Shelah's singular compactness theorem

We present Shelah’s famous theorem in a version for modules, together with a self-contained proof and some examples. This exposition is based on lectures given at CRM in October 2006.

Contribution of electrostatic interactions, compactness and quaternary structure to protein thermostability: lessons from structural genomics of Thermotoga maritima.

Studies of the structural basis of protein thermostability have produced a confusing picture. Small sets of proteins have been analyzed from a variety of thermophilic species, suggesting different structural features as responsible for protein thermostability. Taking advantage of the recent advances in structural genomics, we have compiled a relatively large protein structure dataset, which was constructed very carefully and selectively; that is, the dataset contains only experimentally determined structures of proteins from one specific organism, the hyperthermophilic bacterium Thermotoga maritima, and those of close homologs from mesophilic bacteria. In contrast to the conclusions of previous studies, our analyses show that oligomerization order, hydrogen bonds, and secondary structure play minor roles in adaptation to hyperthermophily in bacteria. On the other hand, the data exhibit very significant increases in the density of salt-bridges and in compactness for proteins from T.maritima. The latter effect can be measured by contact order or solvent accessibility, and network analysis shows a specific increase in highly connected residues in this thermophile. These features account for changes in 96% of the protein pairs studied. Our results provide a clear picture of protein thermostability in one species, and a framework for future studies of thermal adaptation.

Maximal compactness, min- imal Hausdor ness and reversibility of frames and a study on automorphism group of frames

One can do research in pointfree topology in two ways. The rst is the contravariant way where research is done in the category Frm but the ultimate objective is to obtain results in Loc. The other way is the covariant way to carry out research in the category Loc itself directly. According to Johnstone [23], \frame theory is lattice theory applied to topology whereas locale theory is topology itself". The most part of this thesis is written according to the rst view. In this thesis, we make an attempt to study about 1. the frame counterparts of maximal compactness, minimal Hausdor - ness and reversibility, 2. the automorphism groups of a nite frame and its relation with the subgroups of the permutation group on the generator set of the frame

Limit operators, collective compactness, and the spectral theory of infinite matrices

In the first half of this memoir we explore the interrelationships between the abstract theory of limit operators (see e.g. the recent monographs of Rabinovich, Roch and Silbermann (2004) and Lindner (2006)) and the concepts and results of the generalised collectively compact operator theory introduced by Chandler-Wilde and Zhang (2002). We build up to results obtained by applying this generalised collectively compact operator theory to the set of limit operators of an operator (its operator spectrum). In the second half of this memoir we study bounded linear operators on the generalised sequence space , where and is some complex Banach space. We make what seems to be a more complete study than hitherto of the connections between Fredholmness, invertibility, invertibility at infinity, and invertibility or injectivity of the set of limit operators, with some emphasis on the case when the operator is a locally compact perturbation of the identity. Especially, we obtain stronger results than previously known for the subtle limiting cases of and . Our tools in this study are the results from the first half of the memoir and an exploitation of the partial duality between and and its implications for bounded linear operators which are also continuous with respect to the weaker topology (the strict topology) introduced in the first half of the memoir. Results in this second half of the memoir include a new proof that injectivity of all limit operators (the classic Favard condition) implies invertibility for a general class of almost periodic operators, and characterisations of invertibility at infinity and Fredholmness for operators in the so-called Wiener algebra. In two final chapters our results are illustrated by and applied to concrete examples. Firstly, we study the spectra and essential spectra of discrete Schrödinger operators (both self-adjoint and non-self-adjoint), including operators with almost periodic and random potentials. In the final chapter we apply our results to integral operators on .

Hankel and Toeplitz transforms on H 1: continuity, compactness and Fredholm properties

We revisit the boundedness of Hankel and Toeplitz operators acting on the Hardy space H 1 and give a new proof of the old result stating that the Hankel operator H a is bounded if and only if a has bounded logarithmic mean oscillation. We also establish a sufficient and necessary condition for H a to be compact on H 1. The Fredholm properties of Toeplitz operators on H 1 are studied for symbols in a Banach algebra similar to C + H ∞ under mild additional conditions caused by the differences in the boundedness of Toeplitz operators acting on H 1 and H 2.

Effect of pH and temperature on the global compactness, structure, and activity of cellobiohydrolase Cel7A from Trichoderma harzianum

Due to its elevated cellulolytic activity, the filamentous fungus Trichoderma harzianum (T. harzianum) has considerable potential in biomass hydrolysis application. Cellulases from Trichoderma reesei have been widely used in studies of cellulose breakdown. However, cellulases from T. harzianum are less-studied enzymes that have not been characterized biophysically and biochemically as yet. Here, we examined the effects of pH and temperature on the secondary and tertiary structures, compactness, and enzymatic activity of cellobiohydrolase Cel7A from T. harzianum (Th Cel7A) using a number of biophysical and biochemical techniques. Our results show that pH and temperature perturbations affect Th Cel7A stability by two different mechanisms. Variations in pH modify protonation of the enzyme residues, directly affecting its activity, while leading to structural destabilization only at extreme pH limits. Temperature, on the other hand, has direct influence on mobility, fold, and compactness of the enzyme, causing unfolding of Th Cel7A just above the optimum temperature limit. Finally, we demonstrated that incubation with cellobiose, the product of the reaction and a competitive inhibitor, significantly increased the thermal stability of Th Cel7A. Our studies might provide insights into understanding, at a molecular level, the interplay between structure and activity of Th Cel7A at different pH and temperature conditions.

Lions-type compactness and Rubik actions on the Heisenberg group

In this paper we prove a Lions-type compactness embedding result for symmetric unbounded domains of the Heisenberg group. The natural group action on the Heisenberg group TeX is provided by the unitary group U(n) × {1} and its appropriate subgroups, which will be used to construct subspaces with specific symmetry and compactness properties in the Folland-Stein’s horizontal Sobolev space TeX. As an application, we study the multiplicity of solutions for a singular subelliptic problem by exploiting a technique of solving the Rubik-cube applied to subgroups of U(n) × {1}. In our approach we employ concentration compactness, group-theoretical arguments, and variational methods.

Urban sprawl and city compactness. A proposal for regional sustainability indicators. Case study of the towns of Alcorcon and Majadahonda (Comunidad de Madrid, Spain)

The main contribution of this research paper is to display a range of figures and values which could help urban planners to quantify the urban phenomenon of sprawl. In this way, after a rigorous analysis and comparison between a scattered urban fabric (Majadahonda) and a compact urban fabric (Alcorcón), several possible indexes are established and characterized in order to verify the main hypothesis: in what extent land consumption and exploitation of energy resources are higher in a scattered urban fabric than in a compact one.

Relative Compactness for Hyperspaces

The present paper contains results characterizing relatively compact subsets of the space of the closed subsets of a metrizable space, equipped with various hypertopologies. We investigate the hyperspace topologies that admit a representation as weak topologies generated by families of gap functionals defined on closed sets, as well as hit-and-miss topologies and proximal-hit and-miss topologies.

Compactness in the First Baire Class and Baire-1 Operators

For a polish space M and a Banach space E let B1 (M, E) be the space of first Baire class functions from M to E, endowed with the pointwise weak topology. We study the compact subsets of B1 (M, E) and show that the fundamental results proved by Rosenthal, Bourgain, Fremlin, Talagrand and Godefroy, in case E = R, also hold true in the general case. For instance: a subset of B1 (M, E) is compact iff it is sequentially (resp. countably) compact, the convex hull of a compact bounded subset of B1 (M, E) is relatively compact, etc. We also show that our class includes Gulko compact. In the second part of the paper we examine under which conditions a bounded linear operator T : X ∗ → Y so that T |BX ∗ : (BX ∗ , w∗ ) → Y is a Baire-1 function, is a pointwise limit of a sequence (Tn ) of operators with T |BX ∗ : (BX ∗ , w∗ ) → (Y, · ) continuous for all n ∈ N. Our results in this case are connected with classical results of Choquet, Odell and Rosenthal.

Selections, Paracompactness and Compactness

2000 Mathematics Subject Classification: 54C60, 54C65, 54D20, 54D30.

Compactness in quasi-Banach function spaces and applications to compact embeddings of Besov-type spaces

There are two main aims of the paper. The first one is to extend the criterion for the precompactness of sets in Banach function spaces to the setting of quasi-Banach function spaces. The second one is to extend the criterion for the precompactness of sets in the Lebesgue spaces $L_p(\Rn)$, $1 \leq p < \infty$, to the so-called power quasi-Banach function spaces. These criteria are applied to establish compact embeddings of abstract Besov spaces into quasi-Banach function spaces. The results are illustrated on embeddings of Besov spaces $B^s_{p,q}(\Rn)$, $0

Remyelination in experimentally demyelinated connexin 32 KnockOut mice

The aim of this study was to evaluate the role of connexin 32 (Cx 32) during remyelination of the peripheral nervous system, through a local injection of either 0,1% ethidium bromide solution or saline in the sciatic nerve of Cx 32 knockout mice. Euthanasia was performed ranging from 1, 2, 3, 7, 15, 21 to 30 days after injection. Histochemical, immunohistochemical, immunofluorescence and transmission electron microscopical techniques were used to analyze the development of the lesions. Within the sciatic nerves, Schwann cells initially showed signs of intoxication and rejected their sheaths; after seven days, some thin newly formed myelin sheaths with uneven compactness and redundant loops (tomacula) were conspicuous. We concluded that the regeneration of lost myelin sheaths within the PNS followed the pattern already reported for this model in other laboratory species. Therefore, these results suggest that absence of Cx 32 did not interfere with the normal pattern of remyelination in this model in young mice.