Harmonic maps with fixed singular sets

Suppose Ω is a smooth domain in R^m,N is a compact smooth Riemannian manifold, andZ is a fixed compact subset of Ω having finite (m − 3)-dimensional Minkowski content (e.g.,Z ism − 3 rectifiable). We consider various spaces of harmonic mapsu: Ω →N that have a singular setZ and controlled behavior nearZ. We study the structure of such spacesH and questions of existence, uniqueness, stability, and minimality under perturbation. In caseZ = 0,H is a Banach manifold locally diffeomorphic to a submanifold of the product of the boundary data space with a finite-dimensional space of Jacobi fields with controlled singular behavior. In this smooth case, the projection ofu εH tou |ϖΩ is Fredholm of index 0. R. H.’s research partially supported by the National Science Foundation.     

Caratheodory interpolation kernels

Kernels over the unit disk for which a version of Carathéodory interpolation is true are characterized in a simple computationally verifiable manner.     

Methods for parallel integration of stiff systems of odes

This paper presents a class of parallel numerical integration methods for stiff systems of ordinary differential equations which can be partitioned into loosely coupled sub-systems. The formulas are called decoupled backward differentiation formulas, and they are derived from the classical formulas by restricting the implicit part to the diagnonal sub-system. With one or several sub-systems allocated to each processor, information only has to be exchanged after completion of a step but not during the solution of the nonlinear algebraic equations.The main emphasis is on the formula of order 1, the decoupled implicit Euler formula. It is proved that this formula even for a wide range of multirate formulations has an asymptotic global error expansion permitting extrapolation. Besides, sufficient conditions for absolute stability are presented.     

Algebraic models defined over Q of differential manifolds

Here we prove that every compact differential manifold has a smooth algebraic model defined over Q. In dimension 2 we find an algebraic model (may be singular) defined over Q and birational over Q to the projective plane.     

On computing the number of subgroups of a finite Abelian group

We consider the numberN _A (r) of subgroups of orderp ^r ofA, whereA is a finite Abelianp-group of type =₁,₂,..., _l ()), i.e. the direct sum of cyclic groups of order ⁱⁱ. Formulas for computingN _A (r) are well known. Here we derive a recurrence relation forN _A (r), which enables us to prove a conjecture of P. E. Dyubyuk about congruences betweenN _A (r) and the Gaussian binomial coefficient .     

Spline collocation for strongly elliptic equations on the torus

Summary We prove convergence and error estimates in Sobolev spaces for the collocation method with tensor product splines for strongly elliptic pseudodifferential equations on the torus. Examples of applications include elliptic partial differential equations with periodic boundary conditions but also the classical boundary integral operators of potential theory on torus-shaped domains in three or more dimensions. For odd-degree splines, we prove convergence of nodal collocation for any strongly elliptic operator. For even-degree splines and midpoint collocation, we find an additional condition for the convergence which is satisfied for the classical boundary integral operators. Our analysis is a generalization to higher dimensions of the corresponding analysis of Arnold and Wendland [4].     

Ramsey partitions of integers and pair divisions

Ifk ₁ andk ₂ are positive integers, the partitionP = (₁,₂,..., _n ) ofk ₁+k ₂ is said to be a Ramsey partition for the pairk ₁,k ₂ if for any sublistL ofP, either there is a sublist ofL which sums tok ₁ or a sublist ofP –L which sums tok ₂. Properties of Ramsey partitions are discussed. In particular it is shown that there is a unique Ramsey partition fork ₁,k ₂ having the smallest numbern of terms, and in this casen is one more than the sum of the quotients in the Euclidean algorithm fork ₁ andk ₂.An application of Ramsey partitions to the following fair division problem is also discussed: Suppose two persons are to divide a cake fairly in the ratiok ₁k ₂. This can be done trivially usingk ₁+k ₂-1 cuts. However, every Ramsey partition ofk ₁+k ₂ also yields a fair division algorithm. This method yields fewer cuts except whenk ₁=1 andk ₂=1, 2 or 4.     

全变差有界函数列的一致(R)可积性

给出了一致有界单调函数列一致可积性定理 ,由此得出全变差序列有界的收敛函数列的一致可积性 .说明了该结论可判断一些非一致收敛函数列的逐项积分性质 .     

Numerosities of labelled sets: a new way of counting

The notions of “labelled set” and “numerosity” are introduced to generalize the counting process of finite sets. The resulting numbers, called numerosities, are then used to develop nonstandard analysis. The existence of a numerosity function is equivalent to the existence of a selective ultrafilter, hence it is independent of the axioms of ZFC.