Whitney's type theorems for infinite dimensional spaces

It is proved that for any $f$ is an element of $C^k(L,R)$, where k is a natural number and L is a closed linear subspace of a nuclear Frechet space $X$, the function $f$ can be extended to a function of class $C^{k-1}$ defined on the entire space $X$. It is also proved that for any $f$ is an element of $C^k(L, R)$, where $k$ is a natural number of infinity and L is a closed linear subspace of a dual $X$ of a nuclear Frechet space, the function $f$ can be extended to a function of class $C^k$ defined on the entire space $X$. In addition, it is proved that under these conditions, the existence of a linear extension operator is equivalent to the complementability of the subspace.

Investigation of DNA polymorphisms in SMAD genes for genetic predisposition to diabetic nephropathy in patients with type 1 diabetes mellitus

Aims/hypothesis: SMAD proteins are involved in multiple signalling pathways and are key modulators of gene expression. We hypothesised that genetic variation in selected SMAD genes contributes to susceptibility to diabetic nephropathy. Methods: We selected 13 haplotype tag (ht) single nucleotide polymorphisms (SNPs) from 67 variants identified by resequencing the SMAD2 and SMAD3 genes. For SMAD1, SMAD4 and SMAD5 genes, genotype data were downloaded for 217 SNPs from Phase II of the International HapMap project. Of these, 85 SNPs met our inclusion criteria, resulting in the selection of 13 tag SNPs for further investigation. A case-control approach was employed, using 267 nephropathic patients and 442 controls with type 1 diabetes from Ireland. Two further populations (totalling 1,407 patients, 2,238 controls) were genotyped to validate initial findings. Genotyping was conducted using iPLEX, TaqMan and gel electrophoresis.
Results: The distribution of genotypes was in Hardy-Weinberg equilibrium. Analysis by the ? 2 test of genotype and allele frequencies in patients versus controls in the Irish population (n?=?709) revealed evidence for the association of one allele at 5% level of significance (rs10515478, p uncorrected?=?0.006; p corrected?=?0.04). This finding represents a relatively small difference in allele frequency of 6.4% in the patient group compared with 10.7% in the control group; this difference was not supported in subsequent investigations using DNA from European individuals with similar phenotypic characteristics.
Conclusions/interpretation: We selected an appropriate subset of variants for the investigation of common genetic risk factors and assessed SMAD1 to SMAD5 genes for association with diabetic nephropathy. We conclude that common polymorphisms in these genes do not strongly influence genetic susceptibility to diabetic nephropathy in white individuals with type 1 diabetes mellitus.

1-amenability of A(X) for Banach spaces with 1-unconditional bases

The main result of the note is a characterization of 1-amenability of Banach algebras of approximable operators for a class of Banach spaces with 1-unconditional bases in terms of a new basis property. It is also shown that amenability and symmetric amenability are equivalent concepts for Banach algebras of approximable operators, and that a type of Banach space that was long suspected to lack property A has in fact the property. Some further ideas on the problem of whether or not amenability (in this setting) implies property A are discussed.

Generation of metallosupramolecular polymer gels from multiply functionalized grid-type complexes

A ditopic ligand (1), containing two tridentate bis(acylhydrazone) subunits and bearing both long alkyl chains and hydrogen-bonding groups, has been synthesised. Metal cation binding in the presence of a base leads to hierarchical self-assembly, forming first a neutral [2 x 2] grid-type complex (2) that hierarchically assembles into metallosupramolecular polymer gels in toluene.