Wave Invariants for Non-degenerate Closed Geodesics

((Without abstract)) Submitted: January 1997, Revised version: May 1997, Final version: September 1997     

$ \pi _1 $ of Symplectic Automorphism Groups and Invertibles in Quantum Homology Rings

((Without abstract)) Submitted: March 1996, revised: April 1997, final version: September 1997     

Lipschitz maps and nets in Euclidean space

((Without abstract)) Submitted: April 1997, Final version: July 1997     

Complex projective surfaces and infinite groups

((Without abstract)) Submitted: March 1997, Final version: September 1997     

A Geometric Inequality with Applications to the Kakeya Problem in Three Dimensions

((Without abstract)) Submitted: May 1997, Revised version: August 1997     

Bounded and L2 harmonic forms on universal covers

((Without abstract)) Submitted: April 1997, Revision: October 1997     

Blossoming: A Geometrical Approach

A geometrical approach of a notion of blossom for piecewise smooth Chebyshev functions is developed by considering convenient intersections of osculating flats. A subblossoming principle allows us to obtain all the expected properties and leads to the notion of blossom for splines based on a given piecewise smooth Chebyshev function. January 7, 1997. Date revised: October 1, 1997. Date accepted: December 22, 1997.     

Recurrence Formula for Polynomials of Two Variables, Orthogonal with Respect to Rotation Invariant Measures

We give the recurrence formula satisfied by polynomials of two variables, orthogonal with respect to a rotation invariant measure. Moreover, we show that for polynomials satisfying such a recurrence formula, there exists an orthogonality measure which is rotation invariant. We also compute explicitly the recurrence coefficients for the disk polynomials. January 4, 1997. Date revised: October 15, 1997. Date accepted: October 21, 1997.     

Multiplicities of mod p Galois representations

We study the multiplicity with which 2-dimensional mod p Galois representations occur in Jacobians of modular curves. Received: 26 February 1997 / Revised version: 2 July 1997     

On Bernstein-Sato polynomials

We show that for fixed and the set of Bernstein-Sato polynomials of all the polynomials in at most variables of degrees at most is finite. As a corollary, we show that there exists an integer depending only on and such that generates as a module over the ring of the -linear differential operators of , where is an arbitrary field of characteristic 0, is the ring of polynomials in variables over and is an arbitrary non-zero polynomial of degree at most .

     

Congruences between Modular Forms, Cyclic Isogenies of Modular Elliptic Curves, and Integrality of -Functions

Let be a congruence subgroup of type and of level . We study congruences between weight 2 normalized newforms and Eisenstein series on modulo a prime above a rational prime . Assume that , is a common eigenfunction for all Hecke operators and is ordinary at . We show that the abelian variety associated to and the cuspidal subgroup associated to intersect non-trivially in their -torsion points. Let be a modular elliptic curve over with good ordinary reduction at . We apply the above result to show that an isogeny of degree divisible by from the optimal curve in the -isogeny class of elliptic curves containing to extends to an étale morphism of Néron models over if . We use this to show that -adic distributions associated to the -adic -functions of are -valued.

     

A Strict Version of the Non-commutative Urysohn Lemma

Given a pair , of -commuting, hereditary -subalgebras of a unital -algebra , such that is -unital and , there is an element in , with , such that is strictly positive in and is strictly positive in in . Moreover, is strictly positive in in .

     

Isomorphism of lattices of recursively enumerable sets

Let , and for , let be the lattice of subsets of which are recursively enumerable relative to the ``oracle' . Let be , where is the ideal of finite subsets of . It is established that for any , is effectively isomorphic to if and only if , where is the Turing jump of . A consequence is that if , then . A second consequence is that can be effectively embedded into preserving least and greatest elements if and only if .

     

Bilocal derivations of standard operator algebras

In this paper, we shall show the following two results: (1) Let be a standard operator algebra with , if is a linear mapping on which satisfies that maps into for all , then is of the form for some in . (2) Let be a Hilbert space, if is a norm-continuous linear mapping on which satisfies that maps into for all self-adjoint projection in , then is of the form for some in .

     

Regularity of Multivariate Refinable Functions

The regularity of a univariate compactly supported refinable function is known to be related to the spectral properties of an associated transfer operator. In the case of multivariate refinable functions with a general dilation matrix A , although factorization techniques, which are typically used in the univariate setting, are no longer applicable, we derive similar results that also depend on the spectral properties of A . September 30, 1996. Dates revised: December 1, 1996; February 14, 1997; August 1, 1997; November 11, 1997. Date accepted: November 14, 1997.     

Inner derivations on ultraprime normed algebras

We show that, for every ultraprime Banach algebra , there exists a positive number satisfying for all in , where denotes the centre of and denotes the inner derivation on induced by . Moreover, the number depends only on the ``constant of ultraprimeness' of .

     

Invariance of spectrum for representations of -algebras on Banach spaces

Let be a Banach space, a unital -algebra, and an injective, unital homomorphism. Suppose that there exists a function such that, for all , and all ,

(a) ,

(b) ,

(c) .
Then for all , the spectrum of in equals the spectrum of as a bounded linear operator on . If satisfies an additional requirement and is a -algebra, then the Taylor spectrum of a commuting -tuple of elements of equals the Taylor spectrum of the -tuple in the algebra of bounded operators on . Special cases of these results are (i) if is a closed subspace of a unital -algebra which contains as a unital -subalgebra such that , and only if , then for each , the spectrum of in is the same as the spectrum of left multiplication by on ; (ii) if is a unital -algebra and is an essential closed left ideal in , then an element of is invertible if and only if left multiplication by on is bijective; and (iii) if is a -algebra, is a Hilbert -module, and is an adjointable module map on , then the spectrum of in the -algebra of adjointable operators on is the same as the spectrum of as a bounded operator on . If the algebra of adjointable operators on is a -algebra, then the Taylor spectrum of a commuting -tuple of adjointable operators on is the same relative to the algebra of adjointable operators and relative to the algebra of all bounded operators on .

     

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.

     

Commutative group algebras of -summable abelian groups

In this note we study the commutative modular and semisimple group rings of -summable abelian -groups, which group class was introduced by R. Linton and Ch. Megibben. It is proved that is -summable if and only if is -summable, provided is an abelian group and is a commutative ring with 1 of prime characteristic , having a trivial nilradical. If is a -summable -group and the group algebras and over a field of characteristic are -isomorphic, then is a -summable -group, too. In particular provided is totally projective of a countable length.

Moreover, when is a first kind field with respect to and is -torsion, is -summable if and only if is a direct sum of cyclic groups.

     

An Engel condition with derivation for left ideals

We generalize a number of results in the literature by proving the following theorem: Let be a semiprime ring, a nonzero derivation of , a nonzero left ideal of , and let . If for some positive integers , and all , the identity holds, then either or else the ideal of generated by and is in the center of . In particular, when is a prime ring, is commutative.