A generalization of the rogosinski-rogosinski theorem

We establish necessary and sufficient conditions for numerical functions αj(x), j ∈ N, x ∈ X, under which the conditions K(f _j ⊂ K(f ₁) ∀j≥2 and yield The functions fj(x) are uniformly bounded on the set X and take values in a boundedly compact space L, and K(fj) is the kernel of the function fj. The well-known Rogosinski-Rogosinski theorem follows from the proved statements in the case where X = N, α _j (x) ≡ α_j, and the space L is the m-dimensional Euclidean space.     

A generalization of the Riesz-Fischer theorem

An analog is established, in a certain sense, of the Riesz-Fischer theorem for the space L^P, p1, and a corollary derived.Translated from Matematicheskie Zametki, Vol. 12, No. 4, pp. 365–372, October, 1972.     

A generalization of the Razmyslov-Procesi theorem

It is proved that all relations between the invariants of several n x n-matrices over an infinite field of arbitrary characteristic follow from σ_n+1,σ_n+2,... where σ_i is the ith coefficient of a characteristic polynomial extended to matrices of any order ≥i. Similarly, all relations between the concomitants are implied by X_n+1, X_n+2, …, where X_i is a characteristic polynomial in the general n x n-matrix, also extended to matrices of any order. Supported by RFFR grant No. 95-01-00513. Translated fromAlgebra i Logika, Vol. 35, No. 4, pp. 433–457, July–August, 1996.     

A generalization of the Gauss-Lucas theorem

Given a set of points in the complex plane, an incomplete polynomial is defined as the one which has these points as zeros except one of them. The classical result known as Gauss-Lucas theorem on the location of zeros of polynomials and their derivatives is extended to convex linear combinations of incomplete polynomials. An integral representation of convex linear combinations of incomplete polynomials is also given.     

A generalization of Plantholt's theorem

Let K denote the complete graph K_2n+1 with each edge replicated r times and let χ′(G) denote the chromatic index of a multigraph G. A multigraph G is critical if χ′(G) > χ′(G/e) for each edge e of G. Let S be a set of sn – 1 edges of K. We show that, for 0 < s ≦ r, G/S is critical and that χ′ (G/(S ∪{e})) = 2rn + r – s for all e ∈ E(G/S). Plantholt [M. Plantholt, The chromatic index of graphs with a spanning star. J. Graph Theory 5 (1981) 5–13] proved this result in the case when r = 1.     

A generalization of Jung's theorem

Jung's theorem establishes a relation between circumradius and diameter of a convex body. Half of the diameter can be interpreted as the maximum of circumradii of all 1-dimensional sections or 1-dimensional orthogonal projections of a convex body. This point of view leads to two series of j-dimensional circumradii, defined via sections or projections. In this paper we study some relations between these circumradii and by this we find a natural generalization of Jung's theorem.I would like to thank Prof. Dr J. M. Wills, who called my attention to these generalized circumradii.     

A generalization of Barbashin-Krasovski theorem

The classical criterion of asymptotic stability of the zero solution of equations x^′=f(t,x) is that there exists a function V(t,x), a(‖x‖)?V(t,x)?b(‖x‖) for some a,b∈K, such that for some c∈K. In this paper we prove that if f(t,x) is bounded, is uniformly continuous and bounded, then the condition that can be weakened and replaced by and contains no complete trajectory of , t∈[−T,T], where , uniformly for (t,x)∈[−T,T]×BH.     

A generalization of Stark's theorem

Let K and K′ be distinct real quadratic fields, and let L and L′ be totally imaginary quadratic extensions of K and K′, respectively, both L and L′ having class number 1. Then there exists an effectively computable constant c = c(K, K′) such that either |_L| ≤ c or |d_L′| ≤ c, where d_L and d_L′ denote absolute discriminants.     

A generalization of the little Picard theorem

Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 46, pp. 90–100, 1986.