the triviality problem for profinite completions we prove that there is no algorithm that can determine whether or not a finitely presented group has a non trivial finite quotient indeed this property remains undecidable among the fundamental groups of compact non positively curved square complexes we deduce that many other properties of groups are undecidable for hyperbolic groups there cannot exist algorithms to determine largeness the existence of a linear representation with infinite image over any infinite field or the rank of the profinite completion

algorithms determining finite simple images of finitely presented groups we address the question for which collections of finite simple groups does there exist an algorithm that determines the images of an arbitrary finitely presented group that lie in the collection we prove both positive and negative results for a collection of finite simple groups that contains infinitely many alternating groups or contains classical groups of unbounded dimensions we prove that there is no such algorithm on the other hand for families of simple groups of lie type of bounded rank we obtain positive results for example for any fixed untwisted lie type x there is an algorithm that determines whether or not any given finitely presented group has simple images of the form x q for infinitely many q and if there are finitely many the algorithm determines them

asymptotic behaviour of solutions of the fast diffusion equation near its extinction time abstract let n 3 0 m n 2 n 1 0 m 1 n 2 n m and 2 1 1 m for any 0 we will prove the existence and uniqueness for 1 n 2 n m of radially symmetric singular solution g c r n 0 of the elliptic equation v m v x v 0 v 0 in r n 0 satisfying lim x 0 x g x 1 1 m when is sufficiently large we prove the higher order asymptotic behaviour of radially symmetric solutions of the above elliptic equation as x we also obtain an inversion formula for the radially symmetric solution of the above equation as a consequence we will prove the extinction behaviour of the solution u of the fast diffusion equation u t u m in r n 0 t near the extinction time t 0

singular limit of solutions of the very fast diffusion equation we prove that the distribution solutions of the very fast diffusion equation u t um m u t um m u 0u 0 in rn 0 rn 0 u x 0 u0 x u x 0 u0 x in rnrn where m n 2p n 2 and u0 x e x 2 u0 x e x 2 for any x r1 x r1 where e 0e 0 r1 0r1 0 m0 1 m0 n 2p 1 m0 n 2 we prove that the solution of the above problem will converge uniformly on every compact subset of rn 0 rn 0 to the maximal solution of the equation vt logvvt logv v x 0 u0 x v x 0 u0 x as m 0 m 0 for any smooth bounded domain rn rn m0 1 m0 max 1 n 2 p 1 m0 max 1 n 2 we prove the existence and uniqueness of solutions of the dirichlet problem u t um m u t um m u 0u 0 in 0 0 u u0u u0 in u gu g on 0 0 with either finite or infinite positive boundary value gg we also prove a similar convergence result for the solutions of the above dirichlet problem as m 0m 0

an l2 quotient algorithm for finitely presented groups abstract the paper develops algorithmic methods to enumerate all normal subgroups of a finitely presented group such that the factor groups are either isomorphic to psl 2 p n or to pgl 2 p n the case of two generators is treated in detail a range of examples starting with the free group on two generators and ending with groups having only finitely many normal subgroups of this type is discussed

an l2 quotient algorithm for finitely presented groups on arbitrarily many generators abstract we generalize the plesken fabianska l 2 quotient algorithm for finitely presented groups on two or three generators to allow an arbitrary number of generators the main difficulty lies in a constructive description of the invariant ring of gl 2 k on m copies of sl 2 k by simultaneous conjugation by giving this description we generalize and simplify some of the known results in invariant theory an implementation of the algorithm is available in the computer algebra system magma

on the subgroup structure of classical groups in this paper we establish a result on the subgroup structure of classical groups over algebraically closed elds and use this to give a new proof of a fundamental theorem of m aschbacher on subgroups of nite classical groups let v be a nite dimensional vector space over an algebraically closed eld k and let g be one of the classical algebraic groups sl v sp v or so v our result is a reduction theorem concerning the subgroups of g we de ne a certain collection c of natural proper subgroups of g and prove that any closed nite or in nite subgroup of g either lies in a member of c or is roughly speaking a simple group acting irreducibly on v aschbacher s result is an analogous reduction theorem for subgroups of nite classical groups we obtain this as a relatively easy consequence of our main result by taking xed points under the action of a frobenius morphism using a standard process involving lang s theorem the proof of the main result uses elementary linear algebra together with a few basic facts from the theory of algebraic groups various complications which arise in the nite group setting in as become much more straightforward in the algebraic group setting in particular questions involving extension elds do not occur and issues of conjugacy are easily settled when we descend to nite invent math 134 427 453 1998

sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities the goal of this paper is to state the optimal decay rate for solutions of the nonlinear fast diffusion equation and in self similar variables the optimal convergence rates to barenblatt self similar profiles and their generalizations it relies on the identification of the optimal constants in some related hardy poincare inequalities and concludes a long series of papers devoted to generalized entropies functional inequalities and rates for nonlinear diffusion equations

on the sum of laplacian eigenvalues of a signed graph abstract for a signed graph let e denote the number of edges and s k denote the sum of the k largest eigenvalues of the laplacian matrix of we conjecture that for any signed graph with n vertices s k e k 1 2 1 holds for k 1 n we prove the conjecture for any signed graph when k 2 and prove that this conjecture is true for unicyclic and bicyclic signed graphs

on the sum of the laplacian eigenvalues of a tree given an n vertex graph g v e the laplacian spectrum of g is the set of eigenvalues of the laplacian matrix l d a where d and a denote the diagonal matrix of vertex degrees and the adjacency matrix of g respectively in this paper we study the laplacian spectrum of trees more precisely we find a new upper bound on the sum of the k largest laplacian eigenvalues of every n vertex tree where k 1 n this result is used to establish that the n vertex star has the highest laplacian energy over all n vertex trees which answers affirmatively to a question raised by radenkovic and gutman 10

on a conjecture for the signless laplacian eigenvalues abstract let g be a simple graph with n vertices and e g edges and q 1 g q 2 g q n g 0 be the signless laplacian eigenvalues of g let s k g i 1 k q i g where k 1 2 n f ashraf et al conjectured that s k g e g k 1 2 for k 1 2 n in this paper we give various upper bounds for s k g and prove that this conjecture is true for the following cases connected graph with sufficiently large k unicyclic graphs and bicyclic graphs for all k and tricyclic graphs when k 3 finally we discuss whether the upper bound given in this conjecture is tight or not for c cyclic graphs and propose some problems for future research

on the sum of signless laplacian eigenvalues of a graph abstract for a simple graph g let e g denote the number of edges and s k g denote the sum of the k largest eigenvalues of the signless laplacian matrix of g we conjecture that for any graph g with n vertices s k g e g k 1 2 for k 1 n we prove the conjecture for k 2 for any graph and for all k for regular graphs the conjecture is an analogous to a conjecture by a e brouwer with a similar statement but for the eigenvalues of laplacian matrices of graphs

on generalized distance energy of graphs abstract for the distance matrix d g and diagonal matrix of the vertex transmissions t r g of a simple connected graph g the generalized distance matrix d g is the convex combinations of t r g and d g and is defined as d g t r g 1 d g for 0 1 if 1 2 n are the eigenvalues of d g we define the generalized distance energy of the graph g as e d g i 1 n i 2 w g n where w g is the wiener index of g this is analogous to the energies associated with the distance laplacian and distance signless laplacian matrices of g we obtain upper and lower bounds for the generalized distance energy of graphs in terms of various parameters associated with the structure of the graph g we show that for 1 2 1 the complete bipartite graph has the minimum generalized distance energy among all connected bipartite graphs and for 0 2 n 3 n 2 the star graph has the minimum generalized distance energy among all trees

primitive permutation groups of bounded orbital diameter we give a description of infinite families of finite primitive permutation groups for which there is a uniform finite upper bound on the diameter of all orbital graphs this is equivalent to describing families of finite permutation groups such that every ultraproduct of the family is primitive a key result is that in the almost simple case with socle of fixed lie rank apart from very specific cases there is such a diameter bound this is proved using recent results on the model theory of pseudofinite fields and difference fields

residual properties of the modular group and other free products abstract using a probabilistic approach we establish new residual properties of the modular group psl 2 z and of more general free products we prove that the modular group is residually in any infinite collection of finite simple groups not containing a suzuki group sz q or a 4 dimensional symplectic group psp4 q with q a power of 2 or 3 this result is best possible since the groups excluded are not quotients of the modular group we also show that if s is a collection of classical groups of unbounded rank then an arbitrary free product a b of nontrivial finite groups not both 2 groups is residually s and prove results about free products a z

spectral radius and signless laplacian spectral radius of strongly connected digraphs abstract let d be a strongly connected digraph and a d be the adjacency matrix of d let diag d be the diagonal matrix with outdegrees of the vertices of d and q d diag d a d be the signless laplacian matrix of d the spectral radius of q d is called the signless laplacian spectral radius of d denoted by q d in this paper we give a sharp bound on q d where d has a given outdegree sequence and compare the bound with known bounds we establish some sharp upper or lower bound on q d with some given parameter such as clique number girth or vertex connectivity and characterize the extremal graph in addition we also determine the unique digraph which achieves the minimum or maximum the second minimum or maximum the third minimum the fourth minimum spectral radius and signless laplacian spectral radius among all strongly connected digraphs and answer the open problem proposed by lin and shu 14

signless laplacian energy of a graph and energy of a line graph abstract for a simple graph g of order n size m and with signless laplacian eigenvalues q 1 q 2 q n the signless laplacian energy q e g is defined as q e g i 1 n q i d where d 2 m n is the average vertex degree of g we obtain the lower bounds for q e g in terms of first zagreb index m 1 g maximum degree d 1 second maximum degree d 2 minimum degree d n and second minimum degree d n 1 as a consequence of these bounds we obtain several bounds for the energy e l g of the line graph l g of graph g in terms of various graph parameters like m 1 g the clique number n m etc which improve some recently known bounds

rate of convergence to barenblatt profiles for the fast diffusion equation we study the asymptotic behaviour of positive solutions of the cauchy problem for the fast diffusion equation as t approaches the extinction time we find a continuum of rates of convergence to a self similar profile these rates depend explicitly on the spatial decay rates of the initial data