Collineation groups irreducible on the components of a translation plane

We investigate finite translation planes of odd dimension over their kernels in which the translation complement induces on each component l a permutation group whose order is divisible by a p-primitive divisor. Using results of this investigation, we show that rank 3 affine planes of odd dimension over their kernels are either generalized André planes or semi-field planes. A similar result is given for translation planes having a collineation group which is doubly transitive on each affine line; besides the above two possibilities, there is a third possibility; the plane has order 27, the translation complement is doubly transitive on , and SL(2, 13) is contained in the translation complement.We also consider translation planes of odd dimension over their kernels which have a collineation group isomorphic to SL(2, w) with w prime to 5 and the characteristic, and having no affine perspectivity. We show that such planes have order 27, the prime power w=13, and the given group together with the translations forms a doubly transitive collineation group on {ie153-1}. This indicates quite strongly that the Hering translation plane of order 27 is unique with respect to the above properties.Both authors supported in part by NSF Grant No. MCS76-0661 A01.     

New Dimensions on Translations Between Logics

After a brief promenade on the several notions of translations that appear in the literature, we concentrate on three paradigms of translations between logics: (conservative) translations, transfers and contextual translations. Though independent, such approaches are here compared and assessed against questions about the meaning of a translation and about comparative strength and extensibility of a logic with respect to another. Dedicated to the memory of Mário Tourasse Teixeira and Antonio Mário Sette     

Hyperovals and Fano configurations

LetF be any field, finite or infinite, of characteristic 2. Put =PG(2,F). The classification of hyperovals is a difficult open problem. In this note we study the structure of the translation hyperovals and the hyperconics. We determine which quadrangles in a hyperconic have a given line as Fano line. The hyperconics and the translation hyperovals are similar with respect to containing quadrangles with certain Fano lines. We give two axioms satisfied by both. Remarkably, any hyperoval satisfying these must either be a hyperconic or else a translation hyperoval. It would be of great interest if one could find a way to slightly relax these conditions, or to piece together quadrangles with certain lines as Fano lines to obtain a new hyperoval.We are greatful for the help of Aiden Bruen. This research was funded in part by an NSERC Research Grant of Prof. A.A. Bruen, The University of Western Ontario, to whom we are greatful for the support.     

Spreads,ovoids andS 5

We consider translation planes of orderq ² (whereq andq ² - 1 are coprime to 30) such thatS ₅ acts on the line at infinity. It turns out that the Klein correspondence is in particular useful for the investigation of these planes. Representations of the planes, automorphisms and examples of low order are studied in detail. In view of a problem of Ostrom (Math. Z. 156 (1977), 59–71), series of translation planes are constructed with the following property: the translation complement is nonsolvable and has an order coprime to the characteristic of the plane.     

Maximal Baer groups in translation planes and compatibility with homology groups

The translation planes with spreads in PG(3, q) that admit at least two Baer groups of order q–1 are classified.     

Representation and approximation of convex dynamic risk measures with respect to strong–weak topologies

We provide a representation for strong-weak continuous dynamic risk measures from L^p into L^p_t spaces where these spaces are equipped respectively with strong and weak topologies and p is a finite number strictly larger than one. Conversely, we show that any such representation that admits a compact (with respect to the product of weak topologies) sub-differential generates a dynamic risk measure that is strong--weak continuous. Furthermore, we investigate sufficient conditions on the sub-differential for which the essential supremum of the representation is attained. Finally, the main purpose is to show that any convex dynamic risk measure that is strong-weak continuous can be approximated by a sequence of convex dynamic risk measures which are strong--weak continuous and admit compact sub-differentials with respect to the product of weak topologies. Throughout the arguments, no conditional translation invariance or monotonicity assumptions are applied.     

The uniqueness of Hering's translation plane of order 27

It is shown that the following conjecture of Kallaher and Ostrom [2] is correct: Hering's translation plane of order 27 is the only translation plane of odd dimension over its kernel which has a collineation group isomorphic to SL(2, w) with w prime to 5 and to the characteristic, and having no affine perspectivity.     

The property of maximal eigenvectors of trees

Let T be a tree on n vertices and L(T) be its Laplacian matrix. The eigenvalues and eigenvectors of T are respectively referred to those of L(T). With respect to a given eigenvector Y of T, a vertex u of T is called a characteristic vertex if Y [u] = 0 and there is a vertex w adjacent to u with Y [w] ≠ 0; an edge e = (u, w) of G is called a characteristic edge if Y [u]Y [w] < 0. By 𝒞(T, Y) we denote the characteristic set of T with respect to the vector Y, which is defined as the collection of all characteristic vertices and characteristic edges of T corresponding to Y. Merris shows that 𝒞(T, Y) is fixed for all Fiedler vectors of the tree T. An eigenvector of T is called a k-vector (k ≥ 2) of T if this eigenvector corresponds to an eigenvalue λ _k with λ _k > λ _k?1, where λ₁, λ₂, …, λ _n are the eigenvalues of T arranged in non-decreasing order. A k-vector Y of T is called k-maximal if 𝒞(T, Y) has maximum cardinality among all k-vectors of T. We prove that (1) the characteristic set of T with respect to an arbitrary k-vector is contained in that with respect to any k-maximal vector; and consequently (2) the characteristic sets of T with respect to any two k-maximal vectors are same. Our result may be considered as a generalization of Merris' result as Fiedler vectors are 2-maximal.     

Stable Finiteness of Group Rings in Arbitrary Characteristic

We show that every (discrete) group ring D[G] of a free-by-amenable group G over a division ring D of arbitrary characteristic is stably finite, in the sense that one-sided inverses in all matrix rings over D[G] are two-sided. Our methods use Sylvester rank functions and the translation ring of an amenable group.     

Equivelar Polyhedra with Few Vertices

We know that the polyhedra corresponding to the Platonic solids are equivelar. In this article we have classified completely all the simplicial equivelar polyhedra on ≤ 11 vertices. There are exactly 27 such polyhedra. For each n\geq -4 , we have classified all the (p,q) such that there exists an equivelar polyhedron of type {p,q} and of Euler characteristic n . We have also constructed five types of equivelar polyhedra of Euler characteristic -2m , for each m\geq 2 . Received February 14, 2000, and in revised form August 15, 2000. Online publication March 26, 2001.     

COMPOSITION SUPERALGEBRAS

ABSTRACT

The super counterparts of both unital composition algebras and of symmetric composition algebras are classified. It turns out that new objects arise only over fields of characteristic 3. The Principle of Local Triality is considered for this new objects. It is related to some order 3 automorphisms of the simple Liesuperalgebras osp(1,2) and osp(4,2) in characteristic 3.     

Desarguesian partial parallelisms

This article establishes connections between Desarguesian partial parallelisms of deficiency one in PG(3,q) and translation planes of orderq ⁴ admitting a collineation group isomorphic to SL(2,q) which is generated by Baer collineations.The author is grateful to Professor Alan Prince for helpful conversations with respect to this article and, in particular, with respect to the actual bounds for certain maximal partial spreads.The author is grateful to the referee for helpful comments in the writing of this article.     

On the transposed Rao’s flag-transitive plane of order 49

Rao’s flag-transitive plane π of order 49 and π ^t , the plane obtained by transposing matrices of a representative set of π, has been studied. It is shown that π ^t is flag-transitive, π ^t is not isomorphic to π, and π ^t is obtained from π by replacement of a net of degree 25. Further, (1) the flag-transitive planes associated with 1-spread sets S _2b and S _2a in the classified list of translation planes of order 49 enumerated by Mathon et al, are respectively isomorphic to π and π ^t (2) The flag-transitive planes associated with the 1-spread sets of 0an* in the classified list of translation planes of order 49 enumerated by Charnes et al are isomorphic to π and π ^t in some order.     

Meromorphic solutions of equations over non-Archimedean fields

In this paper, we give some conditions to assure that the equation P(X)=Q(Y) has no meromorphic solutions in all K, where P and Q are polynomials over an algebraically closed field K of characteristic zero, complete with respect to a non-Archimedean valuation. In particular, if P and Q satisfy the hypothesis (F) introduced by H. Fujimoto, a necessary and sufficient condition is obtained when deg P=deg Q. The results are presented in terms of parametrization of a projective curve by three entire functions. In this way we also obtain similar results for unbounded analytic functions inside an open disk.      

A class of locally $\varphi $-symmetric contact metric spaces

We prove that a non-Sasakian contact metric manifold such that its characteristic vector field belongs to the (k,m)(k,\mu )-nullity distribution, is locally homogeneous. Further, these spaces are all locally j\varphi -symmetric in the strong sense, i.e., the reflections with respect to characteristic curves are local isometries.     

Translation planes admitting many homologies

The translation planes with spreads inPG(3,2 ^r ) that admit >2 ^r +1 axes (coaxes) of homologies of orderu1 are classified.Partially supported by FONDECYT Project No. 890-1061     

The translation planes admitting a nonsolvable doubly transitive line-sized orbit

The finite translation planes of ordern not in {3⁴,3⁶,11²,19²,29²,59² that admit a non-solvable doubly transitive line-sized orbit are completely classified.     

On Galois Coverings of the Enveloping Algebras of Self-Injective Nakayama Algebras

Abstract

There is constructed a Galois covering F of the enveloping K-algebra A ^e of a self-injective Nakayama K-algebra A such that the right A ^e -module A is of the first kind with respect to F. Then, with the help of the constructed Galois covering, the Auslander-Reiten translation period of A is computed.