the regular spaces with countably based models the regular spaces which may be realized as the set of maximal elements in an continuous dcpo are the polish spaces in addition we give a new and conceptually simple model for complete metric spaces these results enable us to prove that the probabilistic powerdomain of a countably based model of a metric space always contains a copy of the normalized borel measures in their weak topology and to establish the hierarchy for countably based models

average path length in random networks analytic solution for the average path length in a large class of uncorrelated random networks with hidden variables is found we apply the approach to classical random graphs of erdos and renyi er evolving networks introduced by barabasi and albert as well as random networks with asymptotic scale free connectivity distributions characterized by an arbitrary scaling exponent alpha 2 our result for 2 infinity there is a saturation effect for the average path length

generation of complex bipartite graphs by using a preferential rewiring process it is important in computer science sociology and so on to investigate complex bipartite graphs from a viewpoint of statistical physics we propose a model to generate complex bipartite graphs without growing the bipartite graphs are assumed to have two sets of the fixed numbers of nodes and a fixed number of edges between nodes belonging to different sets of nodes in this model essential ingredients are a preferential rewiring process and a fitness distribution function by using the preferential rewiring process we confirm that a bipartite graph reaches a stationary state after a sufficiently long time has passed we find that the obtained bipartite graph has a scale free like property when a suitable fitness distribution is used it turns out that a condensation of edges takes place in the cases of certain fitness distributions

polish group actions dichotomies and generalized elementary embeddings the results in this paper involve two different topics in the descriptive theory of polish group actions the book becker kechris 6 is an introduction to that theory our two topics and two collections of theorems are rather unrelated but the proofs for both topics are essentially the same locally compact polish groups i e second countable locally compact groups are the traditional objects of study in the field known as topological groups more recently there has been interest in nonlocally compact polish groups such as s the symmetric group on n topologized as a subspace of nn much of this paper is concerned with a proper subclass of the class of polish groups namely those polish groups which admit a complete left invariant metric we call these cli groups the class of cli groups includes all locally compact groups proposition 3 c 2 d but also much more for example all solvable groups are cli hjorth solecki see proposition 3 c 2 f and a closed subgroup g of s is cli iff g is closed in nn see example 3 c 3 a we study continuous actions by polish groups on polish spaces and the associated orbit equivalence relation for locally compact groups all of our results are known or trivial or both but some of our results can be viewed as generalizations of known theorems about locally compact groups to the larger class of cli groups one of the two topics considered in this paper is dichotomy theorems for equivalence relations these are theorems which assert that either the quotient space is small or else it contains a copy of a specific large set we prove some dichotomy theorems for the orbit space of actions by cli groups there are two types of dichotomies which we call the silver vaught dichotomy and the glimm effros dichotomy the silver vaught dichotomy asserts that either there are only countably many equivalence classes or else there is a perfect set of pairwise inequivalent elements assuming the negation of the continuum hypothesis the silver vaught dichotomy is equivalent to the proposition that there are either countably many equivalence classes or 2 0 equivalence classes the topological vaught conjecture which is open is for any continuous action by a polish group g on a polish space x the orbit equivalence relation satisfies the silver vaught dichotomy this conjecture implies and was motivated by the vaught conjecture in model theory indeed

the average distances in random graphs with given expected degrees abstract random graph theory is used to examine the small world phenomenon any two strangers are connected through a short chain of mutual acquaintances we will show that for certain families of random graphs with given expected degrees the average distance is almost surely of order log n log d where d is the weighted average of the sum of squares of the expected degrees of particular interest are power law random graphs in which the number of vertices of degree k is proportional to 1 k for some fixed exponent for the case of 3 we prove that the average distance of the power law graphs is almost surely of order log n log d however many internet social and citation networks are power law graphs with exponents in the range 2 3 for which the power law random graphs have average distance almost surely of order log log n but have diameter of order log n provided having some mild constraints for the average distance and maximum degree in particular these graphs contain a dense subgraph which we call the core having nc log log n vertices almost all vertices are within distance log log n of the core although there are vertices at distance log n from the core

spatial low discrepancy sequences spherical cone discrepancy and applications in financial modeling in this paper we introduce a reproducing kernel hilbert space defined on r d 1 as the tensor product of a reproducing kernel defined on the unit sphere s d in r d 1 and a reproducing kernel defined on 0 we extend stolarsky s invariance principle to this case and prove upper and lower bounds for numerical integration in the corresponding reproducing kernel hilbert space the idea of separating the direction from the distance from the origin can also be applied to the construction of quadrature methods an extension of the area preserving lambert transform is used to generate points on s d 1 via lifting sobol points in 0 1 d to the sphere the d th component of each sobol point suitably transformed provides the distance information so that the resulting point set is normally distributed in r d numerical tests provide evidence of the usefulness of constructing quasi monte carlo type methods for integration in such spaces we also test this method on examples from financial applications option pricing problems and compare the results with traditional methods for numerical integration in r d

shortest path discovery of complex networks in this rapid communication we present an analytic study of sampled networks in the case of some important shortest path sampling models we present analytic formulas for the probability of edge discovery in the case of an evolving and a static network model we also show that the number of discovered edges in a finite network scales much more slowly than predicted by earlier mean field models finally we calculate the degree distribution of sampled networks and we demonstrate that they are analogous to a destroyed network obtained by randomly removing edges from the original network

a generalized discrepancy and quadrature error bound an error bound for multidimensional quadrature is derived that includes the koksma hlawka inequality as a special case this error bound takes the form of a product of two terms one term which depends only on the integrand is defined as a generalized variation the other term which depends only on the quadrature rule is defined as a generalized discrepancy the generalized discrepancy is a figure of merit for quadrature rules and includes as special cases the l p star discrepancy and p that arises in the study of lattice rules

on the number of permutatively inequivalent basic sequences in a banach space let x be a banach space with a schauder basis en n n the relation e0 is borel reducible to permutative equivalence between normalized block sequences of en n n or x is c0 or lp saturated for some 1 p if en n n is shrinking unconditional then either it is equivalent to the canonical basis of c0 or lp 1 p or the relation e0 is borel reducible to permutative equivalence between sequences of normalized disjoint blocks of x or of x if en n n is unconditional then either x is isomorphic to l2 or x contains 2 subspaces or 2 quotients which are spanned by pairwise permutatively inequivalent normalized unconditional bases

a glimm effros dichotomy for borel equivalence relations a basic dichotomy concerning the structure of the orbit space of a transformation group has been discovered by glimm g12 in the locally compact group action case and extended by effros e 1 e2 in the polish group action case when additionally the induced equivalence relation is f it is the purpose of this paper to extend the glimm effros dichotomy to the very general context of an arbitrary borel equivalence relation not even necessarily induced by a group action despite the totally classical descriptive set theoretic nature of our result our proof requires the employment of methods of effective descriptive set theory and thus ultimately makes crucial use of computability or recursion theory on the integers

ideal models of spaces ideal domains have an elementary order theoretic structure every element is either compact or maximal despite this we establish that 1 they can model any space currently known to possess a countably based model and 2 the metric spaces with ideal models are exactly the completely metrizable spaces

turbulence orbit equivalence and the classification of nuclear c algebras we bound the borel cardinality of the isomorphism relation for nuclear simple separable c algebras it is turbulent yet borel reducible to the action of the automorphism group of the cuntz algebra o2 on its closed subsets the same bounds are obtained for affine homeomorphism of metrizable choquet simplexes as a by product we recover a result of kechris and solecki namely that homeomorphism of compacta in the hilbert cube is borel reducible to a polish group action these results depend intimately on the classification theory of nuclear simple c algebras by k theory and traces both of necessity and in order to lay the groundwork for further study on the borel complexity of c algebras we prove that many standard c algebra constructions and relations are borel and we prove borel versions of kirchberg s o2 stability and embedding theorems we also find a c algebraic witness for a k hard equivalence relation the authors dedicate this article to the memory of greg hjorth

the smooth ideal we give a classical proof of the generalization of the characterization of smoothness to quotients of polish spaces by borel equivalence relations as an application we describe the extent to which any given borel equivalence relation on a polish space is encoded by the corresponding ideal generated by the family of borel sets on which it is smooth

quasi monte carlo methods in financial engineering an equivalence principle and dimension reduction quasi monte carlo qmc methods are playing an increasingly important role in the pricing of complex financial derivatives for models in which the prices of the underlying assets are driven by brownian motions the performance of qmc methods is known to depend crucially on the construction of brownian motions this paper focuses on the impact of various constructions although the brownian bridge bb construction often yields very good results as papageorgiou pointed out there are financial derivatives for which the bb construction performs badly papageorgiou a 2002 the brownian bridge does not offer a consistent advantage in quasi monte carlo integration j complexity18 1 171 186 in this paper we first extend papageorgiou s analysis to establish an equivalence principle if the bb construction or any other construction is the preferred construction for a particular financial derivative then for any other construction there is another financial derivative for which the latter construction is the preferred one in this sense all methods of construction are equivalent and no method is consistently superior to others it all depends on the particular financial derivative we then show how to find a good construction for a particular class of financial derivatives in practice our strategy is to find a good construction for an easy problem and then apply it to more complicated problems related to the easy one this strategy is applied to the arithmetic asian options including bermudan asian options based on the weighted average of the stock prices we do this by studying a simpler problem namely the geometric asian option for which the best construction is easily available and applying it to the arithmetic asian option numerical experiments confirm the success of this strategy whereas in qmc all the common methods the standard method bb and principal component analysis may lose their power in some situations the new method behaves very well in all cases further large variance reduction can be achieved in combination with a control variate the new method can be interpreted as a practical way of reducing the effective dimension for some class of functions

essential countability of treeable equivalence relations we establish a dichotomy theorem characterizing the circumstances under which a treeable borel equivalence relation e is essentially countable under additional topological assumptions on the treeing we in fact show that e is essentially countable if and only if there is no continuous embedding of e1 into e our techniques also yield the first classical proof of the analogous result for hypersmooth equivalence relations and allow us to show that up to continuous kakutani embeddability there is a minimum borel function which is not essentially countable to one

scaling of average sending time on weighted koch networks random walks on weighted complex networks especially scale free networks have attracted considerable interest in the past but the efficiency of a hub sending information on scale free small world networks has been addressed less in this paper we study random walks on a class of weighted koch networks with scaling factor 0 r 1 we derive some basic properties for random walks on the weighted koch networks based on which we calculate analytically the average sending time ast defined as the average of mean first passage times mfpts from a hub node to all other nodes excluding the hub itself the obtained result displays that for 0 r 1 in large networks the ast grows as a power law function of the network order with the exponent represented by log43r 1r and for r 1 in large networks the ast grows with network order as n ln n which is larger than the linear scaling of the average receiving time defined as the average of mfpts for random walks to a given hub node averaged over all starti

on the concept of largeness in group theory abstract in this paper we will consider the borel complexity of pride s quasi order p and pride s equivalence relation p on the space g of finitely generated groups our main results show that these relations are as complex as they conceivably could be

scaling of average weighted shortest path and average receiving time on weighted hierarchical networks abstract recent work on the networks has focused on the weighted hierarchical networks that are significantly different from the un weighted hierarchical networks in this paper we study a family of weighted hierarchical networks which are recursively defined from an initial uncompleted graph in which the weights of edges have been assigned to different values with certain scale firstly we study analytically the average weighted shortest path awsp on the weighted hierarchical networks using a recursive method we determine explicitly the awsp the obtained rigorous solution shows that the networks grow unbounded but with the logarithm of the network size while the weighted shortest paths stay bounded then depending on a biased random walk we research the mean first passage time mfpt between a hub node and any peripheral node finally we deduce the analytical expression of the average of mfpts for a random walker originating from any node to first visit to a hub node which is named as the average receiving time art the obtained result shows that art is bounded or grows sublinearly with the network order relating to the number of initial nodes and the weighted factor or grows quadratically with the iteration

complete analytic equivalence relations we prove that various concrete analytic equivalence relations arising in model theory or analysis are complete i e maximum in the borel reducibility ordering the proofs use some general results concerning the wider class of analytic quasi orders

the classification problem for torsion free abelian groups of finite rank in 1937 baer 5 introduced the notion of the type of an element in a torsion free abelian group and showed that this notion provided a complete invariant for the classification problem for torsion free abelian groups of rank 1 since then despite the efforts of such mathematicians as kurosh 23 and malcev 25 no satisfactory system of complete invariants has been found for the torsion free abelian groups of finite rank n 2 so it is natural to ask whether the classification problem is genuinely more difficult for the groups of rank n 2 of course if we wish to show that the classification problem for the groups of rank n 2 is intractible it is not enough merely to prove that there are 2 such groups up to isomorphism for there are 2 pairwise nonisomorphic groups of rank 1 and we have already pointed out that baer has given a satisfactory classification for this class of groups in this paper following friedman stanley 11 and hjorth kechris 15 we shall use the more sensitive notions of descriptive set theory to measure the complexity of the classification problem for the groups of rank n 2 recall that up to isomorphism the torsion free abelian groups of rank n are exactly the additive subgroups of the n dimensional vector space qf which contain n linearly independent elements thus the collection of torsion free abelian groups of rank 1 r n can be naturally identified with the set s qn of all nontrivial additive subgroups of qn notice that s qn is a borel subset of the polish space p qn of all subsets of qn and hence s qn can be regarded as a standard borel space i e a polish space equipped with its associated o algebra of borel subsets here we are identifying p qn with the space 2 qn of all functions h qn 0 1 equipped with the product topology furthermore the natural action of gln q on the vector space qn induces a corresponding borel action on s qn and it is easily checked that if a b c s qn then a v b iff there exists an element f e gln q such that p a b it follows that the isomorphism relation on s qn is a countable borel equivalence relation if x is a standard borel space then a borel equivalence relation on x is an equivalence relation e c x2 which is a borel subset of x2 the borel equivalence relation e is said to be countable iff every e equivalence class is countable

good path generation methods in quasi monte carlo for pricing financial derivatives the quasi monte carlo qmc method is a powerful numerical tool for pricing complex derivative securities whose accuracy is affected by the smoothness of the integrands the payoff functions of many financial derivatives involve two types of nonsmooth factors an indicator function called jump structure and a positive part of a smooth function called kink structure this paper develops a good path generation method pgm for recovering the superiority of the qmc method on problems involving multiple such structures this is achieved by realigning these structures such that the associated nonsmooth surfaces are parallel to as many coordinate axes as possible the proposed method has the advantage of addressing different structures according to their importance we also offer a systematic measurement of different structures for quantifying and then ranking their importance numerical experiments demonstrate that the proposed method is more efficient than traditional pgms for pricing exotic options suc

trapping in scale free networks with hierarchical organization of modularity a wide variety of real life networks share two remarkable generic topological properties scale free behavior and modular organization and it is natural and important to study how these two features affect the dynamical processes taking place on such networks in this paper we investigate a simple stochastic process trapping problem a random walk with a perfect trap fixed at a given location performed on a family of hierarchical networks that exhibit simultaneously striking scale free and modular structure we focus on a particular case with the immobile trap positioned at the hub node having the largest degree using a method based on generating functions we determine explicitly the mean first passage time mfpt for the trapping problem which is the mean of the node to trap first passage time over the entire network the exact expression for the mfpt is calculated through the recurrence relations derived from the special construction of the hierarchical networks the obtained rigorous formula corroborated by extensive direct numerical calculations exhibits that the mfpt grows algebraically with the network order concretely the mfpt increases as a power law function of the number of nodes with the exponent much less than 1 we demonstrate that the hierarchical networks under consideration have more efficient structure for transport by diffusion in contrast with other analytically soluble media including some previously studied scale free networks we argue that the scale free and modular topologies are responsible for the high efficiency of the trapping process on the hierarchical networks

deterministic scale free small world networks of arbitrary order abstract in many real life networks both the scale free distribution of degree and small world behavior are important features there are many random or deterministic models of networks to simulate these features separately however there are few models that combine the scale free effect and small world behavior especially in terms of deterministic versions what is more all the existing deterministic algorithms running in the iterative mode generate networks with only several discrete numbers of nodes this contradicts the purpose of creating a deterministic network model on which we can simulate some dynamical processes as widely as possible according to these facts this paper proposes a deterministic network generation algorithm which can not only generate deterministic networks following a scale free distribution of degree and small world behavior but also produce networks with arbitrary number of nodes our scheme is based on a complete binary tree and each newly generated leaf node is further linked to its full brother and one of its direct ancestors analytical computation and simulation results show that the average degree of such a proposed network is less than 5 the average clustering coefficient is high larger than 0 5 even for a network of size 2 million and the average shortest path length increases much more slowly than logarithmic growth for the majority of small world network models

random walks on weighted networks random walks constitute a fundamental mechanism for a large set of dynamics taking place on networks in this article we study random walks on weighted networks with an arbitrary degree distribution where the weight of an edge between two nodes has a tunable parameter by using the spectral graph theory we derive analytical expressions for the stationary distribution mean first passage time mfpt average trapping time att and lower bound of the att which is defined as the average mfpt to a given node over every starting point chosen from the stationary distribution all these results depend on the weight parameter indicating a significant role of network weights on random walks for the case of uncorrelated networks we provide explicit formulas for the stationary distribution as well as att particularly for uncorrelated scale free networks when the target is placed on a node with the highest degree we show that att can display various scalings of network size depending also on the same parameter our findings could pave a way to delicately controlling random walk dynamics on complex networks

turbulence representations and trace preserving actions we establish criteria for turbulence in certain spaces of c algebra repre sentations and apply this to the problem of nonclassiability by countable structures for group actions on a standard atomless probability space x and on the hypernite ii1 factor r we also prove that the conjugacy action on the space of free actions of a countably innite amenable group on r is turbulent and that the conjugacy action on the space of ergodic measure preserving ows on x is generically turbulent

growing network with local rules preferential attachment clustering hierarchy and degree correlations the linear preferential attachment hypothesis has been shown to be quite successful in explaining the existence of networks with power law degree distributions it is then quite important to determine if this mechanism is the consequence of a general principle based on local rules in this work it is claimed that an effective linear preferential attachment is the natural outcome of growing network models based on local rules it is also shown that the local models offer an explanation for other properties like the clustering hierarchy and degree correlations recently observed in complex networks these conclusions are based on both analytical and numerical results for different local rules including some models already proposed in the literature

effect of node deleting on network structure abstract the ever increasing knowledge of the structure of various real world networks has uncovered their complex multi mechanism governed evolution processes therefore a better understanding of the structure and evolution of these networked complex systems requires us to describe such processes in a more detailed and realistic manner in this paper we introduce a new type of network growth rule which comprises addition and deletion of nodes and propose an evolving network model to investigate the effect of node deleting on network structure it is found that with the introduction of node deleting network structure is significantly transformed in particular degree distribution of the network undergoes a transition from scale free to exponential forms as the intensity of node deleting increases at the same time nontrivial disassortative degree correlation develops spontaneously as a natural result of network evolution in the model we also demonstrate that node deleting introduced in the model does not destroy the connectedness of a growing network so long as the increasing rate of edges is not excessively small in addition it is found that node deleting will weaken but not eliminate the small world effect of a growing network and generally it will decrease the clustering coefficient in a network

local rewiring rules for evolving complex networks abstract the effects of link rewiring are considered for the class of directed networks where each node has the same fixed out degree we model a network generated by three mechanisms that are present in various networked systems growth global rewiring and local rewiring during a rewiring phase a node is randomly selected one of its out going edges is detached from its destination then re attached to the network in one of two possible ways either globally to a randomly selected node or locally to a descendant of a descendant of the originally selected node although the probability of attachment to a node increases with its connectivity the probability of detachment also increases the result is an exponential degree distribution with a small number of outlying nodes that have extremely large degree we explain these outliers by identifying the circumstances for which a set of nodes can grow to very high degree

countable abelian group actions and hyperfinite equivalence relations an equivalence relation e on a standard borel space is hyperfinite if e is the increasing union of countably many borel equivalence relations e n where all e n equivalence classs are finite in this article we establish the following theorem if a countable abelian group acts on a standard borel space in a borel manner then the orbit equivalence relation is hyperfinite the proof uses constructions and analysis of borel marker sets and regions in the space 2 mathbb z omega this technique is also applied to a problem of finding borel chromatic numbers for invariant borel subspaces of 2 mathbb z n

qmc designs optimal order quasi monte carlo integration schemes on the sphere we study equal weight numerical integration or quasi monte carlo qmc rules for functions in a sobolev space hs sd with smoothness parameter s d 2 defined over the unit sphere sd in rd 1 focusing on n point configurations that achieve optimal order qmc error bounds as is the case for efficient spherical designs we are led to introduce the concept of qmc designs these are sequences of n point configurations xn on s d such that the worst case error satisfies sup f h s f hs 1 1 n x xn f x sd f x d d x o n s d n with an implied constant that depends on the hs sd norm but is independent of n here d is the normalized surface measure on s d we provide methods for generation and numerical testing of qmc designs an essential tool is an expression for the worst case error in terms of a reproducing kernel for the space hs sd with s d 2 as a consequence of this and a recent result of bondarenko et al on the existence of spherical designs with appropriate number of points we show that minimizers of the n point energy for this kernel form a sequence of qmc designs for hs sd furthermore without appealing to the bondarenko et al result we prove that point sets that maximize the sum of suitable powers of the euclidean distance between pairs of points form a sequence of qmc designs for hs sd with s in the interval d 2 d 2 1 for such spaces there exist reproducing kernels with simple closed forms that are useful for numerical testing of optimal order quasi monte carlo integration numerical experiments suggest that many familiar sequences of point sets on the sphere equal area points spiral points minimal coulomb or logarithmic energy points and fekete points are qmc designs for appropriate values of s for comparison purposes we show that configurations of random points that are independently and uniformly distributed on the sphere do not constitute qmc designs for any s d 2 if xn is a sequence of qmc designs for h s sd we prove that it is also a sequence of qmc designs for hs sd for all s d 2 s this leads to the question of determining the supremum of such s here called the qmc strength of the sequence for which we provide estimates based on computations for the aforementioned sequences received by the editor august 15 2012 and in revised form february 26 2013 2010 mathematics subject classification primary 65d30 65d32 secondary 11k38 41a55

the classification of hypersmooth borel equivalence relations this paper is a contribution to the study of borel equivalence relations in standard borel spaces i e polish spaces equipped with their borel structure a class of such equivalence relations which has received particular attention is the class of hyperfinite borel equivalence relations these can be defined as the increasing unions of sequences of borel equivalence relations all of whose equivalence classes are finite or as it turns out equivalently those induced by the orbits of a single borel auto morphism hyperfinite equivalence relations have been classified in djk under two notions of equivalence borel bi reducibility and borel isomorphism

a dichotomy characterizing analytic digraphs of uncountable borel chromatic number in any dimension we study the extension of the kechris solecki todorcevic dichotomy on analytic graphs to dimensions higher than 2 we prove that the extension is possible in any dimension finite or infinite the original proof works in the case of the finite dimension we first prove that the natural extension does not work in the case of the infinite dimension for the notion of continuous homomorphism used in the original theorem then we solve the problem in the case of the infinite dimension finally we prove that the natural extension works in the case of the infinite dimension but for the notion of baire measurable homomorphism

non classification of cartan subalgebras for a class of von neumann algebras abstract we study the complexity of the classification problem for cartan subalgebras in von neumann algebras we construct a large family of ii1 factors whose cartan subalgebras up to unitary conjugacy are not classifiable by countable structures providing the first such examples additionally we construct examples of ii1 factors whose cartan subalgebras up to conjugacy by an automorphism are not classifiable by countable structures finally we show directly that the cartan subalgebras of the hyperfinite ii1 factor up to unitary conjugacy are not classifiable by countable structures and deduce that the same holds for any mcduff ii1 factor with at least one cartan subalgebra

the classification problem for s local torsion free abelian groups of finite rank abstract suppose that n 2 and that s t are sets of primes then the classification problem for the s local torsion free abelian groups of rank n is borel reducible to the classification problem for the t local torsion free abelian groups of rank n if and only if s t

vaught s conjecture and the glimm effros property for polish transformation groups we extend the original glimm effros theorem for locally compact groups to a class of polish groups including the nilpotent ones and those with an invariant metric for this class we thereby obtain the topological vaught conjecture

organization of growing random networks the organizational development of growing random networks is investigated these growing networks are built by adding nodes successively and linking each to an earlier node of degree k with an attachment probability a k when a k grows more slowly than linearly with k the number of nodes with k links n k t decays faster than a power law in k while for a k growing faster than linearly in k a single node emerges which connects to nearly all other nodes when a k is asymptotically linear n k t ensuremath sim mathrm tk ensuremath ensuremath nu with ensuremath nu dependent on details of the attachment probability but in the range 2l ensuremath nu l ensuremath infty the combined age and degree distribution of nodes shows that old nodes typically have a large degree there is also a significant correlation in the degrees of neighboring nodes so that nodes of similar degree are more likely to be connected the size distributions of the in and out components of the network with respect to a given node namely its descendants and ancestors are also determined the in component exhibits a robust s ensuremath 2 power law tail where s is the component size the out component has a typical size of order mathrm ln t and it provides basic insights into the genealogy of the network

the space of composants of an indecomposable continuum abstract the family of all composants of an indecomposable continuum is studied we investigate the equivalence relation induced on an indecomposable continuum by its partition into composants we show that up to borel bireducibility such an equivalence relation can be of only two types e 0 and e 1 the e 0 type being simple and the e 1 type being complicated as a consequence of this we show that each indecomposable continuum carries a borel probability measure which assigns 0 to each composant and 0 or 1 to each borel set which is the union of a family of composants in particular it follows that there is no borel transversal for the family of all composants this solves an old problem in the theory of continua we prove that all hereditarily indecomposable continua are of the complicated type that is they fall into the e 1 type we analyze the properties of being of type e 1 or of type e 0 we show using effective descriptive set theory that the first of these properties is analytic and so the second one is coanalytic we construct examples of continua of both types in fact we produce a family of indecomposable continua and use it to prove that these properties are complete analytic and complete coanalytic respectively hence non borel so they do not admit simple topological characterizations we also use continua from this family to show that an indecomposable continuum may be of type e 1 only because of the behavior of composants on a small subset of the continuum this in particular shows that certain natural approaches to solving kuratowski s problem on generic ergodicity of the composant equivalence relation will not work we finish with some open problems

descriptive complexity of some isomorphism classes of banach spaces abstract we present a topological frame in which it is possible to estimate the complexity of some borel families of separable banach spaces we use this frame for evaluating the complexity of the isomorphism class of the hilbert space l 2 of certain asymptotically hilbertian spaces and of the l p spaces under the condition 1 p the complexity appears to increase with the distance to the hilbert space commented open problems conclude the article

on the classification of positions and complex structures in banach spaces abstract a topological setting is defined to study the complexities of the relation of equivalence of embeddings or position of a banach space into another and of the relation of isomorphism of complex structures on a real banach space the following results are obtained a if x is not uniformly finitely extensible then there exists a space y for which the relation of position of y inside x reduces the relation e 0 and therefore is not smooth b the relation of position of l p inside l p or inside l p p 2 reduces the relation e 1 and therefore is not reducible to an orbit relation induced by the action of a polish group c the relation of position of a space inside another can attain the maximum complexity e max d there exists a subspace of l p 1 p 2 on which isomorphism between complex structures reduces e 1 and therefore is not reducible to an orbit relation induced by the action of a polish group

on isometry and isometric embeddability between ultrametric polish spaces abstract we study the complexity with respect to borel reducibility of the relations of isometry and isometric embeddability between ultrametric polish spaces for which a set d of possible distances is fixed in advance these are respectively an analytic equivalence relation and an analytic quasi order and we show that their complexity depends only on the order type of d when d contains a decreasing sequence isometry is borel bireducible with countable graph isomorphism and isometric embeddability has maximal complexity among analytic quasi orders if d is well ordered the situation is more complex for isometry we have an increasing sequence of borel equivalence relations of length 1 which are cofinal among borel equivalence relations classifiable by countable structures while for isometric embeddability we have an increasing sequence of analytic quasi orders of length at least 3 we then apply our results to solve various open problems in the literature for instance we answer a long standing question of gao and kechris by showing that the relation of isometry on locally compact ultrametric polish spaces is borel bireducible with countable graph isomorphism

invariantly universal analytic quasi orders we introduce the notion of an invariantly universal pair s e where s is an analytic quasi order and es s 1 is an analytic equivalence relation this means that for any analytic quasi order r there is a borel set b invariant under e such that r is borel equivalent to the restriction of s to b we prove a general result giving a sufficient condition for invariant universality and we demonstrate several applications of this theorem by showing that the phenomenon of invariant universality is widespread in fact it occurs for a great number of complete analytic quasi orders arising in different areas of mathematics when they are paired with natural equivalence relations

uniformisations partielles et criteres a la hurewicz dans le plan we give characterizations of the borel sets potentially in some wadge class among the borel sets with countable vertical sections of a product of two polish spaces to do this we use some partial uniformization results

smooth sets for a borel equivalence relation we study some properties of smooth borel sets with respect to a borel equivalence relation showing some analogies with the collection of countable sets from a descriptive set theoretic point of view we found what can be seen as an analog of the hyperarithmetic points in the context of smooth sets we generalize a theorem of weiss from z actions to actions by arbitrary countable groups we show that the ci ideal of closed smooth sets is ill non borel

monte carlo variance of scrambled net quadrature hybrids of equidistribution and monte carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the latter this paper studies the variance of one such hybrid scrambled nets by applying a multidimensional multiresolution wavelet analysis to the integrand the integrand is assumed to be measurable and square integrable but not necessarily of bounded variation in simple monte carlo every nonconstant term of the multiresolution contributes to the variance of the estimated integral for scrambled nets certain low dimensional and coarse terms do not contribute to the variance for any integrand in l2 the sampling variance tends to zero faster under scrambled net quadrature than under monte carlo sampling as the number of function evaluations n tends to infinity some finite n results bound the variance under scrambled net quadrature by a small constant multiple of the monte carlo variance uniformly over all integrands f latin hypercube sampling is a special case of scrambled net quadrature

decomposing borel sets and functions and the structure of baire class 1 functions all spaces considered are metric separable and are denoted usually by the letters x y or z w stands for the set of all natural numbers if a metric separable space is additionally complete we call it polish if it is a continuous image of ww or equivalently of a polish space it is called souslin the main subject of the present paper is the structure of baire class 1 functions recent developments in banach space theory in particular discoveries of rosenthal and bourgain fremlin and talagrand see r for a survey of these developments stimulated investigations into the structure of baire class 1 functions the interested reader may consult papers by haydon odel rosenthal hor kechris louveau kl rosenthal r2 and references quoted there these investigations have already provided new applications in banach space theory see for example ri in our study of baire class 1 functions we will be interested in two decomposition properties one of them defined by lusin the other one by jayne and rogers first however we want to consider a more general problem of determining how difficult it is to represent a borel set as a union of simpler borel sets or the graph of a borel function as a union of the graphs of simpler borel functions using effective descriptive set theory in particular louveau s theorem we show that if a c x is borel x polish then a e e or there is a continuous injection q ww a such that q 1 b is meager for any b c a which is e2 this gives a new proof of j stern s result that if a borel set a is the union of cov m sets in e then a is itself e cov m is the smallest cardinality of a family of meager sets covering ill we prove similar results for functions put for f x y and a family of functions g

scaling of average receiving time and average weighted shortest path on weighted koch networks abstract in this paper we present weighted koch networks based on classic koch networks a new method is used to determine the average receiving time art whose key step is to write the sum of mean first passage times mfpts for all nodes to absorption at the trap located at a hub node as a recursive relation we show that the art exhibits a sublinear or linear dependence on network order thus the weighted koch networks are more efficient than classic koch networks in receiving information moreover average weighted shortest path awsp is calculated in the infinite network order limit the awsp depends on the scaling factor the weighted koch network grows unbounded but with the logarithm of the network size while the weighted shortest paths stay bounded

orbit equivalence for cantor minimal ℤd systems we show that every minimal action of any finitely generated abelian group on the cantor set is topologically orbit equivalent to an af relation as a consequence this extends the classification up to orbit equivalence of minimal dynamical systems on the cantor set to include af relations and d actions

measure reducibility of countable borel equivalence relations we show that every basis for the countable borel equivalence relations strictly above mathbb e 0 under measure reducibility is uncountable thereby ruling out natural generalizations of the glimm effros dichotomy we also push many known results concerning the abstract structure of the measure reducibility hierarchy to its base using arguments substantially simpler than those previously employed

uniformly recurrent subgroups and simple c algebras we study uniformly recurrent subgroups urs introduced by glasner and weiss 18 answering their query we show that any urs z of a finitely generated group is the stability system of a minimal z proper action we also show that for any sofic urs z there is a z proper action admitting an invariant measure we prove that for a urs z all z proper actions admits an invariant measure if and only if z is coamenable in the second part of the paper we study the separable c algebras associated to urs s we prove that if a urs is generic then its c algebra is simple we give various examples of generic urs s with exact and nuclear c algebras and an example of a urs z for which the associated simple c algebra is not exact and not even locally reflexive in particular it admits both a uniformly amenable trace and a nonuniformly amenable trace

linear algebraic groups and countable borel equivalence relations this paper is a contribution to the study of borel equivalence relations on standard borel spaces i e polish spaces equipped with their borel structure in mathematics one often deals with problems of classification of objects up to some notion of equivalence by invariants frequently these objects can be viewed as elements of a standard borel space x and the equivalence turns out to be a borel equivalence relation e on x a complete classification of x up to e consists of finding a set of invariants i and a map c x i such that xey c x c y for this to be of any interest both i and c must be explicit or definable and as simple and concrete as possible the theory of borel equivalence relations studies the set theoretic nature of possible invariants and develops a mathematical framework for measuring the complexity of such classification problems

orbit equivalence for cantor minimal mathbb z 2 systems we show that every minimal action of any finitely generated abelian group on the cantor set is topologically orbit equivalent to an af relation as a consequence this extends the classification up to orbit equivalence of minimal dynamical systems on the cantor set to include af relations and d actions

on the complexity of the uniform homeomorphism relation between separable banach spaces we investigate the uniform homeomorphism relation between separable banach spaces and the related relation of local equivalence we completely characterize the descriptive complexity of local equivalence in the borel reducibility hierarchy this also provides a lower bound for the complexity of the uniform homeomorphism

borel actions of polish groups we show that a borel action of a polish group on a standard borel space is borel isomorphic to a continuous action of the group on a polish space and we apply this result to three aspects of the theory of borel actions of polish groups universal actions invariant probability measures and the topological vaught conjecture we establish the existence of universal actions for any given polish group extending a result of mackey and varadarajan for the locally compact case we prove an analogue of tarski s theorem on paradoxical decompositions by showing that the existence of an invariant borel probability measure is equivalent to the nonexistence of paradoxical decompositions with countably many borel pieces we show that various natural versions of the topological vaught conjecture are equivalent to each other and in the case of the group of permutations of n with the model theoretic vaught conjecture for infinitary logic this depends on our identification of the universal action for that group

a new community based evolving network model in order to describe the community structure upon the dynamical evolution of complex networks we propose a new community based evolving network cben model having increasing communities with preferential mechanisms of both community sizes and node degrees whose cumulative distribution and raw distribution follow scale invariant power law distributions p s s s and p k k with exponents of 1 and 2 respectively besides complex networks generated by the cben model are hierarchically structured which cover the range from disassortative networks to assortative networks

completeness of the isomorphism problem for separable c algebras this paper studies the descriptive set theoretical complexity of the isomorphism problem for separable c algebras we prove that the isomorphism problem for separable simple ai c algebras is complete in the class of orbit equivalence relations this means that any isomorphism problem arising from a continuous action of a separable completely metrizable group can be reduced to the isomorphism of simple separable ai c algebras

scaling of weighted spectral distribution in deterministic scale free networks abstract scale free networks are abundant in the real world in this paper we investigate the scaling properties of the weighted spectral distribution in several deterministic and stochastic models of evolving scale free networks first we construct a new deterministic scale free model whose node degrees have a unified format using graph structure features we derive a precise formula for the spectral metric in this model this formula verifies that the spectral metric grows sublinearly as network size i e the number of nodes grows additionally the mathematical reasoning of the precise formula theoretically provides detailed explanations for this scaling property finally we validate the scaling properties of the spectral metric using some stochastic models the experimental results show that this scaling property can be retained regardless of local world node deleting and assortativity adjustment

complexity of ramsey null sets abstract we show that the set of codes for ramsey positive analytic sets is 2 1 complete this is an analogue of a theorem of hurewicz saying that the set of uncountable compact subsets of an uncountable polish space is 1 1 complete as a corollary we get that the ideal of ramsey null sets is not zfc correct which answers a question of ikegami pawlikowski and zapletal

the structure of hyperfinite borel equivalence relations we study the structure of the equivalence relations induced by the orbits of a single borel automorphism on a standard borel space we show that any two such equivalence relations which are not smooth i e do not admit borel selectors are borel embeddable into each other this utilizes among other things work of effros and weiss using this and also results of dye varadarajan and recent work of nadkarni we show that the cardinality of the set of ergodic invariant measures is a complete invariant for borel isomorphism of aperiodic nonsmooth such equivalence relations in particular since the only possible such cardinalities are the finite ones countable infinity and the cardinality of the continuum there are exactly countably infinitely many isomorphism types canonical examples of each type are also discussed

equidistributed icosahedral configurations on the sphere abstract we present and analyze a new sequence of equidistributed icosahedral configurations these configurations are created by combining the m n icosahedral nodes of caspar and klug and adapting the azimuthal projection method of snyder to create a low deformation equal area mapping from the regular icosahedron to s 2 we prove an upper bound on the mesh ratio of these configurations and analyze numerically the riesz potential energies of the point sets of size n 50 000

on the complexity of the isomorphism relation for finitely generated groups working within the framework of descriptive set theory we show that the isomorphism relation for finitely generated groups is a universal essentially countable borel equivalence relation we also prove the corresponding result for the conjugacy relation for subgroups of the free group on two generators the proofs are group theoretic and we refer to descriptive set theory only for the relevant definitions and for motivation for the results

on the isomorphism problem for measures on boolean algebras abstract the paper investigates possible generalizations of maharam s theorem to a classification of boolean algebras that support a finitely additive measure we prove that boolean algebras that support a finitely additive non atomic uniformly regular measure are metrically isomorphic to a subalgebras of the jordan algebra with the lebesgue measure we give some partial analogues to be used for a classification of algebras that support a finitely additive non atomic measure with a higher uniform regularity number we show that some naturally induced equivalence relations connected to metric isomorphism are quite complex even on the cantor algebra and therefore probably we cannot hope for a nice general classification theorem for finitely additive measures we present an example of a boolean algebra which supports only separable measures but no uniformly regular one

dichotomy theorems for families of non cofinal essential complexity we prove that for every borel equivalence relation e either e is borel reducible to mathbb e 0 or the family of borel equivalence relations incompatible with e has cofinal essential complexity it follows that if f is a borel equivalence relation and cal f is a family of borel equivalence relations of non cofinal essential complexity which together satisfy the dichotomy that for every borel equivalence relation e either e in cal f or f is borel reducible to e then cal f consists solely of smooth equivalence relations thus the dichotomy is equivalent to a known theorem

generic dynamics and monotone complete c algebras soit r une relation d equivalence generique denombrable ergodique sur un espace parfait polonais x modulo un sous ensemble maigre de x r peut s identifier a la relation d equivalence d orbite suivant une action canonique de z

a lower bound for the worst case cubature error on spheres of arbitrary dimension this paper is concerned with numerical integration on the unit sphere sr of dimension r 2 in the euclidean space r 1 we consider the worst case cubature error denoted by e qm hs sr of an arbitrary m point cubature rule qm for functions in the unit ball of the sobolev space hs sr where s and show that the positive constant cs r in the estimate depends only on the sphere dimension r 2 and the index s of the sobolev space hs sr this result was previously only known for r 2 in which case the estimate is order optimal the method of proof is constructive we construct for each qm a bad function fm that is a function which vanishes in all nodes of the cubature rule and for which our proof uses a packing of the sphere sr with spherical caps as well as an interpolation result between sobolev spaces of different indices

memory and burstiness in dynamic networks a discrete time random process is described which can generate bursty sequences of events a bernoulli process where the probability of an event occurring at time t is given by a fixed probability x is modified to include a memory effect where the event probability is increased proportionally to the number of events that occurred within a given amount of time preceding t for small values of x the interevent time distribution follows a power law with exponent 2 x we consider a dynamic network where each node forms and breaks connections according to this process the value of x for each node depends on the fitness distribution x from which it is drawn we find exact solutions for the expectation of the degree distribution for a variety of possible fitness distributions and for both cases where the memory effect either is or is not present this work can potentially lead to methods to uncover hidden fitness distributions from fast changing temporal network data such as online social communications and fmri scans

minimum structural controllability problems of complex networks abstract controllability of complex networks has been one of the attractive research areas for both network and control community and has yielded many promising and significant results in minimum inputs and minimum driver vertices however few studies have been devoted to studying the minimum controlled vertex set through which control over the network with arbitrary structure can be achieved in this paper we prove that the minimum driver vertices driven by different inputs are not sufficient to ensure the full control of the network when the associated graph contains the inaccessible strongly connected component which has perfect matching and propose an algorithm to identify a minimum controlled vertex set for network with arbitrary structure using convenient graph and mathematical tools and the simulation results show that the controllability of network is correlated to the number of inaccessible strongly connected components which have perfect matching and these results promote us to better understand the relationship between the network s structural characteristics and its control

a new deterministic complex network model with hierarchical structure abstract we introduce a new simple pseudo tree like network model deterministic complex network dcn the proposed dcn model may simulate the hierarchical structure nature of real networks appropriately and have the unique property of skipping the levels which is ubiquitous in social networks our results indicate that the dcn model has a rather small average path length and large clustering coefficient leading to the small world effect strikingly our dcn model obeys a discrete power law degree distribution p k k with exponent approaching 1 0 we also discover that the relationship between the clustering coefficient and degree follows the scaling law c k k 1 which quantitatively determines the dcn s hierarchical structure

borel chromatic numbers we study in this paper graph coloring problems in the context of descriptive set theory we consider graphs g x r where the vertex set x is a standard borel space i e a complete separable metrizable space equipped with its algebra of borel sets and the edge relation r x 2 is definable i e borel analytic co analytic etc a borel n coloring of such a graph where 1 n n 0 is a borel map c x y with card y n such that xry c x c y if such a borel coloring exists we define the borel chromatic number of g in symbols x b g to be the smallest such n otherwise we say that g has uncountable borel chromatic number in symbols x b g n 0