Mazur module for elliptic curves

Under certain assumptions parameters of the Mazur module for an elliptic curve E over a extension K/K₀ are computed. This makes it possible, in particular, to prove in certain cases that the group E(K) is finitely generated without assuming that the groups E(K₀) and III(K₀/K₀, E) are finite.Translated from Matematicheskie Zametki, Vol. 17, No. 2, pp. 319–328, February, 1975.The author thanks Yu. I. Manin for his useful discussions and attention to my work.     

New weighted Rogers-Ramanujan partition theorems and their implications

This paper has a two-fold purpose. First, by considering a reformulation of a deep theorem of Göllnitz, we obtain a new weighted partition identity involving the Rogers-Ramanujan partitions, namely, partitions into parts differing by at least two. Consequences of this include Jacobi's celebrated triple product identity for theta functions, Sylvester's famous refinement of Euler's theorem, as well as certain weighted partition identities. Next, by studying partitions with prescribed bounds on successive ranks and replacing these with weighted Rogers-Ramanujan partitions, we obtain two new sets of theorems - a set of three theorems involving partitions into parts (mod 6), and a set of three theorems involving partitions into parts (mod 7), .

     

Oligomerization of 1,2‐Ethanedithiol: An Expedient Approach to Oligothiaethylenethioglycols

Reactions of ethylenedithioglycol (ETG) with Na₂CO₃, K₂CO₃, and Cs₂CO₃ provided the oligothiaethylenethioglycols (nETG): di‐ (DETG), tri‐ (TrETG), tetra‐ (TETG), and pentathiaethylenethioglycol (PETG), along with higher polymers. The most efficient carbonate was K₂CO₃ and reactions using DETG and TrETG as starting materials—or their mixtures—were also found to afford similar species. This largely unknown oligomerization process was thoroughly explored and potential pathways were put forward. A convenient conversion of ETG to laboratory quantities of the otherwise scarce and/or expensive DETG, TrETG, TETG, and PETG oligomers, in organic or aqueous media was achieved. Notably, this straightforward reaction takes place without the addition of expensive or toxic metal catalysts and with pure water as the solvent, thereby highlighting its potential as a green chemical reaction. Moreover, the process opens up new approaches to dynamic combinatorial libraries (DCLs) of oligomers and macrocycles with manifolded nETG [(SCH₂CH₂)_nS] bridges.     

On the number of subgroups of given structure in a finitep-group

Supported in part by the Ministry of Science and Technology of Israel and the Rashi Foundation.     

On the comparison theorem for étale cohomology of non-Archimedean analytic spaces

Let ϕ:Y →X be a morphism of finite type between schemes of locally finite type over a non-Archimedean fieldk, and letF be an étale constructible sheaf onY. In [Ber2] we proved that if the torsion orders ofF are prime to the characteristic of the residue field ofk then the canonical homomorphisms (R ^Q ϱ*F)^an →R ^q ϱ _* ^an F ^an are isomorphisms. In this paper we extend the above result to the class of sheavesF with torsion orders prime to the characteristic ofk.     

Positivity preserving transformations for -binomial coefficients

Several new transformations for -binomial coefficients are found, which have the special feature that the kernel is a polynomial with nonnegative coefficients. By studying the group-like properties of these positivity preserving transformations, as well as their connection with the Bailey lemma, many new summation and transformation formulas for basic hypergeometric series are found. The new -binomial transformations are also applied to obtain multisum Rogers-Ramanujan identities, to find new representations for the Rogers-Szegö polynomials, and to make some progress on Bressoud's generalized Borwein conjecture. For the original Borwein conjecture we formulate a refinement based on new triple sum representations of the Borwein polynomials.