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.

     

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.

     

Contractions on a manifold polarized by an ample vector bundle

A complex manifold of dimension together with an ample vector bundle on it will be called a generalized polarized variety. The adjoint bundle of the pair is the line bundle . We study the positivity (the nefness or ampleness) of the adjoint bundle in the case . If this was previously done in a series of papers by Ye and Zhang, by Fujita, and by Andreatta, Ballico and Wisniewski.

If is nef then, by the Kawamata-Shokurov base point free theorem, it supports a contraction; i.e. a map from onto a normal projective variety with connected fiber and such that , for some ample line bundle on . We describe those contractions for which . We extend this result to the case in which has log terminal singularities. In particular this gives Mukai's conjecture 1 for singular varieties. We consider also the case in which for every fiber and is birational.

     

Linear isometries between subspaces of continuous functions

We say that a linear subspace of is strongly separating if given any pair of distinct points of the locally compact space , then there exists such that . In this paper we prove that a linear isometry of onto such a subspace of induces a homeomorphism between two certain singular subspaces of the Shilov boundaries of and , sending the Choquet boundary of onto the Choquet boundary of . We also provide an example which shows that the above result is no longer true if we do not assume to be strongly separating. Furthermore we obtain the following multiplicative representation of : for all and all , where is a unimodular scalar-valued continuous function on . These results contain and extend some others by Amir and Arbel, Holszty\'{n}ski, Myers and Novinger. Some applications to isometries involving commutative Banach algebras without unit are announced.

     

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.

     

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 .

     

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.

     

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 .

     

Generalized Weil's reciprocity law and multiplicativity theorems

Fix a one-dimensional group variety with Euler-characteristic , and a quasi-projective variety , both defined over . For any and constructible sheaf on , we construct an invariant , which provides substantial information about the topology of the fiber-structure of and the structure of along the fibers of . Moreover, is a group homomorphism.

     

Restriction of stable bundles in characteristic

Let be an algebraically closed field of characteristic . Let be a nonsingular projective variety defined over and an ample line bundle on . We shall prove that there exists an explicit number such that if is a -stable vector bundle of rank at most three, then the restriction is -stable for all and all smooth irreducible divisors . This result has implications to the geometry of the moduli space of -stable bundles on a surface or a projective space.

     

Essential embedding of cyclic modules in projectives

Let be a ring and its injective envelope. We show that if every simple right -module embeds in and every cyclic submodule of is essentially embeddable in a projective module, then has finite essential socle. As a consequence, we prove that if each finitely generated right -module is essentially embeddable in a projective module, then is a quasi-Frobenius ring. We also obtain several other applications and, among them: a) we answer affirmatively a question of Al-Huzali, Jain, and López-Permouth, by showing that a right CEP ring (i.e., a ring such that every cyclic right module is essentially embeddable in a projective module) is always right artinian; b) we prove that if is right FGF (i.e., any finitely generated right -module embeds in a free module) and right CS, then is quasi-Frobenius.

     

Kaehler structures on

Let be a compact connected semi-simple Lie group, let , and let be an Iwasawa decomposition. To a given -invariant Kaehler structure on , there corresponds a pre-quantum line bundle on . Following a suggestion of A.S. Schwarz, in a joint paper with V. Guillemin, we studied its holomorphic sections as a -representation space. We defined a -invariant -structure on , and let denote the space of square-integrable holomorphic sections. Then is a unitary -representation space, but not all unitary irreducible -representations occur as subrepresentations of . This paper serves as a continuation of that work, by generalizing the space considered. Let be a Borel subgroup containing , with commutator subgroup . Instead of working with , we consider , for all parabolic subgroups containing . We carry out a similar construction, and recover in the unitary irreducible -representations previously missing. As a result, we use these holomorphic sections to construct a model for : a unitary -representation in which every irreducible -representation occurs with multiplicity one.

     

Coloring graphs with fixed genus and girth

It is well known that the maximum chromatic number of a graph on the orientable surface is . We prove that there are positive constants such that every triangle-free graph on has chromatic number less than and that some triangle-free graph on has chromatic number at least . We obtain similar results for graphs with restricted clique number or girth on or . As an application, we prove that an -polytope has chromatic number at most . For specific surfaces we prove that every graph on the double torus and of girth at least six is 3-colorable and we characterize completely those triangle-free projective graphs that are not 3-colorable.

     

The nonexistence of expansive homeomorphisms of a class of continua which contains all decomposable circle-like continua

A homeomorphism of a compactum with metric is expansive if there is such that if and , then there is an integer such that . It is well-known that -adic solenoids () admit expansive homeomorphisms, each is an indecomposable continuum, and cannot be embedded into the plane. In case of plane continua, the following interesting problem remains open: For each , does there exist a plane continuum so that admits an expansive homeomorphism and separates the plane into components? For the case , the typical plane continua are circle-like continua, and every decomposable circle-like continuum can be embedded into the plane. Naturally, one may ask the following question: Does there exist a decomposable circle-like continuum admitting expansive homeomorphisms? In this paper, we prove that a class of continua, which contains all chainable continua, some continuous curves of pseudo-arcs constructed by W. Lewis and all decomposable circle-like continua, admits no expansive homeomorphisms. In particular, any decomposable circle-like continuum admits no expansive homeomorphism. Also, we show that if is an expansive homeomorphism of a circle-like continuum , then is itself weakly chaotic in the sense of Devaney.

     

Integer translation of meromorphic functions

Let be a given open set in the complex plane. We prove that there is an entire function such that its integer translations forms a normal family in a neighborhood of exactly for in if and only if is periodic with period 1, i.e., for all .

     

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