Cohomological approach to asymptotic dimension

We introduce the notion of asymptotic cohomology based on the bounded cohomology and define cohomological asymptotic dimension asdim _Z X of metric spaces. We show that it agrees with the asymptotic dimension asdim X when the later is finite. Then we use this fact to construct an example of a metric space X of bounded geometry with finite asymptotic dimension for which asdim(X × R) = asdim X. In particular, it follows for this example that the coarse asymptotic dimension defined by means of Roe’s coarse cohomology is strictly less than its asymptotic dimension.      

On transfinite extension of asymptotic dimension

We prove that a transfinite extension of the asymptotic dimension asind is trivial. We introduce a transfinite extension of the asymptotic dimension asdim and give an example of a metric proper space which has transfinite infinite dimension.     

A Hurewicz-type theorem for asymptotic dimension and applications to geometric group theory

We prove an asymptotic analog of the classical Hurewicz theorem on mappings that lower dimension. This theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finite-dimensional metric spaces and allows us to prove a useful extension theorem for asymptotic dimension. As applications we find upper bound estimates for the asymptotic dimension of nilpotent and polycyclic groups in terms of their Hirsch length. We are also able to improve the known upper bounds on the asymptotic dimension of fundamental groups of complexes of groups, amalgamated free products and the hyperbolization of metric spaces possessing the Higson property.

     

Assouad-Nagata dimension of locally finite groups and asymptotic cones

In this paper we study two problems concerning Assouad-Nagata dimension:

(1)

Is there a metric space of positive asymptotic Assouad-Nagata dimension such that all of its asymptotic cones are of Assouad-Nagata dimension zero? (Dydak and Higes, 2008 [11, Question 4.5]).

(2)

Suppose G is a locally finite group with a proper left invariant metric dG. If dimAN(G,dG)>0, is dimAN(G,dG) infinite? (Brodskiy et al., preprint [6, Problem 5.3]).

The first question is answered positively. We provide examples of metric spaces of positive (even infinite) Assouad-Nagata dimension such that all of its asymptotic cones are ultrametric. The metric spaces can be groups with proper left invariant metrics.The second question has a negative solution. We show that for each n there exists a locally finite group of Assouad-Nagata dimension n. As a consequence this solves for non-finitely generated countable groups the question about the existence of metric spaces of finite asymptotic dimension whose asymptotic Assouad-Nagata dimension is larger but finite.     

A Metric Space with Transfinite Asymptotic Dimension 2ω + 1*

The authors construct a metric space whose transfinite asymptotic dimension and complementary-finite asymptotic dimension are both 2ω + 1, where ω is the smallest infinite ordinal number. Therefore, an example of a metric space with asymptotic property C is obtained.     

Asymptotic hereditary asphericity of metric spaces of asymptotic dimension 1

Asymptotic hereditary asphericity (AHA) is a coarse property introduced by Januszkiewicz and ?wia?tkowski in the context of systolic complexes and groups. We show, that spaces of asymptotic dimension 1 are all AHA.     

The asymptotic dimension of box spaces of virtually nilpotent groups

We show that every box space of a virtually nilpotent group has asymptotic dimension equal to the Hirsch length of that group.     

On exactness and isoperimetric profiles of discrete groups

We introduce a quasi-isometry invariant related to Property A and explore its connections to various other invariants of finitely generated groups. This allows to establish a direct relation between asymptotic dimension on one hand and isoperimetry and random walks on the other. We apply these results to reprove sharp estimates on isoperimetric profiles of some groups and to answer some questions in dimension theory.     

Asymptotic properties of groups acting on complexes

We examine asymptotic dimension and property A for groups acting on complexes. In particular, we prove that the fundamental group of a finite, developable complex of groups will have finite asymptotic dimension provided the geometric realization of the development has finite asymptotic dimension and the vertex groups are finitely generated and have finite asymptotic dimension. We also prove that property A is preserved by this construction provided the geometric realization of the development has finite asymptotic dimension and the vertex groups all have property A. These results naturally extend the corresponding results on preservation of these large-scale properties for fundamental groups of graphs of groups. We also use an example to show that the requirement that the development have finite asymptotic dimension cannot be relaxed.

     

Asymptotic Behaviour and Artinian Property of Graded Local Cohomology Modules

In this article, considering the difference between the finiteness dimension and cohomological dimension of a finitely generated graded module, we investigate the asymptotic behaviour of grades of components of a graded local cohomology module with respect to the irrelevant ideal. To this end, we study some Artinian and tameness properties of certain graded local cohomology modules.     

Asymptotic expansions of stress tensor for linearly elastic shell

In this paper, the asymptotic expansions of stress tensor for linearly elastic shell have been proposed by new asymptotic analysis method, which is different from the classical asymptotic analysis. The new asymptotic analysis method has two distinguishing features: one is that the displacement is expanded with respect to the thickness variable of the middle surface not to the thickness; another is that the first order term and the second order term of the displacement variable can be algebraically expressed by the leading term. To decompose stress tensor totally into 2-D variable and thickness variable, we have three steps: operator splitting, variables separation and dimension splitting. In the end, a numerical experiment of special hemispherical shell by FEM (finite element method) is provided. We derive the distribution of displacements and stress fields in the middle surface.     

Asymptotics of matrix integrals and tensor invariants of compact Lie groups

In this paper we give an asymptotic formula for a matrix integral which plays a crucial role in the approach of Diaconis et al. to random matrix eigenvalues. The choice of parameter for the asymptotic analysis is motivated by an invariant-theoretic interpretation of this type of integral. For arbitrary regular irreducible representations of arbitrary connected semisimple compact Lie groups, we obtain an asymptotic formula for the trace of permutation operators on the space of tensor invariants, thus extending a result of Biane on the dimension of these spaces.

     

Asymptotic cones and Assouad-Nagata dimension

We prove that the dimension of any asymptotic cone over a metric space does not exceed the asymptotic Assouad-Nagata dimension of . This improves a result of Dranishnikov and Smith (2007), who showed for all separable subsets of special asymptotic cones , where is an exponential ultrafilter on natural numbers.

We also show that the Assouad-Nagata dimension of the discrete Heisenberg group equals its asymptotic dimension.

     

Hyperbolic groups have finite asymptotic dimension

We detail a proof of a result of Gromov, that hyperbolic groups (and metric spaces) have finite asymptotic dimension. This fact has become important in recent work on the Novikov conjecture.

     

Remark on the stabilization of a viscous barotropic medium with a nonmonotonic equation of state

We consider the compressible barotropic Navier-Stokes system in one dimension, with a nonmonotonic equation of state. The associated free boundary problem is investigated, and we prove asymptotic properties of the unique globally defined solution for large time.

We also make some comments on a related model of quantum fluid describing the dynamics of cold nuclear matter (zero temperature).     

Qualitative properties of positive singular solutions to nonlinear elliptic systems with critical exponent

We studied the asymptotic behavior of local solutions for strongly coupled critical elliptic systems near an isolated singularity. For the dimension less than or equal to five we prove that any singular solution is asymptotic to a rotationally symmetric Fowler type solution. This result generalizes the celebrated work due to Caffarelli, Gidas and Spruck [1] who studied asymptotic proprieties to the classic Yamabe equation. In addition, we generalize similar results by Marques [12] for inhomogeneous context, that is, when the metric is not necessarily conformally flat.     

On Hypereuclidean Manifolds

We show that the universal cover of an aspherical manifold whose fundamental groups has finite asymptotic dimension in sense of Gromov is hypereuclidean after crossing with some Euclidean space     

Asymptotic Log-Harnack inequality and applications for stochastic systems of infinite memory

The asymptotic log-Harnack inequality is established for several kinds of models on stochastic differential systems with infinite memory: non-degenerate SDEs, neutral SDEs, semi-linear SPDEs, and stochastic Hamiltonian systems. As applications, the following properties are derived for the associated segment Markov semigroups: asymptotic heat kernel estimate, uniqueness of the invariant probability measure, asymptotic gradient estimate (hence, asymptotically strong Feller property), as well as asymptotic irreducibility.     

Asymptotic dimension of finitely presented groups

We prove that if a finitely presented group is one-ended, then its asymptotic dimension is greater than . It follows that a finitely presented group of asymptotic dimension is virtually free.