Computation of the scattering amplitude for a scattering wave produced by a disc — Approach by a fundamental solution method

Our fundamental solution method gives an analytic representation of the approximate solution for the reduced wave problem in the exterior region of a disc. The asymptotic behavior of this representation yields an approximate formula for the scattering amplitude. An error estimate for this formula is given. We add two numerical tests: the numerical estimate of errors; and profiles of scattering cross sections and the far-field coefficient. Both tests include cases of high wave numbers.     

Approximation and representation of the value for some differential games with asymmetric information

We consider differential games with incomplete information. For special games with dynamics independent of the state of the system and linear payoffs, we give a representation formula for the value similar to the value of repeated games with lack of information on both sides. For general games, this representation formula does not hold and we introduce an approximation of the value: we build a sequence of functions converging to the value function.     

A Cauchy kernel for the Hermitian submonogenic system

Hermitian monogenic functions are the null solutions of two complex Dirac type operators. The system of these complex Dirac operators is overdetermined and may be reduced to constraints for the Cauchy datum together with what we called the Hermitian submonogenic system (see [8], [9]). This last system is no longer overdetermined and it has properties that are similar to those of the standard Dirac operator in Euclidean space, such as a Cauchy–Kowalevski extension theorem and Vekua type solutions. In this paper, we investigate plane wave solutions of the Hermitian submonogenic system, leading to the construction of a Cauchy kernel. We also establish a Stokes type formula that, when applied to the Cauchy kernel provides an integral representation formula for Hermitian submonogenic functions.     

Two-stream counter-flow heat exchanger equation with time-varying velocities

We study the two-stream counter-flow heat exchanger equation with time-varying fluid velocities. Formulating it into a time-varying boundary control system, a representation formula of the solution is obtained.     

Diagonalization of the system of static Lamé equations of isotropic linear elasticity

We find a simplest representation for the general solution to the system of the static Lamé equations of isotropic linear elasticity in the form of a linear combination of the first derivatives of three functions that satisfy three independent harmonic equations. The representation depends on 12 free parameters choosing which it is possible to obtain various representations of the general solution and simplify the boundary value conditions for the solution of boundary value problems as well as the representation of the general solution for dynamic Lamé equations. The system of Lamé equations diagonalizes; i.e., it is reduced to the solution of three independent harmonic equations. The representation implies three conservation laws and some formula for producing new solutions which makes it possible, given a solution, to find new solutions to the static Lamé equations by derivations. In the two-dimensional case of a plane deformation, the so-found solution immediately implies the Kolosov-Muskhelishvili representation for shifts by means of two analytic functions of complex variable. Two examples are given of applications of the proposed method of diagonalization of the two-dimensional elliptic systems.     

(Shifted) Macdonald polynomials: q-Integral representation and combinatorial formula

We extend some results about shifted Schur functions to the general context of shifted Macdonald polynomials. We strengthen some theorems of F. Knop and S. Sahi and give two explicit formulas for these polynomials: a q-integral representation and a combinatorial formula. Our main tool is a q-integral representation for ordinary Macdonald polynomial. We also discuss duality for shifted Macdonald polynomials and Jack degeneration of these polynomials.     

Representation formula for discrete indefinite affine spheres

We present a representation formula for discrete indefinite affine spheres via loop group factorizations. This formula is derived from the Birkhoff decomposition of loop groups associated with discrete indefinite affine spheres. In particular we show that a discrete indefinite improper affine sphere can be constructed from two discrete plane curves.     

INTEGRABLE SYSTEM AND SPACELIKE SURFACES WITH PRESCRIBED MEAN CURVATURE IN MINKOWSKI 3-SPACE

1IntroductionItiswell-knownthattheclassicalWeierstrass-EnneperrepresentationformuladescribeschinalsurfacesinEuclidean3-spaceR3intermsoftheirGaussmapsandauxiliaryholomorphicfunctions[1].Moregenerally,aremarkablerepresentationforlllulahasbeendiscoveredbyKenmotsu[2]forarbitrarysurfaCesinR3withnonvallishingmeancurvature,whichdescribesthesesurfacesintermsoftheirGaussmapsandmeancurvaturefunctions.Recently,Konopelchenko[31rediscoveredthisrepresentationformulaindifferentbutequivalentformillconnect…     

The numerical stability of barycentric Lagrange interpolation

The Lagrange representation of the interpolating polynomialcan be rewritten in two more computationally attractive forms:a modified Lagrange form and a barycentric form. We give anerror analysis of the evaluation of the interpolating polynomialusing these two forms. The modified Lagrange formula is shownto be backward stable. The barycentric formula has a less favourableerror analysis, but is forward stable for any set of interpolatingpoints with a small Lebesgue constant. Therefore the barycentricformula can be significantly less accurate than the modifiedLagrange formula only for a poor choice of interpolating points.This analysis provides further weight to the argument of Berrutand Trefethen that barycentric Lagrange interpolation shouldbe the polynomial interpolation method of choice.     

Spectral transformation of constant mean curvature surfaces in H3 and Weierstrass representation

By using the method of integrable system, we study the deformation of constant mean curvature surfaces in three-dimensional hyperbolic space form H3. We also obtain a Weierstrass representation formula of the constant mean curvature surfaces with mean curvature greater than 1.     

Two New FCT Algorithms Based on Product System

In this paper we present a product system and give a representation for consine functions with the system. Based on the formula two new algorithms are designed for computing the Discrete Cosine Transform. Both algorithms have regular recursive structure and good numerical stability and are easy to parallize.     

Two New FCT Algorithms Based on Product System

In this paper we present a product system and give a representation for consine functions with the system. Based on the formula two new algorithms are designed for computing the Discrete Cosine Transform. Both algorithms have regular recursive structure and good numerical stability and are easy to parallize.     

A representation theorem for smooth Brownian martingales

We show that, under certain smoothness conditions, a Brownian martingale, when evaluated at a fixed time, can be represented via an exponential formula at a later time. The time-dependent generator of this exponential operator only depends on the second order Malliavin derivative operator evaluated along a ‘frozen path’. The exponential operator can be expanded explicitly to a series representation, which resembles the Dyson series of quantum mechanics. Our continuous-time martingale representation result can be proven independently by two different methods. In the first method, one constructs a time-evolution equation, by passage to the limit of a special case of a backward Taylor expansion of an approximating discrete-time martingale. The exponential formula is a solution of the time-evolution equation, but we emphasize in our article that the time-evolution equation is a separate result of independent interest. In the second method, we use the property of denseness of exponential functions. We provide several applications of the exponential formula, and briefly highlight numerical applications of the backward Taylor expansion.     

Timelike Minimal Surfaces via Loop Groups

This work consists of two parts. In Part I, we shall give a systematic study of Lorentz conformal structure from structural viewpoints. We study manifolds with split-complex structure. We apply general results on split-complex structure for the study of Lorentz surfaces.In Part II, we study the conformal realization of Lorentz surfaces in the Minkowski 3-space via conformal minimal immersions. We apply loop group theoretic Weierstrass-type representation of timelike constant mean curvature for timelike minimal surfaces. Classical integral representation formula for timelike minimal surfaces will be recovered from loop group theoretic viewpoint.     

Explicitly defined fractal interpolation functions with variable parameters

We construct an explicit formula for the fractal interpolation function associated to an IFS with variable parameters. The solution is given in terms of the base p representation of numbers. This construction is a consequence of the formulation of the problem in a general functional equation setting. We introduce compatibility conditions as essential hypotheses to ensure problems in the functional system form are well-defined.     

Spectral transformation of constant mean curvature surfaces in H3 and Weierstrass representation

By using the method of integrable system, we study the deformation of constant mean curvature surfaces in three-dimensional hyperbolic space form H₃. We also obtain a Weierstrass representation formula of the constant mean curvature surfaces with mean curvature greater than 1     

Spectral transformation of constant mean curvature surfaces in H3 and Weierstrass representation

By using the method of integrable system, we study the deformation of constant mean curvature surfaces in three-dimensional hyperbolic space form H₃. We also obtain a Weierstrass representation formula of the constant mean curvature surfaces with mean curvature greater than 1     

A stochastic lagrangean picture for the three dimensional navier-stokes equation +

We derive a representation formula for the solutions of the navier-Stokes flow in three dimensions in absence of boundary in terms of stochastic current lines. This pictures generalizes an analogous one given in two dimensions and on the other hand the classical Lafrangian picture for the Euler flow. We prove a Vanishing viscosity limit for the whole structure     

The general solution and reduction to diagonal form of a system of equations of linear isotropic elasticity

A simple representation is obtained of the general solution of the Lame system of equations for an isotropic material in the form of a linear combination of the first derivatives of the three functions satisfying three independent wave or harmonic (in the static case) equations. In the two-dimensional case of plane deformation, the found solution directly implies the Kolosov-Muskhelishvili representation of the displacements by two analytic functions of a complex variable. A formula for generation of new solutions is given.     

Further Results on the Drazin Inverse of Tensors via Einstein Product

In this paper,we give further results on the Drazin inverse of tensors via the Einstein product.We give a limit formula for the Drazin inverse of tensors.By using this formula,the representations for the Drazin inverse of several block tensor are obtained.Further,we give the Drazin inverse of the sum of two tensors based on the representation for the Drazin inverse of a block tensor.