Real elements in groups of type <Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>

Let G be a group. An element x ∈ G is called real if x is conjugate to x ^?1 in G. In this paper we study the structure of real elements in the compact connected Lie group of type F ₄ and algebraic groups of type F ₄ defined over an arbitrary field.     

Automorphisms of <Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript> quadrangles

The goal of this article is to show that the automorphism group of a Moufang quadrangle of type F₄ is, up to field automorphisms, generated by the root groups. Mathematics Subject Classification (2000):51E12, 15A63, 20E42The author is a Research Assistant of the Fund for Scientific Research - Flanders (Belgium) (F.W.O.-Vlaanderen).     

An elliptic hypergeometric integral with <Emphasis Type="Italic">W</Emphasis>(<Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>) symmetry

In this article we give a new transformation between elliptic hypergeometric beta integrals, which gives rise to a Weyl group symmetry of type F ₄. The transformation is a generalization of a series transformation discovered by Langer, Schlosser, and Warnaar (SIGMA 5:055, 2009). Moreover we consider various limits of this transformation to basic hypergeometric functions obtained by letting p tend to 0.     

On <Emphasis Type="Italic">F</Emphasis><Subscript><Emphasis Type="Italic">h</Emphasis></Subscript>-supplemented subgroups of finite groups

Suppose that F is a formation of finite groups. We introduce the concept of F _h -supplemented subgroups and investigate the structure of finite groups on assuming that some maximal subgroups of Sylow subgroups, maximal subgroups, minimal subgroups, and 2-maximal subgroup are F _h -supplemented, respectively. Some available results are generalized.     

Quasirecognition by prime graph of the simple group <Superscript>2</Superscript><Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>)

As the main result, we show that if G is a finite group such that Γ(G) = Γ(² F ₄(q)), where q = 2^2m+1 for some m ≧ 1, then G has a unique nonabelian composition factor isomorphic to ² F ₄(q). We also show that if G is a finite group satisfying |G| =|² F ₄(q)| and Γ(G) = Γ(² F ₄(q)), then G ≅ ² F ₄(q). As a consequence of our result we give a new proof for a conjecture of W. Shi and J. Bi for ² F ₄(q). The third author was supported in part by a grant from IPM (No. 87200022).     

Irreducibility of Hurwitz spaces of coverings with one special fiber and monodromy group a Weyl group of type <Emphasis Type="Italic">D</Emphasis><Subscript><Emphasis Type="Italic">d</Emphasis></Subscript>

Let Y be a smooth, connected, projective complex curve. In this paper, we study the Hurwitz spaces which parameterize branched coverings of Y whose monodromy group is a Weyl group of type D _d and whose local monodromies are all reflections except one. We prove the irreducibility of these spaces when and successively we extend the result to curves of genus g ≥ 1.     

Continuous <Emphasis Type="Italic">l</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">p</Emphasis></Subscript>-symmetric distributions

For p > 0, the l _n,p -generalized surface measure on the l _n,p -unit sphere is studied and used for deriving a geometric measure representation for l _n,p -symmetric distributions having a density.     

Codimensions of Newton strata for <Emphasis Type="Italic">SL</Emphasis><Subscript>3</Subscript>(<Emphasis Type="Italic">F</Emphasis>) in the Iwahori case

We study the Newton stratification on SL ₃(F), where F is a Laurent power series field. We provide a formula for the codimensions of the Newton strata inside each component of the affine Bruhat decomposition on SL ₃(F). These calculations are related to the study of certain affine Deligne–Lusztig varieties. In particular, we describe a method for determining which of these varieties is non-empty in the case of SL ₃(F).     

Automorphisms of Coxeter groups of type <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

A Coxeter system (W, S) is said to be of type K _n if the associated Coxeter graph Γ_S is complete on n vertices and has only odd edge labels. If W satisfies either of: (1) n = 3; (2) W is rigid; then the automorphism group of W is generated by the inner automorphisms of W and any automorphisms induced by Γ_S. Indeed, Aut(W) is the semidirect product of Inn(W) and the group of diagram automorphisms, and furthermore W is strongly rigid. We also show that if W is a Coxeter group of type K _n then W has exactly one conjugacy class of involutions and hence Aut(W) = Spec(W).     

Asymptotics of the Codimensions <Emphasis Type="Italic">c</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> in the Algebra <Emphasis Type="Italic">F</Emphasis><Superscript>(7)</Superscript>

The paper studies the additive structure of the algebra F⁽⁷⁾, i.e., a relatively free associative countably generated algebra with the identity [x₁,..., x₇] = 0 over an infinite field of characteristic ≠ 2, 3. First, the space of proper multilinear polynomials in this algebra is investigated. As an application, estimates for the codimensions c_n = dimF_n⁽⁷⁾ are obtained, where F_n⁽⁷⁾ stands for the subspace of multilinear polynomials of degree n in the algebra F⁽⁷⁾.     

Novikov superalgebras with <Emphasis Type="Italic">A</Emphasis><Subscript>0</Subscript> = <Emphasis Type="Italic">A</Emphasis><Subscript>1</Subscript><Emphasis Type="Italic">A</Emphasis><Subscript>1</Subscript>

Novikov superalgebras are related to quadratic conformal superalgebras which correspond to the Hamiltonian pairs and play a fundamental role in completely integrable systems. In this note we show that the Novikov superalgebras with A ₀ = A ₁ A ₁ and dim A ₁ = 2 are of type N and give a class of Novikov superalgebras of type S with A ₀ = A ₁ A ₁.     

On <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Subscript>-factorization of complete bipartite multigraphs

Let λK _m,n be a complete bipartite multigraph with two partite sets having m and n vertices, respectively. A K _p,q -factorization of λK _m,n is a set of edge-disjoint K _p,q -factors of λK _m,n which partition the set of edges of λK _m,n . When p = 1 and q is a prime number, Wang, in his paper [On K _1,q -factorization of complete bipartite graph, Discrete Math., 126: (1994), 359-364], investigated the K _1,q -factorization of K _m,n and gave a sufficient condition for such a factorization to exist. In papers [K _1,k -factorization of complete bipartite graphs, Discrete Math., 259: 301-306 (2002),; K _p,q -factorization of complete bipartite graphs, Sci. China Ser. A-Math., 47: (2004), 473-479], Du and Wang extended Wang’s result to the case that p and q are any positive integers. In this paper, we give a sufficient condition for λK _m,n to have a K _p,q -factorization. As a special case, it is shown that the necessary condition for the K _p,q -factorization of λK _m,n is always sufficient when p : q = k : (k + 1) for any positive integer k.     

On a new property of <Emphasis Type="Italic">n</Emphasis>-poised and <Emphasis Type="Italic">G</Emphasis><Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> sets

In this paper we consider n-poised planar node sets, as well as more special ones, called G C _n sets. For the latter sets each n-fundamental polynomial is a product of n linear factors as it always holds in the univariate case. A line ? is called k-node line for a node set \(\mathcal X\) if it passes through exactly k nodes. An (n + 1)-node line is called maximal line. In 1982 M. Gasca and J. I. Maeztu conjectured that every G C _n set possesses necessarily a maximal line. Till now the conjecture is confirmed to be true for n ≤ 5. It is well-known that any maximal line M of \(\mathcal X\) is used by each node in \(\mathcal X\setminus M, \)meaning that it is a factor of the fundamental polynomial. In this paper we prove, in particular, that if the Gasca-Maeztu conjecture is true then any n-node line of G C _n set \(\mathcal {X}\) is used either by exactly \(\binom {n}{2}\) nodes or by exactly \(\binom {n-1}{2}\) nodes. We prove also similar statements concerning n-node or (n ? 1)-node lines in more general n-poised sets. This is a new phenomenon in n-poised and G C _n sets. At the end we present a conjecture concerning any k-node line.     

On constant <Emphasis Type="Italic">Uq</Emphasis>(<Emphasis Type="Italic">sl</Emphasis><Subscript>2</Subscript>)-invariant <Emphasis Type="Italic">R</Emphasis>-matrices

We consider the spectral resolution of a Uq (sl ₂)-invariant solution R of the constant Yang–Baxter equation in the braid group form. It is shown that if the two highest coefficients in this resolution are not equal, then R is either the Drinfeld R-matrix or its inverse. Bibliography: 13 titles.     

Cyclic codes over <Emphasis Type="Italic">R</Emphasis><Subscript><Emphasis Type="Italic">k</Emphasis></Subscript>

Cyclic codes over an infinite family of rings are defined. The general properties of cyclic codes over these rings are studied, in particular nontrivial one-generator cyclic codes are characterized. It is also proved that the binary images of cyclic codes over these rings under the natural Gray map are binary quasi-cyclic codes of index 2 ^k . Further, several optimal or near optimal binary codes are obtained from cyclic codes over R _k via this map.     

Exponentially Many Monochromatic <Emphasis Type="Italic">n</Emphasis>-Matchings in <Emphasis Type="Italic">K</Emphasis><Subscript>3<Emphasis Type="Italic">n</Emphasis>-1</Subscript>

Erdös et al and Gerencsér et al had shown that in any 2-edge-coloring of K _3n-1, there is a n-matching containing edges with the same color(we call such matching monochromatic matching). In this paper we show that for any 2-edge-coloring of K _3n-1 there exists a monochromatic subgraph H of K _3n-1 which contains exponentially many monochromatic n-matchings.     

Structure of the semigroup <Emphasis Type="Italic">OT</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

We study the structure of the semigroup OT _n , which is a unique (up to an isomorphism) R-section of the semigroup T _n . For this semigroup, we describe Green relations, determine regular and nilpotent elements, describe maximal nilpotent subsemigroups, and determine the unique irreducible system of generatrices and maximal subsemigroups.     

Characterization of finite simple group <Emphasis Type="Italic">D</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(2)

Let G be a finite group, and let π _e (G) be the spectrum of G, that is, the set of all element orders of G. In 1987, Shi Wujie put forward the following conjecture. If G is a finite group and M is a non-abelian simple group, then G ≅ M if and only if |G| = |M| and π _e (G) = π _e (M). In this short paper, we prove that if G is a finite group, then G ≅ M if and only if |G| = |M| and π _e (G) = π _e (M), where M = D _n (2) and n is even.     

Abelianization of the <Emphasis Type="Italic">F</Emphasis>-divided fundamental group scheme

Let (X , x ₀) be a pointed smooth proper variety defined over an algebraically closed field. The Albanese morphism for (X , x ₀) produces a homomorphism from the abelianization of the F-divided fundamental group scheme of X to the F-divided fundamental group of the Albanese variety of X. We prove that this homomorphism is surjective with finite kernel. The kernel is also described.     

Availability Analysis of Repairable <Emphasis Type="Italic">N</Emphasis>-Server <Emphasis Type="Italic">n</Emphasis><Subscript><Emphasis Type="Italic">i</Emphasis></Subscript>-Replicated Systems

We develop an exact closed-form expression for the steady-state availability of a repairable, N-server system in which the ith server contains n _i identical, reconfigurable, breakdown-prone units. Our approach, which follows from the Markov chain balance equations and the recursive properties of Hessenberg matrix determinants, is simpler than previously proposed matrix geometric approaches, and can readily be adapted to the availability analysis of more complicated structures. We illustrate this by computing the steady-state availability of a mixed parallel-serial gracefully degrading replicated system.