Search

The present paper develops the ideas presented in [1]. Let o be a Dedeking ring, and let Λ be a finitely generated algebra over o. An integral representation in the broad sense of the ring Λ over o is a homomorphism of Λ to the endomorphism ring of a finitely generated module over o. A representation in the restricted sense is a representation by matrices over o. Thus, the problem of describing the integral representations over o is subdivided into the following two problems: the description of representations in the broad sense and the selection of them of representations in the restricted sense. It is proved that any representation of Λ by matrices over the field k of fractions of the ring o is equivalent over k to an integral representation in the broad sense. This fact simplifies the problem of describing the representations in the broad sense. A representation is equivalent to a representation in the restricted sense if its degree over k and the order of the ideal class group of the ring o are relatively prime, or if it is the direct sum of h copies of one and the same representation over k, where h is the exponent of the ideal class group of o. Bibliography:3 titles.

Let σ be a Dedekind ring, let σ be a maximal order in a quadratic extension K of the field k of quotients of the ring σ, let Λ be a subring of the ring σ, containing σ and such that ΛK=K. It is proved that σ/Λ is a cyclic Λ-module. From here there follows, in particular, that each finitely generated torsion-free Λ-module is a direct sum of modules which are isomorphic to the ideals of ring Λ.

Two examples are given, showing the connections between the representations of algebraic numbers of a field by rational symmetric matrices, and the representations of integers of a field by symmetric integer-valued matrices.

The authors' survey paper is devoted to the present state of computational methods in linear algebra. Questions discussed are the means and methods of estimating the quality of numerical solution of computational problems, the generalized inverse of a matrix, the solution of systems with rectangular and poorly conditioned matrices, the inverse eigenvalue problem, and more traditional questions such as algebraic eigenvalue problems and the solution of systems with a square matrix (by direct and iterative methods).

The properties of the system of irreducible complex representations of full linear groups are described, without making use of the characteristics of the representations. The description is given in terms involving the multiplication of irreducible representations, which was introduced by Green. Explicitly, the concept of prime and primary representations is introduced, theorems are proved on the unique decomposition into primary factors, and the structure of the primary representations, belonging to a fixed prime representation, is described.

For pt.I, see Kibernetika, vol.13, no.6, p.28 (1977). Considerable effort was devoted in the surveyed period to automatic decomposition of sequential algorithms, or rather of procedures or subprograms written in the algorithmic languages ALGOL or FORTRAN. The authors do not consider this body of research, they only note that, on the one hand, the available linear algebra subprograms included in Eispack provide convenient objects for testing various approaches to automatic construction of parallel programs and, on the other, an important state in this activity is the development of methods for fast and efficient solution of linear recurrences, which reduce to solving systems of linear equations with band-triangular matrix (in particular, of sufficiently small width). This article reflects the penetration of the parallelism ideas into the computational methods of linear algebra in recent years. 74 references.

Let λ be an associative ring with unity. The main result of the article consists in the proof of the periodicity of the Hochschild cohomologies of λ in the case when λ is a Z-ring with a power basis. The period is equal to 2. This result is proved for maximal orders of fields of algebraic numbers.

One describes the matrices which are inverses to nonsingular irreducible Hessenberg matrices.

A probability measure $\mu $ on a finite set R is called interior if $\mu (a) > 0$ for any $a \in R$. The set of all interior measures on R is denoted by $W(R)$.Theorem. There exists a mapping$\varphi $ of $W(R)$into Euclidean spaceEof suitable dimension with two properties1. All conditional probabilities\[ \mu (\left. a \right|A) = \frac{{\mu (a)}}{{\mu (A)}},\quad a \in A \subset R, \]are uniformly continuous functions$\varphi (\mu )$on the whole set$\varphi (W(R))$in the sense of the metric onE.2. The closure of $\varphi (W(R))$ in Eis homeomorphic to the closed simplex of suitable dimension

An estimate of the deviation of the solution of linear algebraic equations from the solution of a “frozen” system, under sufficiently small variations of the system's coefficients, is given within the framework of linear error theory.

We give an interpretation of A. V. Yakovlev's “additional embeddability condition,” which, together with a compatibility condition, is necessary and sufficient for the solvability of the embedding problem in Galois theory.