Tensor Rank: Matching Polynomials and Schur Rings

We study the polynomial equations vanishing on tensors of a given rank. By means of polarization we reduce them to elements  $A$ of the group algebra ${\mathbb {Q}}[S_n\times S_n]$ and describe explicit linear equations on the coefficients of  $A$ to vanish on tensors of a given rank. Further, we reduce the study to the Schur ring over the group $S_n\times S_n$ that arises from the diagonal conjugacy action of  $S_n$ . More closely, we consider elements of ${\mathbb {Q}}[S_n\times S_n]$ vanishing on tensors of rank $n-1$ and describe them in terms of triples of Young diagrams, their irreducible characters, and nonvanishing of their Kronecker coefficients. Also, we construct a family of elements in ${\mathbb {Q}}[S_n\times S_n]$ vanishing on tensors of rank $n-1$ and illustrate our approach by a sharp lower bound on the border rank of an explicitly produced tensor. Finally, we apply this construction to prove a lower bound $5n^2/4$ on the border rank of the matrix multiplication tensor (being, of course, weaker than the best known one $(2-\epsilon )\cdot n^2$ , due to Landsberg, Ottaviani).     

Revêtements modérés et groupe fondamenttal de graphe de groupes

Let R be a discrete complete valuation ring, with algebraically closedresidue field. Let X be a semi-stable R-curve, with smooth generic fibre. In this paper we study tame coverings of X.     

Inequalities for moments of symmetric polynomial forms in random variables

In the last few years, interest in the properties of distributions of quadratic and polynomial forms in random variables has again grown. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 159–161, Perm, 1993.     

On amorphic <Emphasis Type="Italic">C</Emphasis>-algebras

An amorphic association scheme has the property that any of its fusions also is an association scheme. In this paper, the property of being amorphic is extended to an arbitrary C-algebra, and it is proved that, up to isomorphism, any amorphic C-algebra is determined by the multiset of its diagonal structure constants and an additional integer equal to ±1. It is shown that any amorphic C-algebra with rational structure constants is a fusion of a homogeneous amorphic C-algebra. As a special case, the well-known Ivanov characterization of the intersection numbers of amorphic association schemes is obtained. Bibliography: 13 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 340, 2006, pp. 87–102.     

The Catenary and Tame Degree in Finitely Generated Commutative Cancellative Monoids

Problems involving chains of irreducible factorizations in atomic integral domains and monoids have been the focus of much recent literature. If S is a commutative cancellative atomic monoid, then the catenary degree of S (denoted c(S)) and the tame degree of S (denoted t(S)) are combinatorial invariants of S which describe the behavior of chains of factorizations. In this note, we describe methods to compute both c(S) and t(S) when M is a finitely generated commutative cancellative monoid.     

Synthesis of fluoro-containing substituted 1,3,5-triazines

Conclusions By the method of acylation-cyclodehydration of imidoylamidines we have prepared 1,3,5-triazines with three different substituents, among them reactive unsaturated and nitrile groups.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 4, pp. 928–933, April, 1989.     

Polynomial isomorphism algorithm for graphs which do not pinch to K3,g

In this paper we construct a polynomial algorithm for verifying the isomorphism of graphs which do not pinch to K_3,g. In the construction we use properties of colored graphs from this class and results of Babai-Luks on canonization of graphs. The class cited contains the class of graphs of genus not exceeding g, and we apply the algorithm constructed to the recognition of isomorphism of graphs of bounded valence. The method given is a generalization of the combinatorial and group-theoretic approach to the isomorphism problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 137, pp. 99–114, 1984.     

Cycloaddition of perfluorovinyl ethers to dienes

Conclusions Cycloaddition of perfluorovinyl ethers to butadiene or piperylene led to 1-vinyl- or 1-propenyl-2-perfluoroalkoxy-2,3,3-trifluorocyclobutanes as mixtures of cis and trans isomers.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 10, pp. 2372–2377, October, 1988.     

Point-contact spectroscopy of the electron-phonon interaction in LaNi5

The electron-phonon interaction in LaNi₅ is studied by point-contact spectroscopy. The spectrum corresponding to the electron-phonon interaction function g (the phonon density of states modulated by the matrix element of the electron-phonon interaction) is found. The functiong is used to analyze the point-contact spectra of PrNi₅, where besides the electron-phonon contribution there is also an interaction between the conduction electrons and the crystal electric field levels of the Pr³⁺ ion.     

Preparation of bis-(γ-trifluoropropyl)dichloro

Summary It was shown that (CF₃CH₂CH₂)₂SiCl₂, along with Cl₃SiCH₂CH₂CF₃ and CH₃Si(Cl)₂CH₂CH₂CF₃ may easily be prepared without danger by the addition of H₂SiCl₂ and HSiCl₃ or CH₂SiHCl₂ in the presence of platinohydrochloric acid.