Weakly automorphism invariant modules and essential tightness

While a module is pseudo-injective if and only if it is automorphism-invariant, it was not known whether automorphism-invariant modules are tight. It is shown that weakly automorphism-invariant modules are precisely essentially tight. We give various examples of weakly automorphism-invariant and essentially tight modules and study their properties. Some particular results: (1) R is a semiprime right and left Goldie ring if and only if every right (left) ideal is weakly injective if and only if every right (left) ideal is weakly automorphism invariant; (2) R is a CEP-ring if and only if R is right artinian and every indecomposable projective right R-module is uniform and essentially R-tight.     

Weakly dual automorphism invariant and superfluous cotightness

Abstract

The aim of the present paper is to introduce and study the dual concepts of weakly automorphism invariant modules and essential tightness. These notions are non-trivial generalizations of both weakly projectivity, dual automorphism invariant property and cotightness. We obtain certain relations between weakly projective modules, weakly dual automorphism invariant modules and superfluous cotight modules. It is proved that: (1) for right perfect rings, every module is a direct summand of a weakly dual automorphism invariant module and (2) weakly dual automorphism invariant modules are precisely superfluous cotight modules.     

Synthesis of nonlinear control systems invariant under perturbations

For nonlinear control systems with perturbations, we consider the problem of synthesis of perturbation-invariant characteristics (invariant functions) with the use of feedbacks. The existence of invariant functions is related to a decomposition of the control system for which the quotient system is independent of perturbations. We present conditions for the existence of such quotient systems, which are certain systems of partial differential equations. The synthesizing controls are found from these equations.     

Phase portraits of integrable quadratic systems with an invariant parabola and an invariant straight line

We classify the phase portraits of the quadratic polynomial differential systems having an invariant parabola, an invariant straight line, and a Darboux first integral produced by these two invariant curves.     

Orthogonal systems

An equidistant permutation array (E.P.A.)A(r, v) is av × r array in which every row is a permutation of the integers 1, 2, ,r such that any two distinct rows have precisely columns in common. In this paper we introduce the concept of orthogonality for E.P.A.s. A special case of this is the well known idea of a set of pairwise orthogonal latin squares. We show that a set of these arrays is equivalent to a particular type of resolvable (r, )-design. It is also shown that the cardinality of such a set is bounded byr – with the upper bound being obtained only if = 0. A brief survey of related orthogonal systems is included. In particular, sets of pairwise orthogonal symmetric latin squares, sets of orthogonal Steiner systems and sets of orthogonal skeins.     

Extension of dual subspaces invariant under an algebra

Phillips' known hypothesis concerning the extension of dual pairs of subspaces {£ ₁ ⁰ , £ ₂ ⁰ }, invariant under a commutative J-symmetric algebra R in a Hilbert space , to a dual pair of maximal subspaces {£₁, £₂}, invariant under R is established in the case where a dual pair of maximal subspaces exists {£₁, £₂, invariant under R with , and the pair {£ ₁ ⁰ , £ ₂ ¹ } consists of Jneutral subspaces.Translated from Matematicheskie Zametki, Vol. 3, No. 3, pp. 253–260, March, 1968.Finally, I express my sincere gratitude to M. A. Naimark for the attention he paid to this work.     

Sets invariant under an integral constraint on controls

In this paper we study the invariance of given sets with respect to a system with distributed parameters. The considered system is described by a heat conductivity equation whose right-hand side written in the additive form contains a control. For the initial data we obtain sufficient conditions for the strong and weak invariance of the set that represents the graph of a given multivalued mapping.     

Steiner triple systems with block-transitive automorphism groups

This paper describes the Steiner triple systems which have automorphism groups acting transitively on the blocks but not doubly transitively on the points. In the case of those systems not previously described, there is a group acting regularly on the blocks and the number of points is a prime power congruent to 7 modulo 12. A formula yielding the number (up to isomorphism) of these systems with a given number of points is derived.     

Existence of an invariant form under a linear map

Let F be a field of characteristic different from 2 and V be a vector space over F. Let J: α → α ^J be a fixed involutory automorphism on F. In this paper we answer the following question: given an invertible linear map T: V → V, when does the vector space V admit a T-invariant nondegenerate J-hermitian, resp. J-skew-hermitian, form?     

Orthogonal steiner systems

Aequationes mathematicae -     

Partitioning of invariant imbedding systems

The choice of partitioning the system matrix for a system of N linear ordinary differential equations may determine the ease or difficulty of obtaining a solution by the invariant imbedding method of Scott. This paper shows how the configuration and partitioning of the system matrix is reflected in the fundamental matrix. The partitioning of the fundamental matrix is the key to the ease or difficulty of obtaining a solution. If the fundamental matrix is known for a given system matrix configuration and partitioning, then the fundamental matrix associated with a new system matrix configuration may be derived by the same row and column interchanges that transformed the old system matrix into the new system matrix. The fundamental matrix for the new system matrix does not have to be recalculated anew from the Kronecker delta initial conditions.     

Nilpotent groups related to an automorphism

The aim of this paper is to state some results on an \(\alpha \)-nilpotent group, which was recently introduced by Barzegar and Erfanian (Caspian J. Math. Sci. 4(2) (2015) 271–283), for any fixed automorphism \(\alpha \) of a group G. We define an identity nilpotent group and classify all finitely generated identity nilpotent groups. Moreover, we prove a theorem on a generalization of the converse of the known Schur’s theorem. In the last section of the paper, we study absolute normal subgroups of a finite group.     

Partially translation invariant linear systems

The subject of this paper is the analysis of abstract linear systems in several variables, i.e. continuous linear mappings from Banach space valued test functions as inputs to Banach space valued distributions as outputs, N being partially translation invariant in the sense that it commute with translations from a given closed subgroup G of the additive group. The classical convolution representation of the invariant case generalizes to a Fourier series representation whose coefficients N_k are linear systems commuting with translations from the vector subgroup lin G generated by G. Thus N is essentially a "Fourier superposition" of classical systems. It is shown that causality of N with respect to a subsemigroup P of implies causality of the N_k which in turn corresponds to carrier conditions on the N_k.Dedicated to Professor Heinz König on the occasion of his 50th birthday.     

Parallel algorithm for calculating the invariant sets of high-dimensional linear systems under uncertainty

The development of efficient computational methods for synthesizing controls of high-dimensional linear systems is an important problem in theoretical mathematics and its applications. This is especially true for systems with geometrical constraints imposed on the controls and uncertain disturbances. It is well known that the synthesis of target controls under the indicated conditions is based on the construction of weakly invariant sets (reverse reachable sets) generated by the solving equations of the process under study. Methods for constructing such equations and corresponding invariant sets are described, and the computational features for high-dimensional systems are discussed. The approaches proposed are based on the previously developed theory and methods of ellipsoidal approximations of multivalued functions.