Every order is the endomorphism ring of a projective module over a quasi–hereditary order

We give in this note some new bounds on p _c(G) in terms of σ _c for several classes of finite groups, in particular, we prove that ρ _c ≤4σ _c (G) for any finite solvable group G, which improves some known results. We also pose some related open problems.     

Estimation of isotropic nuclear magnetic shieldings in the CCSD(T) and MP2 complete basis set limit using affordable correlation calculations

A linear correlation between isotropic nuclear magnetic shielding constants for seven model molecules (CH₂O, H₂O, HF, F₂, HCN, SiH₄ and H₂S) calculated with 37 methods (34 density functionals, RHF, MP2 and CCSD(T)), with affordable pcS‐2 basis set and corresponding complete basis set results, estimated from calculations with the family of polarization‐consistent pcS‐n basis sets is reported. This dependence was also supported by inspection of profiles of deviation between CBS estimated nuclear shieldings and shieldings obtained with the significantly smaller basis sets pcS‐2 and aug‐cc‐pVTZ‐J for the selected set of 37 calculation methods. It was possible to formulate a practical approach of estimating the values of isotropic nuclear magnetic shielding constants at the CCSD(T)/CBS and MP2/CBS levels from affordable CCSD(T)/pcS‐2, MP2/pcS‐2 and DFT/CBS calculations with pcS‐n basis sets. The proposed method leads to a fairly accurate estimation of nuclear magnetic shieldings and considerable saving of computational efforts. Copyright © 2013 John Wiley & Sons, Ltd.     

Gain of regularity for semilinear Schrödinger equations

We discuss local existence and gain of regularity for semilinear Schr?dinger equations which generally cause loss of derivatives. We prove our results by advanced energy estimates. More precisely, block diagonalization and Doi's transformation, together with symbol smoothing for pseudodifferential operators with nonsmooth coefficients, apply to systems of Schr?dinger-type equations. In particular, the sharp G?rding inequality for pseudodifferential operators whose coefficients are twice continuously differentiable, plays a crucial role in our proof. Received: 14 December 1998     

SOME OF MY FAVORITE SOLVED AND UNSOLVED PROBLEMS IN GRAPH THEORY

Abstract

During my long life I published many papers with related titles. To keep this paper short I will not give proofs and will restrict myself to problems in graph theory, but I will try to give references and make these as complete as possible.

I will start with Turk type problems in extremal graph theory.     

A generalization of gamma functions and Kronecker's limit formulas

We obtain a “Kronecker limit formula” for the Epstein zeta function. This is done by introducing a generalized gamma function attached to the Epstein zeta function. The methods involve generalizing ideas of Shintani and Stark. We first show that a generalized gamma function appears as the value at s=0 of the first derivative of the associated Epstein zeta function. Then this is used to yield Kronecker's limit formula and its “s=0”-version.     

The rank of the covariance matrix of an evanescent field

Evanescent random fields arise as a component of the 2D Wold decomposition of homogeneous random fields. Besides their theoretical importance, evanescent random fields have a number of practical applications, such as in modeling the observed signal in the space-time adaptive processing (STAP) of airborne radar data. In this paper we derive an expression for the rank of the low-rank covariance matrix of a finite dimension sample from an evanescent random field. It is shown that the rank of this covariance matrix is completely determined by the evanescent field spectral support parameters, alone. Thus, the problem of estimating the rank lends itself to a solution that avoids the need to estimate the rank from the sample covariance matrix. We show that this result can be immediately applied to considerably simplify the estimation of the rank of the interference covariance matrix in the STAP problem.     

On an asymptotically more efficient estimation of the single-index model

In this note, we revisit the single-index model with heteroscedastic error, and recommend an estimating equation method in terms of transferring restricted least squares to unrestricted least squares: the estimator of the index parameter is asymptotically more efficient than existing estimators in the literature in the sense that it is of a smaller limiting variance.     

Highly sensitive non‐isotopic restriction endonuclease fingerprinting of nucleotide variability in the gp60 gene within Cryptosporidium species,genotypes and subgenotypes infective to humans,and its implications

The high‐resolution analysis of genetic variation has major implications for the identification of parasites and micro‐organisms to species and subspecies as well as for population genetic and epidemiological studies. In this study, we critically assessed the effectiveness of a PCR‐based restriction endonuclease fingerprinting (REF) method for the detection of mutations in the 60 kDa glycoprotein gene (gp60) of Cryptosporidium, a genus of parasitic protists of major human and animal health importance globally. This gene displays substantial intraspecific variability in sequence, particularly in a TCA (perfect and imperfect) microsatellite region, is present as a single copy in the nuclear genome and is used widely as a marker in molecular epidemiological studies of Cryptosporidium hominis and C. parvum, the two predominant species that infect humans. The results of this study demonstrated an exquisite capacity of REF to detect nucleotide variability in the gp60 gene within each of the two species. The differentiation of genotypes/subgenotypes based on REF analysis was supported by targeted sequencing, allowing the detection of levels of variation as low as a single‐nucleotide transversion for amplicons of ∼1 kb in size. The high‐throughput potential and relatively low‐cost of REF make it a particularly useful tool for large‐scale genetic analyses of C. hominis and C. parvum. REF could also be utilized for comparative surveys of genetic variability across large nuclear genomic regions. Such analyses of Cryptosporidium in clinical and environmental samples by REF have important implications for identifying sources of infection, modes of transmission and/or possible infectivity to humans, thus assisting in the surveillance and control of cryptosporidiosis. Given its excellent mutation detection capacity, REF should find broad applicability to various single‐copy genes as well as a wide range of other protozoan and metazoan parasites. (The nucleotide sequences reported in this article are available in the GenBank database under accession numbers GU214343–GU214371).     

Rectangularity of products and completions of their subsets

The approach to the problem of the distribution of the functors of the Stone-?ech compactification, the Hewitt realcompactification or the Dieudonné completion with the operation of taking products is discussed using uniform structures on products. In particular, the role of different rectangular conditions is shown. Relative analogues of this question and new examples of (strongly) rectangular products are presented. Characterizations of bounded rectangular subsets of the product are given.     

On dimensional properties of the generalized Smirnov's spaces

We show that the transfinite inductive dimensions modulo PP-trind and P-trInd introduced in M.G. Charalambous (1997) [2] differ by simple spaces, where P is the absolutely additive Borel class A(α) or the absolutely multiplicative Borel class M(α), 0?α<ω₁.