probabilistic generation of finite simple groups ii abstract in earlier work it was shown that each nonabelian finite simple group g has a conjugacy class c such that whenever 1 x g the probability is greater than 1 10 that g x y for a random y c much stronger asymptotic results were also proved here we show that allowing equality the bound 1 10 can be replaced by 13 42 and excluding an explicitly listed set of simple groups the bound 2 3 holds we use these results to show that any nonabelian finite simple group g has a conjugacy class c such that if x 1 x 2 are nontrivial elements of g then there exists y c such that g x 1 y x 2 y similarly aside from one infinite family and a small explicit finite set of simple groups g has a conjugacy class c such that if x 1 x 2 x 3 are nontrivial elements of g then there exists y c such that g x 1 y x 2 y x 3 y we also prove analogous but weaker results for almost simple groups

anatomy of the monster ii abstract we summarise much of what is currently known about the subgroup structure of the fischer griess monster introduction the publication of the atlas of finite groups 1 henceforth referred to as the atlas marked the end of a chapter in the history of group theory in more ways than one the classification of finite simple groups had been completed and using new computational techniques the last of the sporadic groups had been constructed and enough progress had been made in classifying their maximal subgroups to make the rest seem like a tidying up job since the publication of the atlas these computational methods have been pushed further the classification of the maximal subgroups for fi 22 th fi 23 j 4 and fi 24 was completed by kleidman linton parker and wilson in 7 8 9 10 12 13 14 this means that appendix 2 of the atlas of brauer characters 11 when collated with the atlas itself completes the listing for all sporadic groups except the baby monster and monster for the baby monster four new maximal subgroups s 6 l 3 4 2 2 s 6 s 6 4 l 2 31 and l 2 49 2 3 were given in appendix 2 of 11 and wilson has since found four more m 11 l 3 3 l 2 17 2 and l 2 ll 2 for references see 23 24 25 with work on the 2 locals by meierfrankenfeld and shpektorov 15 an end to the baby monster problem is in sight with a lot of work also having been done on maximal subgroups of generic groups this leaves only the monster unfortunately the degree of its smallest representation over any field 196882 is too big for present day computational techniques to make any headway

criteria for solvable radical membership via p elements abstract guralnick kunyavskii plotkin and shalev have shown that the solvable radical of a finite group g can be characterized as the set of all x g such that x y is solvable for all y g we prove two generalizations of this result firstly it is enough to check the solvability of x y for every p element y g for every odd prime p secondly if x has odd order then it is enough to check the solvability of x y for every 2 element y g

hopf bifurcation with cubic symmetry and instability of abc flow we examine the dynamics of generic hopf bifurcation in a system that is symmetric under the action of the rotational symmetries of the cube we classify the generic branches of periodic solutions at bifurcation there are generically 27 branches corresponding to maximal symmetries organized into five symmetry types there are also up to 22 periodic solution branches of two other symmetry types these results are found by examination of the normal form with s 1 normal form symmetry for the bifurcation truncated at the third order in addition to the periodic branches whose branching and stability we find there are several branches of tori homoclinic bifurcations and chaotic attractors in the dynamics of the third order normal form since many of these features are not amenable to analysis we give some numerical examples on breaking the normal form symmetry there may be breakup of the branches of tori but the predictions for the periodic solutions will be reliable for the navier stokes equations with a particular forcing an abc flow is a dynamically stable solution for small reynolds numbers r for the most symmetric case a b c 1 the first instability of this system is a hopf bifurcation at r 13 04 with rotational symmetry of the cube we use our normal form analysis to explain the observed behaviour of solutions at this primary instability numerical simulations show that there is supercritical branching to rotating waves that alternate between the three axes which undergo secondary hopf bifurcation to a 2 torus at approximately r 13 09 the eight symmetrically related tori break up and then merge to form a chaotic attractor with full symmetry we can explain all these features by use of the generic third order normal form and s 1 normal form symmetry breaking terms

the euler equations as a differential inclusion we propose a new point of view on weak solutions of the euler equations describing the motion of an ideal incompressible fluid in r n with n 2 we give a reformulation of the euler equations as a differential inclusion and in this way we obtain transparent proofs of several celebrated results of v scheffer and a shnirelman concerning the non uniqueness of weak solutions and the existence of energy decreasing solutions our results are stronger because they work in any dimension and yield bounded velocity and pressure

subgroups of maximal rank in finite exceptional groups of lie type the purpose of this paper is to investigate a large and natural class of maximal subgroups of the finite exceptional groups of lie type which we call subgroups of maximal rank these subgroups play a prominent role both in the classification of local maximal subgroups in 9 theorem 1 and in the reduction theorem for subgroups in 28 theorem 2 in 9 it is shown that any local maximal subgroup of a finite exceptional group of lie type is either a subgroup of maximal rank or one of a small list of exceptions and in 28 using this result it is proved that any maximal subgroup is either of maximal rank or almost simple or in a known list to describe these subgroups we require the following notation let g be a simple adjoint algebraic group over an algebraically closed field k of characteristic p 0 and let o be an endomorphism of g such that l ga is a finite exceptional simple group of lie type over fq where q p let a be a group such that f x l the group aut l is generated by ga together with field and graph automorphisms see 5 38 all of which extend to morphisms of the abstract group g commuting with o thus there is a subgroup x of caut c a such that x x o and so x acts on the set of a stable subsets of g for a a stable subset y we write nx y for the stabilizer in x of y if d is a a stable closed connected reductive subgroup of g containing a maximal torus t of g and m nx d we call m a subgroup of maximal rank in x in this paper we determine the structure and conjugacy classes of those subgroups of maximal rank which are maximal in x

dynamical behavior of vibro impact machinery near a point of codimension two bifurcation a dual component system with vibro impact is considered local codimension two bifurcation of maps involving a real eigenvalue and a complex conjugate pair escaping the unit circle simultaneously is analyzed by using the center manifold and normal form method for maps the period n single impact motion and poincare map of the vibro impact system are derived analytically stability and local bifurcation of single impact periodic motion are analyzed by using the poincare map a center manifold theorem technique is applied to reduce the poincare map to a three dimensional one and the normal form map associated with the codimension two bifurcation is obtained local behavior of the dual component system with vibro impact near the point of codimension two bifurcation is analyzed it is found that near the point of codimension two bifurcation there exists not only hopf bifurcation of period one single impact motion but also hopf bifurcation of period two double impact motion period doubling bifurcation of period one single impact motion commonly exists near the point of codimension two bifurcation however no period doubling cascade emerges due to change of the type of period two fixed points and occurrence of hopf bifurcation associated with period two fixed points the period two fixed points are symmetrical about the corresponding period one point the results from simulation show that the attracting invariant circles associated with period two points are symmetrical about the corresponding period one point near the value of hopf bifurcation of period two points two actual examples the impact forming machinery and inertial shaker are chosen to analyze further the phenomena of codimension two bifurcation of maps and local bifurcation analysis and numerical simulation are carried out to unfold dynamical behavior of these vibro impact systems near the points of codimension two bifurcations

linear groups with orders having certain large prime divisors in this paper we obtain a classification of those subgroups of the finite general linear group gld q with orders divisible by a primitive prime divisor of qe 1 for some in the course of the analysis we obtain new results on modular representations of finite almost simple groups in particular in the last section we obtain substantial extensions of the results of landazuri and seitz on small cross characteristic representations of some of the finite classical groups 1991 mathematics subject classification primary 20g40 secondary 20c20 20c33 20c34 20e99

quasi genericity of bifurcations to high dimensional invariant tori for maps we consider a family of maps in a banach spacee near the situation when the derivative at the fixed point has two pairs of complex eigenvalues lying on the unit circle the other part of the spectrum being strictly inside the unit disc we focus our attention on the region of the parameter space where the truncated normal form of the maps shows a bifurcation of a family of invariantt 1 circles into a family of invariantt 2 tori we show that this problem needs a 3 dimensional parameter unfolding and that for the complete maps bifurcation occurs at points where is the rotation number on the non normally hyperbolict 1 circle ande 2i are the eigenvalues of the constant matrix conjugated to the non contracting part of the linearization on the normal fiber bundle overt 1 making some non resonance and diophantine assumptions on leading to a positive measure cantor set int 2 we show that in paraboloidal regions of the 3 dim parameter space we have clean bifurcations as for the truncated normal form the complement of these regions forms a set of bubbles such as the ones obtained by chenciner in chen for a codimension 2 problem for maps in 2 the main tool here is a generalization for a matrix function ont 1 close to a constant of the quasi conjugacy to a constant modulo a minimum of additional parameters moved quasi conjugacy for the infinite dimensional case we use ac decoupling result on the angular dependent linear parts into a contraction still angular dependent and another part quasi conjugated to a constant matrix this type of analysis applies for a wide range of problems where truncated normal forms of the maps give bifurcations fromt n tot n 1 tori and this needs a n 1 dimensional parameter unfolding

from thompson to baer suzuki a sharp characterization of the solvable radical abstract we prove that an element g of prime order 3 belongs to the solvable radical r g of a finite or more generally a linear group if and only if for every x g the subgroup generated by g x g x 1 is solvable this theorem implies that a finite or a linear group g is solvable if and only if in each conjugacy class of g every two elements generate a solvable subgroup

a solvable version of the baer suzuki theorem suppose that g is a finite group and x g has prime order p 5 then x is contained in the solvable radical of g o g if and only if x x g is solvable for all g g if g is an almost simple group and x g has prime order p 5 then this implies that there exists g g such that x x g is not solvable in fact this is also true when p 3 with very few exceptions which are described explicitly

the dirichlet problem at infinity for a negatively curved manifold on considere une n variete de riemann complete m qui est connexe et simplement connexe et a des courbures sectionnelles k bornees par 2 constantes finies negatives on etudie l existence de fonctions harmoniques bornees non constantes pour ces varietes

hopf saddle node bifurcation for fixed points of 3d diffeomorphisms analysis of a resonance bubble the dynamics near a hopf saddle node bifurcation of fixed points of diffeomorphisms is analysed by means of a case study a two parameter model map g is constructed such that at the central bifurcation the derivative has two complex conjugate eigenvalues of modulus one and one real eigenvalue equal to 1 to investigate the effect of resonances the complex eigenvalues are selected to have a 1 5 resonance it is shown that near the origin of the parameter space the family g has two secondary hopf saddle node bifurcations of period five points a cone like structure exists in the neighbourhood formed by two surfaces of saddle node and a surface of hopf bifurcations quasi periodic bifurcations of an invariant circle forming a frayed boundary are numerically shown to occur in model g along such cantor like boundary an intricate bifurcation structure is detected near a 1 5 resonance gap subordinate quasi periodic bifurcations are found nearby suggesting the occurrence of a cascade of quasi periodic bifurcations c 2008 elsevier b v all rights reserved

on the nonuniqueness of weak solution of the euler equation weak solution of the euler equations is defined as an l2 vector field satisfying the integral relations expressing the mass and momentum balance their general nature has been quite unclear in this work an example of a weak solution on a 2 dimensional torus is constructed that is identically zero outside a finite time interval this example is simpler and more transparent than the previous example of v scheffer j geom anal 3 4 1993 pp 343 401 1997 john wiley sons inc

products of conjugacy classes in finite and algebraic simple groups abstract we prove the arad herzog conjecture for various families of finite simple groups if a and b are nontrivial conjugacy classes then a b is not a conjugacy class we also prove that if g is a finite simple group of lie type and a and b are nontrivial conjugacy classes either both semisimple or both unipotent then a b is not a conjugacy class we also prove a strong version of the arad herzog conjecture for simple algebraic groups and in particular show that almost always the product of two conjugacy classes in a simple algebraic group consists of infinitely many conjugacy classes as a consequence we obtain a complete classification of pairs of centralizers in a simple algebraic group which have dense product a special case of this has been used by prasad to prove a uniqueness result for tits systems in quasi reductive groups our final result is a generalization of the baer suzuki theorem for p elements with p 5

smoothness of spectral subbundles and reducibility of quasi periodic linear differential systems abstract in this paper we study linear differential systems 1 x a t x where a is an n n matrix valued function defined on the k torus t k and t t is a given irrational twist flow on t k first we show that if a c n t k where n 0 1 2 then the spectral subbundles are of class c n on t k next we assume that a is sufficiently smooth on t k and satisfies a suitable small divisors inequality we show that if 1 satisfies the full spectrum assumption then there is a quasi periodic linear change of variables x p t y that transforms 1 to a constant coefficient system y by finally we study the case where the matrix a t in 1 is the jacobian matrix of a nonlinear vector field x evaluated along a quasi periodic solution x t of 2 x x we give sufficient conditions in terms of smoothness and small divisors inequalities in order that there is a coordinate system z defined in the vicinity of h the hull of so that the linearized system 1 can be represented in the form z dz where d is a constant matrix our results represent substantial improvements over known methods because we do not require that a be close to a constant coefficient system

the maximal subgroups of the chevalley groups g2 q with q odd the ree groups 2g2 q and their automorphism groups

three coupled oscillators mode locking global bifurcations and toroidal chaos abstract many systems exhibit behaviour which can be described by three coupled oscillators provided the coupling is not too strong such systems can be modelled by maps of the two torus to itself in this paper we describe and explain those aspects of the bifurcation diagram for two parameter families of torus maps that involve change of mode locking type we introduce the concepts of partial and full mode locking for the coupled oscillators these notions correspond to the presence of one or two rational relations between the frequencies respectively numerical investigation of a particular family of torus maps reveals an intricate web of global bifurcations in order to explain the results we first show that we can approximate maps in the neighbourhood of a rational translation to arbitrary order by the time 1 map of a flow on the torus then we analyse those condimension 1 and 2 bifurcations for flows on the torus which change the set of frequency ratios we find a large variety of bifurcation diagrams in particular many involving homotopically non trivial saddle connections next we ask how the picture changes when the time 1 maps of a family of flows are perturbed to a general family of diffeomorphisms we find toroidal chaos in the neighbourhood of all these bifurcations meaning that there exist orbits which perform a pseudo random sequence of rotations in different directions around the torus the corresponding behaviour for the coupled oscillators is that the frequency ratios perform a random walk finally we show how all these ingredients can be put together to give global scenarios for bifurcation for families of torus maps which we believe to be of general applicability to physical systems with three weakly coupled modes of oscillation

euler and navier stokes equations on the hyperbolic plane we show that nonuniqueness of the leray hopf solutions of the navier stokes equation on the hyperbolic plane 2 observed by chan and czubak is a consequence of the hodge decomposition we show that this phenomenon does not occur on n whenever n 3 we also describe the corresponding general hamiltonian framework of hydrodynamics on complete riemannian manifolds which includes the hyperbolic setting

dynamical behavior of a class of vibratory systems with symmetrical rigid stops near the point of codimension two bifurcation a multi degree of freedom vibratory system having symmetrically placed rigid stops is considered the system consists of linear components but the maximum displacement of one of the masses is limited to a threshold value by the symmetrical rigid stops repeated impacts usually occur in the vibratory system due to the rigid amplitude constraints such models play an important role in the studies of mechanical systems with clearances or gaps local codimension two bifurcation of maps involving a real eigenvalue and a complex conjugate pair escaping the unit circle simultaneously is analyzed by using the center manifold theorem technique and normal form method for maps symmetrical double impact periodic motion and poincare map of the system are derived analytically a center manifold theorem technique is applied to reduce the poincare map to a three dimensional one and the normal form map associated with the codimension two bifurcation is obtained local behaviors of the vibratory systems with symmetrical rigid stops near the points of codimension two bifurcations are reported by the presentation of results for a two degree of freedom vibratory system with symmetrical stops the existence and stability of symmetrical double impact periodic motion are analyzed explicitly also local bifurcations at the points of change in stability are analyzed near the point of codimension two bifurcation there exists not only hopf bifurcation of period one double impact motion but also pitchfork bifurcation of the motion pitchfork bifurcation of period one double impact symmetrical motion results in the period one double impact unsymmetrical motion the unsymmetrical double impact motion is of two antisymmetrical forms due to different initial conditions and symmetrical stops with change of the forcing frequency the unsymmetrical double impact periodic motion will undergo hopf bifurcation moreover the period one double impact symmetrical motion will undergo hopf bifurcation directly as the forcing frequency is changed in the contrary direction the routes of quasi periodic impact motions to chaos are observed by results from simulation

routes to chaos in high dimensional dynamical systems a qualitative numerical study this paper examines the most probable route to chaos a high dimensional dynamical systems function space time delay neural networks endowed with a probability measure in a computational setting the most probable route to chaos relative to the measure we impose on the function space as the dimension is increased is observed to be a sequence of neimark sacker bifurcations into chaos the analysis is composed of the study of an example dynamical system followed by a probabilistic study of the ensemble of dynamical systems from which the example was drawn a scenario depicting the decoupling of the stable manifolds of the torus leading up to the onset of chaos in high dimensional dissipative dynamical systems is also presented c 2006 elsevier b v all rights reserved

interaction of hopf and period doubling bifurcations of a vibro impact system abstract an inertial shaker as a vibratory system with impact is considered by means of differential equations periodicity and matching conditions the theoretical solution of periodic n 1 impacting motion can be obtained and the poincare map is established dynamics of the system are studied with special attention to interaction of hopf and period doubling bifurcations corresponding to a codimension 2 one when a pair of complex conjugate eigenvalues crosses the unit circle and the other eigenvalue crosses 1 simultaneously for the jacobi matrix the four dimensional map can be reduced to a three dimensional normal form by the center manifold theorem and the theory of normal forms the two parameter unfoldings of local dynamical behavior are put forward and the singularity is investigated it is proved that there exist curve doubling bifurcation a torus doubling bifurcation hopf bifurcation of 2 2 fixed points as well as period doubling bifurcation and hopf bifurcation of 1 1 fixed points near the critical point numerical results indicate that the vibro impact system presents complicated and interesting curve doubling bifurcation and hopf bifurcation as the two controlling parameters vary

further solvable analogues of the baer suzuki theorem and generation of nonsolvable groups abstract let g be an almost simple group we prove that if x g has prime order p 5 then there exists an involution y such that x y is not solvable also if x is an involution then there exist three conjugates of x that generate a nonsolvable group unless x belongs to a short list of exceptions which are described explicitly we also prove that if x has order 6 or 9 then there exist two conjugates that generate a nonsolvable group

factorizations of simple algebraic groups we determine all factorizations of simple algebraic groups as the product of two maximal closed connected subgroups additional results are established which drop the maximality assumption and applications are given to the study of subgroups of classical groups transitive on subspaces of a given dimension

a simple global characterization for normal forms of singular vector fields we derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems using an appropriate scalar product in the space of homogeneous vector polynomials we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator this leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form explicit examples and results obtained when there are additional symmetries are also presented

the dirichlet problem at infinity for manifolds of negative curvature on considere l existence de fonctions harmoniques bornees sur des varietes simplement connexes n n de courbure negative

a generalisation of a theorem of wielandt abstract in 1974 helmut wielandt proved that in a finite group g a subgroup a is subnormal if and only if it is subnormal in a g for all g g in this paper we prove that the subnormality of an odd order nilpotent subgroup a of g is already guaranteed by a seemingly weaker condition a is subnormal in g if for every conjugacy class c of g there exists c c for which a is subnormal in a c we also prove the following property of finite non abelian simple groups if a is a subgroup of odd prime order p in a finite almost simple group g then there exists a cyclic p subgroup of f g which does not normalise any non trivial p subgroup of g that is generated by conjugates of a