A Tridimensional Inverse Shaping Problem

We study the following tridimensional magnetostatic inverse shaping problem: can one find a distribution of currents around a levitating liquid metal bubble so that it takes a given shape? It leads to the resolution of an Eilonal equation on the surface of the bubble which has self-contained interest. We answer the question for closed smooth surface which are homeomorphic to a sphere. We give a necessary and sufficient condition on the data for existence and uniqueness of a C¹ solution. When the desired shape is axisymmetric and analytic, the solution is also analytic and the problem can be completely solved. A counterexample proves that not all analytic perturbations of such surface are shapable.     

Optimal defect corrections on composite nonmatching finite-element meshes

Ahmed-Salah Chibi ^{In this paper, we analyse the ‘defect-correction’technique on a general smooth region, via composite finite-elementmeshes (a Cartesian mesh and a polar mesh) on two overlappingsubdomains (a rectangle and an annulus). Boundary interpolatorymappings of higher degree are used, in the Schwarz method, topass from one mesh to another. An explicit relation is givenbetween the degree of these mappings and the number of optimalcorrections to be computed. Optimal convergence results forthe discrete bilinear basic solution, in higher-order discreteSobolev norms, are obtained on the subdomains. Because the successof the defect-correction technique is based on the uniformityof the discretization and the regularity of the exact solution,the defects are computed on the subdomains in the same way asfor the basic solution. Optimal O(h2) improvement per correctionis obtained. Numerical results are presented to support thetheory.}     

A discrete Galerkin method for a hyperslngular boundary integral equation

Received on 14 August 1995. Revised on 20 August 1996. Consider solving the interior Neumann problem with D a simply-connected planar region and S=D a smooth curve.A double-layer potential is used to represent the solution,and it leads to the problem of solving a hypersingular integralequation. This integral equation is reformulated as a Cauchysingular integral equation. A discrete Galerkin method withtrigonometric polynomials is then given for its solution. Anerror analysis is given, and numerical examples complete thepaper.     

A New Class of Results for the Algebraic Equations of Implicit Runge-Kutta Processes

We are concerned with the solvability of the discrete equationsarising in the use of Runge-Kutta methods. Under suitable assumptionson the RK tableau, we show that in the neighbourhood of a sink(asymptotically stable equilibrium) a unique solution existsfor arbitrarily large stepsizes. Furthermore in the neighbourhoodof a slowly varying integral curve z = z(t) that attracts neighbouringintegral curves of the ODE system, a unique solution of thealgebraic equations exists, provided that the stepsize is suitablyrestricted. This restriction does not depend on the stiffnessof the ODEs being integrated.     

Discrete Variational Calculus for B-Spline Curves

In this work, we study discrete variational problems, for B-spline curves, which are invariant under translation and rotation. We show this approach has advantages over studying smooth variational problems whose solutions are approximated by B-spline curves. The latter method has been well studied in the literature but leads to high order approximation problems. We are particularly interested in Lagrangians that are invariant under the special Euclidean group for which B-spline approximated curves are well suited. The main application we present here is the curve completion problem in 2D and 3D. Here, the aim is to find various aesthetically pleasing solutions as opposed to a solution of a physical problem. Smooth Lagrangians with special Euclidean symmetries involve curvature, torsion, and arc length. Expressions of these, in the original coordinates, are highly complex. We show that, by contrast, relatively simple discrete Lagrangians offer excellent results for the curve completion problem. The novel methods we develop for the discrete curve completion problem are general, and can be used to solve other discrete variational problems for B-spline curves. Our method completely avoids the difficulties of high order smooth differential invariants.     

An algorithm for constrained smoothing functions

In this paper we study a method for the approximation by a monotone smooth function of a discrete set of observations. By this method it is possible to obtain a completely automatic program. We show that the solution is monotone with probability tending to one and by the test examples that it is sufficient in the practical case.This work was supported by the Italian M.U.R.S.T.     

Analytic solutions for unsteady flows over a superhydrophobic surface

A class of unsteady analytic solutions for the evolution of viscous Newtonian fluids over a superhydrophobic surface is derived. The surface is represented by regular rectilinear riblets and the fluid is assumed to flow along the grooves’ direction. A mixed boundary condition is imposed on the surface, consisting in homogeneous Neumann conditions over the riblet voids and homogeneous Dirichlet conditions on the wall intervals. The transition between the above conditions is modelled through a Robin condition with non-constant smooth coefficients in a completely general manner. A global solution is derived and relevant examples, that can be fruitfully adopted as benchmark solutions for testing numerical solvers, are discussed.     

A posteriori error estimates for a time discrete scheme for a phase-field system of Penrose-Fife type

A time discrete scheme is used to approximate the solution toa phase field system of Penrose–Fife type with a non-conservedorder parameter. An a posteriori error estimate is presentedthat allows the estimation of the difference between continuousand semidiscrete solutions by quantities that can be calculatedfrom the approximation and given data.     

l1 solution of linear inequalities

The numerical solution of a possible inconsistent system oflinear inequalities in the l₁ sense is considered. The non-differentiablel₁ norm minimization problem is approximated by a piecewisequadratic Huber smooth function. A continuation algorithm isdesigned to find an l₁ solution of the inequality system. Inthe case where the linear inequality system is consistent, asolution is obtained by solving any smoothed problem. Otherwise,the algorithm is shown to terminate in a finite number of iterations.We also consider an alternative smoothing scheme which sharessimilar properties with the first one, but results in an improvedcomputational performance of the continuation algorithm on inconsistentsystems. Numerical experiments are conducted to test the efficiencyof the algorithm.     

Interpolation of Vector-Valued Real Analytic Functions

Let R^d be an open domain. The sequentially complete DF-spacesE are characterized such that for each (some) discrete sequence(z_n) , a sequence of natural numbers (k_n) and any family the infinite system of equations has an E-valued real analytic solution f.     

A Peak Point Theorem for Uniform Algebras Generated by Smooth Functions on Two-Manifolds

We establish the peak point conjecture for uniform algebrasgenerated by smooth functions on two-manifolds: if A is a uniformalgebra generated by smooth functions on a compact smooth two-manifoldM, such that the maximal ideal space of A is M, and every pointof M is a peak point for A, then A = C(M). We also give an alternativeproof in the case when the algebra A is the uniform closureP(M) of the polynomials on a polynomially convex smooth two-manifoldM lying in a strictly pseudoconvex hypersurface in Cⁿ.     

On Hill's Equation with a Singular Complex-Valued Potential

In this paper Hill's equation y' + qy = Ey, where q is a complex-valuedfunction with inverse square singularities, is studied. Resultson the dependence of solutions to initial value problems onthe parameter E and the initial point x₀, on the structure ofthe conditional stability set, and on the asymptotic distributionof (semi-)periodic and Sturm-Liouville eigenvalues are obtained.It is proved that a certain subset of the set of Floquet solutionsis a line bundle on a certain analytic curve in C². We establishnecessary and sufficient conditions for q to be algebro-geometric,that is, to be a stationary solution of some equation in theKorteweg-de Vries (KdV) hierarchy. To do this a distinctionbetween movable and immovable Dirichlet eigenvalues is employed.Finally, an example showing that the finite-band property doesnot imply that q is algebro-geometric is given. This is in contrastto the case where q is real and non-singular. 1991 MathematicsSubject Classification: 34L40, 14H60.     

Statistical Properties of Unimodal Maps

We consider typical analytic unimodal maps which possess a chaotic attractor. Our main result is an explicit combinatorial formula for the exponents of periodic orbits. Since the exponents of periodic orbits form a complete set of smooth invariants, the smooth structure is completely determined by purely topological data (“typical rigidity”), which is quite unexpected in this setting. It implies in particular that the lamination structure of spaces of analytic unimodal maps (obtained by the partition into topological conjugacy classes, see [ALM]) is not transversely absolutely continuous. As an intermediate step in the proof of the formula, we show that the distribution of the critical orbit is described by the physical measure supported in the chaotic attractor.     

Maximum-norm estimates for resolvents of elliptic finite element operators

Let be a convex domain with smooth boundary in . It has been shown recently that the semigroup generated by the discrete Laplacian for quasi-uniform families of piecewise linear finite element spaces on is analytic with respect to the maximum-norm, uniformly in the mesh-width. This implies a resolvent estimate of standard form in the maximum-norm outside some sector in the right halfplane, and conversely. Here we show directly that such a resolvent estimate holds outside any sector around the positive real axis, with arbitrarily small angle. This is useful in the study of fully discrete approximations based on -stable rational functions, with small.

     

大气运动基本方程组的解析解

在已知大气运动基本方程于光滑函数类中具有最好的稳定性前提下,讨论了它的局部解的解空间构造．根据它的解空间构造,分析了这个方程具有代表性和应用性的第三初值问题,在解析函数类中给出了适定的第三初值问题的解析解的计算方法以及具体的关系表达式,在局部解意义下完整的解决了这一点初值问题的解析解所涉及的理论与计算问题．指出其它类型定解问题都可以仿照文中的计算方法和步骤,求出所需要的稳定的解析解．     

Asymptotic Solution of a Model Non-linear Convective Diffusion Equation

In this paper we consider the large-time solution of the equation for initial data with compact support. With m = 4 and n = 3the equation models the flow of a thin viscous sheet on an inclinedbed while for n m > 1 it has application in porous mediaflow under gravity. The equation can also be regarded as ananalogue of Burgers equation in non-linear diffusion. It isknown that two moving boundaries exist along which certain boundaryconditions are required to hold. The paper extends earlier workand determines the analytic behaviour of the moving boundariesexist along which certain boundary conditions are required tohold. The paper extends earlier work and determines the analyticbehaviour of the moving boundaries together with the structureof the solution which at large times is shown to depend cruciallyon the location in (n, m) parameter space.     

On the existence of Analytic Semigroups Bounded on the Half-Disc in some Banach Algebras

Let G be a locally compact group. We show that L¹(G) does notcontain any analytic semigroups which are bounded on the half-disc,by using some facts about the analytic Radon-Nikodm propertyin Banach spaces. This solves a question raised by J. Esterle.     

Smooth ideals in hyperelliptic function fields

Recently, several algorithms have been suggested for solving the discrete logarithm problem in the Jacobians of high-genus hyperelliptic curves over finite fields. Some of them have a provable subexponential running time and are using the fact that smooth reduced ideals are sufficiently dense. We explicitly show how these density results can be derived. All proofs are purely combinatorial and do not exploit analytic properties of generating functions.

     

Fast iterative methods for symmetric sinc-Galerkin systems

The symmetric sinc-Galerkin method developed by Lund, when appliedto the second-order self-adjoint boundary value problem, givesrise to a symmetric coefficient matrix has a special structureso that it can be advantageously used in solving the discretesystem. In this paper, we employ the preconditioned conjugategradient method with banded matrices as preconditioners. Weprove that the condition number of the preconditioned matrixis uniformly bounded by a constant independent of the size ofthe matrix. In particular, we show that the solution of an n-by-ndiscrete symmetric sinc-Galerkin system can be obtained in O(nlog n) operations. We also extend our method to the self-adjointelliptic partial differential equation. Numerical results aregiven to illustrate the effectiveness of our fast iterativesolvers.