In this study, two-electron one- and two-center Coulomb integrals with the same and different screening parameters are investigated numerically in the real Slater type orbital (STO) basis using ...Fourier transform method. In momentum space firstly, for atomic, i.e. one-center, Coulomb integrals are calculated, and analytical expressions are obtained in terms of binomial coefficients. Then, the solutions of the two-center Coulomb integrals are made with the modified Bessel function of second kind and the results are expressed in terms of binomial and Gaunt coefficients, irregular solid harmonics, and finite sum of STOs. A computer program is written in the MATHEMATICA language to determine the accuracy of the analytical expressions that are highly suitable for programming. The numerical results obtained from the program are given in the tables, and it is shown that the results agree with the literature.
Bu çalışmada, aynı ve farklı perdeleme sabitlerine sahip iki elektronlu bir- ve iki-merkezli Coulomb integralleri, Fourier dönüşüm yöntemi kullanılarak reel Slater tipi orbitaller (STO) bazında sayısal olarak incelenmiştir. Momentum uzayında ilk olarak atomik, yani tek-merkezli, Coulomb integralleri için hesaplama yapılmış ve analitik ifadeler binom katsayıları cinsinden elde edilmiştir. Daha sonra, iki-merkezli Coulomb integrallerinin çözümleri, ikinci tür modifiye edilmiş Bessel fonksiyonları ile yapılmış ve sonuçlar binom ve Gaunt katsayıları, düzensiz katı harmonikler ve STO’ların sonlu toplamı cinsinden ifade edilmiştir. Programlamaya son derece uygun olan analitik ifadelerin doğruluğunu belirlemek için MATHEMATICA dilinde bir bilgisayar programı yazılmıştır. Programdan elde edilen sayısal sonuçlar tablolarda verilmiş ve sonuçların literatür ile uyumlu olduğu gösterilmiştir.
This volume explores integral operators of Hardy type and related Sobolev embeddings from an s-numbers perspective. The text uses proof methods that incorporate the geometry of Banach spaces and ...generalized trigonometric functions.
This article presents a method for assessing the hit probability for stationary shooting targets as a function of the projectile horizontal range (PHR), the target type and the point of aim (POA) ...location. The proposed procedure consists of three blocks. Block I is devoted to the double integral formula taking into account changes in the point of impact (POI) location as a function of the PHR. The characteristics of bullet accuracy and precision versus the PHR are determined in block II. The basis of blocks I and II is the bi-variate uncorrelated Laplace-Gauss probability distribution. The functions of the POI location (ordinate/abscissa) versus the PHR and the functions of the characteristics of the bullet dispersion versus the PHR are represented in the form of polynomials. The description of the target silhouette contour is given in block III. Mathematically, the target contour is a piecewise function which defines the upper and lower edges of the shooting target and it also represents the limit for the double integral formula of block I. The proposed method is built on a modular basis and allows a user to change types of weapons and shooting targets. For demo calculations, the accuracy and precision characteristics of the 5.45×39 Kalashnikov assault rifle model MPi AK-74N were selected. Five types of Swiss military targets imitating an OPFOR combatant's silhouette were used as shooting targets. For illustrating the operability of the proposed method, the hit probabilities for the K, H, G, F, and E shooting targets were evaluated for the PHR from 50 to 400 m. All computations are implemented in the PTC Mathcad v.15. / Оценка вероятности попадания является важной частью процедуры анализа эффективности стрельбы. В статье предложен метод оценки вероятности попадания в статическую стрелковую мишень как функция горизонтальной дальности до мишении (ГДМ), её типа и положения точки прицеливания. Предлагаемая процедура оценки состоит из трех частей. Часть I посвящена двойной интегральной формуле с учётом изменений в местоположении средней точки попадания (СТП) на поверхности мишени в зависимости от ГДМ. Характеристики дисперсии пуль в зависимости от ГДМ определены в части II. Основой части I и части II является двумерное некоррелированное распределение вероятностей Лапласа-Гаусса. Как правило, функции ординат (абсцисс) СТП от аргумента ГДМ и функции ординат (абсцисс) рассеивания пули как функции ГДМ представлены в виде классических полиномов. В части III даётся описание контура мишени. В математическом плане контур цели является кусочной функцией, которая описывает верхний и нижний края стрелковой мишени, а во-вторых, контур цели в формуле двойного интеграла выступает как предел интегральной формулы части I. Все вычисления реализованы в PTC Mathcad. Предлагаемый метод построен на модульной основе и позволяет пользователю изменять типы оружия и виды мишеней. Для демонстрационных расчётов использованы данные характеризующие внешнюю баллистику и характеристики рассеивания 5.45×39 автомата Калашникова. В качестве рассматриваемой модели автомата была выбрана MPi AK-74N. В качестве стрелковых мишеней использовались пять видов швейцарских армейских мишеней, которые имитируют различные части силуэта комбатанта. Для иллюстрации работоспособности предложенного метода оценена вероятность попадания в мишени K, H, G, F, E и для диапазона ГДМ от 50 до 400 м. / U članku je predstavljena metoda za određivanje verovatnoće pogotka stacionarnih meta u zavisnosti od horizontalnog rastojanja do mete (HRM), vrste mete i lokacije tačke cilja. Predloženi postupak sastoji se od tri bloka. Prvi blok se bavi formulom dvostrukog integrala koja uzima u obzir promene u lokaciji tačke pogotka u zavisnosti od HRM. Karakteristike disperzije metka, u zavisnosti od HRM, određuju se u drugom bloku. Bivarijantna nekorelisana Gaus- Laplasova raspodela verovatnoće uzeta je za osnov prva dva bloka. Funkcije lokacije tačke cilja (ordinata/apcisa) u zavisnosti od HRM i funkcije karakteristika disperzije pogodaka u zavisnosti od HRM prikazane su u obliku polinoma. U trećem bloku naveden je opis kontura silueta meta. U matematičkom smislu, kontura mete je hibridna funkcija koja definiše gornje i donje ivice mete i, takođe, predstavlja granicu formule dvostrukog integrala prvog bloka. Svi proračuni su se vršili u PTC Mathcad. Predloženi metod zasnovan je na modularnom principu i omogućava korisniku da menja vrste oružja i meta. Za ilustraciju je izabran prikaz izračunavanja karakteristika disperzije pogotka koji je postignut iz automatske puške 'kalašnjikov', kalibra 5.45×39, model MPi AK-74N. Za mete je izabrano pet tipova švajcarskih vojnih meta koje imitiraju različite delove siluete vojnika. Operabilnost predloženog metoda ilustrovana je određivanjem verovatnoća pogađanja meta K, N, G, F i E na udaljenosti od 50 do 400 m.
A geometric interpretation of single shot hit probability (Phit) is a volume of the 3D space under the surface f(y,z) described by the bivariate normal distribution and bounded from below by the YOZ ...plane with the target’s contour (T-region). The Phit is proposed to be estimated by a method based on the numerical integration of the double integral. The double integral integrand is the 2D normal distribution of the Y, Z system of random variables. The dispersion characteristics and the coordinates of the dispersion center are known in advance.The limits of the first and the second integral are described by the analytic functions which characterize the geometric shape of the T-region. The implementation of the offered method is as follows: the selected shooting target is partitioned into N geometric subregions and then analytic formula(s) for each subregion’s boundaries is/are determined and each double integral is defined. The Phit estimations are produced using a numerical integration in the computer software Mathcad. The results of the calculus of all Phit values (for subregions) are added up (or subtracted) depending on the geometric relationships between the regions. The schema for solving Phit numerically makes it possible to calculate the likelihood for targets with arbitrary geometric shapes and not just for rectangular-shaped silhouettes. For illustrating the operability of the proposed method, the Phit for two kinds of head-type shooting targets has been evaluated. The developed method has been compared with the already existing works. / Вероятность попадания одиночным выстрелом в цель предложено оценивать формулой, основу которой составляет двойной интеграл. Подынтегральная функция описывает двумерное рассеивание системы случайных величин Y, Z с заранее заданными параметрами координат рассеивания и среднеквадратических отклонений по направлениям Y, Z. Пределы интегрирования описывают геометрическую форму стрелковой мишени и являются аналитическими функциями границ мишени. В статье предложен алгоритм решения задачи оценки вероятности попадания, который позволяет вычислять вероятности попадания в мишени произвольной геометрической формы. На первом шаге алгоритма производится разбиение цели на N геометрических подобластей. Далее для каждой из подобластей записывается двойной интеграл и с помощью численного интегрирования получают его количественную оценку. Далее результаты вычислений (вероятности попаданий в подобласти) складываются (вычитаются). С целью численного интегрирования двойного интеграла использована среда математических вычислений Mathcad. Для иллюстрации работоспособности предложенного метода приведены расчёты определения вероятности попадания в головную цель двух видов. / Verovatnoća pogotka cilja jednim hicem predstavlja se geometrijski zapreminom ispod površine f(y,z) koja je opisana bivarijantnom normalnom raspodelom ograničenom konturom cilja (oblast T) ispod ravni YOZ. Predlaže se da se verovatnoća pogotka (Phit) procenjuje metodom zasnovanim na numeričkoj integraciji dvostrukog integrala. Integrand dvostrukog integrala je dvodimenzionalna normalna raspodela sistema slučajnih varijabli Y i Z. Karakteristike rasturanja i koordinate centra rasturanja poznate su unapred. Granice dva integrala opisane su analitičkim funkcijama koje karakterišu geometrijski oblik kontura cilja. Izabrani cilj se prvo deli na N geometrijskih podoblasti, a zatim se za granice svake od njih određuju analitičke formule i piše dvostruki integral. Verovatnoća pogotka procenjuje se numeričkom integracijom u Mathcad-u. Rezultati izračunavanja svih verovatnoća pogotka (svih podoblasti) sabiraju se ili oduzimaju, zavisno od geometrijskih odnosa između podoblasti. Šema numeričkog izračunavanja verovatnoće pogotka omogućava izračunavanje verovatnoće za mete proizvoljnog geometrijskog oblika, a ne samo za pravougaone mete. Da bi se ilustrovala operabilnost predloženog metoda, procenjena je verovatnoća pogotka za dve vrste meta u obliku glave. Predloženi metod upoređen je s rezultatima već postojećih radova.
This book covers Lebesgue integration and its generalizations from Daniell's point of view, modified by the use of seminorms. Integrating functions rather than measuring sets is posited as the main ...purpose of measure theory. From this point of view Lebesgue's integral can be had as a rather straightforward, even simplistic, extension of Riemann's integral; and its aims, definitions, and procedures can be motivated at an elementary level. The notion of measurability, for example, is suggested by Littlewood's observations rather than being conveyed authoritatively through definitions of (sigma)-algebras and good-cut-conditions, the latter of which are hard to justify and thus appear mysterious, even nettlesome, to the beginner. The approach taken provides the additional benefit of cutting the labor in half. The use of seminorms, ubiquitous in modern analysis, speeds things up even further. The book is intended for the reader who has some experience with proofs, a beginning graduate student for example. It might even be useful to the advanced mathematician who is confronted with situations - such as stochastic integration - where the set-measuring approach to integration does not work. --- Reviews This book provides a complete and rapid introduction to Lebesgue integration and its generalizations from Daniell’s point of view, (…) The development is clear and it contains interesting historical notes and motivations, abundant exercises and many supplements. The connection with the historical development of integration theory is also pointed out. - Zentralblatt MATH The material is well motivated and the writing is pleasantly informal. (…) There are numerous exercises, many destined to be used later in the text, and 15 pages of solutions/hints. - Mathematical Reviews.
The Hamilton-Jacobi formalism is used to discuss the path integral quantization of a spinning superparticle model. The equations of motion are obtained as total differential equations in many ...variables. The equations of motion are integrable, and the path integral is obtained as an integration over the canonical phase space coordinates.
The Lagrangian of a system describing the dynamical behaviour of a time-dependent harmonic oscillator is modified and then used to evaluate the Feynman path integral of the oscillator. The path ...integral of the time-dependent oscillator is shown to reduce to the time-independent within certain limits.
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