Calculating a function of a matrix with a real spectrum

Let T be a square matrix with a real spectrum, and let f be an analytic function. The problem of the approximate calculation of f(T) is discussed. Applying the Schur triangular decomposition and the reordering, one can assume that T is triangular and its diagonal entries t_ii are arranged in increasing order. To avoid calculations using the differences t_ii ? t_jj with close (including equal) t_ii and t_jj, it is proposed to represent T in a block form and calculate the two main block diagonals using interpolating polynomials. The rest of the f(T) entries can be calculated using the Parlett recurrence algorithm. It is also proposed to perform some scalar operations (such as the building of interpolating polynomials) with an enlarged number of significant decimal digits.

     

The Witt Ring of a Smooth Projective Curve Over a Finite Field

In this paper, we calculate the Witt ring W(C) of a smooth geometrically connected projective curve C over a finite field with characteristic other than 2. We view W(C) as a subring of W(k(C)) where k(C) is the function field of C. The calculation is then completed using classical results for bilinear spaces over fields.     

Rectangling a rectangle

We show that the following are equivalent: (i) A rectangle of eccentricityv can be tiled using rectangles of eccentricityu. (ii) There is a rational function with rational coefficients,Q(z), such thatv =Q(u) andQ maps each of the half-planes {z | Re(z) < 0} and {z | Re(z) > 0 into itself, (iii) There is an odd rational function with rational coefficients,Q(z), such thatv = Q(u) and all roots ofv = Q(z) have a positive real part. All rectangles in this article have sides parallel to the coordinate axes and all tilings are finite. We letR(x, y) denote a rectangle with basex and heighty. In 1903 Dehn [1 ] proved his famous result thatR(x, y) can be tiled by squares if and only ify/x is a rational number. Dehn actually proved the following result. (See [4] for a generalization to tilings using triangles.) The first and third authors were partially supported by NSF.     

Motion of a container as a nonholonomous system in a curved section of a horizontal pipeline

The problem of the motion of a container in a curved section of a horizontal pipeline is solved using second-order Lagrange equations in the presence of nonholonous couplings. The special case of the motion of a container in a circular curve is examined.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 90–95, 1987.     

Cutting a Convex Polyhedron Out of a Sphere

Given a convex polyhedron P of n vertices inside a sphere Q, we give an O(n ³)-time algorithm that cuts P out of Q by using guillotine cuts and has cutting cost O(log² n) times the optimal.     

Every Simplicial Polytope with at Most d + 4 Vertices Is a Quotient of a Neighborly Polytope

We show that every simplicial d-polytope with d+4 vertices is a quotient of a neighborly (2d+4)-polytope with 2d+8 vertices, using the technique of affine Gale diagrams. The result is extended to matroid polytopes. Received September 27, 1995.     

Static computation of a laminated nonuniform inclined shell in a temperature field

The finite difference method is used to obtain a solution of a nonlinear static problem for a laminated inclined rectangular shell in a plane acted on by a force load and a temperature field. The approximating system of nonlinear equations is obtained using an approximation of the equation of variations or systems of differential equations.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 86–89.     

The stationary problem of thermoelasticity for a plate with a heated slit

We consider the stationary heat conduction and thermoelasticity problem for a plate that whose lateral surfaces are thermally insulated and which has a slit on whose edges a constant temperature is maintained. The heat conduction problem is solved using a modified logarithmic single-layer potential by reducing it to integral equations.Translated fromMatematicheskie Melody i Fiziko-Mekhanicheskie Polya, Issue 31, 1990, pp. 50–54.     

On a boundary value problem for a p‐Dirac equation

The p‐Laplace equation is a nonlinear generalization of the Laplace equation. This generalization is often used as a model problem for special types of nonlinearities. The p‐Laplace equation can be seen as a bridge between very general nonlinear equations and the linear Laplace equation. The aim of this paper is to solve the p‐Laplace equation for 1 < p < 2 and to find strong solutions. The idea is to apply a hypercomplex integral operator and spatial function theoretic methods to transform the p‐Laplace equation into the p‐Dirac equation. This equation will be solved iteratively by using a fixed‐point theorem. Applying operator‐theoretical methods for the p‐Dirac equation and p‐Laplace equation, the existence and uniqueness of solutions in certain Sobolev spaces will be proved. Copyright © 2016 John Wiley & Sons, Ltd.     

An error estimate for a finite volume scheme for a diffusion–convection problem on a triangular mesh

We study here a finite volume scheme for a diffusion-convection equation on an open bounded set Ω of ?², using a triangular mesh for the discretization of Ω. The 4-point numerical scheme is presented along with the geometrical assumptions on the mesh. An error estimate of order h on the discrete L² norm is obtained, where h denotes the “size” of the mesh. The proof uses an estimate of order h of the consistency error on the fluxes and an estimate of the number of edges of the mesh between one given triangle and the boundary Ω. © 1995 John Wiley & Sons, Inc.     

Percolation in a dependent random environment

Draw planes in ℝ³ that are orthogonal to the x axis, and intersect the x axis at the points of a Poisson process with intensity λ; similarly, draw planes orthogonal to the y and z axes using independent Poisson processes (with the same intensity). Taken together, these planes naturally define a randomly stretched rectangular lattice. Consider bond percolation on this lattice where each edge of length 𝓁 is open with probability e^−𝓁, and these events are independent given the edge lengths. We show that this model exhibits a phase transition: for large enough λ there is an infinite open cluster a.s., and for small λ all open clusters are finite a.s. We prove this result using the method of paths with exponential intersection tails, which is not applicable in two dimensions. The question whether the analogous process in the plane exhibits a phase transition is open. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 16, 333–343, 2000     

On a Class of Traceability Codes

Traceability codes are designed to be used in schemes that protect copyrighted digital data against piracy. The main aim of this paper is to give an answer to a Staddon–Stinson–Wei's problem of the existence of traceability codes with q< w ² and b>q. We provide a large class of these codes constructed by using a new general construction method for q-ary codes.     

Approximate solution methods in mechanics analysis of a class of nonlinear seepage problems by a finite-element method

We analyze the nonlinear boundary-value problem of seepage under a subsurface hydrotechnical construction over an inclined rectilinear aquifer. The method of inverse boundary-value problems is applied, using the velocity hodograph plane in which the original problem is reduced to a linear problem. The linear problem is solved in the general case using the finite-element method. A computer program realizing the proposed algorithms has been developed. We have used this program to run a series of numerical experiments, reaching certain conclusions about the behavior of the main seepage characteristics.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 55, pp. 75–80, 1985.     

Optimally Decomposing Coverings with Translates of a Convex Polygon

We show that any k-fold covering using translates of an arbitrary convex polygon can be decomposed into Ω(k) covers. Such a decomposition can be computed using an efficient (polynomial-time) algorithm.     

Free rotation of a circular ring about a diameter

A circular ring rotates about a diameter in space. Deformations of the ring depend on a non-dimensional parameterJ which represents the relative importance of centrifugal forces to flexural rigidity. The solutions are found by three methods: series expansion for smallJ, matched asymptotic expansions for largeJ and exact numerical solution using both quasi-Newton and homotopy methods.

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Zusammenfassung Ein Kreisring rotiert um einen Durchmesser im Raum. Verformungen des Ringes hängen vom dimensionslosen ParameterJ ab, der die relative Bedeutung der zentrifugalen Kräfte im Verhältnis zur Biegesteifigkeit angibt. Die Lösungen wurden durch drei Methoden gefunden: Reihenentwicklung für kleinesJ, asymptotische Expansion für großesJ, und exakte numerische Integration sowohl mit einer quasi-Newtonschen als auch mit einer homotopischen Methode.

     

On a conjecture of Furstenberg

The Hausdorff dimension of the set of numbers which can be written using digits 0, 1,t in base 3 is estimated. For everyt irrational a lower bound 0.767 … is found. Support from NSF grant DMS-0072312 and Penn State in the form of a sabbatical leave is acknowledged. This work was partly done when the author was visiting the Mathematical Institute of the Polish Academy of Sciences in Warsaw.     

A Pedestrian Approach to Ray Shooting: Shoot a Ray, Take a Walk

We propose a very simple ray-shooting algorithm, whose only data structure is a triangulation. The query algorithm, after locating the triangle containing the origin of the ray, walks along the ray, advancing from one triangle to a neighboring one until the polygon boundary is reached. The key result of the paper is a Steiner triangulation of a simple polygon with the property that a ray can intersect at most O(log n) triangles before reaching the polygon boundary. We are able to compute such a triangulation in linear sequential time, or in O(log n) parallel time using O(n/log n) processors. This gives a simple, yet optimal, ray-shooting algorithm for a simple polygon. Using a well-known technique, we can extend our triangulation procedure to a multiconnected polygon with k components and n vertices, so that a ray intersects at most O(√k log n) triangles.     

An a priori error estimate for the finite element modelling of electromagnetic waves interacting with a periodic diffraction grating

An a priori error estimate using a so called α,β‐ periodic transformation to study electromagnetic waves in a periodic diffraction grating is derived. It has been reported for single scattering that there is an instability in numerical methods for high wavenumbers. To address this problem, the analytical solution of the scattering problem when the domain is scatterer free and an unknown function called the α,β‐quasi periodic solution are used to transform the associated Helmholtz problem. The well‐posedness of the resulting continuous problem is analysed before approximating its solution using a finite element discretization. To guarantee the uniqueness of this approximate solution, an a priori error estimate is derived. Finally, numerical results are presented that suggest that the α,β‐quasi periodic method converges at a far lower number of degrees of freedom than the α,0‐quasi periodic method reported previously; especially for high wavenumbers. This is particularly true when the incident wave only undergoes a small perturbation because of the presence of the scatterer. Copyright © 2012 John Wiley & Sons, Ltd.     

Heat transfer in a peristaltic flow of MHD fluid with partial slip

In the present note, we have discussed the effects of partial slip on the peristaltic flow of a MHD Newtonian fluid in an asymmetric channel. The governing equations of motion and energy are simplified using a long wave length approximation. A closed form solution of the momentum equation is obtained by Adomian decomposition method and an exact solution of the energy equation is presented in the presence of viscous dissipation term. The expression for pressure rise is calculated using numerical integration. The trapping phenomena is also discussed. The graphical results are presented to interpret various physical parameter of interest. It is found that the temperature field decreases with the increase in slip parameter L, and magnetic field M, while with the increase in P_r and E_c, the temperature field increases.