character correspondences and irreducible induction and restriction let s and g be linitc groups such that s acts on g and is ici 1 and let c denote the fixed point subgroup c c s it is well known in this situation that there exists a natural bijection from the set irr g of s invariant irreducible characters of g onto irr c the full set of irreducible characters of c when s is solvable this bijection was constructed by g glauberman 2 and in the remaining case where g is solvable of odd order a totally different construction 3 gives the desired character correspondence it was proved by t r wolf g that when both constructions apply when s is solvable and ig is odd the two constructions yield exactly the same map if 31 e irr g we write x e irr c to denote the uniquely determined corresponding character similarly if he g is s invariant and 0 e irr h we write f3 e irr hn c to denote the image of 8 under the character bijection induced by the action of s on h our main result is the following

the generalized buckley leverett system solvability we propose a new approach to the mathematical modeling of the buckley leverett system which describes two phase flows in porous media considering the initial boundary value problem for a deduced model we prove the solvability of the problem the solvability result relies mostly on the kinetic method

convergence of numerical approximations to non linear continuity equations with rough force fields we prove quantitative regularity estimates for the solutions to non linear continuity equations and their discretized numerical approximations on cartesian grids when advected by a rough force field this allows us to not only recover the known optimal regularity for linear transport equations but also to obtain the convergence of a wide range of numerical schemes our proof is based on novel commutator estimates quantifying and extending to the non linear case the classical commutator approach of the theory of renormalized solutions

compactness for nonlinear continuity equations abstract we prove compactness and hence existence for solutions to a class of nonlinear transport equations the corresponding models combine the features of linear transport equations and scalar conservation laws we introduce a new method which gives quantitative compactness estimates compatible with both frameworks

h measures a new approach for studying homogenisation oscillations and concentration effects in partial differential equations new mathematical objects called h measures are introduced for studying oscillations and concentration effects in partial differential equations applications to transport properties and to homogenisation are given as an example of the new results which can be obtained by this approach

anzellotti s pairing theory and the gauss green theorem abstract in this paper we obtain a very general gauss green formula for weakly differentiable functions and sets of finite perimeter this result is obtained by revisiting anzellotti s pairing theory and by characterizing the measure pairing a d u when a is a bounded divergence measure vector field and u is a bounded function of bounded variation

well posedness of general boundary value problems for scalar conservation laws in this paper we investigate well posedness for the problem u t div ph u f on 0 t times om om subset r n with initial condition u 0 cdot u 0 on om and with general dissipative boundary conditions varphi u cdot nu in beta t x u on 0 t times ptl om here for a e t x in 0 t times ptl om beta t x cdot is a maximal monotone graph on r this includes as particular cases dirichlet neumann robin obstacle boundary conditions and their piecewise combinations as for the well studied case of the dirichlet condition one has to interprete the it formal boundary condition given by beta by replacing it with the adequate it effective boundary condition such effective condition can be obtained through a study of the boundary layer appearing in approximation processes such as the vanishing viscosity approximation we claim that the formal boundary condition given by beta should be interpreted as the effective boundary condition given by another monotone graph tilde beta which is defined from beta by the projection procedure we describe we give several equivalent definitions of entropy solutions associated with tilde beta and thus also with beta for the notion of solution defined in this way we prove existence uniqueness and l 1 contraction monotone and continuous dependence on the graph beta convergence of approximation procedures and stability of the notion of entropy solution are illustrated by several results

smoothing effect in bvφ for entropy solutions of scalar conservation laws abstract this paper deals with a sharp smoothing effect for entropy solutions of one dimensional scalar conservation laws with a degenerate convex flux we briefly explain why degenerate fluxes are related with the optimal smoothing effect conjectured by lions perthame tadmor for entropy solutions of multidimensional conservation laws it turns out that generalized spaces of bounded variation b v are particularly suitable better than sobolev spaces to quantify the regularizing effect and to obtain traces as in bv the function in question is linked to the degeneracy of the flux up to the present the lax ole nik formula has provided optimal results for a uniformly convex flux this formula is validated in this paper for the more general class of c 1 strictly convex fluxes which contains degenerate convex fluxes and enables the b v smoothing effect in this class we give a complete proof that for a c 1 strictly convex flux the lax ole nik formula provides the unique entropy solution namely the kruzhkov solution

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

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

an engquist osher type scheme for conservation laws with discontinuous flux adapted to flux connections we consider scalar conservation laws with the spatially varying flux h x f u 1 h x g u where h x is the heaviside function and f and g are smooth nonlinear functions adimurthi mishra and veerappa gowda j hyperbolic differ equ 2 2005 pp 783 837 pointed out that such a conservation law admits many l 1 contraction semigroups one for each so called connection a b here we define entropy solutions of type a b involving kruzkov type entropy inequalities that can be adapted to any fixed connection a b it is proved that these entropy inequalities imply the l 1 contraction property for l infty solutions in contrast to the piecewise smooth setting of adimurthi mishra and veerappa gowda for a fixed connection these entropy inequalities include a single adapted entropy of the type used by audusse and perthame proc roy soc edinburgh sect a 135 2005 pp 253 265 we prove convergence of a new difference scheme that approximates entropy solutions of type a b for any connection a b if a few parameters are varied the scheme relies on a modification of the standard engquist osher flux is simple as no 2 times2 riemann solver is involved and is designed such that the steady state solution connecting a to b is preserved in contrast to most analyses of similar problems our convergence proof is not based on the singular mapping or compensated compactness methods but on standard spatial variation estimates away from the flux discontinuity some numerical examples are presented

endoisomorphisms and character triple isomorphisms abstract this paper concerns aspects of clifford theory of finite groups in earlier papers turull proved that if two finite groups yielded the same element of the brauer clifford group then there was an endoisomorphism from one group to the other and furthermore that associated with each endoisomorphism there was an essentially unique correspondence of modules over many different fields from one group to the other the paper adapts the definition of character triple isomorphism so that it involves ordinary and brauer characters and it preserves fields of definition schur indices decomposition numbers and blocks it is proved that each endoisomorphism yields exactly one character triple isomorphism character triple isomorphisms can be composed restricted produced by direct sums extension of fields and these operations have their parallel for the endoisomorphisms one goal of the paper is to provide tools for the study of the character theory of finite groups in an accessible way suitable for applications

vanishing viscosity solutions for conservation laws with regulated flux abstract in this paper we introduce a concept of regulated function v t x of two variables which reduces to the classical definition when v is independent of t we then consider a scalar conservation law of the form u t f v t x u x 0 where f is smooth and v is a regulated function possibly discontinuous w r t both t and x by adding a small viscosity one obtains a well posed parabolic equation as the viscous term goes to zero the uniqueness of the vanishing viscosity limit is proved relying on comparison estimates for solutions to the corresponding hamilton jacobi equation as an application we obtain the existence and uniqueness of solutions for a class of 2 2 triangular systems of conservation laws with hyperbolic degeneracy

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

decay of periodic entropy solutions to degenerate nonlinear parabolic equations abstract under a precise nonlinearity diffusivity condition we establish the decay of space periodic entropy solutions of a multidimensional degenerate nonlinear parabolic equation

on the chain rule formulas for divergences and applications to conservation laws abstract in this paper we prove a nonautonomous chain rule formula for the distributional divergence of the composite function v x b x u x where b t is a divergence measure vector field and u is a function of bounded variation as an application we prove a uniqueness result for scalar conservation laws with discontinuous flux

structure of solutions of multidimensional conservation laws with discontinuous flux and applications to uniqueness we investigate the structure of solutions of conservation laws with discontinuous flux under quite general assumption on the flux we show that any entropy solution admits traces on the discontinuity set of the coefficients and we use this to prove the validity of a generalized kato inequality for any pair of solutions applications to uniqueness of solutions are then given

a theory of l 1 dissipative solvers for scalar conservation laws with discontinuous flux we propose a general framework for the study of l1 contractive semigroups of solutions to conservation laws with discontinuous flux u t mathfrak f x u x 0 qquad mathfrak f x u left begin array ll f l u x 0 end array right quad quad quad rm cl where the fluxes fl fr are mainly assumed to be continuous developing the ideas of a number of preceding works baiti and jenssen in j differ equ 140 1 161 185 1997 towers in siam j numer anal 38 2 681 698 2000 towers in siam j numer anal 39 4 1197 1218 2001 towers et al in skr k nor vidensk selsk 3 1 49 2003 adimurthi et al in j math kyoto university 43 1 27 70 2003 adimurthi et al in j hyperbolic differ equ 2 4 783 837 2005 audusse and perthame in proc roy soc edinburgh a 135 2 253 265 2005 garavello et al in netw heterog media 2 159 179 2007 burger et al in siam j numer anal 47 1684 1712 2009 we claim that the whole admissibility issue is reduced to the selection of a family of elementary solutions which are piecewise constant weak solutions of the form c x c l11 left x 0 right we refer to such a family as a germ it is well known that cl admits many different l1 contractive semigroups some of which reflect different physical applications we revisit a number of the existing admissibility or entropy conditions and identify the germs that underly these conditions we devote specific attention to the vanishing viscosity germ which is a way of expressing the condition of diehl j hyperbolic differ equ 6 1 127 159 2009 for any given germ we formulate germ based admissibility conditions in the form of a trace condition on the flux discontinuity line x 0 in the spirit of vol pert math ussr sbornik 2 2 225 267 1967 and in the form of a family of global entropy inequalities following kruzhkov math ussr sbornik 10 2 217 243 1970 and carrillo arch ration mech anal 147 4 269 361 1999 we characterize those germs that lead to the l1 contraction property for the associated admissible solutions our approach offers a streamlined and unifying perspective on many of the known entropy conditions making it possible to recover earlier uniqueness results under weaker conditions than before and to provide new results for other less studied problems several strategies for proving the existence of admissible solutions are discussed and existence results are given for fluxes satisfying some additional conditions these are based on convergence results either for the vanishing viscosity method with standard viscosity or with specific viscosities adapted to the choice of a germ or for specific germ adapted finite volume schemes

on the existence uniqueness and stability of entropy solutions to scalar conservation laws abstract we consider one dimensional scalar conservation laws with and without viscosity where the flux function f x t u is only assumed to be absolutely continuous in x locally integrable in t and continuous in u the existence and uniqueness of entropy solutions for the associated initial value problem are obtained through the vanishing viscosity method and the doubling variables technique we also prove the stability of entropy solutions in c 0 t l loc 1 r and in c 0 t l 1 r with respect to both initial data and flux functions

on clifford theory with galois action abstract let g be a finite group n a normal subgroup of g and irr n let f be a subfield of the complex numbers and assume that the galois orbit of over f is invariant in g we show that there is another triple g 1 n 1 1 of the same form such that the character theories of g over and of g 1 over 1 are essentially the same over the field f and such that the following holds g 1 has a cyclic normal subgroup c contained in n 1 such that 1 n 1 for some linear character of c and such that n 1 c is isomorphic to the abelian galois group of the field extension f f 1 more precisely having the same character theory means that both triples yield the same element of the brauer clifford group brcliff g f defined by a turull

endoisomorphisms yield module and character correspondences abstract in an earlier paper turull 2012 12 the author introduced the concept of endoisomorphisms and showed its natural connection with clifford theory of finite groups associated with clifford theory are module and character correspondences in the present paper the author shows that each endoisomorphism produces a unique module and character correspondence with excellent compatibility results

character correspondences in solvable groups our concern in this paper is with the ordinary complex number character theory of solvable groups if the lengths of some of the earlier papers on this subject l 3 8 are a reliable indication this is a surprisingly complicated and nontrivial theory nevertheless some of the important results are considerably less deep than one would be led to believe by the number of pages involved this paper contains no new theorems and few new ideas it is addressed to the reader who while familiar with character theory has not mastered the literature on characters of solvable groups its purpose is to give such a reader a feeling for the subject and to provide him with accessible proofs for a few of the main results although we certainly do not attempt to prove the strongest or most general results possible or known nevertheless the paper does contain a few items which have application outside of solvable groups in particular we have included in section 4 a brief exposition of tensor induction although our main result could have been proved using an ad hoc argument in place of this general technique the author is now convinced after considerable initial resistance that tensor induction is valuable and deserves further exposure naturally a key idea in the study of characters of solvable groups is to consider the relationships between characters and chief sections in particular suppose that k l is an abelian chief section of g and let 9 e irr k be invariant in g then according to the going down theorem theorem 6 18 of 9 one of the following must occur

on the strong pre compactness property for entropy solutions of a degenerate elliptic equation with discontinuous flux abstract under some non degeneracy condition we show that sequences of entropy solutions of a semi linear elliptic equation are strongly pre compact in the general case of a caratheodory flux vector the proofs are based on localization principles for h measures corresponding to sequences of measure valued functions

hyperbolic conservation laws with discontinuous fluxes and hydrodynamic limit for particle systems abstract we study the following class of scalar hyperbolic conservation laws with discontinuous fluxes 0 1 t x f x 0 the main feature of such a conservation law is the discontinuity of the flux function in the space variable x kruzkov s approach for the l 1 contraction does not apply since it requires the lipschitz continuity of the flux function in x an additional jump wave may occur in the solution besides the classical waves and entropy solutions even for the riemann problem are not unique under the classical entropy conditions on the other hand it is known that in statistical mechanics some microscopic interacting particle systems with discontinuous speed parameter x in the hydrodynamic limit formally lead to scalar hyperbolic conservation laws with discontinuous fluxes of the form 0 2 t x x h 0 the natural question arises which entropy solution the hydrodynamic limit selects thereby leading to a suitable physical relevant notion of entropy solutions of this class of conservation laws this paper is a first step and provides an answer to this question for a family of discontinuous flux functions in particular we identify the entropy condition for 0 1 and proceed to show the well posedness by combining our existence result with a uniqueness result of audusse and perthame 2005 for the family of flux functions we establish a compactness framework for the hydrodynamic limit of large particle systems and the convergence of other approximate solutions to 0 1 which is based on the notion and reduction of measure valued entropy solutions and we finally establish the hydrodynamic limit for a zrp with discontinuous speed parameter governed by an l entropy solution to 0 2

a reduction theorem for the alperin weight conjecture the alperin weight conjecture was stated by j alperin in 1986 and constitutes one of the main problems in the representation theory of finite groups in this paper we provide a reduction to simple groups and as an application we prove the conjecture in several cases

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

convergence of an engquist osher scheme for a multi dimensional triangular system of conservation laws we consider a multi dimensional triangular system of conservation laws this system arises as a model of three phase flow in porous media and includes multi dimensional conservation laws with discontinuous coefficients as a special case the system is neither strictly hyperbolic nor symmetric we propose an engquist osher type scheme for this system and show that the approximate solutions generated by the scheme converge to a weak solution numerical examples are also presented

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

boundary data smoothness for solutions of nonlocal boundary value problems for second order differential equations abstract under certain conditions solutions of the boundary value problem y f x y y y x 1 y 1 and y x 2 i 1 m r i y i y 2 are differentiated with respect to boundary conditions where a x 1 1 m x 2 b r 1 r m r and y 1 y 2 r

a pde model for the spatial dynamics of a voles population structured in age abstract we prove existence and stability of entropy weak solutions for a macroscopic pde model for the spatial dynamics of a population of voles structured in age the model consists of a scalar pde depending on time t age a and space x x 1 x 2 supplemented with a non local boundary condition at a 0 the flux is linear with constant coefficient in the age direction but contains a non local term in the space directions also the equation contains a term of second order in the space variables only existence of solutions is established by compensated compactness see panov 2009 and we prove stability by a doubling of variables type argument

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

on the calculation of local invariants of irreducible characters abstract let k be a finite extension of the field q p of p adic numbers let be an irreducible character of a finite group whose values are in k associated with and k is an element of the brauer group of k and therefore by standard results a local invariant in q z the paper defines some classes of finite groups that the paper calls p basic groups and gives formulas to calculate the local invariant of their characters it also shows how one can calculate the local invariant of any irreducible character of any finite group that has values in k by reducing the problem to the case where the groups are p basic and using the explicit formulas given in the paper

well posedness for a slow erosion model abstract we improve in two ways the well posedness results of 2 for a slow erosion model proposed in 11 firstly we study the asymptotic profile when u 0 1 u 0 l where u 0 is the initial datum moreover using a compensated compactness based argument we prove the existence of solutions when u 0 1 u 0 l 3

clifford theory and endoisomorphisms this paper describes the relation between clifford theory of finite groups over any field and the brauer clifford group let g and g be finite groups let g g be a surjective group homomorphism with kernel h and let f be any field let k be any extension field of f and let s be an irreducible kh module we show that to s is associated in a natural way a specific element s f of a brauer clifford group defined over and f as a tool to prove the existence of this association and to study its properties we use endoisomorphisms these are simply certain isomorphisms of related endomorphism algebras as g algebras over f we show that two modules for two different finite groups have the same in an appropriate sense element of the brauer clifford group if and only if there exists an appropriate endoisomorphism

erratum to existence and strong pre compactness properties for entropy solutions of a first order quasilinear equation with discontinuous flux sequences of entropy solutions of a non degenerate first order quasilinear equation are shown to be strongly pre compact in the general case of a caratheodory flux vector existence of the weak and entropy solution to the cauchy problem for such an evolutionary equation is also established the proofs are based on the general localization principle for h measures corresponding to sequences of measure valued functions

a variety of differentiability results for a multi point boundary value problem the goal of this work is to address differentiability of solutions of a nonlinear dynamic system with multi point boundary conditions described by x f t x m 1k mmx tm r where r rn and for each 1 m k mm is an n n constant matrix and tm t the solution of this system is defined on a measure chain t and x denotes a generalized derivative known as the delta derivative