую равномерность распределения точек в многомерном кубе. Это ощутимо повышает точность кубатур. Существенной трудностью методов Монте-Карло является апостериорное подтверждение фактической точности. В данной работе предложен многосеточный алгоритм, позволяющий найти сеточное значение интеграла одновременно со статистически достоверной оценкой его точности. Ранее такие оценки были неизвестны. Проведены расчёты представительных тестовых интегралов с высокой фактической размерностью до 16. В качестве подынтегральной функции выбрана многомерная функция Вейерштрасса, не имеющая производной ни в одной точке. Эти расчёты убедительно показывают преимущества предложенных методов

Novel Optimized Hybrid Terahertz Photoconductive Antennas

We demonstrate the results of on THz emission from hybrid THz photoconductive antenna loaded with silver nanoantenna. The results of experimental investigations are in a good agreement with numerical simulations presented in our recent work. The conversion efficiency reveals over 5-fold improvement at certain frequencies, if compared with similar photoconductive antenna without silver nanoparticles, while previous results for this type of antenna barely exceeded 2-fold conversion efficiency gain. We propose a cost-effective fabrication procedure to realize such hybrid THz antennas with optimized plasmonic nanostructures via thermal dewetting process, which does not require any post processing and makes the proposed solution very attractive for applications

Recurrent and sporadic Forbush decreases during solar cycles 23–24

A comparison has been made between recurrent (associated with high-speed streams from coronal holes) and sporadic (caused by interplanetary coronal mass ejections (ICMEs)) Forbush decreases (FDs) in solar cycles 23 and 24 (as well as in the maxima of these cycles and the minimum between them). Forbush Effects and Interplanetary Disturbances database created and maintained in IZMIRAN provided a large number of events (about 1700 isolated FDs, among them 350 recurrent FDs, and 207 sporadic FDs selected with high reliability), which allowed us to apply statistical methods. The results revealed that sporadic FDs prevailed in the maxima of the cycles; recurrent FDs, in the minimum between the cycles. FD parameters (magnitude, decrease rate, anisotropy) are larger for sporadic events than for recurrent ones, especially in the maxima of the cycles. FD magnitude is greater in the maxima than in the minimum for sporadic events, and it changes weakly for recurrent ones. The solar wind velocity is on average greater for recurrent events than for sporadic ones; it is larger for recurrent FDs in the minimum and for sporadic FDs in the maxima. The magnetic field is stronger for sporadic FDs than for recurrent ones in the maxima and it is approximately equal for both types of events in the minimum. The magnetic field of ICMEs is weaker in the current solar cycle than in the previous one. The duration of the FD main phase is less in the maxima for both types of events; sporadic FDs developed significantly faster than recurrent ones in the maximum of cycle 23

НОВОЕ НАПРАВЛЕНИЕ НАУЧНЫХ ИССЛЕДОВАНИЙ ПО ИЗУЧЕНИЮ ПРОДУКТИВНОСТИ ЖИВОТНЫХ В СВЯЗИ С РАЗНОЙ КОСМОФИЗИЧЕСКОЙ АКТИВНОСТЬЮ

Brief results of research at the Department of Zootechnics of the PFUR in more than 30 years are reported on the relationship between cosmophysical activity and animal productivity (the rhythm of milk yield of cows and the chemical composition of milk, the biochemical composition of cows' blood). It is established that in even and odd 11-year solar cycles, animals react differently to cosmophysical activity. In even cycles, the increase in solar activity contributes to the growth of milk yields. In odd cycles, increasing solar activity reduces milk yield. The data of scientific observations, covering the periods of even and odd cycles, should be grouped and analyzed separately by cycles. Within 2—3 years, at the lowest peak of solar activity, and 2—3 years — at the top peak of activity (with the change of magnetic poles on the Sun) minimal and unreliable correlations are noted. The strongest correlation was noted in the middle of the rise or decrease in solar activity. The rhythm of milk yield was noted regard-less of the lactation phase of cows. Approximately 1/3 of the time, milk yield increased, 1/3 — decreased and 1/3 — remained at the same level, with fluctuations, with periods from 2.7; 3.3; up to 5.5 days. With an increase in milk yield during 31.0% of lactation days, the content of fat and protein in milk decreased, respectively, 25.8 and 25.4% of the time. With a decrease in milk yield during 33.7% of lactation days, the fat and protein content did not change, respectively, 49.8 and 48.1% days. The most pronounced interval of rhythmicity of the concentration of fat and protein in milk from 2 to 2.5 days with an average value of 2.2 days was established. In addition rhythm 2.8; 3.4; 4.4; 8; 32; 41,1 and 72 days was marked. Approxi-mately with the same cycles in the milk, sugar concentrations, density and acidity were changed.Приведены краткие результаты исследований на кафедре зоотехнии РУДН более чем за 30 лет о взаимосвязи космофизической активности и продуктивности животных (ритмичности удоев коров и химического состава молока, биохимическом составе крови коров). Установлено, что в четные и нечетные 11-летние солнечные циклы животные по-разному реагируют на космофизические про-явления. В четные циклы повышение солнечной активности способствует росту удоев. В нечетные циклы повышение солнечной активности снижает удои. Данные научных наблюдений, охватывающих периоды четных и нечетных циклов, следует группировать и анализировать отдельно по циклам. В течение 2-3 лет на нижнем пике солнечной активности и 2-3 лет - на верхнем пике активности (при смене магнитных полюсов на Солнце) отмечаются минимальные и недостоверные корреляционные связи. Наиболее стабильные связи бывают в середине подъема или снижения солнечной активности. При этом ритмичность удоев отмечалась независимо от фазы лактации коров. Примерно 1/3 времени удои росли, 1/3 - снижались и 1/3 - оставались на одном уровне, при колебаниях суточных удоев с периодами от 2,7; 3,3 до 5,5 суток. При увеличении удоев в течение 31,0% дней лактации содержание жира и белка в молоке уменьшалось соответственно 25,8 и 25,4% времени. При снижении же удоев в течение 33,7% дней лактации содержание жира и белка не изменялось соответственно 49,8 и 48,1% дней. Наиболее выражен интервал ритмичности концентрации жира и белка в молоке от 2 до 2,5 суток со средним значением 2,2 дня; затем отмечается ритм 2,8; 3,4; 4,4; 8; 32; 41,1 и 72 суток. Примерно с такими же циклами в молоке изменялись концентрации сахара, плотности и кислотности

Cubic String Field Theory in pp-wave Background and Background Independent Moyal Structure

We study Witten open string field theory in the pp-wave background in the tensionless limit, and construct the N-string vertex in the basis which diagonalizes the string perturbative spectrum. We found that the Witten *-product can be viewed as infinite copies of the Moyal product with the same noncommutativity parameter $\theta=2$. Moreover, we show that this Moyal structure is universal in the sense that, written in the string bit basis, Witten's *-product for any background can always be given in terms of the above-mentioned Moyal structure. We identify some projective operators in this algebra that we argue to correspond to D-branes of the theory.Comment: Latex, 23 pages, reference adde

A Comprehensive Approach to Investigating Fluorine-Bearing Gas Mixtures

 An integrated method is proposed for examining the compositions of fluorine-bearing gaseous mixtures, which allows for determining the concentration of HF, F2, N2, O2, CO2, CF4, and C2F6 in these mixtures. The concentration of hydrogen fluoride is determined by its sorption on sodium fluoride followed by the determination of the fluoride ion by the potentiometric method. The lower threshold of determination of hydrogen fluoride is found to be 0.09 vol.%. The concentrations of N2, O2, CO2, CF4, and C2F6 are determined by the gas chromatographic method using a thermal conductivity detector. The proposed gas-chromatography method offers a quantitative measurement of the concentration of N2, O2, CO2, CF4, and C2F6 at the lower limits of detection of 0.008, 0.012, 0.011, 0.009, and 0.019 vol.%, respectively. Based on the developed method, the compositions of a standard fluorine-nitrogen (10 vol.%) and anodic gas samples, synthesized in a laboratory electrolyzer at the National Research Tomsk Polytechnic University and in an industrial electrochemical reactor at JSC Siberian Chemical Plant (Russia), are studied

Oscillating Solitons Pinned to a Nonmagnetic Impurity in Layered Antiferromagnets

We argue that an oscillatory motion of impurity-pinned solitons may occur in layered antiferromagnetic compounds. The characteristic frequencies of these modes, that may be detected by resonance or inelastic neutron scattering, are estimated analytically and depend on the soliton sizes and types .Comment: 11 pages, 1 figur

An Extreme Solar Event of 20 January 2005: Properties of the Flare and the Origin of Energetic Particles

The extreme solar and SEP event of 20 January 2005 is analyzed from two perspectives. Firstly, we study features of the main phase of the flare, when the strongest emissions from microwaves up to 200 MeV gamma-rays were observed. Secondly, we relate our results to a long-standing controversy on the origin of SEPs arriving at Earth, i.e., acceleration in flares, or shocks ahead of CMEs. All emissions from microwaves up to 2.22 MeV line gamma-rays during the main flare phase originated within a compact structure located just above sunspot umbrae. A huge radio burst with a frequency maximum at 30 GHz was observed, indicating the presence of a large number of energetic electrons in strong magnetic fields. Thus, protons and electrons responsible for flare emissions during its main phase were accelerated within the magnetic field of the active region. The leading, impulsive parts of the GLE, and highest-energy gamma-rays identified with pi^0-decay emission, are similar and correspond in time. The origin of the pi^0-decay gamma-rays is argued to be the same as that of lower energy emissions. We estimate the sky-plane speed of the CME to be 2000-2600 km/s, i.e., high, but of the same order as preceding non-GLE-related CMEs from the same active region. Hence, the flare itself rather than the CME appears to determine the extreme nature of this event. We conclude that the acceleration, at least, to sub-relativistic energies, of electrons and protons, responsible for both the flare emissions and the leading spike of SEP/GLE by 07 UT, are likely to have occurred simultaneously within the flare region. We do not rule out a probable contribution from particles accelerated in the CME-driven shock for the leading GLE spike, which seemed to dominate later on.Comment: 34 pages, 14 Postscript figures. Solar Physics, accepted. A typo corrected. The original publication is available at http://www.springerlink.co

Quantum phase transitions and thermodynamic properties in highly anisotropic magnets

The systems exhibiting quantum phase transitions (QPT) are investigated within the Ising model in the transverse field and Heisenberg model with easy-plane single-site anisotropy. Near QPT a correspondence between parameters of these models and of quantum phi^4 model is established. A scaling analysis is performed for the ground-state properties. The influence of the external longitudinal magnetic field on the ground-state properties is investigated, and the corresponding magnetic susceptibility is calculated. Finite-temperature properties are considered with the use of the scaling analysis for the effective classical model proposed by Sachdev. Analytical results for the ordering temperature and temperature dependences of the magnetization and energy gap are obtained in the case of a small ground-state moment. The forms of dependences of observable quantities on the bare splitting (or magnetic field) and renormalized splitting turn out to be different. A comparison with numerical calculations and experimental data on systems demonstrating magnetic and structural transitions (e.g., into singlet state) is performed.Comment: 46 pages, RevTeX, 6 figure