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We demonstrate in this paper a combination of the Schwarz–Christoffel mapping and Muskhelishvili's approach with fractional function series in solving the elastic fields of a cracked half-plane, and zoom in on two typical problems, a double branched crack with two rays emanating from one point on the edge and two edge cracks spaced by a certain distance. Typical loading conditions are considered, including far-field uniform tensile stress and concentrated loads along either the tangential or the normal direction of the free surface. We supply a semi-analytic solution to those boundary-value problems in the cracked half-plane, and validate the theory by comparing the theoretical results in terms of stress fields, stress intensity factors (SIFs) and crack opening displacement (COD) with those from finite-element simulations. The theoretical approach shows how two edge cracks may shield the stress intensity factors of each other in a quantitative manner. For the typical Kalthoff–Winkler cracks of length a and being spaced by a distance d , their SIFs K I decay with decreasing d , and K I = K I 0 − K I 1 [ 1 − exp (− a / d) ]. It converges to K I 0 —the SIF of a single edge crack when d approaches to infinity. Those observations and the theory approach itself provide a general way to analyze the mechanical consequence of edge cracks in engineering practice. • Solutions to the elasticity of a double branched crack and two edge cracks are given. • For Kalthoff-Winkler cracks, SIFs exponentially decay with their decreasing distance. • The theory is validated by comparing the theoretical results with FE simulations. [ABSTRACT FROM AUTHOR]

We develop a theoretical framework for studying numerical estimation of lower previsions, generally applicable to two-level Monte Carlo methods, importance sampling methods, and a wide range of other sampling methods one might devise. We link consistency of these estimators to Glivenko–Cantelli classes, and for the sub-Gaussian case we show how the correlation structure of this process can be used to bound the bias and prove consistency. We also propose a new upper estimator, which can be used along with the standard lower estimator, in order to provide a simple confidence interval. As a case study of this framework, we then discuss how importance sampling can be exploited to provide accurate numerical estimates of lower previsions. We propose an iterative importance sampling method to drastically improve the performance of imprecise importance sampling. We demonstrate our results on the imprecise Dirichlet model. [ABSTRACT FROM AUTHOR]

In this work, we investigate numerically a model governed by a multidimensional nonlinear wave equation with damping and fractional diffusion. The governing partial differential equation considers the presence of Riesz space-fractional derivatives of orders in (1, 2], and homogeneous Dirichlet boundary data are imposed on a closed and bounded spatial domain. The model under investigation possesses an energy function which is preserved in the undamped regime. In the damped case, we establish the property of energy dissipation of the model using arguments from functional analysis. Motivated by these results, we propose an explicit finite-difference discretization of our fractional model based on the use of fractional centered differences. Associated to our discrete model, we also propose discretizations of the energy quantities. We establish that the discrete energy is conserved in the undamped regime, and that it dissipates in the damped scenario. Among the most important numerical features of our scheme, we show that the method has a consistency of second order, that it is stable and that it has a quadratic order of convergence. Some one- and two-dimensional simulations are shown in this work to illustrate the fact that the technique is capable of preserving the discrete energy in the undamped regime. For the sake of convenience, we provide a Matlab implementation of our method for the one-dimensional scenario. [ABSTRACT FROM AUTHOR]

We analyze a natural parity problem on the divisors associated to Sturmian words. We find a clear bias towards the odd divisors and prove a sharp asymptotic estimate for the average of the difference odd-even function tamed by a weight function, which confirms various experimental results. [ABSTRACT FROM AUTHOR]

Abstract: In the present paper, an exact solution for the two-dimensional boundary layer viscous flow over a semi-infinite flat plate in the presence of magnetic field is given. Generalized similarity transformations are used to convert the governing boundary layer equations into a third order nonlinear differential equation which is the famous MHD Falkner–Skan equation. This equation contains three flow parameters: the stream-wise pressure gradient (), the magnetic parameter (M), and the boundary stretch parameter (). Closed-form analytical solution is obtained for and in terms of error and exponential functions which is modified to obtain an exact solution for general values of and M. We also obtain asymptotic analyses of the MHD Falkner–Skan equation in the limit of large and . The results obtained are compared with the direct numerical solution of the full boundary layer equation, and found that results are remarkably in good agreement between the solutions. The derived quantities such as velocity profiles and skin friction coefficient are presented. The physical significance of the flow parameters are also discussed in detail. [Copyright &y& Elsevier]

Abstract: We describe a simple, yet efficient, procedure for approximating the Lévy measure of a random variable. We use this approximation to derive a finite sum-representation that converges almost surely to Ferguson’s representation of the Dirichlet process. This approximation is written based on arrivals of a homogeneous Poisson process. We compare the efficiency of our approximation to several other well-known approximations of the Dirichlet process and demonstrate a significant improvement. [Copyright &y& Elsevier]

The representative unit cell (RUC) model of elastic heterogeneous solid with imperfectly bonded ellipsoidal inhomogeneities is developed. The periodic displacement solution has been obtained by combining the superposition principle, Papkovich-Neuber representation of displacement in terms of scalar harmonic potentials and expansion of these potentials in terms of periodic ellipsoidal harmonics. By accurate fulfilling the interface conditions, the RUC model boundary value problem is reduced to an infinite system of linear algebraic equations for the series expansion coefficients. The modified Rayleigh homogenization scheme has been extended to the elastic composite with ellipsoidal inhomogeneities and imperfect interface. The scheme takes into account volume content and elastic moduli of constituents, shape, size and arrangement of inhomogeneities, interaction between them and elastic stiffness of interface. Numerical algorithm of the method provides an accurate analysis of the RUC model for a whole range of the structure parameters. The reported numerical data illustrate convergence rate of the series solution and discover an effect of the volume content, elastic properties and arrangement of particles and interface stiffness on the stress field and macroscopic elastic moduli of the ellipsoidal particle composite. [ABSTRACT FROM AUTHOR]

In this brief note an integral expression is presented for the COM–Poisson renormalization constant Z ( λ , ν ) on the real axis. [ABSTRACT FROM AUTHOR]

The article discusses the theorem on the relations of sines and cosines functions of angles in arithmetic progression. It says that the relations follow more general formulas which can be proven without the use of complex numbers. In the journal "Arbelos," Samuel Greitzer used the Fourier series and the Dirichlet kernel in providing proof for this theorem.