a counterexample to the isomorphism problem for integral group rings let x be a finite group and denote its integral group ring by zx a group basis of zx is a subgroup y of the group of units of zx of augmentation 1 such that zx zy and ixi yi an example of a finite group x is given such that zx has a group basis which is not isomorphic to x a main ingredient is the existence of a subgroup g of x which possesses a non inner automorphism which becomes inner in the integral group ring zg the question whether a finite group x is determined by its integral group ring zx is known as the isomorphism problem for integral group rings it was

infiltration experiments demonstrate an explicit connection between heterogeneity and anomalous diffusion behavior transport in systems containing heterogeneity distributed over multiple length scales can exhibit anomalous diffusion behaviors where the time exponent determining the spreading length scale of the transported scalar differs from the expected value of n 12 here we present experimental measurements of the infiltration of glycerin under a fixed pressure head into a hele shaw cell containing a 3 d printed distribution of flow obstacles a system that is an analog for infiltration into a porous medium in support of previously presented direct simulation results we experimentally demonstrate that when the obstacles are distributed as a fractal carpet with fractal dimension h 2 the averaged progress of infiltration exhibits a subdiffusive behavior n 12 we further show that observed values of the subdiffusion time exponent appear to be quadratically related to the fractal dimension of the carpet

an algorithm to construct candidates to counterexamples to the zassenhaus conjecture abstract let g be a finite group n a nilpotent normal subgroup of g and let v z g n denote the group formed by the units of the integral group ring z g of g which map to the identity under the natural homomorphism z g z g n sehgal asked whether any torsion element of v z g n is conjugate in the rational group algebra of g to an element of g this is a special case of the zassenhaus conjecture by results of cliff and weiss and hertweck sehgal s problem has a positive solution if n has at most one non cyclic sylow subgroup we present some algorithms to study sehgal s problem when n has at most one non abelian sylow subgroup they are based on the cliff weiss inequalities introduced by the authors in a previous paper with the help of these algorithms we obtain some positive answers to sehgal s problem and use them to show that for units in v z g n our method is strictly stronger than the well known help method we then present a method to use the output of one of the algorithms to construct explicit metabelian groups which are candidates to a negative solution to sehgal s problem recently eisele and margolis showed that some of the examples proposed in this paper are indeed counterexamples to the zassenhaus conjecture these are the first known counterexamples moreover we prove that every metabelian negative solution of sehgal s problem satisfying some minimal conditions is given by our construction

relating homogeneous cones and positive definite cones via t algebras t algebras are nonassociative algebras defined by vinberg in the early 1960s for the purpose of studying homogeneous cones vinberg defined a cone k cal a for each t algebra cal a and proved that every homogeneous cone is isomorphic to one such k cal a we relate each t algebra cal a with a space of linear operators in such a way that k cal a is isomorphic to the cone of positive definite self adjoint operators together with vinberg s result we conclude that every homogeneous cone is isomorphic to a slice of a cone of positive definite matrices

torsion units in integral group rings of some metabelian groups let g be an extension of an elementary abelian p group a by an abelian group x with faithful and irreducible action then any torsion unit of the integral group ring zg is rationally conjugate to a trivial unit this confirms a conjecture of zassenhaus for these groups g

lattices and cohomological mackey functors for finite cyclic p groups abstract for a finite cyclic p group g and a discrete valuation domain r of characteristic 0 with maximal ideal p r the r g permutation modules are characterized in terms of the vanishing of first degree cohomology on all subgroups cf theorem a as a consequence any r g lattice can be presented by r g permutation modules cf theorem c the proof of these results is based on a detailed analysis of the category of cohomological g mackey functors with values in the category of r modules it is shown that this category has global dimension 3 cf theorem e a crucial step in the proof of theorem e is the fact that a gentle r order category with parameter p has global dimension less than or equal to 2 cf theorem d

pore scale modeling and continuous time random walk analysis of dispersion in porous media 1 we provide a physically based explanation for the complex macroscopic behavior of dispersion in porous media as a function of peclet number pe using a pore scale network model that accurately predicts the experimental dependence of the longitudinal dispersion coefficient dl on pe the asymptotic dispersion coefficient is only reached after the solute has traveled through a large number of pores at high pe this implies that preasymptotic dispersion is the norm even in experiments in statistically homogeneous media interpreting transport as a continuous time random walk we show that 1 the power law dispersion regime is controlled by the variation in average velocity between throats the distribution of local pe giving dl pe with 3 1 2 where is an exponent characterizing the distribution of transit times between pores 2 the crossover to a linear regime dl pe for pe pecrit 400 is due to a transition from a diffusion controlled late time cutoff to transport governed by advective movement and 3 the transverse dispersion coefficient dt pe for all pe 1

influence of image resolution and evaluation algorithm on estimates of the lacunarity of porous media in recent years experience has demonstrated that the classical fractal dimensions are not sufficient to describe uniquely the interstitial geometry of porous media at least one additional index or dimension is necessary lacunarity a measure of the degree to which a data set is translationally invariant is a possible candidate unfortunately several approaches exist to evaluate it on the basis of binary images of the object under study and it is unclear to what extent the lacunarity estimates that these methods produce are dependent on the resolution of the images used in the present work the gliding box algorithm of allain and cloitre phys rev a 44 3552 1991 and two variants of the sandbox algorithm of chappard et al j pathol 195 515 2001 along with three additional algorithms are used to evaluate the lacunarity of images of a textbook fractal the sierpinski carpet of scanning electron micrographs of a thin section of a european soil and of light transmission photographs of a togolese soil the results suggest that lacunarity estimates as well as the ranking of the three tested systems according to their lacunarity are affected strongly by the algorithm used by the resolution of the images to which these algorithms are applied and at least for three of the algorithms producing scale dependent lacunarity estimates by the scale at which the images are observed depending on the conditions under which the estimation of the lacunarity is carried out lacunarity values range from 1 02 to 2 14 for the three systems tested and all three of the systems used can be viewed alternatively as the most or the least lacunar some of this indeterminacy and dependence on image resolution is alleviated in the averaged lacunarity estimates yielded by chappard et al s algorithm further research will be needed to determine if these lacunarity estimates allow an improved unique characterization of porous media

lebesgue mixed norm estimates for bergman projectors from tube domains over homogeneous cones to homogeneous siegel domains of type ii in this paper the weight functions are fundamental compound functions of the homogeneous cone first we use the recent bourgain demeter l 2 decoupling theorem to obtain the optimal weighted lebesgue mixed norm estimates for weighted bergman projectors on tube domains over lorentz cones this settles a conjecture of d debertol secondly we present a transference principle of weighted lebesgue mixed norm estimates for weighted bergman projectors from tube domains over homogeneous cones to homogeneous siegel domains of type ii associated to the same cones so results of c nana for homogeneous siegel domains of type ii can be deduced from earlier results of c nana and b trojan for tube domains over homogeneous cones combining our two theorems we improve these estimates for homogeneous siegel domains of type ii associated with lorentz cones e g the pyateckii shapiro siegel domain of type ii

a counterexample to the first zassenhaus conjecture abstract hans j zassenhaus conjectured that for any unit u of finite order in the integral group ring of a finite group g there exists a unit a in the rational group algebra of g such that a 1 u a g for some g g we disprove this conjecture by first proving general results that help identify counterexamples and then providing an infinite number of examples where these results apply our smallest example is a metabelian group of order 2 7 3 2 5 7 2 19 2 whose integral group ring contains a unit of order 7 19 which in the rational group algebra is not conjugate to any element of the form g

on principal blocks of p constrained groups a theorem of k w roggenkamp and l l scott shows that for a finite group g with a normal p subgroup containing its own centralizer any two group bases of the integral group ring mathbb z g are conjugate in the units of mathbb z p g though the theorem presents itself in the work of others and appears to be needed there is no published account there seems to be a flaw in the proof because a theorem appearing in the survey k w roggenkamp the isomorphism problem for integral group rings of finite groups progress in mathematics 95 birkhauser basel 1991 pp 193 220 where the main ingredients of a proof are given is false in this paper it is shown how to close this gap at least if one is only interested in the conclusion mentioned above therefore some consequences of the results of a weiss on permutation modules are stated the basic steps of which any proof should consist are discussed in some detail in doing so a complete yet short proof of the theorem is given in the case that g has a normal sylow p subgroup 2000 mathematical subject classification primary 16u60 secondary 20c05

on picard groups of blocks of finite groups we show that the subgroup of the picard group of ap block ofa finite group given by bimodules with endopermutation sources modulo theautomorphism group of a source algebra is determined locally in terms of thefusion system on a defect group we show that the picard group of a block overa complete discrete valuation ringoof characteristic zero with an algebraicclosurekoffpas residue field is a colimit of finite picard groups of blocks overp adic subrings ofo we apply the results to blocks with an abelian defectgroup and frobenius inertial quotient and specialise this further to blockswith cyclic or klein four defect groups

zassenhaus conjecture on torsion units holds for psl 2 p with p a fermat or mersenne prime abstract h j zassenhaus conjectured that any unit of finite order in the integral group ring z g of a finite group g is conjugate in the rational group algebra q g to an element of the form g with g g this is known to be true for some series of solvable groups but recently metabelian counterexamples have been constructed the conjecture is still open for non abelian simple groups and has only been proved for thirteen such groups we prove the zassenhaus conjecture for the groups psl 2 p where p is a fermat or mersenne prime this increases the list of non abelian simple groups for which the conjecture is known by probably infinitely many but at least by 50 groups our result is an easy consequence of known results and our main theorem which states that the zassenhaus conjecture holds for a unit in z psl 2 q of order coprime with 2 q for some prime power q

boundedness of bergman projections on tube domains over light cones let gamma be the future light cone in bbb r n and omega bbb r n i gamma be the associated tube domain we prove that the weighted bergman projection p nu

littlewood paley decompositions related to symmetric cones and bergman projections in tube domains starting from a whitney decomposition of a symmetric cone omega analogous to the dyadic partition 2 j 2 j 1 of the positive real line in this paper we develop an adapted littlewood paley theory for functions with spectrum in omega in particular we define a natural class of besov spaces of such functions b p q nu where the role of the usual derivation is now played by the generalized wave operator of the cone delta frac partial partial x we show that b p q nu consists precisely of the distributional boundary values of holomorphic functions in the bergman space a p q nu t omega at least in a good range of indices 1 leq q 2 moreover we show the equivalence of this problem with the boundedness of bergman projectors p nu colon l p q nu to a p q nu for which our result implies a positive answer when q nu p q q nu p this extends to general cones previous work of the authors on the light cone finally we focus on light cones and introduce new necessary and sufficient conditions for our conjecture to hold in terms of inequalities related to the cone multiplier problem in particular using recent work by laba and wolff we establish the validity of our conjecture for light cones when p is sufficiently large

a sylow theorem for the integral group ring of psl 2 q abstract for g psl 2 p f denote by z g the integral group ring over g and by v z g the group of units of augmentation 1 in z g let r be a prime different from p using the so called help method we prove that units of r power order in v z g are rationally conjugate to elements of g as a consequence we prove that subgroups of prime power order in v z g are rationally conjugate to subgroups of g if p 2 or f 1

on morita and derived equivalences for cohomological mackey algebras by results of the second author a source algebra equivalence between two p blocks of finite groups induces an equivalence between the categories of cohomological mackey functors associated with these blocks and a splendid derived equivalence between two blocks induces a derived equivalence between the corresponding categories ofcohomological mackey functors the main result of this paper proves a partial converse an equivalence resp rickard equivalence between the categories of cohomological mackey functors of two blocks of finite groups induces a permeable morita resp derived equivalence between the two block algebras

the proof of the l 2 decoupling conjecture we prove the l 2 decoupling conjecture for compact hypersurfaces with positive denite second fundamental form and also for the cone this has a wide range of important consequences one of them is the validity of the discrete restriction conjecture which implies the full range of expected l p strichartz estimates for both the rational and up to n losses the irrational torus another one is an improvement in the range for the discrete restriction theorem for lattice points on the sphere various applications to additive combinatorics incidence geometry and number theory are also discussed our argument relies on the interplay between linear and multilinear restriction theory

construction of units in integral group rings of finite nilpotent groups let u be the group of units of the integral group ring of a finite group g we give a set of generators of a subgroup b of u this subgroup is of finite index in u if g is an odd nilpotent group we also give an example of a 2 group such that b is of infinite index in u

units of p power order in principal p blocks of p constrained groups abstract let g be a finite group having a normal p subgroup n that contains its centralizer c g n and let r be a p adic ring it is shown that any finite p group of units of augmentation one in rg which normalizes n is conjugate to a subgroup of g by a unit of rg and if it centralizes n it is even contained in n

infinitely generated pseudocompact modules for finite groups and weiss theorem abstract one of the most beautiful results in the integral representation theory of finite groups is a theorem of a weiss that detects a permutation r lattice for the finite p group g in terms of the restriction to a normal subgroup n and the n fixed points of the lattice where r is a finite extension of the p adic integers using techniques from relative homological algebra we generalize weiss theorem to the class of infinitely generated pseudocompact lattices for a finite p group allowing r to be any complete discrete valuation ring in mixed characteristic a related theorem of cliff and weiss is also generalized to this class of modules the existence of the permutation cover of a pseudocompact module is proved as a special case of a more general result the permutation cover is explicitly described

isomorphisms of p adic group rings a long standing problem first posed by graham higman 15 and later by brauer 4 is the isomorphism problem for integral group rings given finite groups g and h is it true that zg zh implies g 2 h many authors have worked on this question but progress has been difficult 30 perhaps the best positive result was that of whitcomb in 1968 37 who showed that the implication g h holds for g metabelian dade 9 showed there were counterexamples even in the metabelian case if z were replaced by the family of all fields some mathematicians came to believe the problem was a kind of grammatical accident that counterexamples for z were surely there if difficult to find we ourselves began this investigation looking for counterexamples but found that they were indeed very difficult to find much more difficult than we had anticipated slowly we began to believe that at least some exciting positive results were possible in this paper we answer the isomorphism problem for finite p groups over the p adic integers z zcp and in a very strong way in the normalized units augmentation 1 of z pg there is only one con gay czu ss of groups of order g 1 this answers the isomorphism problem in the affirmative for finite p groups g over z or zp and in addition computes the entire picard group 2 of the category of z g modules in terms of automorphisms of g similarly we are able to settle the isomorphism problem for finite nilpotent groups and compute the associated semilocal picard groups we also treat more general coefficient domains namely integral domains s of characteristic 0 in which no rational prime divisor of the group order is invertible for the sg isomorphism problem and treat similar semilocal dedekind domains for the picard group computations the zassenhaus conjecture concerning the rational behavior of group ring automorphisms is verified for the nilpotent case in this general setting cf corollary 3 below over z we announce a positive answer to the isomorphism

origins of anomalous transport in heterogeneous media structural and dynamic controls anomalous or non fickian transport is ubiquitous in the context of tracer migration in geological formations we quantitatively identify the origin of anomalous transport in a representative model of a heterogeneous porous medium under uniform in the mean flow conditions we focus on anomalous transport which arises in the complex flow patterns of lognormally distributed hydraulic conductivity k fields with several decades of k values transport in the domains is determined by a particle tracking technique and characterized by breakthrough curves btcs the btc averaged over multiple realizations demonstrates anomalous transport in all cases which is accounted for entirely by a power law distribution t 1 of local transition times the latter is contained in the probability density function t of transition times embedded in the framework of a continuous time random walk ctrw a unique feature of our analysis is the derivation of t as a function of parameters quantifying the heterogeneity of the domain in this context we first establish the dominance of preferential pathways across each domain and characterize the statistics of these pathways by forming a particle visitation weighted histogram hw k of the hydraulic conductivity by converting the ln k dependence of hw k into time we demonstrate the equivalence of hw k and t and delineate the region of hw k that forms the power law of t this thus defines the origin of anomalous transport analysis of the preferential pathways clearly demonstrates the limitations of critical path analysis and percolation theory as a basis for determining the origin of anomalous transport furthermore we derive an expression defining the power law exponent in terms of the hw k parameters the equivalence between hw k and t is a remarkable result particularly given the nature of the k heterogeneity the complexity of the flow field within each realization and the statistics of the particle transitions

cliff weiss inequalities and the zassenhaus conjecture abstract let n be a nilpotent normal subgroup of the finite group g assume that u is a unit of finite order in the integral group ring z g of g which maps to the identity under the linear extension of the natural homomorphism g g n we show how a result of cliff and weiss can be used to derive linear inequalities on the partial augmentations of u and apply this to the study of the zassenhaus conjecture this conjecture states that any unit of finite order in z g is conjugate in the rational group algebra of g to an element in g

analytic besov spaces and hardy type inequalities in tube domains over symmetric cones we give various equivalent formulations to the partially open problem about l p boundedness of bergman projections in tubes over cones namely we show that such boundedness is equivalent to the duality identity between bergman spaces a p 0 a p and also to a hardy type inequality related to the wave operator we introduce analytic besov spaces in tubes over cones for which such hardy inequalities play an important role for p 2 we identify as a besov space the range of the bergman projection acting on l p and also the dual of a p 0 for the bloch space b 1 we give in addition new necessary conditions on the number of derivatives required in its deflnition

some examples of picard groups of blocks abstract we calculate examples of picard groups for 2 blocks with abelian defect groups with respect to a complete discrete valuation ring these include all blocks with abelian 2 groups of 2 rank at most three with the exception of the principal block of j 1 in particular this shows directly that all such picard groups are finite and picent the group of morita auto equivalences fixing the centre is trivial these are amongst the first calculations of this kind further we prove some general results concerning picard groups of blocks with normal defect groups as well as some other cases

on torsion subgroups in integral group rings of finite groups the object of this article are torsion subgroups of the normalized unit group v zg of the integral group ring zg of a finite group g for specific subgroups w we study the gruenberg kegel graph w it is shown that the central elements of an isolated subgroup u of a group basis h of zg are the normalized units of its centralizer ring czg u moreover nv zg u nh u if g has elementary abelian sylow 2 subgroups of order at most 8 each finite 2 subgroup of v zg is rationally conjugate to a subgroup of g finally torsion subgroups of v zg in the case when g is a minimal simple group are considered it follows that if g is a simple group which admits a non trivial partition for each prime p the p rank of a torsion subgroup of v zg is bounded by that one of g

extensions and solutions for nonlinear diffusion equations and random walks we investigate a connection between random walks and nonlinear diffusion equations within the framework proposed by einstein to explain the brownian motion we show here how to properly modify that framework in order to handle different physical scenarios we obtain solutions for nonlinear diffusion equations that emerge from the random walk approach and analyse possible connections with a generalized thermostatistics formalism finally we conclude that fractal and fractional derivatives may emerge in the context of nonlinear diffusion equations depending on the choice of distribution functions related to the spreading of systems

finite groups of matrices over group rings we investigate certain finite subgroups r of gln zf where ii is a finite nilpotent group such a group r gives rise to a z r x fl7 module we study the characters of these modules to limit the structure of r we also exhibit some exotic subgroups r