2-Selmer groups and the Birch-Swinnerton-Dyer Conjecture for the congruent number curves

We take an approach toward counting the number of integers n for which the curve En: y²=x³−n²x has 2-Selmer groups of a given size. This question was also discussed in a pair of papers by Roger Heath-Brown. In contrast to earlier work, our analysis focuses on restricting the number of prime factors of n. Additionally, we discuss the connection between computing the size of these Selmer groups and verifying cases of the Birch and Swinnerton-Dyer Conjecture. The key ingredient for the asymptotic formulae is the “independence” of the Legendre symbol evaluated at the prime divisors of an integer with exactly k prime factors.     

S-types of Global Towers of Spaces and Exterior Spaces

The closed model category of exterior spaces, that contains the proper category, is a useful tool for the study of non compact spaces and manifolds. The notion of exterior weak ℕ-S-equivalences is given by exterior maps which induce isomorphisms on the k-th ℕ-exterior homotopy groups for k ∈ S, where S is a set of non negative integers. The category of exterior spaces with a base ray localized by exterior weak ℕ-S-equivalences is called the category of exterior ℕ-S-types. The existence of closed model structures in the category of exterior spaces permits to establish equivalences between homotopy categories obtained by dividing by exterior homotopy relations, and categories of fractions (localized categories) given by the inversion of classes of week equivalences. The family of neighbourhoods ‘at infinity’ of an exterior space can be interpreted as a global prospace and under the condition of first countable at infinity we can consider a global tower instead of a prospace. The objective of this paper is to use localized categories to find the connection between S-types of exterior spaces and S-types of global towers of spaces. The main result of this paper establishes an equivalence between the category of S-types of rayed first countable exterior spaces and the category of S-types of global towers of pointed spaces. As a consequence of this result, categories of global towers of algebraic models localized up to weak equivalences can be used to give some algebraic models of S-types. The authors acknowledge the financial support given by the projects FOMENTA 2007/03 and MTM2007-65431.     

可除阿贝尔p-群的自同构群

主要探讨了秩大于或者等于p-1的可除阿贝尔p-群的p-自同构群,并且得到这些p-自同构如何作用在该可除阿贝尔p-群上.这些结论有助于进一步理解 ?ernikov p-群的结构.     

Sub-cover-avoidance Properties and the Structure of Finite Groups

A subgroup H of a group G is said to have the sub-cover-avoidance property in G ffthereis a chief series 1 = G0 ≤ G1 ≤…≤ Gn - G, such that Gi-1（H ∩ Gi） G for every i = 1,2,... ,l. In this paper, we give some characteristic conditions for a group to be solvable under the assumptions that some subgroups of a group satisfy the sub-cover-avoidance property.     

Solid-phase oligosaccharide synthesis of a small library of N-glycans

Solid-phase oligosaccharide synthesis is based on a hydroxymethylbenzyl benzoate spacer linker which is connected to the Merrifield resin (1 P). Glycosylation was performed with O-glycosyl trichloroacetimidates of glucosamine, mannose, and galactose permitting chain extension (2e, 5e), branching (4b, 7b, 8b), and chain termination (3t, 6t, 9t) with the use of O-benzyl, O-benzoyl, and N-dimethylmaleoyl as permanent and O-fluorenylmethoxycarbonyl (Fmoc) and O-phenoxyacetyl (PA) as temporary protecting groups. The steps required on solid phase are i) glycosylation under TMSOTf catalysis, ii) selective cleavage of the temporary protecting groups, Fmoc with NEt3 and PA with 0.5 equivalents of NaOMe in CH2Cl2/MeOH, and iii) product cleavage from the resin with 4.0 equivalents of NaOMe in CH2Cl2/MeOH and following O-acetylation for convenient product isolation. Thus a highly successful synthesis of a small library of seventeen N-glycan structures was made possible comprising the N-glycan pentasaccharide core structure 53 and two further chain extended hexa- and heptasaccharide N-glycans with a glucosamine or a lactosamine residue, respectively, which is attached to one of the mannose residues of the core structure (56 and 59).     

Unexpected reactions of ferrocene acetal derived from tartaric acid with alkyllithium: competition between proton abstraction and nucleophilic attack

We wished to prepare planar chiral compounds by the lithiation of acetal 2-ferrocenyl-(4S,5S)-bis(methoxymethyl)-1,3-dioxolane (1) with butyllithium followed by the reaction with an electrophile. However, the desired products were not observed and two unexpected products, 1-ferrocenyl-1-pentanol (4) of the nucleophilic attack product and 2-ferrocenyl-4,5-dimethylene-1,3-dioxolane (5) of the proton abstraction product, were isolated. Because the nucleophilic attack on acetal carbon is rarely reported so far and both products 4 and 5 may have some potential uses in organic synthesis, these unexpected reactions are investigated in detail. The mechanisms of these reactions are discussed.     

Studies into the Early Degradation Stages of Cellulose by Different Iron Gall Ink Components

Selective fluorescence labelling of oxidized cellulose functionalities followed by GPC-MALLS was used to get a deeper insight into ink-induced degradation processes. As the method is very sensitive towards oxidation and molecular weight changes, slight variations at the very beginning of aging processes, e.g. during ink corrosion of cellulose, can be studied. Five different ink modifications were applied on model papers and underwent mild accelerated aging at 55 °C and cycling humidity (7 days) followed by a short period of static humid aging at 80 °C (2 days). Pure ink constituents like tannic acid or iron sulphate do not result in the same degree of oxidation or chain scission as complete inks. Balanced ink degrades paper more than single compounds, but less than unbalanced inks. Interestingly, some degradation occurs already during or shortly after the application process of unbalanced inks on paper. It could be demonstrated that this oxidation proceeded in a rather high Mw area, while the subsequent aging steps affected predominantly regions of shorter cellulose chains.     

The Quasi-Exactly Solvable Problems in Relativistic Quantum Mechanics

We study the quasi-exactly solvable problems in relativistic quantum mechanics. We consider the problems for the two-dimensional Klein-Gordon and Dirac equations with equal vector and scalar potentials, and try to find the general form of the quasi-exactly solvable potential. After obtaining the general form of the potential, we present several examples to give the specific forms. In the examples, we show for special parameters the harmonic potential plus Coulomb potential, Killingbeck potential and a quartic potential plus Cornell potential are quasi-exactly solvable potentials.     

Numerical Schemes for Linear and Non-Linear Enhancement of DW-MRI

We consider the linear and non-linear enhancement of diffusion weighted magnetic resonance images (DW-MRI) to use contextual information in denoising and inferring fiber crossings. We describe the space of DW-MRI images in a moving frame of reference, attached to fiber fragments which allows for convection-diffusion along the fibers. Because of this approach, our method is naturally able to handle crossings in data. We will perform experiments showing the ability of the enhancement to infer information about crossing structures, even in diffusion tensor images (DTI) which are incapable of representing crossings themselves. We will present a novel non-linear enhancement technique which performs better than linear methods in areas around ventricles, thereby eliminating the need for additional preprocessing steps to segment out the ventricles. We pay special attention to the details of implementation of the various numerical schemes.     

三氟丙基修饰的有机-无机杂化二氧化硅膜制备、氢气分离及水热稳定性能

以三氟丙基三甲氧基硅烷(TFPTMS)和1,2-双(三乙氧基硅基)乙烷(BTESE)为前驱体, 通过溶胶-凝胶法在酸性条件下制备三氟丙基修饰的有机-无机杂化SiO₂膜材料, 并深入研究三氟丙基修饰对溶胶粒径和疏水性能的影响以及膜材料的氢气渗透分离性能和长期水热稳定性. 结果表明三氟丙基已成功修饰到有机-无机杂化SiO₂膜材料中, 且随着TFPTMS修饰量的增加, 溶胶粒径有减小趋势, 膜材料的疏水性能逐渐提高. 当n(TFPTMS)/n(BTESE)＝0.6时, 溶胶平均粒径为2.11 nm, 膜材料对水的接触角达到111.6°±0.7°. H₂在修饰后膜材料中的输运主要遵循微孔扩散机理, 300 ℃时H₂的渗透率为8.86×10^-7 mol·m^-2·s^-1·Pa^-1, H₂/CO₂的理想分离系数达到5.4, 且当进气摩尔比例为1∶1时H₂/CO₂的双组分气体分离系数达到了4.82, 均高于Knudsen扩散分离因子(H₂/CO₂＝4.69), 膜材料呈现出良好的分子筛分性能. 膜材料在250 ℃及水蒸气摩尔含量为5%的水热环境中能稳定工作300 h以上.