Dirichlet series solutions of differential equations

In this paper we obtained Dirichlet series solutions, convergent in a half plane, of an ordinary linear differential equation of nth order whose coefficients possess Dirichlet series expansions convergent also in a half plane, by a simple application of Banach's fixed point theorem. We also proved the existence ofn linearly independent solutions of such an equation.     

Continuous Subsonic-Sonic Flows in a Convergent Nozzle

This paper concerns continuous subsonic-sonic potential flows in a two-dimensional convergent nozzle. It is shown that for a given nozzle which is a perturbation of a straight one, a given point on its wall where the curvature is zero, and a given inlet which is a perturbation of an arc centered at the vertex, there exists uniquely a continuous subsonic-sonic flow whose velocity vector is along the normal direction at the inlet and the sonic curve, which satisfies the slip conditions on the nozzle walls and whose sonic curve intersects the upper wall at the given point. Furthermore, the sonic curve of this flow is a free boundary, where the flow is singular in the sense that the speed is only C^1/2 Hölder continuous and the acceleration blows up. The perturbation problem is solved in the potential plane, where the flow is governed by a free boundary problem of a degenerate elliptic equation with two free boundaries and two nonlocal boundary conditions, and the equation is degenerate at one free boundary.     

Scattering by an Inhomogeneous Medium

A method is given for constructing the total field resultingfrom a plane wave scattered by an inhomogeneous medium withcompact support. The problem is reformulated as a Fredholm integralequation of the second kind, whose solutions may be representedas a convergent Neumann series, for small values of the wavenumber. A truncation error estimate of this series is given.To illustrate the method, examples of both a sphere and a prolatespheroid filled with an inhomogeneous medium are considered.In both cases the scattering amplitude is investigated as afunction of the wave number. A comparison is made between themethod presented here for finding the total field and Rorres'method (1970).     

Forced viscosity anisotropy of polymers

For two-dimensional flow the viscosity of an anisotropic medium is determined by two quantities. In relation to polymer melts, one of these corresponds to the viscosity for tension in the direction of orientation and the other to viscosity for shear in planes parallel to the orientation. By solving the problem of the flow of such a medium in a plane convergent channel and making optical-polarization measurements we were able to observe and quantitatively describe this phenomenon.Mekhanika Polimerov, Vol. 1, No. 5, pp. 13–18, 1965     

Birefringence in viscous media subjected to pulsed deformation

A pulsed system for studying forced optical anisotropy, in which a viscous medium flows through a plane convergent channel, is proposed. In this way it is possible to investigate the deformation of the material in pure shear and at high strain rates. The possibility of obtaining characteristic interference patterns, fairly sharp even in the boundary layer, is experimentally demonstrated with reference to the flow of castor oil and a polystyrene solution. The specific possibilities associated with the pulsed operating principle are examined.Mekhanika Polimerov, Vol. 4, No. 1, pp. 164–168, 1968     

Photoelastic investigation of the flow instability of a polystyrene melt

The flow instability of a polystyrene melt in plane convergent channels is investigated by the photoelastic technique. The possible mechanisms leading to loss of stability are analyzed.Institute of Physics, Academy of Sciences of the Belorussian SSR, Minsk. Translated from Mekhanika Polimerov, No. 3, pp. 564–565, May–June, 1969.     

First order limits of sparse graphs: Plane trees and path‐width

Ne?et?il and Ossona de Mendez introduced the notion of first order convergence as an attempt to unify the notions of convergence for sparse and dense graphs. It is known that there exist first order convergent sequences of graphs with no limit modeling (an analytic representation of the limit). On the positive side, every first order convergent sequence of trees or graphs with no long path (graphs with bounded tree‐depth) has a limit modeling. We strengthen these results by showing that every first order convergent sequence of plane trees (trees with embeddings in the plane) and every first order convergent sequence of graphs with bounded path‐width has a limit modeling. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 612–635, 2017     

A mathematical model of Helmholtz type for a parachute profile in the presence of gravity

A model of Helmholtz type for a plane inviscid incompressible and potential fluid flow past a curvilinear obstacle of parachute in the presence of gravity is considered. Assuming that the “attack” (wind) flow is unsteady, it is shown that a bounded cavity zone should occur behind the obstacle. The determination of the fluid flow is reduced to a boundary value problem of Volterra type, for a half plane whose solution is explicitly set up, once the unknown separation (jet) lines are found under some approximation hypotheses.     

An eternal curve flow in centro-affine geometry

In this paper, we investigate the long-term behavior for an invariant plane curve flow, whose evolution process can be expressed as a second-order nonlinear parabolic equation with respect to centro-affine curvature. The forward and backward limits in time are discussed, which shows that a closed convex embedded curve may converge to an ellipse when evolving according to this flow. In addition, we obtain the isoperimetric inequality in centro-affine plane geometry.     

Cutting plane approach for the maximum flow interdiction problem

The maximum flow interdiction is a class of leader–follower optimization problems that seek to identify the set of edges in a network whose interruption minimizes the maximum flow across the network. Particularly, maximum flow interdiction is important in assessing the vulnerability of networks to disruptions. In this paper, the problem is formulated as a bi-level mixed-integer program and an iterative cutting plane algorithm is proposed as a solution methodology. The cutting planes are implemented in a branch-and-cut approach that is computationally effective. Extensive computational results are presented on 336 different instances with varying parameters and with networks of sizes up to 50 nodes, 1200 edge, and 800 commodities. The computational results show that the proposed cutting plane approach has significant computational advantage over the direct solution of the monolithic formulation of the maximum flow interdiction problem for the majority of the tested instances.     

On the use of cuts in reverse convex programs

A cutting plane method, using the idea of Tuy cuts, has been suggested in earlier papers as a possible means of solving reverse convex programs. However, the method is fraught with theoretical and numerical difficulties. Stringent sufficient conditions for convergence inn dimensions are given. However, examples of nonconvergence are given and reasons for this nonconvergence are developed. A result of the discussion is a convergent algorithm which combines the idea of the cutting plane method with vertex enumeration procedures in order to numerically improve upon the edge search procedure of Hillestad.     

A note on problems in 3D boundary layer computations in streamline coordinates

Turbulent boundary layers with convergent and divergent external streamlines over a flat plate in the neighbourhood of a plane of symmetry have been computed using a finite-difference method based on streamline coordinates. While the results for the divergent case are generally satisfactory, error growth has been observed for the convergent flowfield. This is most pronounced near the lateral boundary of the computational domain, but also occurs in the plane of symmetry. As an ad-hoc engineering solution, a modified and more restrictive definition of the domain of dependence is proposed, which eliminates the part of the computational domain where the largest error growth occurs. The observed tendency to instability in the convergent case is confirmed by a simplified stability analysis after von Neumann of the uncoupled governing equations.     

Calculating the total resistance for a tube with a variable cross section

A study has been made on the decisive effect from the exit cone on the total resistance of a variable cross section tube composed of alternating divergent and convergent parts, whose radii vary on a cosine law. Numerical calculations have been performed for steady-state stabilized laminar flow of a viscous incompressible liquid. The frictional and pressure resistance coefficients have been calculated. The contribution from those resistances to the total resistance has been examined.Donetsk. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 120–124, 1990.     

Motion of a nonlinear viscoelastic liquid in cylindrical channels of triangular cross section

The problem of the nonrectilinear steady-state flow of a nonlinear viscoelastic liquid in cylindrical channels is considered. It is established that in channels whose cross sections form an equilateral triangle or an isosceles right-angled triangle there are six transverse currents (eddies) in a plane perpendicular to the main longitudinal flow. In cylinders with cross sections in the form of an arbitrary triangle there may be four or six eddies, depending on the shape of the triangle.     

On the convergence of cutting plane algorithms for a class of nonconvex mathematical programs

We discuss the convergence of cutting plane algorithms for a class of nonconvex programs called the Generalized Lattice Point Problems (GLPP). A set of sufficient conditions which guarantee finite convergence are presented. Although these conditions are usually difficult to enforce in a practical implementation, they do illustrate the various factors that must be involved in a convergent rudimentary cutting plane algorithm. A striking example of nonconvergence (in which no subsequence converges to a feasible solution, even when seemingly strong cutting planes are used), is presented to show the effect of neglecting one such factor. We give an application of our analysis to problems with multiple choice constraints and finally discuss a modification of cutting plane algorithms so as to make finite convergence more readily implementable.     

A robust finite difference scheme for strongly coupled systems of singularly perturbed convection‐diffusion equations

This paper is devoted to developing an Il'in‐Allen‐Southwell (IAS) parameter‐uniform difference scheme on uniform meshes for solving strongly coupled systems of singularly perturbed convection‐diffusion equations whose solutions may display boundary and/or interior layers, where strong coupling means that the solution components in the system are coupled together mainly through their first derivatives. By decomposing the coefficient matrix of convection term into the Jordan canonical form, we first construct an IAS scheme for 1D systems and then extend the scheme to 2D systems by employing an alternating direction technique. The robustness of the developed IAS scheme is illustrated through a series of numerical examples, including the magnetohydrodynamic duct flow problem with a high Hartmann number. Numerical evidence indicates that the IAS scheme appears to be formally second‐order accurate in the sense that it is second‐order convergent when the perturbation parameter ϵ is not too small and when ϵ is sufficiently small, the scheme is first‐order convergent in the discrete maximum norm uniformly in ϵ.     

On low Reynolds number optimization of non-linear partial differential equations for convergent–divergent engulfment: A numerical result

The flow field in a convergent–divergent engulfment along with the installation of infinite cylinders as an obstacle results in non-linear partial differential equations and the scientific computation in this regard remains a challenging task. The present attempt is the numerical motivation in this direction to evaluate the flowing liquid stream in the convergent–divergent channel at a low Reynolds number. From the left wall, the liquid stream move with the parabolic profile and have interaction with the case-wise installation of infinite cylinders in the left vicinity of the convergent–divergent throat. The differential system is constructed for the flow field in the channel and hybrid meshed finite element method is utilized to report the numerical solution. A comparative study is enclosed for the hydrodynamic forces faced by obstructions in the left region of the convergent–divergent throat. The drag coefficient for a triangular cylinder acting as an obstruction is higher than that of a circular hitch. In comparison to both triangular and circular hitches, the square-shaped obstacle suffered the most drag force. Considering drag coefficient one can extend this work to obtain information for the real behavior of the vehicle toward air flow and may conclude findings toward reduction of fuel consumption.     

Sensitivity of convergent matrices and polynomials

A matrix A is said to be convergent if and only if all its characteristic roots have modulus less than unity. When A is real an explicit expression is given for real matrices B such that A + B is also convergent, this expression depending upon the solution of a quadratic matrix equation of Riccati type. If A and A + B are taken to be in companion form, then the result becomes one of convergent polynomials (i.e., polynomials whose roots have modulus less then unity), and is much easier to apply. A generalization is given for the case when A and A + B are complex and have the same number of roots inside and outside a general circle.     

Complex symmetric matrices with strongly stable iterates

We study complex-valued symmetric matrices. A simple expression for the spectral norm of such matrices is obtained, by utilizing a unitarily congruent invariant form. Consequently, we provide a sharp criterion for identifying those symmetric matrices whose spectral norm does not exceed one: such strongly stable matrices are usually sought in connection with convergent difference approximations to partial differential equations. As an example, we apply the derived criterion to conclude the strong stability of a Lax-Wendroff scheme.     

Effective numerical—analytical solution of isoperimetric variational problems of mechanics by an accelerated convergence method

A new numerical-analytical method for solving non-linear variational problems of mechanics is presented. The method enables additional isoperimetric conditions and boundary conditions of different types to be taken into account. Unlike existing approaches, the method is based on the use of residuals of the unknown functions corresponding to abscissae at which they satisfy the required conditions. The method has a clear geometrical interpretation and provides more confident idea as to the convergence of the iteration algorithm which is quadratic in nature. The algorithm constructed is used to compute single-mode and multi-mode viscous incompressible flows in a plane convergent channel (Jeffrey-Hamel flow) for a broad range of governing parameters (the aperture angle and Reynolds number).