Theorem of extension to regular measure

Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 32, No. 1, pp. 59–70, January–March, 1992.     

Extremal extension of a finitely additive measure

Translated from Matematicheskie Zametki, Vol. 43, No. 1, pp. 25–30, January, 1988.     

An extension of regular supermagic graphs

A graph is called supermagic if it admits a labelling of the edges by pairwise different consecutive positive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. In this paper we consider an extension of regular supermagic graphs and apply it to some constructions of supermagic graphs. Using the extension we prove that for any graph G there is a supermagic regular graph which contains an induced subgraph isomorphic to G.     

On the extension of real regular embedding

Let X be a real nonsingular affine algebraic variety of dimensionk. It is proved that any two regular (algebraic) embeddingsX ⁿ are regularly equivalent, provided that n 4k + 2.     

Uniqueness for algebraically regular solutions to pathological differential equations

A uniqueness theorem is proved for algebraically regular solutions to the unbounded initial value problem P′ = AP, P(0) = diag(1, 1, 1,…) in the real Banach algebra of infinite matrices $\text{M}$ with standard norm. It is not assumed that $\text{A}$ ∈ $\text{M}$, but it is required that A have an inverse in $\text{M}$, a property which is seen to be implied quite naturally by certain divergent or pathological systems. The conditions for the theorem are motivated by a particular system, previously considered by Hille and Feller, which arises from a divergent, purebirth, time dependent stochastic process, although no restriction requiring the solution matrix to be either stochastic or substochastic is necessary.The theorem may be easily generalized to any Banach algebra with identity.     

Every finitely generated regular field extension has a stable transcendence base

A fieldK is called stable if every finitely generaed regular field extensionF/K has a transcendence basex ₁, …,x _n with the following properties: The field extensionF/K(x ₁,…,x _n ) is separable and the Galois hull ofF/K(x ₁,…,x _n ) remains regular overK, i.e.K is algebraically closed in . We prove in this paper thatevery field is stable. This generalizes results from [FJ1] and [GJ] which prove that fields of characteristic 0 and infinite perfect fields are stable, respectively. [G] showed that finite fields are stable in dimension 1, i.e. every finitely generated regular field extension of transcendence degree 1 over a finite field has a stable transcendence base. In the last section of this paper we apply the theorem to the construction of PAC fields with additional properties. A fieldK is called PAC if every absolutely irreducible variety overK has at least oneK-rational point.     

Completion of an additive measure to a σ-additive measure by an extension of the space

For an algebra of subsets of a set X there is constructed a set and an algebra of its subsets so that the mapping is a one-to-one correspondence between and and for each additive measure on the measure on defined by the equation is countably additive.Translated from Matematicheskie Zametki, Vol. 3, No. 1, pp. 71–76, January, 1968.The author wishes to express his deep appreciation to S. V. Fomin, under whose guidance this paper was written.     

Weyl groups as Galois groups of a regular extension of the field ℚ

For a finite group G, Gal_T(G) denotes the property that there exists a regular Galois extension of the rational function field ℚ(T) over the field of rationals ℚ, with a Galois group G. This property is established to be satisfied by all Weyl groups except the type F₄, from which it follows that it holds also for Chevalley groups C₃(2) and D₄(2). Translated fromAlgebra i Logika, Vol. 34, No. 3, pp. 311-315, May-June, 1995.     

Extendability of quadratic modules over a polynomial extension of an equicharacteristic regular local ring

We prove that a quadratic A[T]$A[T]$-module Q   with Witt index (Q/TQ)?d$(Q/TQ)?d$, where d is the dimension of the equicharacteristic regular local ring A, is extended from A. This improves a theorem of the second named author who showed it when A is the local ring at a smooth point of an affine variety over an infinite field. To establish our result, we need to establish a local–global principle (of Quillen) for the Dickson–Siegel–Eichler–Roy (DSER) elementary orthogonal transformations.     

An extension of the basic Palm measure correspondence

The notion of Palm measure provides a one-to-one correspondence between the distributions of simple, stationary point processes on ℝ with finite intensity and those of stationary sequences of positive random variables with finite mean. Here we extend this result to any stationary random measures on ℝ, using a new notion of spacing measure based on an elementary measure inversion. Our approach is symmetric, in the sense that an iteration of the spacing measure construction leads back to the original distribution. The classical results for point processes arise as simple special cases. Received: 30 September 1998 / Revised version: 9 June 1999 / Published online: 30 March 2000     

An extension of weighted generalized cumulative past measure of information

Ricerche di Matematica - In this paper, we consider a shift-dependent measure of generalized cumulative entropy and its dynamic (past) version in the case where the weight is a general non-negative...