-equivalence in adjoint classical groups over fields of virtual cohomological dimension

Let be a field of characteristic not whose virtual cohomological dimension is at most . Let be a semisimple group of adjoint type defined over . Let denote the normal subgroup of consisting of elements -equivalent to identity. We show that if is of classical type not containing a factor of type , . If is a simple classical adjoint group of type , we show that if and its multi-quadratic extensions satisfy strong approximation property, then . This leads to a new proof of the -triviality of -rational points of adjoint classical groups defined over number fields.

     

On the unique representation of families of sets

Let and be uncountable Polish spaces. represents a family of sets provided each set in occurs as an -section of . We say that uniquely represents provided each set in occurs exactly once as an -section of . is universal for if every -section of is in . is uniquely universal for if it is universal and uniquely represents . We show that there is a Borel set in which uniquely represents the translates of if and only if there is a Vitali set. Assuming there is a Borel set with all sections sets and all non-empty sets are uniquely represented by . Assuming there is a Borel set with all sections which uniquely represents the countable subsets of . There is an analytic set in with all sections which represents all the subsets of , but no Borel set can uniquely represent the sets. This last theorem is generalized to higher Borel classes.

     

Asymptotics for discrete weighted minimal Riesz energy problems on rectifiable sets

Given a closed -rectifiable set embedded in Euclidean space, we investigate minimal weighted Riesz energy points on ; that is, points constrained to and interacting via the weighted power law potential , where is a fixed parameter and is an admissible weight. (In the unweighted case () such points for fixed tend to the solution of the best-packing problem on as the parameter .) Our main results concern the asymptotic behavior as of the minimal energies as well as the corresponding equilibrium configurations. Given a distribution with respect to -dimensional Hausdorff measure on , our results provide a method for generating -point configurations on that are ``well-separated' and have asymptotic distribution as .

     

Large orbits in coprime actions of solvable groups

Let be a solvable group of automorphisms of a finite group . If and are coprime, then there exists an orbit of on of size at least . It is also proved that in a -solvable group, the largest normal -subgroup is the intersection of at most three Hall -subgroups.

     

Generalized Seifert surfaces and signatures of colored links

In this paper, we use `generalized Seifert surfaces' to extend the Levine-Tristram signature to colored links in . This yields an integral valued function on the -dimensional torus, where is the number of colors of the link. The case corresponds to the Levine-Tristram signature. We show that many remarkable properties of the latter invariant extend to this -variable generalization: it vanishes for achiral colored links, it is `piecewise continuous', and the places of the jumps are determined by the Alexander invariants of the colored link. Using a -dimensional interpretation and the Atiyah-Singer -signature theorem, we also prove that this signature is invariant by colored concordance, and that it provides a lower bound for the `slice genus' of the colored link.

     

Decomposition numbers for weight three blocks of symmetric groups and Iwahori-Hecke algebras

Let be a field, a non-zero element of and the Iwahori-Hecke algebra of the symmetric group . If is a block of of -weight and the characteristic of is at least , we prove that the decomposition numbers for are all at most . In particular, the decomposition numbers for a -block of of defect are all at most .

     

Quantum symmetric derivatives

For , a one-parameter family of symmetric quantum derivatives is defined for each order of differentiation as are two families of Riemann symmetric quantum derivatives. For , symmetrization holds, that is, whenever the th Peano derivative exists at a point, all of these derivatives of order also exist at that point. The main result, desymmetrization, is that conversely, for , each symmetric quantum derivative is a.e. equivalent to the Peano derivative of the same order. For and , each th symmetric quantum derivative coincides with both corresponding th Riemann symmetric quantum derivatives, so, in particular, for and , both th Riemann symmetric quantum derivatives are a.e. equivalent to the Peano derivative.

     

Equipartitions of measures in

A well-known problem of B. Grünbaum (1960) asks whether for every continuous mass distribution (measure) on there exist hyperplanes dividing into parts of equal measure. It is known that the answer is positive in dimension (see H. Hadwiger (1966)) and negative for (see D. Avis (1984) and E. Ramos (1996)). We give a partial solution to Grünbaum's problem in the critical dimension by proving that each measure in admits an equipartition by hyperplanes, provided that it is symmetric with respect to a -dimensional affine subspace of . Moreover we show, by computing the complete obstruction in the relevant group of normal bordisms, that without the symmetry condition, a naturally associated topological problem has a negative solution. The computation is based on Koschorke's exact singularity sequence (1981) and the remarkable properties of the essentially unique, balanced binary Gray code in dimension ; see G. C. Tootill (1956) and D. E. Knuth (2001).

     

Completely isometric representations of

Let be a locally compact group. It is shown that there exists a natural completely isometric representation of the completely bounded Fourier multiplier algebra , which is dual to the representation of the measure algebra , on . The image algebras of and in are intrinsically characterized, and some commutant theorems are proved. It is also shown that for any amenable group , there is a natural completely isometric representation of on , which can be regarded as a duality result of Neufang's completely isometric representation theorem for .

     

Profinite and pro- completions of Poincaré duality groups of dimension 3

We establish some sufficient conditions for the profinite and pro- completions of an abstract group of type (resp. of finite cohomological dimension, of finite Euler characteristic) to be of type over the field for a fixed natural prime (resp. of finite cohomological -dimension, of finite Euler -characteristic).

We apply our methods for orientable Poincaré duality groups of dimension 3 and show that the pro- completion of is a pro- Poincaré duality group of dimension 3 if and only if every subgroup of finite index in has deficiency 0 and is infinite. Furthermore if is infinite but not a Poincaré duality pro- group, then either there is a subgroup of finite index in of arbitrary large deficiency or is virtually . Finally we show that if every normal subgroup of finite index in has finite abelianization and the profinite completion of has an infinite Sylow -subgroup, then is a profinite Poincaré duality group of dimension 3 at the prime .

     

Equilibriums of some non-Hölder potentials

We consider one-sided subshifts with some potential functions which satisfy the Hölder condition everywhere except at a fixed point and its preimages. We prove that the systems have conformal measures and invariant measures absolutely continuous with respect to , where may be finite or infinite. We show that the systems are exact, and are weak Gibbs measures and equilibriums for . We also discuss uniqueness of equilibriums and phase transition.

These results can be applied to some expanding dynamical systems with an indifferent fixed point.

     

The kernels of radical homomorphisms and intersections of prime ideals

We establish a necessary condition for a commutative Banach algebra so that there exists a homomorphism from into another Banach algebra such that the prime radical of the continuity ideal of is not a finite intersection of prime ideals in . We prove that the prime radical of the continuity ideal of an epimorphism from onto another Banach algebra (or of a derivation from into a Banach -bimodule) is always a finite intersection of prime ideals. Under an additional cardinality condition (and assuming the Continuum Hypothesis), this necessary condition is proved to be sufficient. En route, we give a general result on norming commutative semiprime algebras; extending the class of algebras known to be normable. We characterize those locally compact metrizable spaces for which there exists a homomorphism from into a radical Banach algebra whose kernel is not a finite intersection of prime ideals.

     

Koszul homology and extremal properties of Gin and Lex

For every homogeneous ideal in a polynomial ring and for every we consider the Koszul homology with respect to a sequence of of generic linear forms. The Koszul-Betti number is, by definition, the dimension of the degree part of . In characteristic , we show that the Koszul-Betti numbers of any ideal are bounded above by those of the gin-revlex of and also by those of the Lex-segment of . We show that iff is componentwise linear and that and iff is Gotzmann. We also investigate the set of all the gin of and show that the Koszul-Betti numbers of any ideal in are bounded below by those of the gin-revlex of . On the other hand, we present examples showing that in general there is no is such that the Koszul-Betti numbers of any ideal in are bounded above by those of .

     

Finiteness of cousin cohomologies

The notion of the Cousin complex of a module was given by Sharp in 1969. It wasn't known whether its cohomologies are finitely generated until recently. In 2001, Dibaei and Tousi showed that the Cousin cohomologies of a finitely generated -module  are finitely generated if the base ring  is local, has a dualizing complex, satisfies Serre's -condition and is equidimensional. In the present article, the author improves their result. He shows that the Cousin cohomologies of  are finitely generated if is universally catenary, all the formal fibers of all the localizations of  are Cohen-Macaulay, the Cohen-Macaulay locus of each finitely generated -algebra is open and all the localizations of  are equidimensional. As a consequence of this, he gives a necessary and sufficient condition for a Noetherian ring to have an arithmetic Macaulayfication.

     

Algebroid prestacks and deformations of ringed spaces

For a ringed space , we show that the deformations of the abelian category of sheaves of -modules (Lowen and Van den Bergh, 2006) are obtained from algebroid prestacks, as introduced by Kontsevich. In case is a quasi-compact separated scheme the same is true for , the category of quasi-coherent sheaves on . It follows in particular that there is a deformation equivalence between and .

     

Generically there is but one self homeomorphism of the Cantor set

We describe a self homeomorphism of the Cantor set and then show that its conjugacy class in the Polish group of all homeomorphisms of forms a dense subset of . We also provide an example of a locally compact, second countable topological group which has a dense conjugacy class.

     

On homeomorphic Bernoulli measures on the Cantor space

Let be the Bernoulli measure on the Cantor space given as the infinite product of two-point measures with weights and . It is a long-standing open problem to characterize those and such that and are topologically equivalent (i.e., there is a homeomorphism from the Cantor space to itself sending to ). The (possibly) weaker property of and being continuously reducible to each other is equivalent to a property of and called binomial equivalence. In this paper we define an algebraic property called ``refinability' and show that, if and are refinable and binomially equivalent, then and are topologically equivalent. Next we show that refinability is equivalent to a fairly simple algebraic property. Finally, we give a class of examples of binomially equivalent and refinable numbers; in particular, the positive numbers and such that and are refinable, so the corresponding measures are topologically equivalent.

     

On embedding all -manifolds into a single -manifold

For each composite number , there does not exist a single connected closed -manifold such that any smooth, simply-connected, closed -manifold can be topologically flatly embedded into it. There is a single connected closed -manifold such that any simply-connected, -manifold can be topologically flatly embedded into if is either closed and indefinite, or compact and with non-empty boundary.