an existence result for homoclinic solutions to a nonlinear second order ode through differential inequalities abstract in this paper an existence result of homoclinic solutions to a nonlinear second order ode is presented to this end a method based on differential inequalities is used

the equivalence of some conjectures of dade and robinson we demonstrate that conjecture 4 1 of g r robinson proc london math soc 3 72 1996 312 330 and dade s projective conjecture are equivalent in a way which is compatible with the p local rank further we consider refinements of these conjectures similar to those of isaacs navarro and uno show their equivalence and demonstrate that in order to verify them it suffices to consider only those groups with no non central normal p subgroup

the mckay conjecture and galois automorphisms the main problem of representation theory of finite groups is to find proofs of several conjectures stating that certain global invariants of a finite group g can be computed locally the simplest of these conjectures is the mckay conjecture which asserts that the number of irreducible complex characters of g of degree not divisible by p is the same if computed in a p sylow normalizer of g in this paper we propose a much stronger version of this conjecture which deals with galois automorphisms in fact the same idea can be applied to the celebrated alperin and dade conjectures

perfect generalized characters inducing the alperin mckay conjecture it is well known that the perfect isometries predicted in broue s conjecture do not always exist when the defect groups are non abelian even when the blocks have equivalent brauer categories we consider perfect generalized characters which induce bijections between the sets of irreducible characters of height zero of a block and of its brauer correspondent in the normalizer of a defect group in this way the perfect isometries predicted in broue s conjecture for blocks with abelian defect groups are generalized whilst such generalized characters do not exist in general we show that they do exist when the defect groups are non abelian trivial intersection subgroups of order p 3 as well as for 2b 2 q for q a power of two and psu 3 q for all q further we show that these blocks satisfy a generalized version of an isotypy

a robust petrov galerkin discretisation of convection diffusion equations a petrov galerkin discretisation is studied of an ultra weak variational formulation of the convection diffusion equation in a mixed form to arrive at an implementable method the truly optimal test space has to be replaced by its projection onto a finite dimensional test search space to prevent that this latter space has to be taken increasingly large for vanishing diffusion a formulation is constructed that is well posed in the limit case of a pure transport problem numerical experiments show approximations that are very close to the best approximations to the solution from the trial space uniformly in the size of the diffusion term

precise damping conditions for global asymptotic stability for nonlinear second order systems global asymptotic stability of the rest point for nonlinear equations has been treated by a number of authors in this paper we unify and extend previous work by obtaining various necessary and sucient conditions for the asymptotic stability as t 1 of solutions of quasi variational systems of the form rl t u u 0 0 r ul t u u 0 q t u u 0

a reduction theorem for the mckay conjecture the mckay conjecture asserts that for every finite group g and every prime p the number of irreducible characters of g having p degree is equal to the number of such characters of the normalizer of a sylow p subgroup of g although this has been confirmed for large numbers of groups including for example all solvable groups and all symmetric groups no general proof has yet been found in this paper we reduce the mckay conjecture to a question about simple groups we give a list of conditions that we hope all simple groups will satisfy and we show that the mckay conjecture will hold for a finite group g if every simple group involved in g satisfies these conditions also we establish that our conditions are satisfied for the simple groups psl2 q for all prime powers q 4 and for the suzuki groups sz q and ree groups r q where q 2 e or q 3 e respectively and e 1 is odd since our conditions are also satisfied by the sporadic simple group j 1 it follows that the mckay conjecture holds for all primes p for every finite group having an abelian sylow 2 subgroup

refinements of dade s projective conjecture for p solvable groups dade s projective conjecture is known to be true for finite p solvable groups thanks to work of g r robinson but remains open in general work of isaacs and navarro suggested to uno and boltje refinements of this conjecture these refinements were studied for finite p solvable groups by glesser in the present paper inspired by earlier work of turull we propose further refinements of the conjecture that take into account the schur indices and the elements of the brauer group we prove that all these refinements of dade s projective conjecture hold for all finite p solvable groups in particular we obtain that the version of dade s projective conjecture which involves character degree residues modulo p fields of definition and schur indices as well as the full strength of boltje s conjecture both hold for all finite p solvable groups the proof develops a clifford theory for normalizers of chains of p subgroups which allows one to reduce the calculation of the relevant sums to simpler groups

the mckay conjecture is true for the sporadic simple groups abstract the mckay conjecture states that the number of irreducible complex characters of a group g that have degree prime to p is equal to the same number for the sylow p normalizer in g we verify this conjecture for the 26 sporadic simple groups

strengthening the mckay conjecture to include local fields and local schur indices abstract a central problem in the representation theory of finite groups is given a prime p the study of relationships between the representations of g and those of p local subgroups of g the simplest of these is perhaps the mckay conjecture which concerns the number of irreducible characters of degree prime to p even though this conjecture remains open in an effort to understand it a number of strengthenings of it have been proposed recently isaacs and navarro and later navarro have proposed interesting stronger conjectures indicating that congruences modulo p of the degree of the characters and the invariance under certain type of fixed galois automorphism could be taken into account as well we propose an even further strengthening of mckay s conjecture which takes into account in addition the p local field of definition of the irreducible characters as well as their p local schur indices

homoclinic orbits for second order hamiltonian systems possessing superquadratic potentials received by the editors november 27 1990 1991 mathematics subject classification primary 34c99 58e99 58f99 the first author was supported by ministero p i gruppo 40 calcolo delle variazioni the second author s research was sponsored in part by the u s army research office under contract daal03 87 k 0043 the national science foundation under grant mcs 8110556 and the office of naval research under grant n00014 88 ko134 any reproduction for the purpose of the united states govemment is permitted

some refinements of dade s projective conjecture abstract in this paper a further refinement of dade s projective conjecture due to boltje is presented this new statement includes ideas first published by isaacs and navarro as well as the recent contractibility version of alperin s conjecture introduced by boltje leaning heavily on the work of robinson weaker forms of the conjecture are proved in the case of p solvable groups

uno s invariant conjecture for the general linear and unitary groups in nondefining characteristics abstract uno s invariant conjecture which is a combination of dade s invariant conjecture and the isaacs navarro conjecture is proved for general linear and unitary groups in nondefining characteristics

height zero characters and normal subgroups in p solvable groups abstract fix a prime number p and let n be a normal subgroup of a finite p solvable group g let b be a p block of n and suppose b is a p block of g covering b let d be a defect group for the fong reynolds correspondent of b with respect to b and let b be the unique p block of n n g d having defect group d and inducing b suppose further that irr b and let irr 0 b be the set of irreducible characters in b of height zero that lie over we show that the number of characters in irr 0 b is equal to the number of characters in t irr 0 b t where t runs through the inertial group t of b in g this result generalizes a theorem of t okuyama and m wajima which confirms the alperin mckay conjecture for p solvable groups

existence of homoclinic solutions to periodic orbits in a center manifold abstract consider a lagrangian of the form l x x q q 1 2 x 2 x 2 1 2 q 2 1 x v q where x q r assuming that is bounded and v periodic in q is such that v 0 0 we prove existence of infinitely many solutions homoclinic to periodic orbits in the center manifold q 0 q 0 of the corresponding system

the strengthened alperin weight conjecture for p solvable groups abstract the alperin weight conjecture is a well known conjecture and it is central to the modern representation theory of finite groups it is known to be true for many types of finite groups but it remains open in particular the conjecture is known to be true for finite p solvable groups in this paper we prove that a very strong form of the conjecture holds for all finite p solvable groups it follows that the strengthening of the alperin weight conjecture that includes galois automorphisms over the p adic numbers holds for all finite p solvable groups this strengthening of the alperin weight conjecture was suggested in an earlier work of navarro on the mckay conjecture the present paper provides the first direct evidence beyond the calculation of individual cases and the case of groups of odd order for this strengthening

character correspondences in solvable groups let g be a finite solvable group p be some prime let p be a sylow p subgroup of g and let n be its normalizer in g it is known that the number of irreducible characters of g of degree prime to p equals the number of irreducible characters of n of degree prime to p let f be any field of characteristic zero such that if g n is even then f contains qp q 2 the intersection of the field of p adic numbers with the field of rational numbers extended by all roots of unity whose order is a power to two in this paper we show that there exists a bijection from the set of all irreducible characters of g of degree prime to p to the set of all the irreducible characters of degree prime to p of n such that it preserves the degrees modulo p all the field of values over f and the schur index over every field containing f

the mckay conjecture and brauer s induction theorem let g be an arbitrary finite group the mckay conjecture asserts that g and the normaliser ng p of a sylow p subgroup p in g have the same number of characters of degree not divisible by p that is of p degree we propose a new refinement of the mckay conjecture which suggests that one may choose a correspondence between the characters of p degree of g and ng p to be compatible with induction and restriction in a certain sense this refinement implies in particular a conjecture of isaacs and navarro we also state a corresponding refinement of the broue abelian defect group conjecture we verify the proposed conjectures in several special cases

dade s invariant conjecture for steinberg s triality groups d43 2n in defining characteristic abstract in this paper we verify dade s invariant conjecture for steinberg s triality groups d 4 3 2 n in the defining characteristic i e in characteristic 2 together with the results in j an dade s conjecture for steinberg triality groups d 4 3 q in non defining characteristics math z 241 2002 445 469 and j an f himstedt s huang uno s invariant conjecture for steinberg s triality groups in defining characteristic in preparation this completes the proof of dade s conjecture for steinberg s triality groups furthermore we show that the isaacs malle navarro version of the mckay conjecture holds for d 4 3 2 n in the defining characteristic i e d 4 3 2 n is good for the prime 2 in the sense of isaacs malle and navarro

uno s invariant conjecture for chevalley groups g2 q in nondefining characteristics abstract uno s invariant conjecture for the chevalley groups g 2 q has been verified when the characteristic of the modular representation is distinct from the defining characteristic of the groups thus dade s reductive conjecture and the isaacs navarro conjecture both hold for g 2 q in non defining characteristics

uno s invariant conjecture for steinberg s triality groups in defining characteristic abstract we verify uno s invariant conjecture for steinberg s triality groups d 4 3 q q a power of an odd prime p in the defining characteristic p uno s invariant conjecture is a refinement of dade s invariant conjecture

above the glauberman correspondence let s be a finite solvable group and suppose s acts on the finite group n and they have coprime orders then the celebrated glauberman correspondence provides a natural bijection from the set irrs n of irreducible characters of n which are invariant under the action of s to the set irr cn s of all irreducible characters of the centralizer of s in n suppose further that the semidirect product sn is a normal subgroup of a finite group g let irrs n and let irr cn s be its glauberman correspondent we prove that there is a bijection with good compatibility properties from the set irr g of the irreducible characters of g above to irr ng s such that in the case when s is a p group for some prime p it preserves fields of values and schur indices over qp the field of p adic numbers using this result we also prove a strengthening of the mckay conjecture for all p solvable groups

tilting equivalences from hereditary algebras to symmetric groups abstract we study koszul duality for finite dimensional hereditary algebras and various generalisations to trivial extension algebras to schur algebras to doubles of schur bialgebras and to deformations of doubles of schur bialgebras we describe applications to the modular representation theory of symmetric groups

new refinements of the mckay conjecture for arbitrary finite groups let g be an arbitrary finite group and fix a prime number p the mckay conjecture asserts that g and the normalizer in g of a sylow p subgroup have equal numbers of irreducible characters with degrees not divisible by p the alperin mckay conjecture is version of this as applied to individual brauer p blocks of g we offer evidence that perhaps much stronger forms of both of these conjectures are true

isometries and extra special sylow groups of order p3 abstract a new type of isometry between the sets of irreducible characters of blocks is introduced and a conjecture is stated using the classifications of finite simple groups and saturated fusion systems over an extra special group p of order p 3 and exponent p we check that the conjecture is affirmatively answered for the principal blocks of finite groups with sylow p subgroup p

the isaacs navarro conjecture for the alternating groups abstract a recent refinement of the mckay conjecture is verified for the case of the alternating groups the argument builds upon the verification of the conjecture for the symmetric groups p fong the isaacs navarro conjecture for symmetric groups j algebra 250 1 2003 154 161

stability for a damped nonlinear oscillator abstract the stability of the null solution of eq 1 1 below is discussed under some unusual assumptions we obtain new stability results for this classical oscillator equation our approach allows extensions to both the vector case and the case of the whole real line

groups whose real irreducible characters have degrees coprime to p abstract in this paper we study groups for which every real irreducible character has degree not divisible by some given odd prime p