Some results on finite Drinfeld modules

Let be a global function field, a degree one prime divisor of and let be the Dedekind domain of functions in regular outside . Let be the Hilbert class field of , the integral closure of in . Let be a rank one normalized Drinfeld -module and let be a prime ideal in . We explicitly determine the finite -module structure of . In particular, if , is an odd prime number and is the Carlitz -module, then the finite -module is always cyclic.

     

On scrambled sets and a theorem of Kuratowski on independent sets

The measure of scrambled sets of interval self-maps was studied by many authors, including Smítal, Misiurewicz, Bruckner and Hu, and Xiong and Yang. In this note, first we introduce the notion of ``-chaos" which is related to chaos in the sense of Li-Yorke, and we prove a general theorem which is an improvement of a theorem of Kuratowski on independent sets. Second, we apply the result to scrambled sets of higher dimensional cases. In particular, we show that if a map of the unit -cube is -chaotic on , then for any there is a map such that and are topologically conjugate, and has a scrambled set which has Lebesgue measure 1, and hence if , then there is a homeomorphism with a scrambled set satisfying that is an -set in and has Lebesgue measure 1.

     

Derived lengths and character degrees

Let be a finite solvable group. Assume that the degree graph of has exactly two connected components that do not contain . Suppose that one of these connected components contains the subset , where and are coprime when . Then the derived length of is less than or equal to .

     

Discriminants of convex curves are homeomorphic

For a given real generic curve let denote the ruled hypersurface in consisting of all osculating subspaces to of codimension 2. In this note we show that for any two convex real projective curves and the pairs and are homeomorphic.

     

The moduli of substructures of a compact complex space

We construct a space of fine moduli for the substructures of an arbitrary compact complex space . A substructure of is given by a subalgebra of the structure sheaf with the additional feature that is also a complex space; and are called equivalent if and only if and are isomorphic as subalgebras of .

Since substructures are quotients, it is only natural to start with the fine moduli space of all complex-analytic quotients of . In order to obtain a representable moduli functor of substructures, we are forced to concentrate on families of quotients which satisfy some flatness condition for relative differential modules of higher order. Considering the corresponding flatification of , we realize that its open subset consisting of all substructures turns out to be a complex space which has the required universal property.

     

Duality and local group cohomology

Let be a group, let be a field, and let be a local system - an upwardly directed collection of subgroups whose union is . In this paper we give a short, elementary proof of the following result: If either is a --bimodule, or else is finite dimensional over its center, then . From this we deduce as easy corollaries some recent results of Meierfrankenfeld and Wehrfritz on the cohomology of a finitary module.

     

Some properties of ordinary sense slice 1-links: some answers to problem (26) of Fox

We prove that, for any ordinary sense slice 1-link , we can define the Arf invariant, and Arf()=0. We prove that, for any -component 1-link , there exists a -component ordinary sense slice 1-link of which is a sublink.

     

A generalization of Carleman's uniqueness theorem and a discrete Phragmén-Lindelöf theorem

Let be a Borel measure on and be its moments. T. Carleman found sharp conditions on the magnitude of for to be uniquely determined by its moments. We show that the same conditions ensure a stronger property: if are the moments of another measure, with then the measure is supported on the interval This result generalizes both the Carleman theorem and a theorem of J. Mikusi\'{n}ski. We also present an application of this result by establishing a discrete version of a Phragmén-Lindelöf theorem.

     

Relative to any nonrecursive set

There is a countable first order structure such that for any set of integers , is not recursive if and only if there is a presentation of which is recursive in .

     

On certain character sums over

Let be the finite field with elements and let denote the ring of polynomials in one variable with coefficients in . Let be a monic polynomial irreducible in . We obtain a bound for the least degree of a monic polynomial irreducible in ( odd) which is a quadratic non-residue modulo . We also find a bound for the least degree of a monic polynomial irreducible in which is a primitive root modulo .

     

Hankel operators on the Bergman space of the unit ball

We characterize the bounded holomorphic functions in the unit ball of for which the operator is compact. For the result was obtained by Axler and Gorkin in 1988 and by Zheng in 1989.

     

On uniqueness of invariant means

The following results on uniqueness of invariant means are shown:

(i) Let be a connected almost simple algebraic group defined over . Assume that , the group of the real points in , is not compact. Let be a prime, and let be the compact -adic Lie group of the -points in . Then the normalized Haar measure on is the unique invariant mean on .

(ii) Let be a semisimple Lie group with finite centre and without compact factors, and let be a lattice in . Then integration against the -invariant probability measure on the homogeneous space is the unique -invariant mean on .

     

Lie Incidence Systems from Projective Varieties

The homogeneous space , where is a simple algebraic group and a parabolic subgroup corresponding to a fundamental weight (with respect to a fixed Borel subgroup of in ), is known in at least two settings. On the one hand, it is a projective variety, embedded in the projective space corresponding to the representation with highest weight . On the other hand, in synthetic geometry, is furnished with certain subsets, called lines, of the form where is a preimage in of the fundamental reflection corresponding to and . The result is called the Lie incidence structure on . The lines are projective lines in the projective embedding. In this paper we investigate to what extent the projective variety data determines the Lie incidence structure.

     

Mean exit time from convex hypersurfaces

L. Karp and M. Pinsky proved that, for small radius , the mean exit time function of an extrinsic -ball in a hypersurface is bounded from below by the corresponding function defined on an extrinsic -ball in . A counterexample given by C. Mueller proves that this inequality doesn't holds in the large. In this paper we show that, if is convex, then the inequality holds for all radii. Moreover, we characterize the equality and show that analogous results are true in the sphere.

     

Bordism of two commuting involutions

In this paper we obtain conditions for a Whitney sum of three vector bundles over a closed manifold, , to be the fixed data of a -action; these conditions yield the fact that if is the fixed data of a -action, where is the trivial one dimensional bundle, then the same is true for . The results obtained, together with techniques previously developed, are used to obtain, up to bordism, all possible -actions fixing the disjoint union of an even projective space and a point.

     

Optimal estimation of shell thickness in Cutland's construction of Wiener measure

In Cutland's construction of Wiener measure, he used the product of Gaussian measures on , where is an infinite integer. It is mentioned by Cutland and Ng that for the product measure ,

where and with any positive infinite number. We prove here that may be replaced by with any positive infinite number. This is the optimal estimation for the shell thickness. It is also proved that . And for the *Lebesgue measure , is finite and not infinitesimal iff with finite, while for the *Lebesgue area of the sphere , should be .

     

A remark on normal derivations

Given a Hilbert space , let be operators on . Anderson has proved that if is normal and , then for all operators . Using this inequality, Du Hong-Ke has recently shown that if (instead) , then for all operators . In this note we improve the Du Hong-Ke inequality to for all operators . Indeed, we prove the equivalence of Du Hong-Ke and Anderson inequalities, and show that the Du Hong-Ke inequality holds for unitarily invariant norms.

     

Sufficient conditions for one domain to contain another in a space of constant curvature

As an application of the analogue of C-S. Chen's kinematic formula in the 3-dimensional space of constant curvature , that is, Euclidean space , -sphere , hyperbolic space (, respectively), we obtain sufficient conditions for one domain to contain another domain in either an Euclidean space , or a -sphere or a hyperbolic space .

     

Continuity of K-theory: An example in equal characteristics

If is a perfect field of characteristic , we show that the Quillen K-groups are uniquely -divisible for . In fact, the Milnor K-groups are uniquely -divisible for all . This implies that is -connected after profinite completion for a complete discrete valuation ring with perfect residue field.

     

Degrees of high-dimensional subvarieties of determinantal varieties

Let , where is a prime, and . In , let be the variety defined by . We show that any subvariety of of codimension less than must have degree a multiple of . We also show that the bounds on the codimension in our results are strict by exhibiting subvarieties of the appropriate codimension whose degrees are prime to .