Fixed points of the mapping class group in the moduli spaces

Let be a compact oriented surface with or without boundary components. In this note we prove that if then there exist infinitely many integers such that there is a point in the moduli space of irreducible flat connections on which is fixed by any orientation preserving diffeomorphism of . Secondly we prove that for each orientation preserving diffeomorphism of and each there is some such that has a fixed point in the moduli space of irreducible flat connections on . Thirdly we prove that for all there exists an integer such that the 'th power of any diffeomorphism fixes a certain point in the moduli space of irreducible flat connections on .

     

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 .

     

The structure of hypersurfaces with some curvature conditions

Let be a hypersurface in , and let denote the mean curvature and the scalar curvature of respectively. We show that if is compact and , then is diffeomorphic to . Also we prove that if is complete, is constant and , then is or or .

     

An invariance of domain result for multi-valued maximal monotone operators whose domains do not necessarily contain any open sets

Let be a real, reflexive, locally uniformly convex Banach space with locally uniformly convex. Let be a maximal monotone operator and open and bounded. Assume that is pathwise connected and such that and Then If, moreover, is of type () on then may be replaced above by The significance of this result lies in the fact that it holds for multi-valued mappings which do not have to satisfy It has also been used in this paper in order to establish a general ``invariance of domain' result for maximal monotone operators, and may be applied to a greater variety of problems involving partial differential equations. No degree theory has been used. In addition to the above, necessary and sufficient conditions are given for the existence of a zero (in an open and bounded set ) of a completely continuous perturbation of a maximal monotone operator such that is locally monotone on

     

A note on the relative class number in function fields

Let be a finite field, , and . Let be the field extension of obtained by adjoining the -torsion on the Carlitz module. The class number of can be written as a product . The number is called the relative class number. In this paper a formula for is derived which is the analogue of the Maillet determinant formula for the relative class number of the cyclotomic field of -th roots of unity. Some consequences of this formula are also derived.

     

The expected value of the number of real zeros of a random sum of Legendre polynomials

It is known that the expected number of zeros in the interval of the sum , in which is the normalized Legendre polynomial of degree and the coefficients are independent normally distributed random variables with mean 0 and variance 1, is asymptotic to for large . We improve this result and show that this expected number is for any positive .

     

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 .

     

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 .

     

A note on the fundamental groups of manifolds with almost nonnegative curvature

We show that given and , there exists a positive number such that if a closed -manifold satisfies and , then is almost abelian.

     

Less Saturated Ideals

We prove the following:

(1)

If is weakly inaccessible then is not -saturated.

(2)

If is weakly inaccessible and is regular then is not -saturated.

(3)

If is singular then is not -saturated.

Combining this with previous results of Shelah, one obtains the following:

(A)

If then is not -saturated.

(B)

If then is not -saturated.

     

Finite linear groups and theorems of Minkowski and Schur

Let be a finite group with a faithful rational valued character of degree . A theorem of I. Schur gives a bound for the order of in terms of , generalizing an earlier result of H. Minkowski who showed that the same bound holds if . This note contains strengthened versions of these results which in particular show that a -subgroup of of maximum possible order contains a reflection.

     

Gröbner duality and multiple points in linearly general position

It is proved for each , , that a primary -dimensional scheme in of degree in linearly general position lies in a rational normal scroll of dimension .

     

A convolution estimate for a measure on a curve in

Let and fix an interval . If is the operator on defined by , then maps into .

     

Weakly coupled bound states in quantum waveguides

We study the eigenvalue spectrum of Dirichlet Laplacians which model quantum waveguides associated with tubular regions outside of a
bounded domain. Intuitively, our principal new result in two dimensions asserts that any domain obtained by adding an arbitrarily small ``bump' to the tube (i.e., , open and connected, outside a bounded region) produces at least one positive eigenvalue below the essential spectrum of the Dirichlet Laplacian . For sufficiently small ( abbreviating Lebesgue measure), we prove uniqueness of the ground state of and derive the ``weak coupling' result using a Birman-Schwinger-type analysis. As a corollary of these results we obtain the following surprising fact: Starting from the tube with Dirichlet boundary conditions at , replace the Dirichlet condition by a Neumann boundary condition on an arbitrarily small segment , , of . If denotes the resulting Laplace operator in , then has a discrete eigenvalue in no matter how small is.

     

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 .

     

On the curves of contact on surfaces in a projective space. III

Suppose a smooth curve is a set-theoretic complete intersection of two surfaces and with the multiplicity of along less than or equal to the multiplicity of along . One obtains a relation between the degrees of , and , the genus of , and the multiplicity of along in case has only ordinary singularities. One obtains (in the characteristic zero case) that a nonsingular rational curve of degree 4 in is not set-theoretically an intersection of 2 surfaces, provided one of them has at most ordinary singularities. The same result holds for a general nonsingular rational curve of degree .

     

On a class of subalgebras of and the intersection of their free maximal ideals

Let be a Tychonoff space and a subalgebra of containing . Suppose that is the set of all functions in with compact support. Kohls has shown that is precisely the intersection of all the free ideals in or in . In this paper we have proved the validity of this result for the algebra . Gillman and Jerison have proved that for a realcompact space , is the intersection of all the free maximal ideals in . In this paper we have proved that this result does not hold for the algebra , in general. However we have furnished a characterisation of the elements that belong to all the free maximal ideals in . The paper terminates by showing that for any realcompact space , there exists in some sense a minimal algebra for which becomes -compact. This answers a question raised by Redlin and Watson in 1987. But it is still unsettled whether such a minimal algebra exists with respect to set inclusion.

     

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.

     

Commensurators of parabolic subgroups of Coxeter groups

Let be a Coxeter system, and let be a subset of . The subgroup of generated by is denoted by and is called a parabolic subgroup. We give the precise definition of the commensurator of a subgroup in a group. In particular, the commensurator of in is the subgroup of in such that has finite index in both and . The subgroup can be decomposed in the form where is finite and all the irreducible components of are infinite. Let be the set of in such that for all . We prove that the commensurator of is . In particular, the commensurator of a parabolic subgroup is a parabolic subgroup, and is its own commensurator if and only if .