Tauberian theorems and stability of solutions of the Cauchy problem

Let be a bounded, strongly measurable function with values in a Banach space , and let be the singular set of the Laplace transform in . Suppose that is countable and uniformly for , as , for each in . It is shown that

as , for each in ; in particular, if is uniformly continuous. This result is similar to a Tauberian theorem of Arendt and Batty. It is obtained by applying a result of the authors concerning local stability of bounded semigroups to the translation semigroup on , and it implies several results concerning stability of solutions of Cauchy problems.

     

Generalized Hestenes' Lemma and extension of functions

Suppose we have an -jet field on which is a Whitney field on the nonsingular part of . We show that, under certain hypotheses about the relationship between geodesic and euclidean distance on , if the field is flat enough at the singular part , then it is a Whitney field on (the order of flatness required depends on the coefficients in the hypotheses). These hypotheses are satisfied when is subanalytic. In Section II, we show that a function on can be extended to one on if the differential goes to faster than the order of divergence of the principal curvatures of and if the first covariant derivative of is sufficiently flat. For the general case of functions with , we give a similar result for in Section III.

     

Factorisation in nest algebras. II

The main result of this paper is Theorem 5, which provides a necessary and sufficient condition on a positive operator for the existence of an operator in the nest algebra of a nest satisfying (resp. . In Section 3 we give a new proof of a result of Power concerning outer factorisation of operators. We also show that a positive operator has the property that there exists for every nest an operator in satisfying (resp. ) if and only if is a Fredholm operator. In Section 4 we show that for a given operator in there exists an operator in satisfying if and only if the range of is equal to the range of some operator in . We also determine the algebraic structure of the set of ranges of operators in . Let be the set of positive operators for which there exists an operator in satisfying . In Section 5 we obtain information about this set. In particular we discuss the following question: Assume and are positive operators such that and belongs to . Which further conditions permit us to conclude that belongs to ?

     

The average edge order of triangulations of 3-manifolds with boundary

Feng Luo and Richard Stong introduced the average edge order of a triangulation and showed in particular that for closed 3-manifolds being less than 4.5 implies that is on . In this paper, we establish similar results for 3-manifolds with non-empty boundary; in particular it is shown that being less than 4 implies that is on the 3-ball.

     

Equivalence of norms on operator space tensor products of -algebras

The Haagerup norm on the tensor product of two -algebras and is shown to be Banach space equivalent to either the Banach space projective norm or the operator space projective norm if and only if either or is finite dimensional or and are infinite dimensional and subhomogeneous. The Banach space projective norm and the operator space projective norm are equivalent on if and only if or is subhomogeneous.

     

An intersection number for the punctual Hilbert scheme of a surface

We compute the intersection number between two cycles and of complementary dimensions in the Hilbert scheme parameterizing subschemes of given finite length of a smooth projective surface . The -cycle corresponds to the set of finite closed subschemes the support of which has cardinality 1. The -cycle consists of the closed subschemes the support of which is one given point of the surface. Since is contained in , indirect methods are needed. The intersection number is , answering a question by H. Nakajima.

     

Approximation by polynomials with nonnegative coefficients and the spectral theory of positive operators

For a compact subset of symmetric with respect to conjugation and a continuous function, we obtain sharp conditions on and that insure that can be approximated uniformly on by polynomials with nonnegative coefficients. For a real Banach space, a closed but not necessarily normal cone with , and a bounded linear operator with , we use these approximation theorems to investigate when the spectral radius of belongs to its spectrum . A special case of our results is that if is a Hilbert space, is normal and the 1-dimensional Lebesgue measure of is zero, then . However, we also give an example of a normal operator (where is unitary and ) for which and .

     

The

In this paper we analyze the localization of , the fiber of the double suspension map , with respect to . If four cells at the bottom of , the th extended power spectrum of the Moore spectrum, are collapsed to a point, then one obtains a spectrum . Let be the James-Hopf map followed by the collapse map. Then we show that the secondary suspension map has a lifting to the fiber of and this lifting is shown to be a -periodic equivalence, hence an -equivalence.

     

The Classification of the Simple Modular Lie Algebras: VI. Solving the Final Case

We investigate the structure of simple Lie algebras over an algebraically closed field of characteristic . Let denote a torus in the -envelope of in of maximal dimension. We classify all for which every 1-section with respect to every such torus is solvable. This settles the remaining case of the classification of these algebras.

     

Induction theorems on the stable rationality of the center of the ring of generic matrices

Following Procesi and Formanek, the center of the division ring of generic matrices over the complex numbers is stably equivalent to the fixed field under the action of , of the function field of the group algebra of a -lattice, denoted by . We study the question of the stable rationality of the center over the complex numbers when is a prime, in this module theoretic setting. Let be the normalizer of an -sylow subgroup of . Let be a -lattice. We show that under certain conditions on , induction-restriction from to does not affect the stable type of the corresponding field. In particular, and are stably isomorphic and the isomorphism preserves the -action. We further reduce the problem to the study of the localization of at the prime ; all other primes behave well. We also present new simple proofs for the stable rationality of over , in the cases and .

     

Iterations of holomorphic Collet-Eckmann maps: conformal and invariant measures. Appendix: On non-renormalizable quadratic polynomials

We prove that for every rational map on the Riemann sphere , if for every -critical point whose forward trajectory does not contain any other critical point, the growth of is at least of order for an appropriate constant as , then . Here is the so-called essential, dynamical or hyperbolic dimension, is Hausdorff dimension of and is the minimal exponent for conformal measures on . If it is assumed additionally that there are no periodic parabolic points then the Minkowski dimension (other names: box dimension, limit capacity) of also coincides with . We prove ergodicity of every -conformal measure on assuming has one critical point , no parabolic, and . Finally for every -conformal measure on (satisfying an additional assumption), assuming an exponential growth of , we prove the existence of a probability absolutely continuous with respect to , -invariant measure. In the Appendix we prove also for every non-renormalizable quadratic polynomial with not in the main cardioid in the Mandelbrot set.

     

Composition factors of indecomposable modules

Let be a connected, basic finite dimensional algebra over an algebraically closed field. Our main aim is to prove that if is biserial, its ordinary quiver has no loop and every indecomposable -module is uniquely determined by its composition factors, then each indecomposable -module is multiplicity-free.

     

Morita equivalence for crossed products by Hilbert -bimodules

We introduce the notion of the crossed product of a -algebra by a Hilbert -bimodule . It is shown that given a -algebra which carries a semi-saturated action of the circle group (in the sense that is generated by the spectral subspaces and ), then is isomorphic to the crossed product . We then present our main result, in which we show that the crossed products and are strongly Morita equivalent to each other, provided that and are strongly Morita equivalent under an imprimitivity bimodule satisfying as Hilbert -bimodules. We also present a six-term exact sequence for -groups of crossed products by Hilbert -bimodules.

     

Solutions of nonlinear differential equations on a Riemannian manifold and their trace on the Martin boundary

Let be a second order elliptic differential operator on a Riemannian manifold with no zero order terms. We say that a function is -harmonic if . Every positive -harmonic function has a unique representation

where is the Martin kernel, is the Martin boundary and is a finite measure on concentrated on the minimal part of . We call the trace of on . Our objective is to investigate positive solutions of a nonlinear equation

for [the restriction is imposed because our main tool is the -superdiffusion, which is not defined for ]. We associate with every solution of (*) a pair , where is a closed subset of and is a Radon measure on . We call the trace of on . is empty if and only if is dominated by an -harmonic function. We call such solutions moderate. A moderate solution is determined uniquely by its trace. In general, many solutions can have the same trace. In an earlier paper, we investigated the case when is a second order elliptic differential operator in and is a bounded smooth domain in . We obtained necessary and sufficient conditions for a pair to be a trace, and we gave a probabilistic formula for the maximal solution with a given trace. The general theory developed in the present paper is applicable, in particular, to elliptic operators with bounded coefficients in an arbitrary bounded domain of , assuming only that the Martin boundary and the geometric boundary coincide.

     

Baire and

Let be a locally compact Hausdorff space and let be the Banach space of all bounded complex Radon measures on . Let and be the -rings generated by the compact subsets and by the compact subsets of , respectively. The members of are called Baire sets of and those of are called -Borel sets of (since they are precisely the -bounded Borel sets of ). Identifying with the Banach space of all Borel regular complex measures on , in this note we characterize weakly compact subsets of in terms of the Baire and -Borel restrictions of the members of . These characterizations permit us to give a generalization of a theorem of Dieudonné which is stronger and more natural than that given by Grothendieck.

     

A -deformation of a trivial symmetric group action

Let be the arrangement of hyperplanes consisting of the reflecting hyperplanes for the root system . Let be the Varchenko matrix for this arrangement with all hyperplane parameters equal to . We show that is the matrix with rows and columns indexed by permutations with entry equal to where is the number of inversions of . Equivalently is the matrix for left multiplication on by

Clearly commutes with the right-regular action of on . A general theorem of Varchenko applied in this special case shows that is singular exactly when is a root of for some between and . In this paper we prove two results which partially solve the problem (originally posed by Varchenko) of describing the -module structure of the nullspace of in the case that is singular. Our first result is that

in the case that where Lie denotes the multilinear part of the free Lie algebra with generators. Our second result gives an elegant formula for the determinant of restricted to the virtual -module with characteristic the power sum symmetric function .

     

Test ideals in quotients of -finite regular local rings

Let be an -finite regular local ring and an ideal contained in . Let . Fedder proved that is -pure if and only if . We have noted a new proof for his criterion, along with showing that , where is the pullback of the test ideal for . Combining the the -purity criterion and the above result we see that if is -pure then is also -pure. In fact, we can form a filtration of , that stabilizes such that each is -pure and its test ideal is . To find examples of these filtrations we have made explicit calculations of test ideals in the following setting: Let , where is either a polynomial or a power series ring and is generated by monomials and the are regular. Set . Then .

     

Picard groups and infinite matrix rings

We describe a connection between the Picard group of a ring with local units and the Picard group of the unital overring . Using this connection, we show that the three groups , , and are isomorphic for any unital ring . Furthermore, each element of arises from an automorphism of , which yields an isomorphsm between and . As one application we extend a classical result of Rosenberg and Zelinsky by showing that the group is abelian for any commutative unital ring .

     

Widths of Subgroups

We say that the width of an infinite subgroup in is if there exists a collection of essentially distinct conjugates of such that the intersection of any two elements of the collection is infinite and is maximal possible. We define the width of a finite subgroup to be . We prove that a quasiconvex subgroup of a negatively curved group has finite width. It follows that geometrically finite surfaces in closed hyperbolic -manifolds satisfy the -plane property for some .