Fourth order approximation of harmonic maps from surfaces

Let be a compact Riemannian surface and let be a compact Riemannian manifold, both without boundary, and assume that N is isometrically embedded into some ℝ ^l . We consider a sequence of critical points of the functional with uniformly bounded energy. We show that this sequence converges weakly in and strongly away from finitely many points to a smooth harmonic map. One can perform a blow-up to show that there separate at most finitely many non-trivial harmonic two-spheres at these finitely many points. Finally we prove the so called energy identity for this approximation in the case that ↪ ℝ ^l . Mathematics Subject Classification (2000) 58E20, 35J60, 53C43     

High order compact schemes for gradient approximation

In this paper, we propose three gradient recovery schemes of higher order for the linear interpolation. The first one is a weighted averaging method based on the gradients of the linear interpolation on the uniform mesh, the second is a geometric averaging method constructed from the gradients of two cubic interpolation on macro element, and the last one is a local least square method on the nodal patch with cubic polynomials. We prove that these schemes can approximate the gradient of the exact solution on the symmetry points with fourth order. In particular, for the uniform mesh, we show that these three schemes are the same on the considered points. The last scheme is more robust in general meshes. Consequently, we obtain the superconvergence results of the recovered gradient by using the aforementioned results and the supercloseness between the finite element solution and the linear interpolation of the exact solution. Finally, we provide several numerical experiments to illustrate the theoretical results.     

Completions of implicitly defined linear time varying vector fields

The stabilization of constraints through such techniques as Baumgarte stabilization has been used in the simulation community for some time. This and a number of control problems can be viewed as either extending, or modifying, a vector field off of some manifold. Generally these approaches required the equations to have a special structure. Motivated by numerical simulation there has recently been new progress on doing this stabilization in numerically robust ways for larger classes of systems. In this paper we point out how earlier linear time invariant results do not immediately apply to the linear time varying case and then analyze the linear time varying case.     

Inverse limits and an implicitly defined difference equation from economics

The purpose of the paper is to introduce mathematicians to a cash-in-advance model from economics. We show that tools from inverse limits and dynamical systems developed in the last forty or so years are applicable to it.     

Cremona maps defined by monomials

Cremona maps defined by monomials of degree 2 are thoroughly analyzed and classified via integer arithmetic and graph combinatorics. In particular, the structure of the inverse map to such a monomial Cremona map is made very explicit as is the degree of its monomial defining coordinates. As a special case, one proves that any monomial Cremona map of degree 2 has inverse of degree 2 if and only if it is an involution up to permutation in the source and in the target. This statement is subsumed in a recent result of L. Pirio and F. Russo, but the proof is entirely different and holds in all characteristics. One unveils a close relationship binding together the normality of a monomial ideal, monomial Cremona maps and Hilbert bases of polyhedral cones.     

A new proof of the Euler-Maclaurin expansion for quadrature over implicitly defined curves

In this paper we describe and justify a method for integrating over implicitly defined curves. This method does not require that the Jacobian be known explicitly. We give a proof of an asymptotic error expansion for this method which is a modification of that of Lyness [4].     

Two properties of condition numbers for convex programs via implicitly defined barrier functions

We study two issues on condition numbers for convex programs: one has to do with the growth of the condition numbers of the linear equations arising in interior-point algorithms; the other deals with solving conic systems and estimating their distance to infeasibility.?These two issues share a common ground: the key tool for their development is a simple, novel perspective based on implicitly-defined barrier functions. This tool has potential use in optimization beyond the context of condition numbers. Received: October 2000 / Accepted: October 2001?Published online March 27, 2002     

Enumeration of order preserving maps

Three results are obtained concerning the number of order preserving maps of an n-element partially ordered set to itself. We show that any such ordered set has at least 2 ^2n/3 order preserving maps (and 2 ² in the case of length one). Precise asymptotic estimates for the numbers of self-maps of crowns and fences are also obtained. In addition, lower bounds for many other infinite families are found and several precise problems are formulated.Supported by ONR Contract N00014-85-K-0769.Supported by NSF Grant DMS-9011850.Supported by NSERC Grants 69-3378 and 69-0259.     

Extension of multilinear maps defined on subspaces

We study the problem of whether every multilinear form defined on the product of n closed subspaces has an extension defined on the product of the entire Banach spaces. We prove that the property derived from this condition (the Multilinear Extension Property) is local. We use this to prove that, for a wide variety of Banach spaces, there exist a product of closed subspaces and a multilinear form defined on it, which has no extension to the product of the entire spaces. We show that the ℓ _p spaces, with 1 ≤p ≤ ∞ and p ≠ 2, are among them and, more generally, every Banach space which fails to have type p for some p < 2 or cotype q for some q > 2.     

The approximation order of polysplines

We show that the scaling spaces defined by the polysplines of order provide approximation order For that purpose we refine the results on one-dimensional approximation order by -splines obtained by de Boor, DeVore, and Ron (1994).

     

Optimal homotopy analysis and control of error for implicitly defined fully nonlinear differential equations

Numerical Algorithms - Implicitly defined fully nonlinear differential equations can admit solutions which have only finitely many derivatives, making their solution via analytical or numerical...     

Linear approximation of approximately biorthogonality preserving maps

We investigate linear properties of mappings from a bounded domain of an n-dimensional normed space into another n-dimensional normed space such that the image of some almost biorthogonal system is almost biorthogonal. In this way we generalize a result of the author on stability of orthogonality in Euclidean spaces.     

Coincidence of classes of functions defined by the generalized shift operator or by the order of best polynomial approximation

In this paper the coincidence of two classes of functions is proved. One of them is determined by the power order of best polynomial approximation. To define the other class, first a new nonsymmetric generalized shift operator is introduced, and next, with its help we introduce a generalized modulus of continuity whose power order determines the second class of functions. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 242–257, August, 1999.     

On feasible sets defined through Chebyshev approximation

LetZ be a compact set of the real space with at leastn + 2 points;f,h1,h2:Z continuous functions,h1,h2 strictly positive andP(x,z),x(x ₀,...,x _n ) ⁿ⁺¹,z , a polynomial of degree at mostn. Consider a feasible setM {x ⁿ⁺¹z Z, –h ₂(z) P(x, z)–f(z)h ₁(z)}. Here it is proved the null vector 0 of ⁿ⁺¹ belongs to the compact convex hull of the gradients ± (1,z,...,z ⁿ ), wherez Z are the index points in which the constraint functions are active for a givenx* M, if and only ifM is a singleton.This work was partially supported by CONACYT-MEXICO.     

Regular maps of 2-power order

Journal of Algebraic Combinatorics - In this paper, we consider the possible types of regular maps of order $$2^n$$ , where the order of a regular map is the order of automorphism group of the map....     

Geometric heuristics for rural radio maps approximation

Given a terrain T and an antenna A located on it, we would like to approximate the Radio Map of A over T, namely, to associate a signal strength for each point p∈T as received from A. This work presents a new Radio Map approximation algorithm using an adaptive radial sweep-line technique. The suggested radar-like algorithm (RLA) uses a pipe-line method for computing the signal strength along points on a ray, and an adaptive method for interpolating the signal strength over regions between two consecutive rays. Whenever the difference between two consecutive rays is above a certain threshold, a middle ray is created. Thus, the density of the sampling rays is sensitive to the shape of the terrain. Finally, we report on an experiment which compares the new algorithm with other well-known methods. The main conclusion is that the new RLA is significantly faster than the others, i.e., its running time is 3–15 times faster for the same approximation accuracy.     

Covering maps that are not compositions of covering maps of lesser order

In 1995 J.W. Heath asked which exactly -to-one maps are compositions of exactly -to-one maps with . This paper deals with compositions of covering maps. Exactly -to-one covering maps on locally arcwise connected continua that are not factorable into covering maps of order are constructed for all 's, and characterized in algebraic terms (fundamental groups). They are not proper compositions of exactly -to-one maps, open maps, or locally one-to-one maps.     

Metric differentiability of Lipschitz maps defined on Wiener spaces

This note is devoted to the differentiability properties of -Lipschitz maps defined on abstract Wiener spaces and with values in metric spaces (Y,d _Y ). We prove the existence γ-a.e. of local seminorms in the Cameron-Martin space which allow to compute the metric derivative of the map. Second Author partially supported by DGES (Spain) MTM2006-03531.