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.

     

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.

     

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).

     

Equivalence of quotient Hilbert modules-II

For any open, connected and bounded set , let be a natural function algebra consisting of functions holomorphic on . Let be a Hilbert module over the algebra and let be the submodule of functions vanishing to order on a hypersurface . Recently the authors have obtained an explicit complete set of unitary invariants for the quotient module in the case of . In this paper, we relate these invariants to familiar notions from complex geometry. We also find a complete set of unitary invariants for the general case. We discuss many concrete examples in this setting. As an application of our equivalence results, we characterise certain homogeneous Hilbert modules over the bi-disc algebra.

     

Free lines for homeomorphisms of the open annulus

Let be a homeomorphism of the open annulus isotopic to the identity and let be a lift of to the universal cover without fixed point. Then we show that admits a Brouwer line which is a lift of a properly imbedded line joining one end to the other in the annulus or admits a free essential simple closed curve.

     

Conformal energy, conformal Laplacian, and energy measures on the Sierpinski gasket

On the Sierpinski Gasket (SG) and related fractals, we define a notion of conformal energy and conformal Laplacian for a given conformal factor , based on the corresponding notions in Riemannian geometry in dimension . We derive a differential equation that describes the dependence of the effective resistances of on . We show that the spectrum of (Dirichlet or Neumann) has similar asymptotics compared to the spectrum of the standard Laplacian, and also has similar spectral gaps (provided the function does not vary too much). We illustrate these results with numerical approximations. We give a linear extension algorithm to compute the energy measures of harmonic functions (with respect to the standard energy), and as an application we show how to compute the dimensions of these measures for integer values of . We derive analogous linear extension algorithms for energy measures on related fractals.

     

-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.

     

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.

     

Gromov-Witten invariants of jumping curves

Given a vector bundle on a smooth projective variety , we can define subschemes of the Kontsevich moduli space of genus-zero stable maps parameterizing maps such that the Grothendieck decomposition of has a specified splitting type. In this paper, using a ``compactification' of this locus, we define Gromov-Witten invariants of jumping curves associated to the bundle . We compute these invariants for the tautological bundle of Grassmannians and the Horrocks-Mumford bundle on . Our construction is a generalization of jumping lines for vector bundles on . Since for the tautological bundle of the Grassmannians the invariants are enumerative, we resolve the classical problem of computing the characteristic numbers of unbalanced scrolls.

     

Free boundary regularity close to initial state for parabolic obstacle problem

In this paper we study the behavior of the free boundary , arising in the following complementary problem:

Here denotes the parabolic boundary, is a parabolic operator with certain properties, is the upper half of the unit cylinder in , and the equation is satisfied in the viscosity sense. The obstacle is assumed to be continuous (with a certain smoothness at , ), and coincides with the boundary data at time zero. We also discuss applications in financial markets.

     

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.

     

Finite dimensional representations of invariant differential operators

Let be an algebraically closed field of characteristic , , and let be an algebraic torus acting diagonally on the ring of algebraic differential operators . We give necessary and sufficient conditions for to have enough simple finite dimensional representations, in the sense that the intersection of the kernels of all the simple finite dimensional representations is zero. As an application we show that if is a representation of a reductive group and if zero is not a weight of a maximal torus of on , then has enough finite dimensional representations. We also construct examples of FCR-algebras with any integer GK dimension .

     

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 .

     

Commutative ideal theory without finiteness conditions: Primal ideals

Our goal is to establish an efficient decomposition of an ideal of a commutative ring as an intersection of primal ideals. We prove the existence of a canonical primal decomposition: , where the are isolated components of that are primal ideals having distinct and incomparable adjoint primes . For this purpose we define the set of associated primes of the ideal to be those defined and studied by Krull. We determine conditions for the canonical primal decomposition to be irredundant, or residually maximal, or the unique representation of as an irredundant intersection of isolated components of . Using our canonical primal decomposition, we obtain an affirmative answer to a question raised by Fuchs, and also prove for that an ideal is an intersection of -primal ideals if and only if the elements of are prime to . We prove that the following conditions are equivalent: (i) the ring is arithmetical, (ii) every primal ideal of is irreducible, (iii) each proper ideal of is an intersection of its irreducible isolated components. We classify the rings for which the canonical primal decomposition of each proper ideal is an irredundant decomposition of irreducible ideals as precisely the arithmetical rings with Noetherian maximal spectrum. In particular, the integral domains having these equivalent properties are the Prüfer domains possessing a certain property.

     

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 .