A Characterization of Finitely Decidable Congruence Modular Varieties

For every finitely generated, congruence modular variety of finite type we find a finite family of finite rings such that the variety is finitely decidable if and only if is congruence permutable and residually small, all solvable congruences in finite algebras from are Abelian, each congruence above the centralizer of the monolith of a subdirectly irreducible algebra from is comparable with all congruences of , each homomorphic image of a subdirectly irreducible algebra with a non-Abelian monolith has a non-Abelian monolith, and, for each ring from , the variety of -modules is finitely decidable.

     

Mean-boundedness and Littlewood-Paley for separation-preserving operators

Suppose that is a -finite measure space, , and is a bounded, invertible, separation-preserving linear operator such that the linear modulus of is mean-bounded. We show that has a spectral representation formally resembling that for a unitary operator, but involving a family of projections in which has weaker properties than those associated with a countably additive Borel spectral measure. This spectral decomposition for is shown to produce a strongly countably spectral measure on the ``dyadic sigma-algebra' of , and to furnish with abstract analogues of the classical Littlewood-Paley and Vector-Valued M. Riesz Theorems for .

     

The Mizohata complex

This paper deals with the local solvability of systems of first order linear partial differential equations defined by a germ at of a -valued, formally integrable (), 1-form with nondegenerate Levi form. More precisely, the size of the obstruction to the solvability, for -forms , of the equation

where is a given -form satisfying is estimated in terms of the De Rham cohomology relative to

     

Signed Quasi-Measures

Let be a compact Hausdorff space and let denote the subsets of which are either open or closed. A quasi-linear functional is a map which is linear on singly generated subalgebras and such that for some . There is a one-to-one correspondence between the quasi-linear functional on and the set functions such that i) , ii) If with and disjoint, then , iii) There is an such that whenever are disjoint open sets, , and iv) if is open and , there is a compact such that whenever is open, then . The space of quasi-linear functionals is investigated and quasi-linear maps between two spaces are studied.

     

Only single twists on unknots can produce composite knots

Let be a knot in the -sphere , and a disc in meeting transversely more than once in the interior. For non-triviality we assume that over all isotopy of . Let () be a knot obtained from by cutting and -twisting along the disc (or equivalently, performing -Dehn surgery on ). Then we prove the following: (1) If is a trivial knot and is a composite knot, then ; (2) if is a composite knot without locally knotted arc in and is also a composite knot, then . We exhibit some examples which demonstrate that both results are sharp. Independently Chaim Goodman-Strauss has obtained similar results in a quite different method.

     

Herz-Schur multipliers and weakly almost periodic functions on locally compact groups

For a locally compact group and , let be the Herz-Figà-Talamanca algebra and the Herz-Schur multipliers of , and the multipliers of . Let be the algebra of continuous weakly almost periodic functions on . In this paper, we show that (1), if is a noncompact nilpotent group or a noncompact [IN]-group, then contains a linear isometric copy of ; (2), for a noncommutative free group is a proper subset of .

     

Differential operators on Stanley-Reisner rings

Let be an algebraically closed field of characteristic zero, and let be a polynomial ring. Suppose that is an ideal in that may be generated by monomials. We investigate the ring of differential operators on the ring , and , the idealiser of in . We show that and are always right Noetherian rings. If is a square-free monomial ideal then we also identify all the two-sided ideals of . To each simplicial complex on there is a corresponding square-free monomial ideal , and the Stanley-Reisner ring associated to is defined to be . We find necessary and sufficient conditions on for to be left Noetherian.

     

Existence and nonexistence of global positive solutions to nonlinear diffusion problems with nonlinear absorption through the boundary

This paper deals with the existence and nonexistence of global positive solutions to in ,

and in . Here is a parameter, is a bounded smooth domain. After pointing out the mistakes in Global behavior of positive solutions to nonlinear diffusion problems with nonlinear absorption through the boundary, SIAM J. Math. Anal. 24 (1993), 317-326, by N. Wolanski, which claims that, for the ball of , the positive solution exists globally if and only if , we reconsider the same problem in general bounded domain and obtain that every positive solution exists globally if and only if .

     

Wavelet transform and orthogonal decomposition of

Let be the Weyl-Poincaré group and be the Iwasawa decomposition of with . Then the ``affine Weyl-Poincaré group' can be realized as the complex tube domain or the classical Cartan domain . The square-integrable representations of and give the admissible wavelets and wavelet transforms. An orthogonal basis of the set of admissible wavelets associated to is constructed, and it gives an orthogonal decomposition of space on (or the Cartan domain
) with every component being the range of wavelet transforms of functions in with .

     

The quantum analog of a symmetric pair: a construction in type

Let be the ideal in the enveloping algebra of generated by the maximal compact subalgebra of . In this paper we construct an analog of in the quantized enveloping algebra corresponding to a type diagram at generic . We find generators for and explicit bases for .

     

Boundary limits and non-integrability of )

In this paper we consider weighted non-tangential and tangential boundary limits of non-negative functions on the unit ball in that are subharmonic with respect to the Laplace-Beltrami operator on . Since the operator is invariant under the group of holomorphic automorphisms of , functions that are subharmonic with respect to are usually referred to as -subharmonic functions. Our main result is as follows: Let be a non-negative -subharmonic function on satisfying

for some and some , where is the -invariant measure on . Suppose . Then for a.e. ,

uniformly as in each , where for ( when )

We also prove that for the only non-negative -subharmonic function satisfying the above integrability criteria is the zero function.

     

Bloch constants of bounded symmetric domains

Let and be two irreducible bounded symmetric domains in the complex spaces and respectively. Let be the Euclidean metric on and the Bergman metric on . The Bloch constant is defined to be the supremum of , taken over all the holomorphic functions and , and nonzero vectors . We find the constants for all the irreducible bounded symmetric domains and . As a special case we answer an open question of Cohen and Colonna.

     

Symmetric powers of complete modules over a two-dimensional regular local ring

Let be a two-dimensional regular local ring with infinite residue field. For a finitely generated, torsion-free -module , write for the th symmetric power of , mod torsion. We study the modules , , when is complete (i.e., integrally closed). In particular, we show that , for any minimal reduction and that the ring is Cohen-Macaulay.

     

Coherent functors, with application to torsion in the Picard group

Let be a commutative noetherian ring. We investigate a class of functors from commutative -algebras to sets, which we call coherent. When such a functor in fact takes its values in abelian groups, we show that there are only finitely many prime numbers such that is infinite, and that none of these primes are invertible in . This (and related statements) yield information about torsion in . For example, if is of finite type over , we prove that the torsion in is supported at a finite set of primes, and if is infinite, then the prime is not invertible in . These results use the (already known) fact that if such an is normal, then is finitely generated. We obtain a parallel result for a reduced scheme of finite type over . We classify the groups which can occur as the Picard group of a scheme of finite type over a finite field.

     

Automorphism Groups and Invariant Subspace Lattices

Let be a - dynamical system and let be the analytic subalgebra of . We extend the work of Loebl and the first author that relates the invariant subspace structure of for a -representation on a Hilbert space , to the possibility of implementing on We show that if is irreducible and if lat is trivial, then is ultraweakly dense in We show, too, that if satisfies what we call the strong Dirichlet condition, then the ultraweak closure of is a nest algebra for each irreducible representation Our methods give a new proof of a ``density' theorem of Kaftal, Larson, and Weiss and they sharpen earlier results of ours on the representation theory of certain subalgebras of groupoid -algebras.

     

On the second adjunction mapping. The case of a 1-dimensional image

Let be a very ample line bundle on an -dimensional projective manifold , i.e., assume that for some embedding . In this article, a study is made of the meromorphic map, , associated to in the case when the Kodaira dimension of is , and has a -dimensional image. Assume for simplicity that . The first main result of the paper shows that is a morphism if either or . The second main result of this paper shows that if , then the genus, , of a fiber, , of the map induced by on hyperplane sections is . Moreover, if then , a connected component of a general fiber of is either a surface or the blowing up at one point of a surface, and . Finally the structure of the finite to one part of the Remmert-Stein factorization of is worked out.

     

Hecke algebras,

The Donald-Flanigan conjecture asserts that the integral group ring of a finite group can be deformed to an algebra over the power series ring with underlying module such that if is any prime dividing then is a direct sum of total matric algebras whose blocks are in natural bijection with and of the same dimensions as those of We prove this for using the natural representation of its Hecke algebra by quantum Yang-Baxter matrices to show that over localized at the multiplicatively closed set generated by and all , the Hecke algebra becomes a direct sum of total matric algebras. The corresponding ``canonical" primitive idempotents are distinct from Wenzl's and in the classical case (), from those of Young.

     

The generalized Berg theorem and BDF-theorem

Let be a separable simple -algebra with finitely many extreme traces. We give a necessary and sufficient condition for an essentially normal element , i.e., is normal ( is the quotient map), having the form for some normal element and We also show that a normal element can be quasi-diagonalized if and only if the Fredholm index for all In the case that is a simple -algebra of real rank zero, with stable rank one and with continuous scale, and has countable rank, we show that a normal element with zero Fredholm index can be written as

where is an (increasing) approximate identity for consisting of projections, is a bounded sequence of numbers and with for any given

     

Randomness and semigenericity

Let contain only the equality symbol and let be an arbitrary finite symmetric relational language containing . Suppose probabilities are defined on finite structures with `edge probability' . By , the almost sure theory of random -structures we mean the collection of -sentences which have limit probability 1. denotes the theory of the generic structures for (the collection of finite graphs with hereditarily nonnegative). . , the almost sure theory of random -structures, is the same as the theory of the -generic model. This theory is complete, stable, and nearly model complete. Moreover, it has the finite model property and has only infinite models so is not finitely axiomatizable.