Analytical two-center integrals over Slater geminal functions

Let P = P(z,d/dz) be a linear ordinary differential operator with holomorphic coefficients in a neighbourgood X of the origin 0 in ℂ. Let Pu = f be an inhomogeneous differential equation in the space O {0} or in O {0}, where 0 {0} is the ring of convergent power series and O {0} is the ring of formal power series at the origin 0.

This paper shows that the method of ‘reduction of order’ is both a simpler way of teaching and of obtaining solutions for inhomogeneous linear ordinary differential equations than is the normally espoused method of ‘variation of parameters’. Details are given of the results of the method for the general nth order equation and the two methods are compared for n = 2, 3. It will be noted that the proposed method takes a particularly simple form when the right‐hand side of the equation is a multiple of one of the solutions to the related homogeneous equation.

Solutions u(x) to the class of inhomogeneous nonlinear ordinary differential equations taking the form for parameter α are studied. The problem is defined on the x line with decay of both the solution u(x) and the imposed forcing f(x) as |x| → ∞. The rate of decay of f(x) is important and has a strong influence on the structure of the solution space. Three particular forcings are examined primarily: a rectilinear top-hat, a Gaussian, and a Lorentzian, the latter two exhibiting exponential and algebraic decay, respectively, for large x. The problem for the top hat can be solved exactly, but for the Gaussian and the Lorentzian it must be computed numerically in general. Calculations suggest that an infinite number of solution branches exist in each case. For the top-hat and the Gaussian the solution branches terminate at a discrete set of α values starting from zero. A general asymptotic description of the solutions near to a termination point is constructed that also provides information on the existence of local fold behaviour. The solution branches for the Lorentzian forcing do not terminate in general. For large α the asymptotic analysis of Keeler, Binder and Blyth (2018 J. Fluid Mech. 832 73–96) is extended to describe the behaviour on any given solution branch using a method for glueing homoclinic connections.

We compute the two-loop master integrals for leading-color QCD scattering amplitudes including a closed light-quark loop in tt¯H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ t\\overline{t}H $$\\end{document} production at hadron colliders. Exploiting numerical evaluations in modular arithmetic, we construct a basis of master integrals satisfying a system of differential equations in ϵ-factorized form. We present the analytic form of the differential equations in terms of a minimal set of differential one-forms. We explore properties of the function space of analytic solutions to the differential equations in terms of iterative integrals which can be exploited for studying the analytic form of related scattering amplitudes. Finally, we solve the differential equations using generalized series expansions to numerically evaluate the master integrals in physical phase space. As the first computation of a set of two-loop seven-scale master integrals, our results provide valuable input for analytic studies of scattering amplitudes in processes involving massive particles and a large number of kinematic scales.

A numerical routine for the crossed vertex diagram with a massive-particle loop

A formulation is developed to predict the vibration response of a finite length, submerged plate due to a line drive. The formulation starts by describing the fluid in terms of elliptic cylinder coordinates, which allows the fluid loading term to be expressed in terms of Mathieu functions. By moving the fluid loading term to the right-hand side of the equation, it is considered to be a force. The operator that remains on the left-hand side is the same as that of the in vacuo plate: a fourth-order, constant coefficient, ordinary differential equation. Therefore, the problem appears to be an inhomogeneous ordinary differential equation. The solution that results has the same form as that of the in vacuo plate: the sum of a forced solution, and four homogeneous solutions, each of which is multiplied by an arbitrary constant. These constants are then chosen to satisfy the structural boundary conditions on the two ends of the plate. Results for the finite plate are compared to the infinite plate in both the wave number and spatial domains. The theoretical predictions of the plate velocity response are also compared to results from finite element analysis and show reasonable agreement over a large frequency range.

The two loop crossed ladder vertex diagram with two massive exchanges

Damaged mosaic laminate model of woven fabric composites with transverse yarn cracking and interface debonding

Extension of an exponential discretization scheme to multidimensional convection–diffusion problems

We consider the optimal portfolio problem of a power investor who wishes to allocate her wealth between several credit default swaps (CDSs) and a money market account. We model contagion risk among the reference entities in the portfolio using a reduced‐form Markovian model with interacting default intensities. Using the dynamic programming principle, we establish a lattice dependence structure between the Hamilton‐Jacobi‐Bellman equations associated with the default states of the portfolio. We show existence and uniqueness of a classical solution to each equation and characterize them in terms of solutions to inhomogeneous Bernoulli type ordinary differential equations. We provide a precise characterization for the directionality of the CDS investment strategy and perform a numerical analysis to assess the impact of default contagion. We find that the increased intensity triggered by default of a very risky entity strongly impacts size and directionality of the investor strategy. Such findings outline the key role played by default contagion when investing in portfolios subject to multiple sources of default risk.

Price impact of a trade is an important element in pre-trade and post-trade analyses. We introduce a framework to analyze the market price of liquidity risk, which allows us to derive an inhomogeneous Bernoulli ordinary differential equation. We obtain two closed form solutions, one of which reproduces the linear function of the order flow in Kyle (1985) for informed traders. However, when traders are not as asymmetrically informed, an S-shape function of the order flow is obtained. We perform an empirical intra-day analysis on Nikkei futures to quantify the price impact of order flow and compare our results with industry’s heuristic price impact functions. Our model of order flow yields a rich framework to not only estimate the liquidity risk parameters, but also to provide a plausible cause of why volatility and correlation are stochastic in nature. Finally, we find that the market depth encapsulates the market price of liquidity risk.

This paper reports a method to solve ordinary fourth-order differential equations in the form of ordinary power series and, for the case of regular special points, in the form of generalized power series. An algorithm has been constructed and a program has been developed in the MAPLE environment (Waterloo, Ontario, Canada) in order to solve the fourth-order differential equations. All types of solutions depending on the roots of the governing equation have been considered. The examples of solutions to the fourth-order differential equations are given; they have been compared with the results available in the literature that demonstrate excellent agreement with the calculations reported here, which confirms the effectiveness of the developed programs. A special feature of this work is that the accuracy of the results is controlled by the number of terms in the power series and the number of symbols (up to 20) in decimal mantissa in numerical calculations. Therefore, almost any accuracy allowed for a given electronic computing machine or computer is achievable. The proposed symbolic-numerical method and the work program could be successfully used for solving eigenvalue problems, in which controlled accuracy is very important as the eigenfunctions are extremely (exponentially) sensitive to the accuracy of eigenvalues found. The developed algorithm could be implemented in other known computer algebra packages such as REDUCE (Santa Monica, CA), MATHEMATICA (USA), MAXIMA (USA), and others. The program for solving ordinary fourth-order differential equations could be used to construct Green’s functions of boundary problems, to solve differential equations with private derivatives, a system of Hamilton’s differential equations, and other problems related to mathematical physics.

In this paper a numerical-analytical study on a stress-strain state of a steel pipeline section with a repair fiberglass viscoelastic layer is described. The layer is considered long enough that allows to solve the problem in a flat axisymmetric statement. A mathematical model of viscoelastic behavior of a fiber-glass is proposed, moreover this model allows to simulate an order of anisotropy of viscoelastic proper-ties that exceeds an order of anisotropy of elastic ones. To mirror viscoelastic behavior of the construc-tion more correctly an orthotropical viscoelasticity is used in the mathematical model. As a result of transformation of the original system of equations of elasticity theory it becomes necessary to solve the inhomogeneous integral equation. In order to solve it quadrature formulas are used. With help of them an iteration process in time was built where an inhomogeneous differential equation with viscoelastic prehistory in right side was solved. For determining values of integration constants of differential equa-tions boundary conditions are rearranged using quadrature formulas. For the case of definite values of physical and geometrical parameters of the construction the right rectangle formula was chosen. To evaluate an error of the mathematical model a discrepancy between the analytical results form previous research and the results according to the given model for the simplified isotropic set of viscoelastic parameters was estimated. It is shown that for 500 iterations the discrepancy is less than 0.1% that al-lows to use this values for further calculations. Calculation results for the case of orthotropic viscolas-ticity are presented as displacement and stress distributions versus time curves. Also a comparison be-tween this results and the stress strain state of the construction in case of isotropic viscoelasticity was made. It is shown that even for a slight deviation of viscoelastic model from isotropic to orthotropic a change of the main parameters of the stress strain state is approximately 20% considering their time varying parts. Based on this fact a conclusion about a necessity of using the analytical model of orthtropic viscoelasticity in calculation or, at least, evaluation of the stress strain state of constructions with viscoelastic parts was made. Also built numerical-analytical model allows to disseminate the re-sults on cases of more complex geometrical and physical conditions without changing in a main idea of the method.

In this paper we consider periodic functional differential and integral equations. Massera proved that the existence of a bounded solution implies the existence of a periodic solution for linear (inhomogeneous) ordinary differential equations. Chow proved a similar result for linear functional differential equations with finite delay. We give a new proof for Chow's result that works for linear functional differential equations with infinite delay and also for integral equations. We also give examples that do not satisfy uniform boundedness or uniform ultimate boundedness, the conditions frequently used to prove that there is a periodic solution.

The unrestrained torsional analysis of bars is an important theme in elasticity theory, first solved by Saint-Venant using semi-inverse methods. It has been considered and solved by several others using analytical methods and numerical procedures due to the importance in the design of machine parts under torsional moments. In this paper, the Saint Venant torsion problem is solved for rectangular prismatic bars using Galerkin-Kantorovich variational method (GKVM). The work presents a detailed theoretical framework of the problem, deriving using first principles considerations the stress compatibility equation in terms of the Prandtl stress function ϕ(x,y). The derived domain equation which is required to be satisfied over the rectangular cross-sectional domain is a partial differential equation of the Poisson type. GKVM is adopted as the solution method for finding the solution to the domain equation. The unknown Prandtl stress function ϕ(x,y) is assumed, following Kantorovich method to be a product of an unknown function for fx sought to minimize the Galerkin-Kantorovich variational functional (integral) (GKVF) and a known function (y2-b2) which satisfies the boundary conditions at all boundary points in the y-direction, that is, at y=±b. The resulting GKVF is a simplified functional whose integral is a second order inhomogeneous ordinary differential equation (ODE) in fx. The integrand is solved to find fx leading in a full determination of the Prandtl stress function. The expression for stresses, torsional moments and torsional parameters are then found and they satisfy the boundary conditions and the domain equation. The results for the torsional moments and torsional parameters are identical to previous results obtained using double finite sine transform method (DFSTM), and analytical methods. The merit of GKVM is that it has led to the exact solution of the unrestrained torsion problems.