Topological hochschild homology of extensions of ?/p? by polynomials, and of ?[x]/(xn) and ?[x]/(xn?1)

Topological Hochschild homology is calculated for the rings /p[x]/(f(x)) (where p is prime and f(x) /p[x] any polynomial), [x]/(x ⁿ) and [x]/(x ⁿ–1). A spectral sequence argument is used for calculating the homology of the topological Hochschild homology spectrum, from which its stable homotopy structure can be read off since the spectrum is known for a priori reasons to be a restricted product of Eilenberg-MacLane spectra.     

Two combinatorial problems on posets

Bialostocki proposed the following problem: Let nk2 be integers such that k|n. Let p(n, k) denote the least positive integer having the property that for every poset P, |P|p(n, k) and every Z _k -coloring f: P Z _k there exists either a chain or an antichain A, |A|=n and _aA f(a) 0 (modk). Estimate p(n, k). We prove that there exists a constant c(k), depends only on k, such that (n+k–2)²–c(k) p(n, k) (n+k–2)²+1. Another problem considered here is a 2-dimensional form of the monotone sequence theorem of Erdös and Szekeres. We prove that there exists a least positive integer f(n) such that every integral square matrix A of order f(n) contains a square submatrix B of order n, with all rows monotone sequences in the same direction and all columns monotone sequences in the same direction (direction means increasing or decreasing).     

Time-asymptotic boson states from infinite mean field quantum systems coupled to the boson gas

By means of cocycle techniques in a recent paper, the global dynamics of mean field-boson couplings has been studied. Here, by restricting to the bosonic system the infinite time limit (t ) for very general initial states, one obtains time-asymptotic states on the bosonicC ^*-Weyl algebra, in which one partially rediscovers the collective ordering of the infinite mean field lattice.     

Cyclic actions on some Klein bottle bundles over S1

Leth be a cyclic action of periodn onM, whereM is eitherS ¹×K, K is the Klein bottle or on , the twisted Klein bottle bundle overS ¹, such that there is a fiberingq:MS ¹ with fiber a Klein bottleK or a torusT with respect to which the action is fiber preserving. We classify all such actions and show that they might be distinguished by their fixed points or by their orbit spaces.     

Perturbed unstable variational inequalities with unbounded operators on approximately given sets

The stability and convergence of the solutions of perturbed and regularized variational inequality to the solutions of the primary (unstable a priori) variational inequality with proper monotone operator are investigated. All the objects of inequality: the operatorA, the right-hand partf and the set of constrains are to be perturbed. At the same time no assumptions of boundedness and smoothness of the operatorA are used. The connection between the parameters of perturbations, which guarantees strong convergence of approximate solutions, is established. It is proved that the existence of the solution to the unperturbed variational inequality is necessary and sufficient condition for convergence of the regularized perturbed inequality solutions.This research was supported in part by the Ministry of Science Grant 3481-1-91 and by the Ministry of Absorption Center for Absorption in Science.     

Asymptotics for the remainder of a class of positive quadratures for integrands with an interior singularity

Summary For functions with an interior singularity, the errors of a class of positive quadrature formulae with high algebraic degree are reduced to those of the much simpler Euler-Maclaurin type formulae. Applying this method to certain classes of functions, such as, for example,f(x)=h(x)|x-u| , where >–1, with a sufficiently smooth functionh, we obtain the main term of the error expansion for quadrature rules of ultraspherical type.     

The gauge theorem for a class of additive functionals of zero energy

Summary In earlier works, the gauge theorem was proved for additive functionals of Brownian motion of the form ₀ ^t q(B _s )ds, whereq is a function in the Kato class. Subsequently, the theorem was extended to additive functionals with Revuz measures in the Kato class. We prove that the gauge theorem holds for a large class of additive functionals of zero energy which are, in general, of unbounded variation. These additive functionals may not be semi-martingales, but correspond to a collection of distributions that belong to the Kato class in a suitable sense. Our gauge theorem generalizes the earlier versions of the gauge theorem.Research supported in part by NSA grant MDA-92-H-30324     

Rigidity of harmonic maps of maximum rank

Thehomotopical rank of a mapf:M →N is, by definition, min{dimg(M) ¦g homotopic tof}. We give upper bounds for this invariant whenM is compact Kähler andN is a compact discrete quotient of a classical symmetric space, e.g., the space of positive definite matrices. In many cases the upper bound is sharp and is attained by geodesic immersions of locally hermitian symmetric spaces. An example is constructed (Section 9) to show that there do, in addition, exist harmonic maps of quite a different character. A byproduct is construction of an algebraic surface with large and interesting fundamental group. Finally, a criterion for lifting harmonic maps to holomorphic ones is given, as is a factorization theorem for representations of the fundamental group of a compact Kähler manifold. The technique for the main result is a combination of harmonic map theory, algebra, and combinatorics; it follows the path pioneered by Siu in his ridigity theorem and later extended by Sampson.