18,822 research outputs found
Short relaxation times but long transient times in both simple and complex reaction networks
When relaxation towards an equilibrium or steady state is exponential at
large times, one usually considers that the associated relaxation time ,
i.e., the inverse of that decay rate, is the longest characteristic time in the
system. However that need not be true, and in particular other times such as
the lifetime of an infinitesimal perturbation can be much longer. In the
present work we demonstrate that this paradoxical property can arise even in
quite simple systems such as a chain of reactions obeying mass action kinetics.
By mathematical analysis of simple reaction networks, we pin-point the reason
why the standard relaxation time does not provide relevant information on the
potentially long transient times of typical infinitesimal perturbations.
Overall, we consider four characteristic times and study their behavior in both
simple chains and in more complex reaction networks taken from the publicly
available database "Biomodels." In all these systems involving mass action
rates, Michaelis-Menten reversible kinetics, or phenomenological laws for
reaction rates, we find that the characteristic times corresponding to
lifetimes of tracers and of concentration perturbations can be much longer than
Subgroup separability in residually free groups
We prove that the finitely presentable subgroups of residually free groups
are separable and that the subgroups of type are virtual
retracts. We describe a uniform solution to the membership problem for finitely
presentable subgroups of residually free groups.Comment: 8 pages, no figure
On the difficulty of presenting finitely presentable groups
We exhibit classes of groups in which the word problem is uniformly solvable
but in which there is no algorithm that can compute finite presentations for
finitely presentable subgroups. Direct products of hyperbolic groups, groups of
integer matrices, and right-angled Coxeter groups form such classes. We discuss
related classes of groups in which there does exist an algorithm to compute
finite presentations for finitely presentable subgroups. We also construct a
finitely presented group that has a polynomial Dehn function but in which there
is no algorithm to compute the first Betti number of the finitely presentable
subgroups.Comment: Final version. To appear in GGD volume dedicated to Fritz Grunewal
The isomorphism problem for profinite completions of residually finite groups
We consider pairs of finitely presented, residually finite groups
. We prove that there is no algorithm that, given an
arbitrary such pair, can determine whether or not the associated map of
profinite completions is an
isomorphism. Nor do there exist algorithms that can decide whether is
surjective, or whether is isomorphic to .Comment: 12 page
Mining SOM expression portraits: Feature selection and integrating concepts of molecular function
Background: 
Self organizing maps (SOM) enable the straightforward portraying of high-dimensional data of large sample collections in terms of sample-specific images. The analysis of their texture provides so-called spot-clusters of co-expressed genes which require subsequent significance filtering and functional interpretation. We address feature selection in terms of the gene ranking problem and the interpretation of the obtained spot-related lists using concepts of molecular function.

Results: 
Different expression scores based either on simple fold change-measures or on regularized Students t-statistics are applied to spot-related gene lists and compared with special emphasis on the error characteristics of microarray expression data. The spot-clusters are analyzed using different methods of gene set enrichment analysis with the focus on overexpression and/or overrepresentation of predefined sets of genes. Metagene-related overrepresentation of selected gene sets was mapped into the SOM images to assign gene function to different regions. Alternatively we estimated set-related overexpression profiles over all samples studied using a gene set enrichment score. It was also applied to the spot-clusters to generate lists of enriched gene sets. We used the tissue body index data set, a collection of expression data of human tissues, as an illustrative example. We found that tissue related spots typically contain enriched populations of gene sets well corresponding to molecular processes in the respective tissues. In addition, we display special sets of housekeeping and of consistently weak and highly expressed genes using SOM data filtering. 

Conclusions:
The presented methods allow the comprehensive downstream analysis of SOM-transformed expression data in terms of cluster-related gene lists and enriched gene sets for functional interpretation. SOM clustering implies the ability to define either new gene sets using selected SOM spots or to verify and/or to amend existing ones
Limit groups, positive-genus towers and measure equivalence
By definition, an -residually free tower is positive-genus if all
surfaces used in its construction are of positive genus. We prove that every
limit group is virtually a subgroup of a positive-genus -residually
free tower. By combining this with results of Gaboriau, we prove that
elementarily free groups are measure equivalent to free groups.Comment: 10 pages; no figures. Minor changes; now to appear in Ergod. Th. &
Dynam. Sy
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