An Efficient Enthalpy-type Method for the Stefan Problem

The paper presents a new finite-difference method for solvingthe one-dimensional two-phase Stefan problem. Under assumptionson the data which guarantee the temperature u and the movingboundary s to belong to and , respectively, we obtain L₂-errorestimates of order O(h + h^–?) provided the time step is chosen such that Numerical aspects are discussed.     

Stability Analysis of Runge-Kutta Methods for Volterra Integral Equations of the Second Kind

Stability analysis of Volterra-Runge-Kutta methods based onthe basic test equation of the form where is a complex parameter, and on the convolution test equation where and are real parameters, is presented. General stabilityconditions are derived and applied to construct numerical methodswith good stability properties. In particular, a family of second-orderV^o-stable Volterra-Runge-Kutta methods is obtained. No V^o-stablemethods of order greater than one have been presented previouslyin the literature.     

Reducing Subspaces for a Class of Multiplication Operators

Let D be the open unit disk in the complex plane C. The Bergmanspace is the Hilbert space of analytic functions f in D such that where dA is the normalized area measure on D. If are two functions in , then the inner product of f and g is given by We study multiplication operators on induced by analytic functions. Thus for H (D), the space ofbounded analytic functions in D, we define by It is easy to check that M is a bounded linear operator on with ||M||=||||=sup{|(z)|:zD}.     

Near-minimax Interpolation by a Polynomial in z and z-1 on a Circular Annulus

A polynomial of degree n in z^–1 and n–1 in z isdefined by an interpolation projection from the space A(N_p) of functions f analytic in the circular annulusp^–1 < <p and continuous on itsboundaries = p^–1, p. The points ofinterpolation are chosen to be spaced at equal angles aroundthe two boundaries, with arguments on the inner boundary midwaybetween those on the outer boundary. By calculating the Lebesgueconstants numerically, is found to be close to a minimax approximation for all p 1and all degrees n in the range 1 n 15. In the limiting casesp = 1 and, it is proved that is asymptotic to 2^–1 log n. More specifically and , where _nis the Lebesgueconstant of Gronwall for equally spaced interpolation on a circleby a polynomial of degree n. It is also demonstrated that is not in general monotonic in p, and that is not everywhere differentiable in p.     

Multipliers on Banach spaces of functions on a locally compact Abelian group

Let E be a Banach space of functions on a locally compact Abeliangroup G satisfying certain conditions. It has been proved thatfor every bounded operator M on E commuting with translationsthere exists such that , where is a suitable subset of the group of the continuous morphismsfrom G into * and is a generalized Fourier transform of g defined on .     

Finite-Element Approximation of a Plasma Equilibrium Problem

The plasma problem studied is: given R⁺ find (, d, u) R ?R ? H¹() such that Let ₁ < ₂ be the first two eigenvalues of the associatedlinear eigenvalue problem: find $$\left(\lambda ,\phi \right)\in\mathrm{R;}\times {\hbox{ H }}_{0}^{1}\left(\Omega \right)$$such that For ₀(0,₂) it is well known that there exists a unique solution(₀, d₀, u₀) to the above problem. We show that the standard continuous piecewise linear Galerkinfinite-element approximatinon $$\left({\lambda }_{0},{\hbox{d }}_{0}^{k},{u}_{0}^{h}\right)$$, for ₀(0,₂), converges atthe optimal rate in the H¹, L², and L norms as h, the mesh length,tends to 0. In addition, we show that dist (, ^h)Ch² ln 1/h,where $${\Gamma }^{\left(h\right)}=\left\{x\in \Omega :{u}_{0}^{\left(h\right)}\left(x\right)=0\right\}$$.Finally we consider a more practical approximation involvingnumerical integration.     

Cohomology of Quantized Function Algebras at Roots of Unity

Let G be a simply-connected, semisimple algebraic group overk, an algebraically closed field of characteristic zero. Let O[G] be the quantized function algebra of G at a primitivelth root of unity , and let be the ‘restricted’ quantized function algebra at, a finite-dimensional k-algebra obtained from O[G] by factoringout a centrally generated ideal. It is known that is a Hopf algebra. We study the cohomology ring, a graded commutative algebra, and, for any finite-dimensional -module M, the -module . We prove that for sufficiently large l there isan isomorphism of graded algebras where each X_i is homogeneous of degree $2$, and $2N$ equalsthe number of roots associated to G. Moreover we show that inthis case is a finitely generated -module. We also show under much less restrictive conditions on l that continues to be a finitely generated graded commutativealgebra over which is a finitely generated module. 1991 Mathematics Subject Classification: 16W30,17B37, 17B56.     

Total Chromatic Number of Graphs of Order 2n + l having Maximum Degree 2n 1

Let G be a graph of order 2n + l having maximum degree 2n –1. We prove that the total chromatic number of G is 2n if andonly if e + ' n, where w is a vertex of minimum degree in G, is the complement of G —w, e is the size of , and ' is the edge independence number of .     

The Bicanonical Map of Surfaces with pg = 0 and K2 >= 7

A minimal surface of general type with p_g(S) = 0 satisfies 1 K² 9, and it is known that the image of the bicanonical map is a surface for , whilst for , the bicanonical map is always a morphism. In this paper it is shown that is birationalif , and that the degree of is at most 2 if or By presenting two examples of surfaces S with and 8 and bicanonical map of degree 2, it is alsoshown that this result is sharp. The example with is, to our knowledge, a new example of a surfaceof general type with p_g = 0. The degree of is also calculated for two other known surfacesof general type with p_g = 0 and . In both cases, the bicanonical map turns out to be birational.     

On Mason's conjecture concerning interpolation by polynomials in z and z-1 on an annulus

A polynomial of degree n in z^–1 and n – 1 in z isdefined by an interpolation projection from the space A(N) of functions f analytic in thecircular annulus ^–1<|z| and continuous on its boundaries|z|=^–1, . The points of interpolation are chosen to coincidewith the n roots of zⁿ=–n and the n roots of zⁿ=^–n.We prove Mason's conjecture that the corresponding Lebesguefunction attains its maximal value on the inner circle. We alsoestimate the bound of the Lebesgue constant . It is proved that the following estimate for theoperator norm holds: where _n, is the Lebesgue constant of Gronwall for equally spacedinterpolation on a circle by a polynomial of degree n.     

Global Branch of Solutions for Non-Linear Schrodinger Equations with Deepening Potential Well

We consider the stationary non-linear Schrödinger equation where > 0 and the functionsf and g are such that and for some bounded open set R^N. We use topological methods to establish the existenceof two connected sets D^± of positive/negative solutionsin R x W², ^p R^N where that cover the interval (, ()) in the sense that and furthermore, The number () is characterized as the unique value of in theinterval (, ) for which the asymptotic linearization has a positiveeigenfunction. Our work uses a degree for Fredholm maps of indexzero. 2000 Mathematics Subject Classification 35J60, 35B32,58J55.     

Finite-Element Approximation of Elliptic Equations with a Neumann or Robin Condition on a Curved Boundary

This paper considers a finite-element approximation of a second-orderself adjoint elliptic equation in a region Rⁿ (with n=2 or 3)having a curved boundary on which a Neumann or Robin conditionis prescribed. If the finite-element space defined over , a union of elements, has approximation power h^kin the L² norm, and if the region of integration is approximatedby ^h with dist (, ^h)Ch^k, then it is shown that one retains optimalrates of convergence for the error in the H¹ and L² norms, whetherQ^h is fitted or unfitted , provided that the numerical integration scheme has sufficientaccuracy.     

Oscillation Criteria for First-Order Delay Equations

This paper is concerned with the oscillatory behaviour of first-orderdelay differential equations of the form (1) where is non-decreasing, (t)< t for t t₀ and . Let the numbers k andL be defined by It is proved here that when L < 1 and 0 < k 1/e all solutionsof equation (1) oscillate in several cases in which the condition holds, where ₁ is the smaller root of the equation = e^k. 2000Mathematics Subject Classification 34K11 (primary); 34C10 (secondary).     

Some Relations Between Packing Premeasure and Packing Measure

Let K be a compact subset of Rⁿ, 0 s n. Let , P^s denote s-dimensional packing premeasure andmeasure, respectively. We discuss in this paper the relationbetween and P^s. We prove:if , then ; and if , then for any > 0, there exists a compact subset F of K such that and P^s(F) P^s(K) – .1991 Mathematics Subject Classification 28A80, 28A78.     

Some Compactness Results in Real Interpolation for Families of Banach Spaces

The paper shows that, if the operator T:A()B() is compact foralmost every , then is compact when or is the interpolation functor constructed for infinitefamilies of Banach spaces and S satisfies certain conditions.     

A Radial Uniqueness Theorem for Sobolev Functions

We show that continuous functions u in the Sobolev space , 1 < p n, which have the limitzero in a certain weak sense in a set of positive p-capacityon B with where B is the open unit ball of Rⁿ and for 0 > > , are identically zero. Conversely, we produce for each 1 > p n and each positive a non-constant function u in , continuous in , and a compact set EB of positive p-capacity such that u = 0 in E and the aboveinequality holds with exponent p – l + .     

Asymptotic Expansions of Multiple Zeta Functions and Power Mean Values of Hurwitz Zeta Functions

Let (s, ) be the Hurwitz zeta function with parameter . Powermean values of the form are studied, where q and h are positive integers. These mean valuescan be written as linear combinations of , where _r(s₁,...,s_r;) is a generalization of Euler–Zagiermultiple zeta sums. The Mellin–Barnes integral formulais used to prove an asymptotic expansion of , with respect to q. Hence a general way of deducingasymptotic expansion formulas for is obtained. In particular, the asymptotic expansion of with respect to q is written down.     

Weighted Hardy-Type Inequalities for Differences and the Extension Problem for Spaces with Generalized Smoothness

It is well known that there are bounded domains Rⁿ whose boundaries are not smooth enough for there to exist a bounded linear extensionfor the Sobolev space into , but the embedding is nevertheless compact. For the Lipboundaries (0<<1) studied in [3, 4], there does not existin general an extension operator of into but there is a bounded linear extension of into and the smoothness retained by thisextension is enough to ensure that the embedding is compact. It is natural to ask if this is typicalfor bounded domains which are such that is compact, that is, that there exists a boundedextension into a space of functions in Rⁿ which enjoy adequatesmoothness. This is the question which originally motivatedthis paper. Specifically we study the ‘extension by zero’operator on a space of functions with given ‘generalized’smoothness defined on a domain with an irregular boundary, anddetermine the target space with respect to which it is bounded.     

Almost Everywhere Convergence of Bochner-Riesz Means On The Heisenberg Group and Fractional Integration On The Dual

Let L denote the sub-Laplacian on the Heisenberg group H_n and the corresponding Bochner-Riesz operator. Let Q denote the homogeneous dimension and D the Euclideandimension of H_n. We prove convergence a.e. of the Bochner-Rieszmeans as r 0 for > 0and for all f L^p(H_n), provided that . Our proof is based on explicit formulas for the operators with a C, defined on the dual ofH_n by , which may be of independent interest. Here is given by for all (z,u) H_n. 2000 Mathematical Subject Classification: 22E30, 43A80.     

On the Discrete Hardy Inequality

Necessary and sufficient conditions for the boundedness of thediscrete Hardy operator of the form , from to when 0 < q < 1 <p , is given.