Parallelisms & Lie Connections

The aim of this article is to study rational parallelisms of algebraic varieties by means of the transcendence of their symmetries. The nature of this transcendence is measured by a Galois group built from the Picard-Vessiot theory of principal connections

Timelike duality, M′M'-theory and an exotic form of the Englert solution

Through timelike dualities, one can generate exotic versions of $M$-theory with different spacetime signatures. These are the $M^*$-theory with signature $(9,2,-)$, the $M'$-theory, with signature $(6,5,+)$ and the theories with reversed signatures $(1,10, -)$, $(2,9, +)$ and $(5,6, -)$. In $(s,t, \pm)$, $s$ is the number of space directions, $t$ the number of time directions, and $\pm$ refers to the sign of the kinetic term of the $3$ form. The only irreducible pseudo-riemannian manifolds admitting absolute parallelism are, besides Lie groups, the seven-sphere $S^7 \equiv SO(8)/SO(7)$ and its pseudo-riemannian version $S^{3,4} \equiv SO(4,4)/SO(3,4)$. [There is also the complexification $SO(8,\mathbb{C})/SO(7, \mathbb{C})$, but it is of dimension too high for our considerations.] The seven-sphere $S^7\equiv S^{7,0}$ has been found to play an important role in $11$-dimensional supergravity, both through the Freund-Rubin solution and the Englert solution that uses its remarkable parallelizability to turn on non trivial internal fluxes. The spacetime manifold is in both cases $AdS_4 \times S^7$. We show that $S^{3,4}$ enjoys a similar role in $M'$-theory and construct the exotic form $AdS_4 \times S^{3,4}$ of the Englert solution, with non zero internal fluxes turned on. There is no analogous solution in $M^*$-theory.Comment: 18 pages, v2: typos fixe

Two types of Cahuilla kinship expressions: inherent and establishing

In my Cahuilla Grammar (Seiler 1977:276-282) and in a subsequent paper (Seiler 1980:229-236) I have drawn attention to the fact that many kin terms in this language, especially those that have a corresponding reciprocal term in the ascending direction – like niece or nephew in relation to aunt – occur in two expressions of quite different morphological shape. The following remarks are intended to furnish an explanation of this apparent duplicity

A Method for Classification of Doubly Resolvable Designs and Its Application

This article presents the principal results of the Ph.D. thesis Investigation and classification of doubly resolvable designs by Stela Zhelezova (Institute of Mathematics and Informatics, BAS), successfully defended at the Specialized Academic Council for Informatics and Mathematical Modeling on 22 February 2010.The resolvability of combinatorial designs is intensively investigated because of its applications. This research focuses on resolvable designs with an additional property - they have resolutions which are mutually orthogonal. Such designs are called doubly resolvable. Their specific properties can be used in statistical and cryptographic applications.Therefore the classification of doubly resolvable designs and their sets of mutually orthogonal resolutions might be very important. We develop a method for classification of doubly resolvable designs. Using this method and extending it with some theoretical restrictions we succeed in obtaining a classification of doubly resolvable designs with small parameters. Also we classify 1-parallelisms and 2-parallelisms of PG(5,2) with automorphisms of order 31 and find the first known transitive 2-parallelisms among them. The content of the paper comprises the essentials of the author’s Ph.D. thesis

Parallelism

EnProblems involving the idea of parallelism occur in finite geometry and in graph theory. This article addresses the question of constructing parallelisms with some degree of "symmetry". In particular, can we say anything on parallelisms admitting an automorphism group acting doubly transitively on "parallel classes"

Conjugates for Finding the Automorphism Group and Isomorphism of Design Resolutions

Consider a combinatorial design D with a full automorphism group G D. The automorphism group G of a design resolution R is a subgroup of G D. This subgroup maps each parallel class of R into a parallel class of R. Two resolutions R 1 and R 2 of D are isomorphic if some automorphism from G D maps each parallel class of R 1 to a parallel class of R 2. If G D is very big, the computation of the automorphism group of a resolution and the check for isomorphism of two resolutions might be difficult. Such problems often arise when resolutions of geometric designs (the designs of the points and t-dimensional subspaces of projective or affine spaces) are considered. For resolutions with given automorphisms these problems can be solved by using some of the conjugates of the predefined automorphisms. The method is explained in the present paper and an algorithm for construction of the necessary conjugates is presented. ACM Computing Classification System (1998): F.2.1, G.1.10, G.2.1

Real parallelisms

A garden of parallelisms in $PG(3,R)$ are constructed,where $R$ is the field of real numbers