Asymptotic Approximation of the Apostol-Tangent Polynomials Using Fourier Series

Asymptotic approximations of the Apostol-tangent numbers and polynomials were established for non-zero complex values of the parameter λ. Fourier expansion of the Apostol-tangent polynomials was used to obtain the asymptotic approximations. The asymptotic formulas for the cases λ=1 and λ=−1 were explicitly considered to obtain asymptotic approximations of the corresponding tangent numbers and polynomials.

Approximations of Tangent Polynomials, Tangent –Bernoulli and Tangent – Genocchi Polynomials in terms of Hyperbolic Functions

Asymptotic approximations of Tangent polynomials, Tangent-Bernoulli, and Tangent-Genocchi polynomials are derived using saddle point method and the approximations are expressed in terms of hyperbolic functions. For each polynomial there are two approximations derived with one having enlarged region of validity.

Generalisation of the Frobenius Formula in the Theory of Block Operators on Normed Spaces

The efficient construction and employment of block operators are vital for contemporary computing, playing an essential role in various applications. In this paper, we prove a generalisation of the Frobenius formula in the setting of the theory of block operators on normed spaces. A system of linear equations with the block operator acting in Banach spaces is considered. Existence theorems are proved, and asymptotic approximations of solutions in regular and irregular cases are constructed. In the latter case, the solution is constructed in the form of a Laurent series. The theoretical approach is illustrated with an example, the construction of solutions for a block equation leading to a method of solving some linear integrodifferential system.

Asymptotic formulae for estimating statistical significance in particle physics analyses

Within the framework of likelihood-based statistical tests for particle physics measurements, we derive expressions for estimating the statistical significance of discovery using the asymptotic approximations of Wilks and Wald for four measurement models. These models include arbitrary numbers of signal regions, control regions, and Gaussian constraints. We extend our expressions to use the representative or "Asimov" dataset proposed by Cowan et al. such that they are made data-free. While many of the expressions are complicated and often involve solving systems of coupled, multivariate equations, we show these expressions reduce to closed-form results under simplifying assumptions. We also validate the predicted significances using toy-based data in select cases and show the asymptotic formulae to be more computationally efficient than the toy-based approach. Additionally, different parameters within each measurement model are varied in order to understand their effect on the predicted significance.

Analytical Calculation of Deformations and Kinematic Analysis of a Flat Truss with an Arbitrary Number of Panels

Постановка задачи. Разыскиваются аналитические зависимости прогиба и смещения опоры плоской фермы решетчатого вида от числа панелей. Ферма имеет сдвоенную решетку, прямолинейный нижний и приподнятый в средней части верхний пояс. Результаты. Для двух видов нагружения по формуле Максвелла-Мора получены аналитические зависимости прогибов конструкции от нагрузки, размеров и числа панелей. Для обобщения серии частных решений с различным числом панелей ферм на произвольный случай использован метод индукции и аналитические возможности системы компьютерной математики Maple. Для некоторых решений получены асимптотические приближения. Показано распределение усилий в элементах фермы. Выводы. Полученные формулы могут быть использованы в задачах оптимизации и как тестовые для оценки приближенных численных решений. Выявлены случаи геометрической изменяемости фермы при числе панелей, кратном трем. Приведен алгоритм выявления соответствующего распределения возможных скоростей шарниров. Statement of the problem. Analytical dependences of the deflection and displacement of the support of a flat lattice truss on the number of panels are being sought. The truss has a double lattice, a rectilinear lower belt and an upper belt raised in the middle part. Results. For two types of loading, according to the Maxwell-Mohr formula, analytical dependences of the deflections of the structure on the load, dimensions and number of panels are obtained. To generalize a series of particular solutions for trusses with different numbers of panels for an arbitrary case, the induction method and the analytical capabilities of the Maple computer mathematics system were used. For some solutions, asymptotic approximations are obtained. The distribution of forces in the rods of the structure is shown. Conclusions. The obtained formulas can be used in optimization problems and as test ones for evaluating approximate numerical solutions. Cases of geometric variability of the truss with the number of panels being a multiple of three are revealed. An algorithm for identifying the corresponding distribution of possible velocities of the joints is presented.

Pandemic-type failures in multivariate Brownian risk models

Modelling of multiple simultaneous failures in insurance, finance and other areas of applied probability is important especially from the point of view of pandemic-type events. A benchmark limiting model for the analysis of multiple failures is the classical d-dimensional Brownian risk model (Brm), see Delsing et al. (Methodol. Comput. Appl. Probab. 22(3), 927–948 2020). From both theoretical and practical point of view, of interest is the calculation of the probability of multiple simultaneous failures in a given time horizon. The main findings of this contribution concern the approximation of the probability that at least k out of d components of Brm fail simultaneously. We derive both sharp bounds and asymptotic approximations of the probability of interest for the finite and the infinite time horizon. Our results extend previous findings of Dȩbicki et al. (J. Appl. Probab. 57(2), 597–612 2020) and Dȩbicki et al. (Stoch. Proc. Appl. 128(12), 4171–4206 2018).

Asymptotic Approximations of Apostol-Genocchi Numbers and Polynomials

Asymptotic approximations of the Apostol-Genocchi numbers andpolynomials are derived using Fourier series and ordering of poles ofthe generating function. Asymptotic formulas for the Apostol-Eulernumbers and polynomials are obtained as consequence. Asymptoticformulas for special cases which include the Genocchi numbers andpolynomials are also explicitly stated.

A Bayesian solution to the Behrens–Fisher problem

A simple solution to the Behrens–Fisher problem based on Bayes factors is presented, and its relation with the Behrens–Fisher distribution is explored. The construction of the Bayes factor is based on a simple hierarchical model, and has a closed form based on the densities of general Behrens–Fisher distributions. Simple asymptotic approximations of the Bayes factor, which are functions of the Kullback–Leibler divergence between normal distributions, are given, and it is also proved to be consistent. Some examples and comparisons are also presented.