uncorrelated random networks we define a statistical ensemble of nondegenerate graphs i e graphs without multiple connections and self connections between nodes the node degree distribution is arbitrary but the nodes are assumed to be uncorrelated this completes our earlier publication phys rev 64 046118 2001 where trees and degenerate graphs were considered an efficient algorithm generating nondegenerate graphs is constructed the corresponding computer code is available on request finite size effects in scale free graphs i e those where the tail of the degree distribution falls like n beta are carefully studied we find that in the absence of dynamical internode correlations the degree distribution is cut at a degree value scaling like n gamma with gamma min 1 2 1 beta 1 where n is the total number of nodes the consequence is that independently of any specific model the internode correlations seem to be a necessary ingredient of the physics of scale free networks observed in nature

p blocks relative to a character of a normal subgroup abstract let g be a finite group let n g and let irr n be a g invariant character we fix a prime p and we introduce a canonical partition of irr g relative to p we call each member b of this partition a block and to each block b we naturally associate a conjugacy class of p subgroups of g n which we call the defect groups of b if n is trivial then the blocks are the brauer p blocks using blocks we can unify the gluck wolf navarro tiep theorem and brauer s height zero conjecture in a single statement which after work of b sambale turns out to be equivalent to the height zero conjecture we also prove that the k b conjecture is true if and only if every block b has size less than or equal the size of any of its defect groups thus bringing normal subgroups to the k b conjecture

jamming at zero temperature and zero applied stress the epitome of disorder we have studied how two and three dimensional systems made up of particles interacting with finite range repulsive potentials jam i e develop a yield stress in a disordered state at zero temperature and zero applied stress at low packing fractions phi the system is not jammed and each particle can move without impediment from its neighbors for each configuration there is a unique jamming threshold phi c at which particles can no longer avoid each other and the bulk and shear moduli simultaneously become nonzero the distribution of phi c values becomes narrower as the system size increases so that essentially all configurations jam at the same packing fraction in the thermodynamic limit this packing fraction corresponds to the previously measured value for random close packing in fact our results provide a well defined meaning for random close packing in terms of the fraction of all phase space with inherent structures that jam the jamming threshold point j occurring at zero temperature and applied stress and at the random close packing density has properties reminiscent of an ordinary critical point as point j is approached from higher packing fractions power law scaling is found for the divergence of the first peak in the pair correlation function and in the vanishing of the pressure shear modulus and excess number of overlapping neighbors moreover near point j certain quantities no longer self average suggesting the existence of a length scale that diverges at j however point j also differs from an ordinary critical point the scaling exponents do not depend on dimension but do depend on the interparticle potential finally as point j is approached from high packing fractions the density of vibrational states develops a large excess of low frequency modes indeed at point j the density of states is a constant all the way down to zero frequency all of these results suggest that point j is a point of maximal disorder and may control behavior in its vicinity perhaps even at the glass transition

geometrical implications of certain infinite dimensional decompositions we investigate the connections between the global structure of a banach space i e the existence of certain finite and infinite dimensional de compositions and the geometrical properties of the closed convex bounded sub sets of such a space i e the existence of extremal and other topologically distin guished points the global structures of various supposedly pathological examples of banach spaces constructed by r c james turn out to be more universal than expected for instance james tree type resp james matrix type decompositions characterize banach spaces with the point of continuity property resp the radon nikodym property moreover the convex point of continuity property is stable under the formation of james infinitely branching tree type sums of infinite dimensional factors we also give several counterex amples to various questions relating some topological and geometrical concepts in banach spaces

percolation critical exponents in scale free networks we study the behavior of scale free networks having connectivity distribution p k k 2l close to the percolation threshold we show that for networks with 3 l 4 known to undergo a transition at a finite threshold of dilution the critical exponents are different than the expected mean field values of regular percolation in infinite dimensions networks with 2 l 3 possess only a percolative phase nevertheless we show that in this case percolation critical exponents are well defined near the limit of extreme dilution where all sites are removed and that also then the exponents bear a strong l dependence the regular mean field values are recovered only for l 4

automorphisms of groups abstract the main purpose of the present paper is to give an explicit description of the wells map of a given group extension to the case of automorphisms acting trivially on the quotient group from this we obtain some of new necessary and sufficient conditions for an automorphism of a normal subgroup of a group to extend to the group itself with trivial action on the quotient group

brauer s height conjecture for p solvable groups we complete the proof of the height conjecture for p solvable groups using the classification of finite simple groups introduction the height conjecture is the statement that a p block of a finite group has an abelian defect group if and only if all ordinary irreducible characters in the block have height zero while a proof of this conjecture for general finite groups seems remote considerable progress has been made toward proving it for p solvable groups fong 5 proved that all characters in a block with abelian defect group have height zero in a p solvable group and he proved the converse direction for the principal block 5 and for solvable groups in the case that p is the largest prime divisor of the group order 6 recently 24 8 the converse direction has been established for all solvable groups in this paper we prove the converse direction for all p solvable groups assuming the classification of finite simple groups in its general outline this paper resembles 8 where we proved the height conjecture for solvable groups the reader is assumed to have some familiarity with 8 now we state our main results the analogs of the main results of 8 theorem a suppose that n g that g n is p solvable that t e irr n and that p 1 x i w l for all x e irr g i p then the p sylow subgroups of g n are abelian theorem b let b be a p block of a p solvable group with defect group d if every ordinary irreducible character in b has height zero then d is abelian theorem c suppose that n g that g n is p solvable and that w e irr n suppose that e is an integer such that pe l does not divide x i w l for all x e irr g ip then the derived length of a p sylow subgroup of g n is at most 2e 1 theorem d let b and d be as in theorem b if every ordinary irreducible character in b has height at most e then the derived length of d is at most 2e 1 received by the editors january 10 1983 1980 mathematics subject classification primary 20c20 20c 15 1984 american mathematical society 0002 9947 84 1 00 25 per page

characters of relative p degree over normal subgroups let z be a normal subgroup of a finite group g let irr z be an irreducible complex character of z and let p be a prime number if p does not divide the integers 1 1 for all irr g lying over then we prove that the sylow p subgroups of g z are abelian this theorem which generalizes the gluck wolf theorem to arbitrary finite groups is one of the principal obstacles to proving the celebrated brauer height zero conjecture

epidemic dynamics in finite size scale free networks many real networks present a bounded scale free behavior with a connectivity cutoff due to physical constraints or a finite network size we study epidemic dynamics in bounded scale free networks with soft and hard connectivity cutoffs the finite size effects introduced by the cutoff induce an epidemic threshold that approaches zero at increasing sizes the induced epidemic threshold is very small even at a relatively small cutoff showing that the neglection of connectivity fluctuations in bounded scale free networks leads to a strong overestimation of the epidemic threshold we provide the expression for the infection prevalence and discuss its finite size corrections the present paper shows that the highly heterogeneous nature of scale free networks does not allow the use of homogeneous approximations even for systems of a relatively small number of nodes

on block induction abstract block induction is a correspondence between certain p blocks of subgroups and blocks of the whole group it is a tool with the help of which one often can reduce the solution of problems in the whole group to smaller groups the literature lists several ways to define induced blocks in the sense of brauer alperin and burry green and wheeler it is well known that any two of these concepts of block induction coincide in their common domain of definition for example each of them is defined if the subgroup h of the group g satisfies dc g d h n g d for a p subgroup d of g we study some other features of blocks which are compatible with the above induction concepts in this sense and formulate them as induction concepts so that one could deal with these and the above concepts in a unified way we compare their fields of definition to the others in general and in the special cases of defect zero blocks and of p groups we give some sufficient conditions so that induction would not be defined in any sense we show that unlike in the case of brauer sense induction block induction in the sense of alperin burry is not always defined from the normalizer of a chain of p subgroups our example also shows that admissible induction and being of multiplicity one are not transitive in the last part of the paper we study the connection of the defect groups of the induced and the original block in the case where the latter group is abelian

on hilbert divisors of brauer characters abstract the hilbert divisor p a of an irreducible p brauer character of a finite group g carries deep information about respectively the module affording in 8 we conjectured that belongs to a p block of defect 0 if and only if its hilbert divisor is 1 in this note we continue our investigations

on the order of uniprimitive permutation groups one of the central problems of 19th century group theory was the estimation of the order of a primitive permutation group g of degree n where g x an we prove i g i exp 4v n log2 n for the case when g is not doubly transitive this result is best possible apart from an o log n factor in the exponent the best result previously known was i g i el h wielandt a similar estimate for the doubly transitive case follows in a subsequent paper for rank 3 groups with subdegree p n 2 we obtain i gi exp 4 n p log2 n in the proof we develop some new combinatorial properties of coherent configurations the results also have relevance to theoretical estimates on the computational complexity of graph isomorphism testing

organization of modular networks we examine the global organization of heterogeneous equilibrium networks consisting of a number of well distinguished interconnected parts char22 communities or modules we develop an analytical approach allowing us to obtain the statistics of connected components and the intervertex distance distribution in these modular networks and to describe their global organization and structure in particular we study the evolution of the intervertex distance distribution with an increasing number of interlinks connecting two infinitely large uncorrelated networks we demonstrate that even a relatively small number of shortcuts unite the networks into one in more precise terms if the number of interlinks is any finite fraction of the total number of connections then the intervertex distance distribution approaches a ensuremath delta function peaked form and so the network is united

heterogeneous k core versus bootstrap percolation on complex networks we introduce the heterogeneous k core which generalizes the k core and contrast it with bootstrap percolation vertices have a threshold r i that may be different at each vertex if a vertex has fewer than r i neighbors it is pruned from the network the heterogeneous k core is the subgraph remaining after no further vertices can be pruned if the thresholds r i are 1 with probability f or k ensuremath geqslant 3 with probability 1 ensuremath f the process can be thought of as a pruning process counterpart to ordinary bootstrap percolation which is an activation process we show that there are two types of transitions in this heterogeneous k core process the giant heterogeneous k core may appear with a continuous transition and there may be a second discontinuous hybrid transition we compare critical phenomena critical clusters and avalanches at the heterogeneous k core and bootstrap percolation transitions we also show that the network structure has a crucial effect on these processes with the giant heterogeneous k core appearing immediately at a finite value for any fg0 when the degree distribution tends to a power law p q q ensuremath ensuremath gamma with ensuremath gamma l3

scale free models for the structure of business firm networks we study firm collaborations in the life sciences and the information and communication technology sectors we propose an approach to characterize industrial leadership using k shell decomposition with top ranking firms in terms of market value in higher k shell layers we find that the life sciences industry network consists of three distinct components a nucleus which is a small well connected subgraph tendrils which are small subgraphs consisting of small degree nodes connected exclusively to the nucleus and a bulk body which consists of the majority of nodes industrial leaders i e the largest companies in terms of market value are in the highest k shells of both networks the nucleus of the life sciences sector is very stable once a firm enters the nucleus it is likely to stay there for a long time at the same time we do not observe the above three components in the information and communication technology sector we also conduct a systematic study of these three components in random scale free networks our results suggest that the sizes of the nucleus and the tendrils in scale free networks decrease as the exponent of the power law degree distribution lambda increases and disappear for lambda or 3 we compare the k shell structure of random scale free model networks with two real world business firm networks in the life sciences and in the information and communication technology sectors we argue that the observed behavior of the k shell structure in the two industries is consistent with the coexistence of both preferential and random agreements in the evolution of industrial networks

the radon nikodým property and the kreĭn milman property are equivalent for strongly regular sets the result announced in the title is proved as corollaries we obtain that rnp and kmp are equivalent for subsets of spaces with an unconditional basis and for k convex banach spaces we also obtain a sharpening of a result of r huff and p morris a dual space has the rnp iff all separable subspaces have the kmp

hall subgroups and fully extendible characters abstract we say that an irreducible character of a subgroup h of g is fully extendible to g if every irreducible constituent of the induced character g is actually an extension of now let n g where g n is separable for some set of primes and let h n be a hall subgroup of g n if irr h is fully extendible to g we show that a hall subgroup k n of g n must be abelian also if is an irreducible constituent of n we show that is k invariant for some hall subgroup k n of g n and furthermore is fully extendible to k conversely if irr n is extendible to k where k n is a hall subgroup of g n we show that some irreducible character of h lying over is extendible to g

clique percolation in random networks the notion of k clique percolation in random graphs is introduced where k is the size of the complete subgraphs whose large scale organizations are analytically and numerically investigated for the erdos renyi graph of n vertices we obtain that the percolation transition of k cliques takes place when the probability of two vertices being connected by an edge reaches the threshold p c k k 1 n 1 k 1 at the transition point the scaling of the giant component with n is highly nontrivial and depends on k we discuss why clique percolation is a novel and efficient approach to the identification of overlapping communities in large real networks

weak polish banach spaces abstract a banach space x which is a subspace of the dual of a banach space y is said to be weak polish proviced x is separable and the closed unit ball of x is polish in the weak topology x is said to be polish provided it is weak polish when regarded as a subspace of x a variety of structural results are obtained for banach spaces with the point of continuity property pcp using the unifying concept of weak polish banach spaces it is proved that every weak polish banach space contains an infinite dimensional weak closed subspace this yields the result of edgar wheeler that every polish banach space has a reflexive subspace it also yields the result of ghoussoub maurey that every banach space with the pcp has a subspace isomorphic to a separable dual in virtue of the following known result linking the pcp and weak polish banach spaces a separable banach space has the pcp if edgar wheeler and only if ghoussoub maurey it is isometric to a weak polish subspace of the dual of some separable banach space for completeness a proof of this result is also given it is proved that if x and y are banach spaces with x y then x is weak polish if and only if x has a weak continuous boundedly complete skipped blocking decomposition combined with the above mentioned linkage this characterization of weak polish banach spaces gives the known result that a separable banach space has the pcp if bourgain rosenthal and only if ghoussoub maurey it has the boundedly complete skipped blocking property the proof of the characterization is a variation of the original arguments for this result the characterization also yields that a banach space is polish if and only if it has the reflexive skipped blocking property it is moreover demonstrated that a banach space has the pcp provided it has the pc skipped blocking property in particular this yields an exposition of the result of bourgain that a space has the pcp provided each of its subspaces with an fdd has the pcp it is proved that the pcp is a three space property thus completing the solution to a problem posed by edgar wheeler that is if y and x are banach spaces with y x then x has the pcp provided both y and x y have the pcp it is also shown that a banach space x has a separable dual provided it has the dual separable skipped blocking property the proof yields the dividend that x is separable provided x j is separable for every basic sequence x j in x

bootstrap percolation on complex networks we consider bootstrap percolation on uncorrelated complex networks we obtain the phase diagram for this process with respect to two parameters f the fraction of vertices initially activated and p the fraction of undamaged vertices in the graph we observe two transitions the giant active component appears continuously at a first threshold there may also be a second discontinuous hybrid transition at a higher threshold avalanches of activations increase in size as this second critical point is approached finally diverging at this threshold we describe the existence of a special critical point at which this second transition first appears in networks with degree distributions whose second moment diverges but whose first moment does not we find a qualitatively different behavior in this case the giant active component appears for any fg0 and pg0 and the discontinuous transition is absent this means that the giant active component is robust to damage and also is very easily activated we also formulate a generalized bootstrap process in which each vertex can have an arbitrary threshold

quasi isolated blocks and brauer s height zero conjecture this paper has two main results firstly we complete the parametrisation of all p blocks of nite quasi simple groups by nding the so called quasi isolated blocks of exceptional groups of lie type for bad primes this relies on the explicit decomposition of lusztig induction from suitable levi subgroups our second major result is the proof of one direction of brauer s long standing height zero conjecture on blocks of nite groups using the

arithmetic results on orbits of linear groups the first four authors acknowledge the support of an arc discovery grant the first author acknowledges the support of an australian research fellowship the third author acknowledges the support of a federation fellowship the fifth author acknowledges the support of the nsf grants dms 0901241 and dms 1201374

first order rigidity on cayley trees tree models for rigidity percolation in systems with only central forces are introduced and solved a probability vector describes the propagation of rigidity outward from a rigid border all components of this vector order parameter are singular at the same rigidity threshold mathrm p mathrm c the infinite cluster probability mathrm p mathrm ensuremath infty is usually first order at mathrm p mathrm c except in those cases which are equivalent to connectivity percolation in many cases mathrm p mathrm ensuremath infty ensuremath sim ensuremath delta mathrm p mathrm ensuremath infty p mathrm p mathrm c 1 mathrm 2 indicating critical fluctuations superimposed on the first order jump ensuremath delta mathrm p mathrm ensuremath infty our tree models for rigidity are in qualitative disagreement with constraint counting mean field theories in an important subclass of tree models bootstrap percolation and rigidity percolation are equivalent

perturbing general uncorrelated networks this paper is a direct continuation of an earlier work where we studied erd os r enyi random graphs perturbed by an interaction hamiltonian favouring the formation of short cycles here we generalize these results we keep the same interaction hamiltonian but let it act on general graphs with uncorrelated nodes and an arbitrary given degree distribution it is shown that the results obtained for erd os r enyi graphs are generic at the qualitative level however scale free graphs are an exception to this general rule and exhibit a singular behaviour studied thoroughly in this paper both analytically and numerically

infection dynamics on scale free networks we discuss properties of infection processes on scale free networks relating them to the node connectivity distribution that characterizes the network considering the epidemiologically important case of a disease that confers permanent immunity upon recovery we derive analytic expressions for the final size of an epidemic in an infinite closed population and for the dependence of infection probability on an individual s degree of connectivity within the population as in an earlier study r pastor satorras and a vesipignani phys rev lett 86 3200 2001 phys rev e 63 006117 2001 for an infection that did not confer immunity upon recovery the epidemic process in contrast with many traditional epidemiological models does not exhibit threshold behavior and we demonstrate that this is a consequence of the extreme heterogeneity in the connectivity distribution of a scale free network finally we discuss effects that arise from finite population sizes showing that networks of finite size do exhibit threshold effects infections cannot spread for arbitrarily low transmission probabilities

k core architecture and k core percolation on complex networks abstract we analytically describe the architecture of uncorrelated complex networks with arbitrary degree distributions as a set of successively enclosed substructures k cores we find the structure of k cores their sizes and their birth points the bootstrap percolation thresholds we show that in networks with a finite mean number z 2 of the second nearest neighbours the emergence of a k core is a unique hybrid phase transition with a jump emergence of the k core as at a first order phase transition but also with a critical singularity as at a continuous phase transition in contrast if z 2 diverges the networks contain an infinite sequence of k cores which are ultra robust against random damage we explain the meaning of the order parameter for k core percolation and reveal the origin of the specific critical phenomena we demonstrate that a so called corona of the k core plays a crucial role it turns out that the k core percolation threshold is at the same time the percolation threshold of finite corona clusters the mean separation of vertices in corona clusters plays the role of the correlation length and diverges at the critical point we demonstrate that a random removal of even one vertex from the k core may result in the collapse of a vast region of the k core around the removed vertex the mean size of this region diverges at the critical point we find an exact mapping of k core percolation to a model of cooperative relaxation this model undergoes critical relaxation with a divergent rate at some critical moment

weight conjectures for fusion systems abstract many of the conjectures of current interest in the representation theory of finite groups in characteristic p are local to global statements in that they predict consequences for the representations of a finite group g given data about the representations of the p local subgroups of g the local structure of a block of a group algebra is encoded in the fusion system of the block together with a compatible family of kulshammer puig cohomology classes motivated by conjectures in block theory we state and initiate investigation of a number of seemingly local conjectures for arbitrary triples s f consisting of a saturated fusion system f on a finite p group s and a compatible family

on the structure of non dentable closed bounded convex sets a self contained proof is given of the following result theorem let k be a non dentable closed bounded convex nonempty subset of a banach space x so that k equals the closed convex hull of its weak to norm points of continuity there exists a nonempty subset w of k satisfying w is non dentable closed convex and the weak and norm topologies on w coincide moreover there exists a closed linear subspace y of x so that y has a finite dimensional decomposition and a nonempty convex subset w satisfying any bounded set w satisfying has no extreme points this yields the recent discovery of w schachermayer that a closed bounded convex subset of a banach space has the radon nikod m property rnp provided it has the convex point of continity property cpcp and the krein milman property kmp and the cpcp case of the earlier discovery of j bourgain that a banach space has the rnp provided every subspace with a finite dimensional decomposition has the rnp the proof is a variation and crystallization of the original arguments for these discoveries in particular the proof uses the concept of convex sets with small combinations of slices this is a refinement of the concept of strong regularity introduced by n ghoussoub g godefroy and b maurey both concepts crystallize ideas appearing in bourgain s work the proof also yields further results concerning the structure of non dentable convex sets

k core bootstrap percolation on complex networks critical phenomena and nonlocal effects we develop the theory of the k core bootstrap percolation on uncorrelated random networks with arbitrary degree distributions we show that the k core percolation is an unusual hybrid phase transition with a jump emergence of the k core as at a first order phase transition but also with a critical singularity as at a continuous transition we describe the properties of the k core explain the meaning of the order parameter for the k core percolation and reveal the origin of the specific critical phenomena we demonstrate that a so called corona of the k core plays a crucial role corona is a subset of vertices in the k core which have exactly k neighbors in the k core it turns out that the k core percolation threshold is at the same time the percolation threshold of finite corona clusters the mean separation of vertices in corona clusters plays the role of the correlation length and diverges at the critical point we show that a random removal of even one vertex from the k core may result in the collapse of a vast region of the k core around the removed vertex the mean size of this region diverges at the critical point we find an exact mapping of the k core percolation to a model of cooperative relaxation this model undergoes critical relaxation with a divergent rate at some critical moment

iterative decomposition of barabasi albert scale free networks we study a decomposition process where all nodes with a targeted degree are removed from the network each removal step results in changes in the degrees of the remaining nodes and other nodes may attain the targeted degree the processes continue iteratively until no more nodes with the targeted degree are present in the decomposed network the network model used in our study is the well known barabasi albert network that is built with an iterative growth based on preferential attachment our results show an exponential decay of the number of nodes removed at each step the total number of nodes removed in the whole process depends on the targeted degree and decay with a power law controlled by the same exponent as the degree distribution of the network

giant strongly connected component of directed networks we describe how to calculate the sizes of all giant connected components of a directed graph including the strongly connected one in particular the world wide web is a directed network the results are obtained for graphs with statistically uncorrelated vertices and an arbitrary joint in and out degree distribution p k i k o we show that if p k i k o does not factorize the relative size of the giant strongly connected component deviates from the product of the relative sizes of the giant in and out components the calculations of the relative sizes of all the giant components are demonstrated using the simplest examples we explain that the giant strongly connected component may be less resilient to random damage than the giant weakly connected one

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

on stegall s smooth variational principle abstract assume that every coercive continuous and bounded below function defined on a real banach space attains after a suitable linear perturbation a not necessarily strong minimum then the space is dentable