the migdal kadanoff approximation optimisation generalisation ising model in external fields and the migdal kadanoff hierarchies the simplest migdal kadanoff approximation is optimised on hypercubic lattices generalisation is made to all regular lattices leading to recursion relations continuous in lattice coordination one variety of these relations shows universality whilst another variety shows duality on planar lattices the ising model in external fields both normal and staggered is investigated phase diagrams and exponents are found under a complete treatment of the ambiguity in the handling of external fields by bond moving finally the underlying exactly solved family of hierarchical lattices is considered it is shown that these lattices exhibit a complete breakdown in the concept of universality

hyperbolic geometry of complex networks we develop a geometric framework to study the structure and function of complex networks we assume that hyperbolic geometry underlies these networks and we show that with this assumption heterogeneous degree distributions and strong clustering in complex networks emerge naturally as simple reflections of the negative curvature and metric property of the underlying hyperbolic geometry conversely we show that if a network has some metric structure and if the network degree distribution is heterogeneous then the network has an effective hyperbolic geometry underneath we then establish a mapping between our geometric framework and statistical mechanics of complex networks this mapping interprets edges in a network as noninteracting fermions whose energies are hyperbolic distances between nodes while the auxiliary fields coupled to edges are linear functions of these energies or distances the geometric network ensemble subsumes the standard configuration model and classical random graphs as two limiting cases with degenerate geometric structures finally we show that targeted transport processes without global topology knowledge made possible by our geometric framework are maximally efficient according to all efficiency measures in networks with strongest heterogeneity and clustering and that this efficiency is remarkably robust with respect to even catastrophic disturbances and damages to the network structure

oscillatory critical amplitudes in hierarchical models we study the oscillatory critical amplitudes of theq states potts model on a diamond hierarchical lattice we consider an example of the generic case finite critical index as well as the degenerate case essential singularity in both cases we compare the magnitude of the oscillations with geometrical characteristics of the julia set of zeroes of the partition function

determinantal formulae for matrices with sparse inverses ii asymmetric zero patterns abstract in an earlier paper formulae for det a as a ratio of products of principal minors of a were exhibited for any given symmetric zero pattern of a 1 these formulae may be presented in terms of a spanning tree of the intersection graph of certain index sets associated with the zero pattern of a 1 however just as the determinant of a diagonal and of a triangular matrix are both the product of the diagonal entries the symmetry of the zero pattern is not essential for these formulae we describe here how analogous formulae for det a may be obtained in the asymmetric zero pattern case by introducing a directed spanning tree we also examine the converse question of determining all possible zero patterns of a 1 which guarantee that a certain determinantal formula holds

minimum deviation quasi lu factorization of nonsingular matrices abstract not all matrices enjoy the existence of an lu factorization for those that do not a number of repairs are possible for nonsingular matrices we offer here a permutation free repair in which the matrix is factored l u with l and u collectively as near as possible to lower and upper triangular in a natural sense defined herein such factorization is not generally unique in any sense in the process we investigate further the structure of matrices without lu factorization and permutations that produce an lu factorization

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

group invariant analogues of hadamard s inequality a wide class of inequalities for the determinant and other real valued functions of an n n complex hermitian or real symmetric matrix h hjk may be obtained by generalizing marshall and olkin s proof of hadamard s inequality deth j 1nhjj for positive definite pd h we shall see that each subgroup g of the group un of n x n unitary matrices not only determines an analogue of 1 for det h but also provides inequalities for a large family of unitarily invariant functions of h not necessarily pd

inequalities for m matrices and inverse m matrices in this paper we establish some determinantal inequalities concerning m matrices and inverse m matrices the main results are as follows 1 if a aij is either an n n m matrix or inverse m matrix then for any permutation i1 i2 in of 1 2 n a deta i 1naii s 2n1 ai1i2 ais 1isaisi1 ai1i1ai2i2 aisis b deta i 1naii if and only if a is essentially triangular 2 if a aij is an n n m matrix b bij is an n n inverse m matrix a b denotes the hadamard product of a and b then a b is an m matrix and for any permutation i1 i2 in of 1 2 n det a b det ab s 2naisisdeta i1 i2 is 1 deta i1 is 1 is bisisdetb i1 i2 is 1 detb i1 is 1 is 1

infinite susceptibility at high temperatures in the migdal kadanoff scheme the migdal kadanoff real space renormalisation group scheme applied to the ising and classical xy and probably other ferromagnetic models yields an infinite susceptibility at all temperatures above the critical temperature for a hypercubic lattice of dimension d or 2

potts ferromagnet transformations and critical exponents in planar hierarchical lattices we prove that the duality transformation for a potts ferromagnet on two rooted planar hierarchical lattices hl preserves the thermal eigenvalue this leads to a relation between the correlation length critical exponents of a hl and its corresponding dual lattice using hyperscaling we show that their specific heat critical exponents coincide for a smaller class of hl namely of diamond and tress types we prove that another transformation also preserves and

spanning tree extensions of the hadamard fischer inequalities abstract all possible graph theoretic generalizations of a certain sort for the hadamard fischer determinantal inequalities are determined these involve ratios of products of principal minors which dominate the determinant furthermore the cases of equality in these inequalities are characterized and equality is possible for every set of values which can occur for the relevant minors this relates recent work of the authors on positive definite completions and determinantal identities when applied to the same collections of principal minors earlier generalizations give poorer more difficult to compute bounds than the present inequalities thus this work extends and in a certain sense completes a series of generalizations of hadamard fischer begun in the 1960s

finding metastable states in real world time series with recurrence networks in the framework of time series analysis with recurrence networks we introduce a self adaptive method that determines the elusive recurrence threshold and identifies metastable states in complex real world time series as initial step we introduce a way to set the embedding parameters used to reconstruct the state space from the time series we set them as the ones giving the maximum shannon entropy of the diagonal line length distribution for the first simultaneous minima of recurrence rate and shannon entropy to identify metastable states as well as the transitions between them we use a soft partitioning algorithm for module finding which is specifically developed for the case in which a system shows metastability we illustrate our method with a complex time series example finally we show the robustness of our method for identifying metastable states our results suggest that our method is robust for identifying metastable states in complex time series even when introducing considerable levels of noise and missing data points

renormalisation groups with periodic and aperiodic orbits motivated by recent results for some frustrated hierarchical models the authors analyse the consequences of renormalisation group transformations with periodic and aperiodic orbits they find that the free energy has in general an infinite number of singularities this occurs already when the transformation has a stable periodic orbit they present an idealised example and a new class of hierarchical potts models to which the theory applies

random rectangular graphs a generalization of the random geometric graph rgg model is proposed by considering a set of points uniformly and independently distributed on a rectangle of unit area instead of on a unit square 0 1 2 the topological properties of the random rectangular graphs rrgs generated by this model are then studied as a function of the rectangle sides lengths a and b 1 a and the radius r used to connect the nodes when a 1 we recover the rgg and when a the very elongated rectangle generated resembles a one dimensional rgg we obtain here analytical expressions for the average degree degree distribution connectivity average path length and clustering coefficient for rrg these results provide evidence that show that most of these properties depend on the connection radius and the side length of the rectangle usually in a monotonic way the clustering coefficient however increases when the square is transformed into a slightly elongated rectangle and after this maximum it decays with the increase of the elongation of the rectangle we support all our findings by computational simulations that show the goodness of the theoretical models proposed for rrgs

some determinantal inequalities for hadamard product of matrices abstract the main result of this paper is the following if both a aij and b bij are m matrices or positive definite real symmetric matrices of order n the hadamard product of a and b is denoted by a b and ak and bk k 1 2 n are the k k leading principal submatrices of a and b respectively then det a b det ab k 2 n a kk det a k 1 det a k b kk det b k 1 det b k 1

submodular spectral functions of principal submatrices of a hermitian matrix extensions and applications we extend the multiplicative submodularity of the principal determinants of a nonnegative definite hermitian matrix to other spectral functions we show that if f is the primitive of a function that is operator monotone on an interval containing the spectrum of a hermitian matrix a then the function i mapsto rm tr f a i is supermodular meaning that rm tr f a i rm tr f a j leq rm tr f a i cup j rm tr f a i cap j where a i denotes the i times i principal submatrix of a we discuss extensions to self adjoint operators on infinite dimensional hilbert space and to m matrices we discuss an application to cur approximation of nonnegative hermitian matrices

potential theory and analytic properties of self similar fractal and multifractal distributions by the use of recursion relations and analytic techniques we deduce general analytic results pertaining to the electrostatic potential moments and fourier transform of exactly self similar fractal and multifractal charge distributions three specific examples are given the binomial distribution on the middle third cantor set which is a multifractal distribution the uniform distribution on the menger sponge which illustrates the added complication of higher dimensionality and the uniform distribution on the von koch snowflake which illustrates the effect of rotations in the defining transformations

extremes on trees this paper considers the asymptotic distribution of the longest edge of the minimal spanning tree and nearest neighbor graph on x1 xnn where x1 x2 are i i d in 2 with distribution f and nn is independent of the xi and satisfies nn n p1 a new approach based on spatial blocking and a locally orthogonal coordinate system is developed to treat cases for which f has unbounded support the general results are applied to a number of special cases including elliptically contoured distributions distributions with independent weibull like margins and distributions with parallel level curves

fractal structure of zeros in hierarchical models we consider an ising and aq state potts model on a diamond hierarchical lattice we give pictures of the distribution of zeros of the partition function in the complex plane of temperatures for several choices ofq these zeros are just the julia set corresponding to the renormalization group transformation

determinantal formulae for matrices with sparse inverses abstract the determinant of a matrix is expressed in terms of certain of its principal minors by a formula which can be read off from the graph of the inverse of the matrix the only information used is the zero pattern of the inverse and each zero pattern yields one or more corresponding formulae for the determinant

isolation and connectivity in random geometric graphs with self similar intensity measures random geometric graphs consist of randomly distributed nodes points with pairs of nodes within a given mutual distance linked in the usual model the distribution of nodes is uniform on a square and in the limit of infinitely many nodes and shrinking linking range the number of isolated nodes is poisson distributed and the probability of no isolated nodes is equal to the probability the whole graph is connected here we examine these properties for several self similar node distributions including smooth and fractal uniform and nonuniform and finitely ramified or otherwise we show that nonuniformity can break the poisson distribution property but it strengthens the link between isolation and connectivity it also stretches out the connectivity transition finite ramification is another mechanism for lack of connectivity the same considerations apply to fractal distributions as smooth with some technical differences in evaluation of the integrals and analytical arguments

inverses of banded matrices abstract we establish a correspondence between the vanishing of a certain set of minors of a matrix a and the vanishing of a related set of minors of a 1 in particular inverses of banded matrices are characterized we then use our results to find patterns for toeplitz matrices with banded inverses finally we give an interesting determinant formula for inverses of banded matrices and show that in general a banded partial matrix may be completed in a unique way to give a banded inverse of the same bandwidth

asymptotic expansions in the poisson limit theorem asymptotic expansions for the distributions of sums of independent nonnegative integer random variables in the neighbourhood of the poisson distribution are derived together with explicit estimates of the truncation error expansions are also derived for the expectations of at most polynomially growing functions of such sums applications to the poisson binomial and poisson negative binomial approximations are considered the method used is an adaptation of the stein chen approach 1 introduction in chen 1975 stein s method of obtaining error bounds for normal approximations was introduced in the poisson context and was used to obtain rates of convergence in total variation to the poisson distribution for sums of stationary sequences of 0 1 random variables chen also observed that stein s method is in principle suited to developing asymptotic expansions as well as obtaining error estimates and used it for independent 0 1 summands to derive the second term in such an expansion together with an estimate of the remaining error kerstan 1964 had previously derived a similar result with a sharper error estimate by a quite different technique the stein chen method was refined in barbour and hall 1984 to yield the best known upper bounds for the discrepancy between the poisson distribution and that of a sum of independent 0 1 random variables as well as complementary lower bounds and new estimates for the error remaining after the second term in the asymptotic expansion were also established however although stein s method was potentially applicable also to higher order expansions it proved in practice too cumbersome to use in this paper it is shown how by means of a simple identity the stein chen method can be made to yield a full asymptotic expansion with a minimum of difficulty the error estimates obtained perform well in comparison with other estimates that are available for sums of independent 0 1 random variables they agree with those of barbour and hall when the series is truncated after one term or after two for the more general case of sums of independent nonnegative integer valued random variables the error estimates obtained after two terms in the expansion differ only slightly from those in barbour and hall comparison with kerstan s estimates for integral nonnegative summands of the error after one term of the expansion is more complicated by and large the techniques of this paper give better approximations except when the summands themselves are almost precisely poisson distributed in section 3 expansions are also obtained for the expectations of at most polynomially growing functions of the

random geometric graphs with general connection functions in the original 1961 gilbert model of random geometric graphs nodes are placed according to a poisson point process and links formed between those within a fixed range motivated by wireless ad hoc networks soft or probabilistic connection models have recently been introduced involving a connection function h r that gives the probability that two nodes at distance r are linked directly connect in many applications not only wireless networks it is desirable that the graph is connected that is every node is linked to every other node in a multihop fashion here the connection probability of a dense network in a convex domain in two or three dimensions is expressed in terms of contributions from boundary components for a very general class of connection functions it turns out that only a few quantities such as moments of the connection function appear good agreement is found with special cases from previous studies and with numerical simulations

an inertia formula for hermitian matrices with sparse inverses abstract let a mn be a nonsingular hermitian matrix let g be a chordal graph on vertices 1 n and suppose that g a 1 g then the inertia of a may be expressed in a simple way in terms of the inertias of those principal submatrices of a corresponding to certain readily identifiable sets of vertices of g specifically we prove that the inertia i a satisfies the identity i a c i a s m i a in which c denotes the collection of maximal cliques of g s denotes the collection of minimal vertex separators of g and m is the multiplicity of a separator

spatially embedded random networks many real world networks analyzed in modern network theory have a natural spatial element e g the internet social networks neural networks etc yet aside from a comparatively small number of somewhat specialized and domain specific studies the spatial element is mostly ignored and in particular its relation to network structure disregarded in this paper we introduce a model framework to analyze the mediation of network structure by spatial embedding specifically we model connectivity as dependent on the distance between network nodes our spatially embedded random networks construction is not primarily intended as an accurate model of any specific class of real world networks but rather to gain intuition for the effects of spatial embedding on network structure nevertheless we are able to demonstrate in a quite general setting some constraints of spatial embedding on connectivity such as the effects of spatial symmetry conditions for scale free degree distributions and the existence of small world spatial networks we also derive some standard structural statistics for spatially embedded networks and illustrate the application of our model framework with concrete examples

positive p matrices completion problem with zeros in the inverse completion in this paper based on the completion problem of partial p matrix for 1 chordal graph we obtain the partial positive p matrix completions of k chordal graphs and k chordal block graphs under some conditions and solve partial positive p matrix completion problems with zeros in the inverse completion for k chordal graphs and k chordal block graphs k 2

positive definite completions and determinant maximization abstract a method is described for determining whether a positive definite completion of a given partial hermitian matrix exists and if so for finding the determinant maximizing positive definite completion no assumption is made about the arrangement of the specified entries the method employs iterative application to a modified problem of an explicit formula for the maximum determinant in case there is only one symmetrically placed pair of unspecified entries a robust and fast algorithm based upon this approach is shown to have global convergence to the necessarily unique solution

extensions of band matrices with band inverses abstract let r ij be a given set of i j matrices for i j 1 n and i j m where 0 m n 1 necessary and sufficient conditions are established for the existence and uniqueness of an invertible block matrix f ij i j 1 n such that f ij r ij for i j m f admits either a left or right block triangular factorization and f 1 ij 0 for i j m the well known conditions for an invertible block matrix to admit a block triangular factorization emerge for the particular choice m n 1 the special case in which the given r ij are positive definite in an appropriate sense is explored in detail and an inequality which corresponds to burg s maximal entropy inequality in the theory of covariance extension is deduced the block toeplitz case is also studied

spatial network surrogates for disentangling complex system structure from spatial embedding of nodes networks with nodes embedded in a metric space have gained increasing interest in recent years the effects of spatial embedding on the networks structural characteristics however are rarely taken into account when studying their macroscopic properties here we propose a hierarchy of null models to generate random surrogates from a given spatially embedded network that can preserve certain global and local statistics associated with the nodes embedding in a metric space comparing the original network s and the resulting surrogates global characteristics allows one to quantify to what extent these characteristics are already predetermined by the spatial embedding of the nodes and links we apply our framework to various real world spatial networks and show that the proposed models capture macroscopic properties of the networks under study much better than standard random network models that do not account for the nodes spatial embedding depending on the actual performance of the proposed null models the networks are categorized into different classes since many real world complex networks are in fact spatial networks the proposed approach is relevant for disentangling the underlying complex system structure from spatial embedding of nodes in many fields ranging from social systems over infrastructure and neurophysiology to climatology

determinantal formulae for matrix completions associated with chordal graphs abstract let a be a partial positive definite hermitian matrix and let g a be the undirected graph of the specified off diagonal entries of a when g a is a chordal graph we present an explicit determinantal formula for the unique determinant maximizing positive definite completion maximum entropy completion of a in order to do this we first give a characterization of a chordal graph in terms of the existence of a proper spanning tree t of the intersection graph of its set of maximal cliques the formula for the maximum determinant is then a quotient of products of specified principal minors of a in which each minor in the numerator corresponds to a maximal clique of g a while each minor in the denominator corresponds to the intersection of two maximal cliques associated with an edge of t we next show that the denominator terms can be identified graph theoretically with the minimal vertex separators of g a and that these terms are independent of the particular proper spanning tree t then an alternative formula is given for the maximum determinant as an inclusion exclusion like ratio of principal minors associated with the maximal cliques of g a finally we show that the entries of the determinant maximizing completion may be calculated by solving a sequence of simple one variable maximization problems

mapping between hierarchical lattices by renormalisation and duality the renormalisation group scheme of migdal 1976 and kadanoff 1976 is applied to a ferromagnetic potts model on two hierarchical lattices in the two cases the renormalisation transformation is split into two elementary processes parallel bonds or series bonds and each lattice can be mapped onto the other one it is then easy to deduce the characteristic parameters critical exponents repulsive fixed points etc of one lattice from those of the other lattice moreover the classical geometrical duality offers a third possibility of mapping and the relationship between the renormalisation and the duality methods is presented in the peculiar case of a two dimensional lattice

entropy determinants and l 2 torsion we show that for any amenable group and any z module m of type fl with vanishing euler characteristic the entropy of the natural action on the pontryagin dual of m is equal to the l 2 torsion of m as a particular case the entropy of the principal algebraic action associated with the module z z f is equal to the logarithm of the fuglede kadison determinant of f whenever f is a non zero divisor in z this conrms a conjecture of deninger as a key step in the proof we provide a general szeg o type approximation theorem for the fuglede kadison determinant on the group von neumann algebra of an amenable group as a consequence of the equality between l 2 torsion and entropy we show that the l 2 torsion of a non trivial amenable group with nite classifying space vanishes this was conjectured by luck finally we establish a milnor turaev formula for the l 2 torsion of a nite acyclic chain complex

rank inequalities for positive semidefinite matrices abstract several inequalities relating the rank of a positive semidefinite matrix with the ranks of various principal submatrices are presented these inequalities are analogous to known determinantal inequalities for positive definite matrices such as fischer s inequality koteljanskii s inequality and extensions of these associated with chordal graphs

finite clusters in high density continuous percolation compression and sphericality a percolation process inrd is considered in which the sites are a poisson process with intensity and the bond between each pair of sites is open if and only if the sites are within a fixed distancer of each other the distribution of the number of sites in the clusterc of the origin is examined and related to the geometry ofc it is shown that when andk are large there is a characteristic radius such that conditionally on c k the convex hull ofc closely approximates a ball of radius with high probability when the normal volumek thatk points would occupy is small the cluster is compressed in that the number of points per unit volume in this ball is much greater than the ambient density for larger normal volumes there is less compression this can be compared to bernoulli bond percolation on the square lattice in two dimensions where an analog of this compression is known not to occur

combinatorial matrix analysis an overview abstract we survey recent work in some components of combinatorial matrix analysis including qualitative matrix theory matrix completion problems bounding of matrix parameters the role of the longest simple circuit and attenuation of matrix products some definitional thoughts historical perspectives and unification challenges etc are offered the style is informal following the talk on which this writeup is based