-analyticity

Let be a finite-dimensional commutative algebra over and let , and be the ring of -differentiable functions of class , the ring of real analytic mappings with values in and the ring of -analytic functions, respectively, defined on an open subset of . We prove two basic results concerning -differentiability and -analyticity: ) , ) if and only if is defined over .

     

Amenability and weak amenability of second conjugate Banach algebras

For a Banach algebra , amenability of necessitates amenability of , and similarly for weak amenability provided is a left ideal in . For a locally compact group, indeed more generally, is amenable if and only if is finite. If is weakly amenable, then is weakly amenable.

     

Representations of the Gupta-Sidki group

If is an odd prime, then the Gupta-Sidki group is an infinite -generated -group. It is defined in a recursive manner as a particular subgroup of the automorphism group of a regular tree of degree . In this note, we make two observations concerning the irreducible representations of the group algebra with an algebraically closed field. First, when , we obtain a lower bound for the number of irreducible representations of any finite degree . Second, when , we show that if has one nonprincipal irreducible representation, then it has infinitely many. The proofs of these two results use similar techniques and eventually depend on the fact that the commutator subgroup of has a normal subgroup of finite index isomorphic to the direct product of copies of .

     

On purely inseparable extensions and their generators

Let be a field of characteristic and a polynomial algebra in two variables. By a -generator of we mean an element of for which there exist and such that . We also define a -line of to mean any element of whose coordinate ring is that of a -generator. Then we prove that if is such that is a -line of (where is an indeterminate over ), then is a -generator of . This is analogous to the well-known fact that if is such that is a line of , then is a variable of . We also prove that if is a -line of for which there exist and such that , then is in fact a -generator of .

     

Cohomological detection and regular elements in group cohomology

Suppose that is a compact Lie group or a discrete group of finite virtual cohomological dimension and that is a field of characteristic . Suppose that is a set of elementary abelian -subgroups such that the cohomology is detected on the centralizers of the elements of . Assume also that is closed under conjugation and that is in whenever some subgroup of is in . Then there exists a regular element in the cohomology ring such that the restriction of to an elementary abelian -subgroup is not nilpotent if and only if is in . The converse of the result is a theorem of Lannes, Schwartz and the second author. The results have several implications for the depth and associated primes of the cohomology rings.

     

Congruence lattices of algebras--- the signed labelling

For an arbitrary algebra a new labelling, called the signed labelling, of the Hasse diagram of is described. Under the signed labelling, each edge of the Hasse diagram of receives a label from the set . The signed labelling depends completely on a subset of the unary polynomials of and its inspiration comes from semigroup theory. For finite algebras, the signed labelling complements the labelled congruence lattices of tame congruence theory (TCT). It provides a different kind of information about those algebras than the TCT labelling particularly with regard to congruence semimodularity. The main result of this paper shows that the congruence lattice of any algebra admits a natural join congruence, denoted , such that satisfies the semimodular law. In an application of that result, it is shown that for a regular semigroup , for which in , is actually a lattice congruence, coincides with , and satisfies the semimodular law.

     

Basic differential forms for actions of Lie groups

A section of a Riemannian -manifold is a closed submanifold which meets each orbit orthogonally. It is shown that the algebra of -invariant differential forms on which are horizontal in the sense that they kill every vector which is tangent to some orbit, is isomorphic to the algebra of those differential forms on which are invariant with respect to the generalized Weyl group of , under some condition.

     

Almost linearity of -bi-Lipschitz maps between real Banach spaces

Let and be real Banach spaces. A map between and is called an -bi-Lipschitz map if for all . In this note we show that if is an -bi-Lipschitz map with from onto , then is almost linear. We also show that if is a surjective -bi-Lipschitz map with , then there exists a linear isomorphism such that

where as and .

     

On Voiculescu's double commutant theorem

For a separable infinite-dimensional Hilbert space , we consider the full algebra of bounded linear transformations and the unique non-trivial norm-closed two-sided ideal of compact operators . We also consider the quotient -algebra with quotient map

For any -subalgebra of , the relative commutant is given by for all in . It was shown by D. Voiculescu that, for any separable unital -subalgebra of ,

In this note, we exhibit a non-separable unital -subalgebra of for which (VDCT) fails.

     

Degree of irrationality of a product of two elliptic curves

We consider the degree of irrationality of some algebraic surface . Firstly we give an estimate of for a surface with a structure of a fiber space. Secondly we prove the existence of a nonsingular curve of genus 3 on for a certain elliptic curve with complex multiplications. As a corollary, we obtain that .

     

The second iterate of a map with dense orbit

Suppose that is a Hausdorff topological space having no isolated points and that is continuous. We show that if the orbit of a point under is dense in while the orbit of under is not, then the space decomposes into three sets relative to which the dynamics of are easy to describe. This decomposition has the following consequence: suppose that has dense orbit under and that the closure of the set of points of having odd period under has nonempty interior; then has dense orbit under .

     

Orientation-preserving self-homeomorphisms of the surface of genus two have points of period at most two

We show that for any orientation-preserving self-homeomorphism of the double torus there exists a point of such that . This answers a question raised by Jakob Nielsen in 1942.

     

-regular maps on smooth manifolds

A continuous map is said to be -regular if whenever are distinct points of , then are linearly independent over . For smooth manifolds we obtain new lower bounds on the minimum for which a -regular map can exist in terms of the dual Stiefel-Whitney classes of .

     

Rational nodal curves with no smooth Weierstrass points

Let denote the rational curve with nodes obtained from the Riemann sphere by identifying 0 with and with for , where is a primitive th root of unity. We show that if is even, then has no smooth Weierstrass points, while if is odd, then has smooth Weierstrass points.

     

On the prime model property

Assume is superstable, is a formula over , is countable and is countable and . We investigate models in assuming has the prime model property. We prove some corollaries on the number of models in . We show an example of an -stable and with having exactly 3 models.

     

Integer-valued polynomials on Krull rings

If is a subring of a Krull ring such that is a valuation ring for every finite index , in Spec, we construct polynomials that map into the maximal possible (for a monic polynomial of fixed degree) power of , for all in Spec simultaneously. This gives a direct sum decomposition of Int, the -module of polynomials with coefficients in the quotient field of that map into , and a criterion when Int has a regular basis (one consisting of 1 polynomial of each non-negative degree).

     

and the associated line bundles

Let be a compact semi-simple Lie group, and let be a maximal unipotent subgroup of the complexified group . In this paper, we classify all the -invariant Kaehler structures on . For each Kaehler structure , let be the line bundle with connection whose curvature is . We then study the holomorphic sections of , which constitute a -representation space.

     

A -dimensional extension of a lemma of Huneke's and formulas for the Hilbert coefficients

A -dimensional version is given of a -dimensional result due to C. Huneke. His result produced a formula relating the length to the difference , where is primary for the maximal ideal of a -dimensional Cohen-Macaulay local ring , is a minimal reduction of , , and is the Hilbert-Samuel polynomial of . We produce a formula that is valid for arbitrary dimension, and then use it to establish some formulas for the Hilbert coefficients of . We also include a characterization, in terms of the Hilbert coefficients of , of the condition .

     

On a polynomial inequality of Kolmogoroff's type

We prove an inequality of the form

for polynomials of degree and any fixed . Here is the -norm on with a weight . The coefficients and are given explicitly and depend on and only. The equality is attained for the Hermite orthogonal polynomials .

     

A topological characterization of linearity for quasi-traces

Let be a -algebra, and let be a (local) quasi-trace on . Then is linear if, and only if, the restriction of to the closed unit ball of is uniformly weakly continuous.