Characterization of simple symplectic groups of degree 4 over locally finite fields of characteristic 2 in the class of periodic groups

Suppose that each finite subgroup of even order of a periodic group containing an element of order 2 lies in a subgroup isomorphic to a simple symplectic group of degree 4 over some finite field of characteristic 2. We prove that in that case the group is isomorphic to a simple symplectic group S ₄(Q) over some locally finite field Q of characteristic 2.     

On the Characters of the Maximal Subgroup <f 1> Sp 4 ( q ) of the Symplectic Group Sp 4(q), q-Even

The aim of this paper is to study the characters of the maximal subgroup of the symplectic group Sp ₄(q)q-even, where is the stabilizer of the one-dimensional space <f ₁> in Sp ₄(q).     

Primitive permutation groups with a regular subgroup

This paper starts the classification of the primitive permutation groups (G,Ω) such that G contains a regular subgroup X. We determine all the triples (G,Ω,X) with soc(G) an alternating, or a sporadic or an exceptional group of Lie type. Further, we construct all the examples (G,Ω,X) with G a classical group which are known to us. Our particular interest is in the 8-dimensional orthogonal groups of Witt index 4. We determine all the triples (G,Ω,X) with . In order to obtain all these triples, we also study the almost simple groups G with GPΩ_2n+1(q). The case GU_n(q) is started in this paper and finished in [B. Baumeister, Primitive permutation groups of unitary type with a regular subgroup, Bull. Belg. Math. Soc. 112 (5) (2006) 657–673]. A group X is called a Burnside-group (or short a B-group) if each primitive permutation group which contains a regular subgroup isomorphic to X is necessarily 2-transitive. In the end of the paper we discuss B-groups.     

Polarizations on Abelian Varieties and Self-dual ℓ-adic Representations of Inertia Groups

It is well known that every finite subgroup of GL _d (Q ) is conjugate to a subgroup of GL _d (Z ). However, this does not remain true if we replace general linear groups by symplectic groups. We say that G is a group of inertia type of G is a finite group which has a normal Sylow-p-subgroup with cyclic quotient. We show that if >d+1, and G is a subgroup of Sp _2d (Q ) of inertia type, then G is conjugate in GL _2d (Q ) to a subgroup of Sp _2d (Z ). We give examples which show that the bound is sharp. We apply these results to construct, for every odd prime , isogeny classes of Abelian varieties all of whose polarizations have degree divisible by ². We prove similar results for Euler characteristic of invertible sheaves on Abelian varieties over fields of positive characteristic.     

The Value Regions of the Systems {f(z 1), f′(z 1)} and {f(z 1), f(z 2)} in the Class of Typically Real Functions

In the class T consisting of regular and typically real functions in the disk |x| < 1, the value regions of the system {f(z ₁), f(z ₁)} and {f(z ₁), f(z ₂)} are found for fixed z ₁ and z ₂. As an application, the value regions of f(z ₁) and f(z ₂) are found for f T with fixed value f(z ₁). Bibliography: 11 titles.     

Action on flag varieties: 2-Dimensional case

LetA be a finite dimensional commutative semisimple algebra over a fieldk and letV be a finitely generatedA-module. We examine the action of the general linear group GL _A (V) on the set of flags ofk-subspaces ofV. Also, let (V, B) be a finitely generated symplectic module overA. We also investigate the action of the symplectic group Sp _A (V, B) on the set of flags ofB-isotropick-subspaces ofV, whereB=°B is thek-symplectic form induced by a nonzerok-linear map :A k. In both cases, the orbits are completely classified in terms of certain integer invariants provided that dim _{k A}=2.This work is partially supported by a KOSEF research grant.     

Optimal quotients of modular Jacobians

Let N be a positive integer, and f a normalized newform of weight two on ₀(N). Attached to f is an optimal quotient A _f of the Jacobian J ₀(N). We prove two theorems concerning such optimal quotients. (A) Let denote the algebra of endomorphisms of J ₀(N) generated by the Hecke operators, and let _f denote the ideal of fusion of f in . If  _f denotes the dual abelian variety to A _f , then the canonical polarization of J ₀(N) induces a polarization _f :  _f A _f . We show that there is an embedding ker( _f )J ₀ (N) [ _f ] whose cokernel is supported at maximal ideals of for which J ₀ (N)[] is not two-dimensional. (B) If N is prime, let C denote the subgroup of J ₀(N) generated by the divisor 0– on X ₀(N). Mazur has shown that C is equal to the full torsion subgroup of J ₀(N)(), and that specialization modulo N induces an isomorphism of C with the group of connected components J ₀ (N) of the characteristic N fibre of the Néron model of J ₀(N). We prove that analogous results hold for every optimal quotient of prime conductor, thereby generalizing results of Mestre and Oesterlé (who treated the case of strong Weil curves) and confirming William Steins refined Eisenstein conjecture. The key idea in the proof of these two theorems is encapsulated in corollary 2.5 below, which allows us to apply multiplicity one results in a novel way to the study of optimal quotients.This work was partially supported by NSF grant DMS-0296095     

On bandwidth sums of graphs

ＯＮＢＡＮＤＷＩＤＴＨＳＵＭＳＯＦＧＲＡＰＨＳＹＡＯＢＩＮＧ（姚兵）;ＷＡＮＧＪＩＡＮＦＡＮＧ（王建方）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ,ＮｏｒｉｈｗｅｓｔｅｔｅｒｎＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ,Ｌａｎｚｈｏｕ７３００７０,Ｃｈｉ...     

On Infinite Groups with Given Properties of the Norm of Infinite Subgroups

We investigate the relationship between the norm N _G() of infinite subgroups of an infinite group G and the structure of this group. We prove that N _G() is Abelian in the nonperiodic case, and a locally finite group is a finite extension of a quasicyclic subgroup if N _G() is a non-Dedekind group. In both cases, we describe the structure of the group G under the condition that the subgroup N _G() has finite index in G.     

On the greatest length of a dead-end disjunctive normal form for almost all Boolean functions

The maximal numberl(f) of conjunctions in a dead-end disjunctive normal form (d.n.f.) of a Boolean functionf and the number (f) of dead-end d.n.f. are important parameters characterizing the complexity of algorithms for finding minimal d.n.f. It is shown that for almost all Boolean functionsl(f)2^n–1, log₂ (f)2^n–1log₂nlog₂log₂n (n).Translated from Matematicheskie Zametki, Vol. 4, No. 6, pp. 649–658, December, 1968.     

On factorization representations for Avakumović--Karamata functions with nondegenerate groups of regular points

Avakumovi-Karamata functions f are generalized regularly varying functions (so--called ORV functions) such that f*()= limsup _x f(x)/f(x) is finite for all >0. In this paper, we investigate classes of ORV functions with "nondegenerate groups of regular points", that is, having points 1, for which f*() exists as a positive and finite limit (instead of limsup) on a nontrivial subgroup of the positive real axis. Certain factorization representations, characterizations and uniform convergence theorems are proved, describing both the structure of ORV functions f as well as that of their limit functions f*. Some well-known results from regular variation theory are covered by this general approach.     

Continuity properties of the normal cone to the level sets of a quasiconvex function

Two main properties of the subgradient mapping of convex functions are transposed for quasiconvex ones. The continuity of the functionxf(x)^–1f(x) on the domain where it is defined is deduced from some continuity properties of the normal coneN to the level sets of the quasiconvex functionf. We also prove that, under a pseudoconvexity-type condition, the normal coneN(x) to the set {x:f(x)f(x)} can be expressed as the convex hull of the limits of type {N(x _n)}, where {x _n} is a sequence converging tox and contained in a dense subsetD. In particular, whenf is pseudoconvex,D can be taken equal to the set of points wheref is differentiable.This research was completed while the second author was on a sabbatical leave at the University of Montreal and was supported by a NSERC grant. It has its origin in the doctoral thesis of the first author (Ref. 1), prepared under the direction of the second author.The authors are grateful to an anonymous referee and C. Zalinescu for their helpful remarks on a previous version of this paper.     

On Modified Strong Dyadic Integral and Derivative

For functions f L(R ₊), we define a modified strong dyadic integral J(f) L(R ₊) and a modified strong dyadic derivative D(f) L(R ₊). We establish a necessary and sufficient condition for the existence of the modified strong dyadic integral J(f). Under the condition f(x)dx = 0, we prove the equalities J(D(f)) = f and D(J(f)) = f. We find a countable set of eigenfunctions of the operators J and D. We prove that the linear span L of this set is dense in the dyadic Hardy space H(R ₊). For the functions f H(R ₊), we define a modified uniform dyadic integral J(f) L (R ₊).     

Likelihood Ratio Statistic for Exponential Mixtures

Let f ₀(x) be the exponential density and f (x) the translation model. Let (X _i) _i=1,n be i.i.d. random variables, with density g. We test that g is f ₀ against g is a simple mixture, using the LRT statistic. We prove that the LRT diverges to infinity with probability 1/2 and it is equal to 0 with probability 1/2. Therefore, the classical likelihood limiting theory does not hold.     

Functions of normal operators under perturbations

In Peller (1980) [27], Peller (1985) [28], Aleksandrov and Peller (2009) [2], Aleksandrov and Peller (2010) [3], and Aleksandrov and Peller (2010) [4] sharp estimates for f(A)−f(B) were obtained for self-adjoint operators A and B and for various classes of functions f on the real line R. In this paper we extend those results to the case of functions of normal operators. We show that if a function f belongs to the Hölder class Λα(R²), 0<α<1, of functions of two variables, and N₁ and N₂ are normal operators, then ‖f(N₁)−f(N₂)‖?const‖fΛα_‖‖N₁−N₂α^‖. We obtain a more general result for functions in the space for an arbitrary modulus of continuity ω. We prove that if f belongs to the Besov class , then it is operator Lipschitz, i.e., . We also study properties of f(N₁)−f(N₂) in the case when f∈Λα(R²) and N₁−N₂ belongs to the Schatten–von Neumann class Sp.     

Quadratic linear Keller maps of nilpotency index three

Let be a homogeneous polynomial map of degree d2 and a power linear map such that f and F_A are a generalized Gorni-Zampieri pair. We discuss the relation between the nilpotency indices of JH and J(Ay)^(d) and we show that f is linearly triangularizable if and only if F_A is linearly triangularizable. As a consequence, we show that a quadratic linear Keller map F_A=y+(Ay)⁽²⁾ with nilpotency index three, i.e., (J(Ay)⁽²⁾)³=0, is linearly triangularizable.     

Symplectic geometry of semisimple orbits

Let G be a complex semisimple group, T G a maximal torus and B a Borel subgroup of G containing T. Let Ω be the Kostant-Kirillov holomorphic symplectic structure on the adjoint orbit O = Ad(G)c G/Z(c), where c Lie(T), and Z(c) is the centralizer of c in G. We prove that the real symplectic form Re Ω (respectively, Im Ω) on O is exact if and only if all the eigenvalues ad(c) are real (respectively, purely imaginary). Furthermore, each of these real symplectic manifolds is symplectomorphic to the cotangent bundle of the partial flag manifold G/Z(cc)B, equipped with the Liouville symplectic form. The latter result is generalized to hyperbolic adjoint orbits in a real semisimple Lie algebra.     

Fonctorialité en K-Théorie bivariante pour les variétés lipschitziennes

Résumé Soit f: M V/F un morphisme continu orienté d'une variété lipschitzienne M dans l'espace des feuilles d'une variété lipschitzienne feuilletée (V,F), et soit C ^* (V,F) la C ^*-algèbre du feuilletage d'A. Connes. On construit un élèment (f) dans le groupe de K-théorie bivariante KK(C ₀ (M); C ^* (V,F)) de G. G. Kasparov et on montre la fonctorialité de cette construction. On utilise l'opérateur de signature de N. Teleman ([42]). Ceci répond pour les variétés lipschitziennes à une conjecture d'A. Connes ([11]) qui a été résolue pour les variétés différentiables dans [13, 8, 19].

---

Let M be a Lipschitz manifold, (V, F) a foliated Lipschitz manifold and let f M V/F be an oriented morphism. Let C ^* (V,F) be the foliation's C ^*-algebra of A. Connes. We then construct an element (f) of the K-theory bivariant group KK(C ₀(M); C ^* (V, F)) of G. G. Kasparov which depends functoriality on f. This uses the signature operator of N. Teleman [42]. It gives a positive answer for Lipschitz manifolds to a conjecture of A. Connes [11] which has been proved for differentiable manifolds in [13, 8, 19].

     

Inequalities involving independence domination, f-domination, connected and total f-domination numbers

Let f be an integer-valued function defined on the vertex set V(G) of a graph G. A subset D of V(G) is an f-dominating set if each vertex x outside D is adjacent to at least f(x) vertices in D. The minimum number of vertices in an f-dominating set is defined to be the f-domination number, denoted by _f (G). In a similar way one can define the connected and total f-domination numbers _c,f (G) and _t,f (G). If f(x) = 1 for all vertices x, then these are the ordinary domination number, connected domination number and total domination number of G, respectively. In this paper we prove some inequalities involving _f (G), _c,f (G), _t,f (G) and the independence domination number i(G). In particular, several known results are generalized.     

Sequences of normal states on maximal op*-algebras

It is proved that a sequence (f_n) of normal states on a maximal Op^*-algebra L⁺(D) converges to a normal state if f_n(A) is a Cauchy sequence for allA L ⁺(D), while D satisfies some additional condition.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 50, pp. 131–136, 1988.