Geometry of distributions of flags on Grassmann manifolds of a projective space. II

We consider intrinsic normalizations of distributions of flags of type (m, m+1) on the Grassmann manifold ofm-planes of a (2m+2)-dimensional projective space. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 335–349, July–September, 2000. Translated by V. Mackevičius     

Geometry of distributions of flags on Grassmann manifolds of a projective space. I

We consider intrinsic normalizations of distributions of flags of type (m, m+1) on the Grassmann manifold ofm-planes of a (2m+2)-dimensional projective space. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 2, pp. 214–227, April–June, 2000. Translated by V. Mackevičius     

On the global theory of projective mappings

We consider the theory of constant rank projective mappings of compact Riemannian manifolds from the global point of view. We study projective immersions and submersions. As an example of the results, letf:(M, g) → (N, g′) be a projective submersion of anm-dimensional Riemannian manifold (M, g) onto an (m−1)-dimensional Riemannian manifold (N, g′). Then (M, g) is locally the Riemannian product of the sheets of two integrable distributions Kerf _* and (Kerf _*)^⊥ whenever (M, g) is one of the two following types: (a) a complete manifold with Ric ≥ 0; (b) a compact oriented manifold with Ric ≤ 0. Translated fromMatematicheskie Zametki, Vol. 58, No. 1, pp. 111–118, July, 1995. This work was partially supported by the Russian Foundation for Basic Research grant No. 94-01-0195.     

Note on elongations of totally projective p-groups by p ω+n -projective p-groups

It is proved that any Σ-group, which is a special elongation of a totally projective abelian p-group by a p ^ω+1-projective abelian p-group, is totally projective. In particular, each p ^ω+1-projective abelian Σ-p-group is a direct sum of countable p-groups of lengths not exceeding ω + 1. This strengthens our recent result published in Comment. Math. Univ. St. Pauli (2006). Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 180–185, April–June, 2006.     

On the stability of <Emphasis Type="Italic">θ</Emphasis>-derivations on <Emphasis Type="Italic">JB</Emphasis>*-triples

We introduce the concept of θ-derivations on JB*-triples and prove the Hyers–Ulam-Rassias stability of θ-derivations on JB*-triples. We deal with the Hyers-Ulam-Rassias stability that was first introduced by Th. M. Rassias in the paper “On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300”. The first author was supported by Korea Research Foundation Grant KRF-2005-070-C00009.     

The semilattice tensor product of projective distributive lattices

Grant A. Fraser defined the semilattice tensor productA ⊗B of distributive latticesA, B and showed that it is a distributive lattice. He proved that ifA ⊗B is projective then so areA andB, that ifA andB are finite and projective thenA ⊗B is projective, and he gave two infinite projective distributive lattices whose semilattice tensor product is not projective. We extend these results by proving that ifA andB are distributive lattices with more than one element thenA ⊗B is projective if and only if bothA andB are projective and both have a greatest element. Presented by W. Taylor.     

Polarities in n-uniform projective Hjelmslev planes

In [9] the author has studied polarities in finite 2-uniform projective Hjelmslev planes. The present paper deals with polarities in finite n-uniform projective Hjelmslev planes (n 2).The author's research was supported by IWONL grant no. 840037.     

Configurations in finite projective planes

We find necessary conditions that should be imposed on a number n for the existence of an arbitrary configuration in at least one finite projective plane of order n. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 10, pp. 1345–1359, October, 1997.     

Corrigendum and addendum to « Multifunctions on abstract measurable spaces and applications to stochastic decision theory »

Summary This note points out that the ? only if ? part of Theorem 1′ (i) of the above paper [Annali di Matematica pura ed applicata, (IV) 101 (1974), pp. 229–236] is false as is part (ii) of Theorem 2. Counterexamples are given and a weak form of the ? only if ? part of Theorem 1′ (i) is established. Entrata in Redazione il 16 novembre 1977. Partially supported by National Science Foundation Grant MCS 76-24436.     

On discrete α-unimodality

A continuous composition semigroup of probability generating functionsF≔(F _t ,t≥0) and the corresponding multiplication ⊙ _F of van Harnet al. (1982,Z. Wahrsch. Verw. Gebiete,61, 97–118) are used to introduce the concept of [F; α]-unimodality which generalizes the discrete α-unimodality due to Abouammoh (1987,Statist. Neerlandica,41, 239–244) and Alamatsaz (1993,Statist. Neerlandica,47, 245–252). We offer various characterizations and other properties of [F;α]-unimodality. Notably, several convolution results are presented. Moreover, we explore the relationship between [F;α]-unimodality and the concepts of discrete self-decomposability and stability. Finally, lower bounds for variances of [F;α]-monotone and [F;α]-unimodal random variables are derived and some examples are also mentioned. Research supported by Grant SS024 of the Research Center of Kuwait University.     

On Rees weakly projective right acts

In this article, we present an extension of the concept of weakly projective acts to the socalled (S/I, S/J)-projective acts. Retracts, coproducts, and products of acts as well as Rees factor acts are considered from the point of view of these properties. They are used to describe monoids that are disjoint unions of a group with a left zero semigroup or with a disjoint union of simple right ideals. We suggest concepts of weak QF-monoids. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 10, No. 3, pp. 85–96, 2004.     

Superintuitionistic logics and the projective beth property

It is proved that in superintuitionistic logics, the projective Beth property follows from the Craig interpolation property, but the converse does not hold. A criterion is found which allows us to reduce the problem asking whether the projective Beth property is valid in superintuitionistic logics to suitable properties of varieties of pseudoboolean algebras. It is shown that the principle of variable separation follows from the projective Beth property. On the other hand, the interpolation property in a logic L implies the projective Beth property in Δ(L). Supported by RFFR grants No. 96-01-01552 and No. 99-01-00600. Translated fromAlgebra i Logika, Vol. 38, No. 6, pp. 680–696, November–December, 1999.     

On structure of small contractions of odd dimensional projective varieties

Let X be a smooth projective variety of dimension 2k-1 (k≥3) over the complex number field. Assume that fR: X→Y is a small contraction such that every irreducible component Ei of the exceptional locus of fR is a smooth subvariety of dimension k. It is shown that each Ei is isomorphic to the k-dimensional projective space Pk, the k-dimensional hyperquadric surface Qk in Pk 1, or a linear Pk-1-bundle over a smooth curve.     

Dominions of universal algebras and projective properties

Let A be a universal algebra and H its subalgebra. The dominion of H in A (in a class {ie304-01}) is the set of all elements a ∈ A such that every pair of homomorphisms f, g: A → ∈ {ie304-02} satisfies the following: if f and g coincide on H, then f(a) = g(a). A dominion is a closure operator on a set of subalgebras of a given algebra. The present account treats of closed subalgebras, i.e., those subalgebras H whose dominions coincide with H. We introduce projective properties of quasivarieties which are similar to the projective Beth properties dealt with in nonclassical logics, and provide a characterization of closed algebras in the language of the new properties. It is also proved that in every quasivariety of torsion-free nilpotent groups of class at most 2, a divisible Abelian subgroup H is closed in each group 〈H, a〉 generated by one element modulo H. Translated from Algebra i Logika, Vol. 47, No. 5, pp. 541–557, September–October, 2008.     

A classification of projective homogeneous varieties up to motivic isomorphism

A complete classification is given for anisotropic projective homogeneous varieties of dimension less than 6 up to motivic isomorphism. Several criteria are presented for anisotropic flag varieties of type A_n to have isomorphic motives. Bibliography: 22 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 330, 2006, pp. 158–172.     

The fundamental theorem about weak relative [C, H]-Hopf modules

In this paper, we mainly introduce the concept of weak relative [C, H]-Hopf modules and give the fundamental theorem of weak relative right [C, H]-Hopf modules. Published in Russian in Matematicheskie Zametki, 2007, Vol. 82, No. 4, pp. 530–537. The text was submitted by the authors in English.     

Injective and projective objects in the category of locally compact modules over the ring of integral values of a global field

In the paper injective and projective objects in the category of locally compact modules over the ring of integral values of a global field are described together with the objects of this category possessing injective and projective resolvents. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 118–123, July, 1997. Translated by A. I. Shtern     

On the ϕ-structure on the projective groupL 2(q)

In the paper it is proved that the projective groupL ₂(q) cannot be the automorphism group of a finite left distributive quasigroup. This is a special case of the conjecture according to which the automorphism group of a left distributive quasigroup is solvable. Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 725–728, May, 1998.