An Algorithm for Linear L1 Approximation of Continuous Functions

The problem of solving linear continuous L₁ approximation problemsis considered. An algorithm is given for approximation on aninterval of the real line, which is globally convergent, usuallyat a second-order rate. The method is illustrated by numericalexamples.     

Best Approximation in an Asymmetrically Weighted L1 Measure

Let f(x) be a given, real-valued, continuous function definedon an interval [a,b]of the real line. Given a set of m real-valued,continuous functions _j(x) defined on [a,b], a linear approximatingfunction can be formed with any real setA = {a₁, a₂,..., a_m}. We present results for determining A sothat F(A, x) is a best approximation to(x) when the measureof goodness of approximation is a weighted sum of |F(A, x)–f(x)|,the weights being positive constants, w, when F(A, x) f(x)and w2 otherwise (when w, = w2 = 1, the measure is the L₁, norm).The results are derived from a linear programming formulationof the problem. In particular, we give a theorem which shows when such bestapproximations interpolate the function at fixed ordinates whichare independent of f(x). We show how the fixed points can becalculated and we present numerical results to indicate thatthe theorem is quite robust.     

Algebraic Isomorphisms and Finite Distributive Subspace Lattices

Let L₁ and L₂ be finite distributive subspace lattices on realor complex Banach spaces. It is shown that every rank-preservingalgebraic isomorphism of AlgL₁ onto AlgL₂ is quasi-spatiallyinduced. If the algebraic isomorphism in question is known onlyto preserve the rank of rank one operators, then it inducesa lattice isomorphism between L₁ and L₂.     

Isomorphisms of Group Algebras

Let G₁ and G₂ be locally compact groups. If T is an algebraisomorphism of L¹(G₁) onto L¹(G₂) with ||T|| (1+3), then G₁and G₂ are isomorphic. This improves on earlier results, and,in a certain sense, is best possible. However, the main conjecturethat the groups are isomorphic if ||T|| < 2 remains unsolvedexcept for abelian groups and for connected groups. Similarresults are given for the measure algebra M(G) and for the algebraC(G) of continuous functions when the group G is compact.     

Properties of a Semi-discrete Approximation to the Beam Equation

The solution of the equation w(x)u_tt+[p(x)u_xx]_xx = 0, 0 < x < L, t > 0, where it is assumed that w and p are positive on the interval[O, L], is approximated by using the method of straight lines.The resulting approximation is a linear system of differentialequations with coefficient matrix S. The matrix S is studiedunder very general boundary conditions which result in a conservativesystem. In all cases the matrix S is either an oscillation matrixor possesses nearly all the properties of an oscillation matrix.     

Oscillation Properties of a Semi-discrete Approximation to the Beam Equation with a Second Order Term

The solution of the equation w(x)u_tt+[p(x)u_xx]_xx–[p(x)u_x]_x=0, 0< x < L, t > 0, where it is assumed that w, p,and q are positive on the interval [0, L], is approximated bythe method of straight lines. The resulting approximation isa linear system of differential equations with coefficient matrixS. The matrix S is studied under a variety of boundary conditionswhich result in a conservative system. In all cases the matrixS is shown to be similar to an oscillation matrix.     

Near-best Lp Approximations by Real and Complex Chebyshev Series

The expansion of a real or complex function in a series of Chebyshevpolynomials of the first and second kinds is discussed in thecontext of near-best approximation. The discussion covers realand complex approximation on the real interval [–1, 1]as a special example of the complex elliptical contour , as well as complex approximationon an elliptical domain, an ellipse exterior, and an ellipticalannulus (including special cases in which part of the boundarycollapses into a "crack"). Two distinct types of function spacesare considered, namely appropriately weighted L_p measure spacesand analytic function spaces, and resulting approximations areshown in all cases to be near-best in the L_p norm within a relativedistance asymptotic to (4^-2 log n)^2p-1–1 for all p (1p ), where relates to the order of approximation.     

A Version of the Poincare-Birkhoff-Witt Theorem

In this note I shall prove that if L is a finite-dimensionalLie algebra over a field F of characteristic zero which is generatedas an algebra by a set of elements {e₁, e₂,...,e_k}, then theuniversal enveloping algebra U(L) of L is linearly generatedby monomials spanned by the elements {e_i} of an a priori boundedwidth. As an application, a criterion of Kostant for a leftideal of U(L) to be of finite codimension is proved by purelyalgebraic means.     

On the relationship between functions with the same optimal knots in spline and piecewise polynomial approximation

The relationship between functions with the same optimal knots for L₂[0, 1] approximation by kth order splines or piecewise polynomials is investigated. It is shown that if two functions have positive continuous kth derivatives they will have the same optimal knots if and only if they differ by a polynomial of order k. An application to design selection for continuous time regression is considered and extensions to L_p approximation are also provided.     

Link Cobordism and the Intersection of Slice Discs

We work in the smooth category. An (oriented) (ordered) m-component n-(dimensional) link isa smooth oriented submanifold L = {K₁, ..., K_m} of Sⁿ⁺² whichis the ordered disjoint union of m manifolds, each PL-homeomorphicto the standard n-sphere. If m = 1, then L is called a knot. We say that m-component n-dimensional links L₀ and L₁ are (link-)concordantor (link-)cobordant if there is a smooth oriented submanifoldC = {C₁, ..., C_m} of Sⁿ⁺² x [0, 1] which meets the boundarytransversely in C, is PL-homeomorphic to L₀ x [0, 1], and meetsSⁿ⁺² x {l} in L_l (l = 0, 1). If m = 1, then we say that n-knotsL₀ and L_l are (knot-)concordant or (knot-)cobordant. Then wecall C a concordance-cylinder of the two n-knots L₀ and L_l. If an n-link L is concordant to the trivial link, then we callL a slice link. If an n-link L = {K₁, ..., K_m} Sⁿ⁺² = Bⁿ⁺³ Bⁿ⁺³ is slice,then there is a disjoint union of (n + 1)-discs in Bⁿ⁺³ such that is called a set of slice discs for L. If m = 1, then is called a slice disc for the knotL. 1991 Mathematics Subject Classification 57M25, 57Q45.     

量子环面李代数的自同构群, 泛中心扩张与导子

C_q:=C_q[x^±1₁, x^±1₂] 为复数域上的量子环面, 其中q≠ 0是一个非单位根, D(C_q) 为C_q的导子李代数. 记L_q 为C_q ㈩ D(C_q)的导出子代数. 该文研究李代数L_q的自同构群, 泛中心扩张和导子李代数.     

Generalized Alternating Polynomials, some Properties and Numerical Applications

In the present paper we introduce the generalized alternatingpolynomials , the coefficients of which are defined together with the parameter W_n by the linear system where T_n(x) = cos (n arc cos x), is a set of n+2 distinct points in the interval [–1, 1], and fis a continuous function on [–1, 1], For the set of nodesx_k = cos [k/(n+1)] the w_n-polynomials coincide with the polynomialsof equiamplitude alternation introduced by de La Vall?e-Poussinand discussed in the literature earlier (Eterman, Malozemov,Meinardus, Cheney & Rivlin, Phillips & Taylor, Brutman,and others). It is shown that the generalized alternating polynomials arerelated to the polynomials of interpolation through the Lanczoseconomization process. Some approximation properties of w_n -polynomialsand W_n -parameters are studied. The application of w_n -polynomialsto function approximation and to the estimate of remainder termsfor quadrature formulas is discussed.     

On the L-Function of the Curves y2 = x5 + A

Let C_A/Q be the curve y² = x⁵ + A, and let L(s, J_A) denote theL-series of its Jacobian. Under the assumption that the signin the functional equation for L(s, J_A) is +1, the criticalvalue L(1, J_A) is evaluated in terms of the value of a thetaseries for depending on Aat a complex multiplication point coming from Q(₅).     

Lie Powers of Modules for Groups of Prime Order

Let L(V) be the free Lie algebra on a finite-dimensional vectorspace V over a field K, with homogeneous components Lⁿ(V) forn 1. If G is a group and V is a KG-module, the action of Gextends naturally to L(V), and the Lⁿ(V) become finite-dimensionalKG-modules, called the Lie powers of V. In the decompositionproblem, the aim is to identify the isomorphism types of indecomposableKG-modules, with their multiplicities, in unrefinable directdecompositions of the Lie powers. This paper is concerned withthe case where G has prime order p, and K has characteristicp. As is well known, there are p indecomposables, denoted hereby J₁,...,J_p, where J_r has dimension r. A theory is developedwhich provides information about the overall module structureof LV) and gives a recursive method for finding the multiplicitiesof J₁,...,J_p in the Lie powers Lⁿ(V). For example, the theoryyields decompositions of L(V) as a direct sum of modules isomorphiceither to J₁ or to an infinite sum of the form J_r J_{p-1} J_{p-1} ... with r 2. Closed formulae are obtained for the multiplicitiesof J₁,..., J_p in Lⁿ(J_p and Lⁿ(J_{p-1). For r < p-1, the indecomposableswhich occur with non-zero multiplicity in Lⁿ(J_r) are identifiedfor all sufficiently large n. 2000 Mathematical Subject Classification:17B01, 20C20.     

Some Approximations of the Characteristics of Second Order Hyperbolic Partial Differential Equations

The characteristic curves of the quasi-linear hyperbolic partialdifferential equation of the second order, having unequallyspaced, discreate initial conditions, are approximated by L₂parabolas, L₁ parabolas and cubic splines.     

Local Cohomology of BP*BP-Comodules

Given a spectrum X, we construct a spectral sequence of BP_*BP-comodulesthat converges to BP_*(L_nX), where L_nX is the Bousfield localizationof X with respect to the Johnson–Wilson theory E(n)_*.The E₂-term of this spectral sequence consists of the derivedfunctors of an algebraic version of L_n. We show how to calculatethese derived functors, which are closely related to local cohomologyof BP_*-modules with respect to the ideal I_n+1. 2000 MathematicsSubject Classification 55N22, 55P60, 16W30.     

Best simultaneous L1 approximations

Three possible definitions are proposed for best simultaneous L₁ approximation to n continuous real-valued functions, and the relation between best simultaneous approximations and best L₁ approximations to the arithmetic mean of the n functions is discussed.     

Error Control with Polynomial Approximations

A detailed analysis is given of the accumulation of errors whichmay occur in evaluating a polynomial approximation to a givenfunction. Both backward recursion using untransformed Chebyshevexpansions and the much faster nested multiplication using thetransformed simple polynomial form are treated. Two types ofarithmetic are dealt with covering most current machines. Forthe case of polynomials with coefficients of the same sign orstrictly alternating signs, a situation which is of considerablepractical importance in polynomial approximation of mathematicalfunctions, we show that controlling relative error requiresthe ratios |y|_max/|y|_min and |y'|_max/|y|_min to be kept small.Experimental verification of these effects is given based onexpansions available in the literature or produced by the authorsfor the Bessel function I₀(x).     

The dimension monoid of a lattice

We introduce the dimension monoid of a lattice L, denoted by Dim L. The monoid Dim L is commutative and conical, the latter meaning that the sum of any two nonzero elements is nonzero. Furthermore, Dim L is given along with the dimension map, D\Delta from L2L to Dim L, which has the intuitive meaning of a distance function. The maximal semilattice quotient of Dim L is isomorphic to the semilattice Con_c L of compact congruences of L; hence Dim L is a precursor of the congruence lattice of L. Here are some additional features of this construction: ¶¶ (1) Our dimension theory provides a generalization to all lattices of the von Neumann dimension theory of continuous geometries. In particular, if L is an irreducible continuous geometry, then Dim L is either isomorphic to \Bbb Z⁺\Bbb Z^+ or to \Bbb R⁺\Bbb R^+.¶ (2) If L has no infinite bounded chains, then Dim L embeds (as an ordered monoid) into a power of \Bbb Z⁺è{￥}.{\Bbb Z}^{+}\cup \{\infty\}.¶ (3) If L is modular or if L has no infinite bounded chains, then Dim L is a refinement monoid.¶ (4) If L is a simple geometric lattice, then Dim L is isomorphic to \Bbb Z⁺\Bbb Z^+, if L is modular, and to the two-element semilattice, otherwise.¶ (5) If L is an à₀\aleph_0-meet-continuous complemented modular lattice, then both Dim L and the dimension function D\Delta satisfy (countable) completeness properties.¶¶ If R is a von Neumann regular ring and if L is the lattice of principal right ideals of the matrix ring M₂ (R), then Dim L is isomorphic to the monoid V (R) of isomorphism classes of finitely generated projective right R-modules. Hence the dimension theory of lattices provides a wide lattice-theoretical generalization of nonstable K-theory of regular rings.     

The Fourier-finite-element method with Nitsche mortaring

^** Email: bernd.heinrich{at}mathematik.tu-chemnitz.de The paper deals with a combination of the Fourier-finite-elementmethod with the Nitsche-finite-element method (as a mortar method).The approach is applied to the Dirichlet problem for the Poissonequation in 3D axisymmetric domains with non-axisymmetric data.The approximating Fourier method yields a splitting of the 3Dproblem into 2D problems on the meridian plane of the givendomain. For solving these 2D problems, the Nitsche-finite-elementmethod with non-matching meshes is applied. Some important propertiesof the approximation scheme are derived and the rate of convergencein an H¹-like norm as well as in the L₂-norm is estimated fora regular solution. Finally, some numerical results are presented.