Critical points of real entire functions and a conjecture of Pólya

Let be a nonconstant real entire function of genus and assume that all the zeros of are distributed in some infinite strip , . It is shown that (1) if has nonreal zeros in the region , and has nonreal zeros in the same region, and if the points and are located outside the Jensen disks of , then has exactly critical zeros in the closed interval , (2) if is at most of order , , and minimal type, then for each positive constant there is a positive integer such that for all has only real zeros in the region , and (3) if is of order less than , then has just as many critical points as couples of nonreal zeros.

     

On free group algebras in division rings with uncountable center

Let be a division algebra with uncountable center . If contains a noncommutative free -algebra, then also contains the -group algebra of the free group of rank 2.

     

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.

     

Parametrizing maximal compact subvarieties

For the Lie group , let be the open orbit of Lagrangian planes of signature in the generalized flag variety of Lagrangian planes in . For a suitably chosen maximal compact subgroup of and a base point we have that the orbit of is a maximal compact subvariety of . We show that for the connected component containing in the space of translates of which lie in is biholomorphic to , where denotes with the opposite complex structure.

     

A theorem of Briançon-Skoda type for regular local rings containing a field

Let be a regular local ring containing a field. We give a refinement of the Briançon-Skoda theorem showing that if is a minimal reduction of where is -primary, then where and is the largest ideal such that . The proof uses tight closure in characteristic and reduction to characteristic for rings containing the rationals.

     

Some remarks on the variation of curve length and surface area

Consider the curve , where is absolutely continuous on . Then has finite length, and if is the -neighborhood of in the uniform norm, we compare the length of the shortest path in with the length of . Our main result establishes necessary and sufficient conditions on such that the difference of these quantities is of order as . We also include a result for surfaces.

     

Non-normal, standard subgroups of the Bianchi groups

Let be a subgroup of , where is a Dedekind ring, and let be the -ideal generated by , where . The subgroup is called standard iff contains the normal subgroup of generated by the -elementary matrices. It is known that, when , is standard iff is normal in . It is also known that every standard subgroup of is normal in when is an arithmetic Dedekind domain with infinitely many units. The ring of integers of an imaginary quadratic number field, , is one example (of three) of such an arithmetic domain with finitely many units. In this paper it is proved that every Bianchi group has uncountably many non-normal, standard subgroups. This result is already known for related groups like .

     

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.

     

Comparative probability on von Neumann algebras

We continue here the study begun in earlier papers on implementation of comparative probability by states. Let be a von Neumann algebra on a Hilbert space and let denote the projections of . A comparative probability (CP) on (or more correctly on is a preorder on satisfying:

with for some .

If , then either or .

If , and are all in and , , then .

A state on is said to implement a on if for , . In this paper, we examine the conditions for implementability of a CP on a general von Neumann algebra (as opposed to only type I factors). A crucial tool used here, as well as in earlier results, is the interval topology generated on by . A will be termed continuous in a given topology on if the interval topology generated by is weaker than the topology induced on by the given topology. We show that uniform continuity of a comparative probability is necessary and sufficient if the von Neumann algebra has no finite direct summand. For implementation by normal states, weak continuity is sufficient and necessary if the von Neumann algebra has no finite direct summand of type I. We arrive at these results by constructing an appropriate additive measure from the CP.

     

Chains of strongly non-reflexive dual groups of integer-valued continuous functions

Answering a question of Eklof-Mekler (Almost free modules, set-theoretic methods, North-Holland, Amsterdam, 1990), we prove: (1) If there exists a non-reflecting stationary set of consisting of ordinals of cofinality for each , then there exist abelian groups such that and for each . (2) There exist abelian groups such that for each and for each . The groups are the groups of -valued continuous functions on a topological space and their dual groups.

     

On fully invariant ideals of the free group algebra

Let be the group algebra of a free noncyclic group over an integral domain . It is proved that if is not a field, then there exists a fully invariant ideal of such is torsion-free but not projective as an -module. In other words, there exists a pure nonprojective variety of group representations over .

     

-Fréchet-Urysohn property of Franklin compact spaces

Franklin compact spaces defined by maximal almost disjoint families of subsets of are considered from the view of its -sequentiality and -Fréchet-Urysohn-property for ultrafilters . Our principal results are the following: CH implies that for every -point there are a Franklin compact -Fréchet-Urysohn space and a Franklin compact space which is not -Fréchet-Urysohn; and, assuming CH, for every Franklin compact space there is a -point such that it is not -Fréchet-Urysohn. Some new problems are raised.

     

Topologically conjugate Kleinian groups

Two Kleinian groups and are said to be topologically conjugate when there is a homeomorphism such that . It is conjectured that if two Kleinian groups and are topologically conjugate, one is a quasi-conformal deformation of the other. In this paper generalizing Minsky's result, we shall prove that this conjecture is true when is finitely generated and freely indecomposable, and the injectivity radii of all points of and are bounded below by a positive constant.

     

Representation of continuous functions as sums of Green functions

Let , where is polar and compact and is a domain with Green function . We characterize those subsets of which have the following property: Every positive continuous function on can be written as , where and for each .

     

On contravariant finiteness of subcategories of modules of projective dimension

Let be an artin algebra. This paper presents a sufficient condition for the subcategory of to be contravariantly finite in , where is the subcategory of consisting of --modules of projective dimension less than or equal to . As an application of this condition it is shown that is contravariantly finite in for each when is stably equivalent to a hereditary algebra.

     

The local cohomology modules of Matlis reflexive modules are almost cofinite

We show that if and are Matlis reflexive modules over a complete Gorenstein local domain and is an ideal of such that the dimension of is one, then the modules are Matlis reflexive for all and if . It follows that the Bass numbers of are finite. If is not a domain, then the same results hold for .

     

Rational surfaces with

The main but not all of the results in this paper concern rational surfaces for which the self-intersection of the anticanonical class is positive. In particular, it is shown that no superabundant numerically effective divisor classes occur on any smooth rational projective surface with . As an application, it follows that any 8 or fewer (possibly infinitely near) points in the projective plane are in good position. This is not true for 9 points, and a characterization of the good position locus in this case is also given. Moreover, these results are put into the context of conjectures for generic blowings up of . All results are proven over an algebraically closed field of arbitrary characteristic.

     

Transformations conjugate to their inverses have even essential values

Let be an ergodic automorphism defined on a standard Borel probability space for which and are isomorphic. We study the structure of the conjugating automorphisms and attempt to gain information about the structure of . It was shown in Ergodic transformations conjugate to their inverses by involutions by Goodson et al. (Ergodic Theory and Dynamical Systems 16 (1996), 97--124) that if is ergodic having simple spectrum and isomorphic to its inverse, and if is a conjugation between and (i.e. satisfies ), then , the identity automorphism. We give a new proof of this result which shows even more, namely that for such a conjugation , the unitary operator induced by on must have a multiplicity function whose essential values on the ortho-complement of the subspace are always even. In particular, we see that can be weakly mixing, so the corresponding must have even maximal spectral multiplicity (regarding as an even number).