forecasting multifractal volatility this paper develops analytical methods to forecast the distribution of future returns for a new continuous time process the poisson multifractal the process captures the thick tails volatility persistence and moment scaling exhibited by many financial time series it can be interpreted as a stochastic volatility model with multiple frequencies and a markov latent state we assume for simplicity that the forecaster knows the true generating process with certainty but only observes past returns the challenge in this environment is long memory and the corresponding infinite dimension of the state space we introduce a discretized version of the model that has a finite state space and an analytical solution to the conditioning problem as the grid step size goes to zero the discretized model weakly converges to the continuous time process implying the consistency of the density forecasts

semistability at the end of a group extension a 1 ended cw complex q is semistable at oo if all proper maps r 0 oo q are properly homotopic if xi and x2 are finite cw complexes with isomorphic fundamental groups then the universal cover of xi is semistable at oo if and only if the universal cover of x2 is semistable at oo hence the notion of a finitely presented group being semistable at oo is well defined we prove main theorem let 1 h g k 1 be a short exact sequence of finitely generated infinite groups if g is finitely presented then g is semistable at cc theorem if a and b are locally compact connected noncompact cw complexes then a x b is semistable at oo theorem x y xy xi y c b and c nonzero integers is semistable at oo the proofs are geometrical in nature and the main tool is covering space theory

connectedness properties of limit sets we study convergence group actions on continua and give a criterion which ensures that every global cut point is a parabolic fixed point we apply this result to the case of boundaries of relatively hyperbolic groups and consider implications for connectedness properties of such spaces

a topological characterisation of hyperbolic groups we characterise word hyperbolic groups as those groups which act properly discontinuously and cocompactly on the space of distinct triples of a compact metrisable space this is in turn equivalent to a convergence group for which every point of the space is a conical limit point

free products in r thompson s group v we investigate some product structures in r thompson s group v primarily by studying the topological dynamics associated with v s action on the cantor set c we draw attention to the class d v c of groups which have embeddings as demonstrative subgroups of v which class can be used to assist in forming various products note that d v c contains all finite groups the free group on two generators and q z and is closed under passing to subgroups and under taking direct products of any member by any finite member if g v and h d v c then g oh embeds into v finally if g h d v c then g h embeds in v using a dynamical approach we also show the perhaps surprising result that z z does not embed in v even though v has many embedded copies of z and has many embedded copies of free products of various pairs of its subgroups

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

linear relaxation times of stochastic processes driven by non gaussian noises the linear relaxation time lrt associated with steady state correlation functions is studied for langevin equations with non gaussian noises dichotomous markov noise and poissonian white shot noise exact results for arbitrary models are obtained and compared with results for gaussian noises some general features of lrts are discussed the concept of dynamic effective diffusion is introduced and the existence of an optimal effective fokker planck approximation is discussed explicit examples for prototype models are presented and briefly compared with the analogs for gaussian noises

kirchhoff indexes of a network in this work we define the effective resistance between any pair of vertices with respect to a value 0 and a weight on the vertex set this allows us to consider a generalization of the kirchhoff index of a finite network it turns out that is the lowest eigenvalue of a suitable semi definite positive schrodinger operator and is the associated eigenfunction we obtain the relation between the effective resistance and hence between the kirchhoff index with respect to and and the eigenvalues of the associated schrodinger operator however our main aim in this work is to get explicit expressions of the above parameters in terms of equilibrium measures of the network from these expressions we derive a full generalization of foster s formulae that incorporate a positive probability of remaining in each vertex in every step of a random walk finally we compute the effective resistances and the generalized kirchhoff index with respect to a and for some families of networks with symmetries specifically for weighted wagon wheels and circular ladders

on the relation between white shot noise gaussian white noise and the dichotomic markov process it is shown that the dichotomic markov process converges to a white shot noise interpreted according to the stratonovich integration rule in the joint limit in which the average duration of one of the states goes to zero and the value at this state goes to infinity a further limit procedure allows us to obtain gaussian white noise from white shot noise these results are applied to the problem of noise induced transitions it is shown that white shot noise can give rise to transitions which do not occur for gaussian white noise the above results are finally generalized in introducing compound dichotomic markov processes

geometry of complex networks and topological centrality abstract we explore the geometry of complex networks in terms of an n dimensional euclidean embedding represented by the moore penrose pseudo inverse of the graph laplacian l the squared distance of a node i to the origin in this n dimensional space l i i yields a topological centrality index defined as c i 1 l i i in turn the sum of reciprocals of individual node centralities i 1 c i i l i i or the trace of l yields the well known kirchhoff index k an overall structural descriptor for the network to put into context this geometric definition of centrality we provide alternative interpretations of the proposed indices that connect them to meaningful topological characteristics first as forced detour overheads and frequency of recurrences in random walks that has an interesting analogy to voltage distributions in the equivalent electrical network and then as the average connectedness of i in all the bi partitions of the graph these interpretations respectively help establish the topological centrality c i of node i as a measure of its overall position as well as its overall connectedness in the network thus reflecting the robustness of i to random multiple edge failures through empirical evaluations using synthetic and real world networks we demonstrate how the topological centrality is better able to distinguish nodes in terms of their structural roles in the network and along with kirchhoff index is appropriately sensitive to perturbations re wirings in the network

random walks for image segmentation a novel method is proposed for performing multilabel interactive image segmentation given a small number of pixels with user defined or predefined labels one can analytically and quickly determine the probability that a random walker starting at each unlabeled pixel will first reach one of the prelabeled pixels by assigning each pixel to the label for which the greatest probability is calculated a high quality image segmentation may be obtained theoretical properties of this algorithm are developed along with the corresponding connections to discrete potential theory and electrical circuits this algorithm is formulated in discrete space i e on a graph using combinatorial analogues of standard operators and principles from continuous potential theory allowing it to be applied in arbitrary dimension on arbitrary graphs

markov fundamental tensor and its applications to network analysis abstract we first present a comprehensive review of various markov metrics used in the literature and express them in a consistent framework we then introduce the fundamental tensor a generalization of the well known fundamental matrix and show that classical markov metrics can be derived from it in a unified manner we provide a collection of useful relations for markov metrics that are useful and insightful for network studies to demonstrate the usefulness and efficacy of the proposed fundamental tensor in network analysis we present four important applications 1 unification of network centrality measures 2 characterization of generalized network articulation points 3 identification of network s most influential nodes and 4 fast computation of network reachability after failures

commute times for a directed graph using an asymmetric laplacian the expected commute times for a strongly connected directed graph are related to an asymmetric laplacian matrix as a direct extension to similar well known formulas for undirected graphs we show the close relationships between the asymmetric laplacian and the so called fundamental matrix we give bounds for the commute times in terms of the stationary probabilities for a random walk over the graph together with the asymmetric laplacian and show how this can be approximated by a symmetrized laplacian derived from a related weighted undirected graph

forest matrices around the laplacian matrix abstract we study the matrices q k of in forests of a weighted digraph and their connections with the laplacian matrix l of the i j entry of q k is the total weight of spanning converging forests in forests with k arcs such that i belongs to a tree rooted at j the forest matrices q k can be calculated recursively and expressed by polynomials in the laplacian matrix they provide representations for the generalized inverses the powers and some eigenvectors of l the normalized in forest matrices are row stochastic the normalized matrix of maximum in forests is the eigenprojection of the laplacian matrix which provides an immediate proof of the markov chain tree theorem a source of these results is the fact that matrices q k are the matrix coefficients in the polynomial expansion of adj i l thereby they are precisely faddeev s matrices for l

the moore penrose inverse of the normalized graph laplacian abstract we prove a formula that relates the moore penrose inverses of two matrices a b such that a n 1 b m 1 and discuss some applications in particular to the representation of the moore penrose inverse of the normalized laplacian of a graph the laplacian matrix of an undirected graph is symmetric and is strictly related to its connectivity properties however our formula applies to asymmetric matrices so that we can generalize our results for asymmetric laplacians whose importance for the study of directed graphs is increasing

kemeny s constant and the effective graph resistance kemeny s constant and its relation to the effective graph resistance has been established for regular graphs by palacios et al 1 based on the moore penrose pseudo inverse of the laplacian matrix we derive a new closed form formula and deduce upper and lower bounds for the kemeny constant furthermore we generalize the relation between the kemeny constant and the effective graph resistance for a general connected undirected graph

detecting free splittings in relatively hyperbolic groups we describe an algorithm which determines whether or not a group which is hyperbolic relative to abelian groups admits a nontrivial splitting over a finite group

the boundary of negatively curved groups gromov s article gr contains fundamental properties of negatively curved groups several sets of seminar notes are available frn swn usn that contain more detailed accounts of and further expand on gromov s article a sequence xi e r i 1 2 is convergent at infinity provided xi x t 00 as i j t 00 two sequences xi yj convergent at infinityare equivalent if xi yi t 00 as i t 00 a point in the boundary ar of r is an equivalence class of sequences convergent at infinity now r u ar has a natural topology in which r is a discrete subspace and a typical not necessarily open neighborhood of a point a e ar represented by a sequence xj is given by y e r u arj a y n where a y lim xi y if y e r and a y lim xi yi if year is represented by a sequence yj ar is a compact metrizable finite dimensional space gr swn the important tool that relates ar with the cohomological properties of r is the rips complex pd r for every d 0 pd r is the simplicial complex whose vertices are elements of r and a collection xi xk e r spans a simplex if d xi x d for all i j the natural group action of r on itself by left translations gives rise to an action on pd r the key observation of rips is that when d is sufficiently large pd r is contractible and therefore when r is torsion free provides a model for er more precisely we have proposition 1 1 gr lemma 1 7 a swn 4 2 proposition 9 suppose that r is a negatively curved group and let t5 be as in the definition choose an integer

peripheral splittings of groups we define the notion of a peripheral splitting of a group this is essentially a representation of the group as the fundamental group of a bipartite graph of groups where all the vertex groups of one colour are held fixed the peripheral subgroups we develop the theory of such splittings and prove an accessibility result the theory mainly applies to relatively hyperbolic groups with connected boundary where the peripheral subgroups are precisely the maximal parabolic subgroups we show that if such a group admits a non trivial peripheral splitting then its boundary has a global cut point moreover the non peripheral vertex groups of such a splitting are themselves relatively hyperbolic these results together with results from elsewhere show that under modest constraints on the peripheral subgroups the boundary of a relatively hyperbolic group is locally connected if it is connected in retrospect one further deduces that the set of global cut points in such a boundary has a simplicial treelike structure

a generalized inverse for graphs with absorption abstract we consider weighted directed graphs with a notion of absorption on the vertices related to absorbing random walks on graphs we define a generalized inverse of the graph laplacian called the absorption inverse that reflects both the graph structure as well as the absorption rates on the vertices properties of this generalized inverse are presented including a basic relationship between the absorption inverse and the group inverse of a related graph a forest theorem for interpreting the entries of the absorption inverse as well as relationships between the absorption inverse and the fundamental matrix of the absorbing random walk applications of the absorption inverse for describing the structure of graphs with absorption are given including a directed distance metric spectral partitioning algorithm and centrality measure

hyperbolic groups with planar boundaries we prove that the class of convex cocompact kleinian groups is quasi isometrically rigid we also establish that a word hyperbolic group with a planar boundary different from the sphere is virtually a convex cocompact kleinian group provided that its boundary has ahlfors regular conformal dimension strictly less than 2 or if it acts geometrically on a cat 0 cube complex

the generalised word problem in hyperbolic and relatively hyperbolic groups abstract we prove that for a finitely generated group hyperbolic relative to virtually abelian subgroups the generalised word problem for a parabolic subgroup is the language of a real time turing machine then for a hyperbolic group we show that the generalised word problem for a quasiconvex subgroup is a real time language under either of two additional hypotheses on the subgroup by extending the muller schupp theorem we show that the generalised word problem for a finitely generated subgroup of a finitely generated virtually free group is context free conversely we prove that a hyperbolic group must be virtually free if it has a torsion free quasiconvex subgroup of infinite index with context free generalised word problem

complexity volumes of splittable groups abstract using graph of groups decompositions of finitely generated groups we define euler characteristic type invariants which are non zero in many interesting classes of hyperbolic limit and csa groups including elementarily free groups and one ended torsion free hyperbolic groups whose jsj decomposition contains a maximal hanging fuchsian vertex group

on resistance distance of markov chain and its sum rules abstract motivated by the notion of resistance distance on graph we define a new resistance distance between two states on a given finite ergodic markov chain based on its fundamental matrix we prove a few equivalent formulations and discuss its relation with other parameters of the markov chain such as its group inverse stationary distribution eigenvalues or hitting time in addition building upon existing sum rules for the hitting time of markov chain we give sum rules of this new resistance distance of markov chains that resembles the sum rules of the resistance distance on graph this yields markov chain counterparts of various classical formulae such as foster s first formula or the kirchhoff index formulae

deciding isomorphy using dehn fillings the splitting case we solve dehn s isomorphism problem for virtually torsion free relatively hyperbolic groups with nilpotent parabolic subgroups we do so by reducing the isomorphism problem to three algorithmic problems in the parabolic subgroups namely the isomorphism problem separation of torsion in their outer automorphism groups by congruences and the mixed whitehead problem an automorphism group orbit problem the first step of the reduction is to compute canonical jsj decompositions dehn fillings and the given solutions of the algorithmic problems in the parabolic groups are then used to decide if the graphs of groups have isomorphic vertex groups and if so whether a global isomorphism can be assembled for the class of finitely generated nilpotent groups we give solutions to these algorithmic problems by using the arithmetic nature of these groups and of their automorphism groups

spectral partitioning works planar graphs and finite element meshes abstract spectral partitioning methods use the fiedler vector the eigenvector of the second smallest eigenvalue of the laplacian matrix to find a small separator of a graph these methods are important components of many scientific numerical algorithms and have been demonstrated by experiment to work extremely well in this paper we show that spectral partitioning methods work well on bounded degree planar graphs and finite element meshes the classes of graphs to which they are usually applied while naive spectral bisection does not necessarily work we prove that spectral partitioning techniques can be used to produce separators whose ratio of vertices removed to edges cut is o n for bounded degree planar graphs and two dimensional meshes and o n1 d for well shaped d dimensional meshes the heart of our analysis is an upper bound on the second smallest eigenvalues of the laplacian matrices of these graphs we prove a bound of o 1 n for bounded degree planar graphs and o 1 n2 d for well shaped d dimensional meshes

laplacian matrices and alexandrov topologies of digraphs abstract we explore the spectral properties of digraph laplacians and how they relate to topological properties of digraphs such as openness closure and strong connectedness under the alexandrov topology

stochastic processes driven by dichotomous markov noise some exact dynamical results stochastic processes defined by a general langevin equation of motion where the noise is the non gaussian dichotomous markov noise are studied a non fokker planck master differential equation is deduced for the probability density of these processes two different models are exactly solved in the second one a nonequilibrium bimodal distribution induced by the noise is observed for a critical value of its correlation time critical slowing down does not appear in this point but in another one

on fast computation of directed graph laplacian pseudo inverse abstract the laplacian matrix and its pseudo inverse for a strongly connected directed graph is fundamental in computing many properties of a directed graph examples include random walk centrality and betweenness measures average hitting and commute times and other connectivity measures these measures arise in the analysis of many social and computer networks in this short paper we show how a linear system involving the laplacian may be solved in time linear in the number of edges times a factor depending on the separability of the graph this leads directly to the column by column computation of the entire laplacian pseudo inverse in time quadratic in the number of nodes i e constant time per matrix entry the approach is based on off the shelf iterative methods for which global linear convergence is guaranteed without recourse to any matrix elimination algorithm

complexity combinatorial group theory and the language of palutators abstract after a brief survey of the known results about group languages we prove that many of the familiar classes of groups have word problems whose complexity is linear time we also consider the complexity of extensions and hnn extensions finally in an attempt to find languages that have complexity which is greater than linear time we discuss the language of palutators where a palutator is defined as a generalization of a palindrome and a commutator

groups quasi isometric to right angled artin groups we characterize groups quasi isometric to a right angled artin group raag g with finite outer automorphism group in particular all such groups admit a geometric action on a cat 0 cube complex that has an equivariant fibering over the davis building of g this characterization will be used in forthcoming work of the first author to give a commensurability classification of the groups quasi isometric to certain raags

divergence in lattices in semisimple lie groups and graphs of groups divergence functions of a metric space estimate the length of a path connecting two points a b at distance 3 s is a finite set of valuations of a number field k including all infinite valuations and o s is the corresponding ring of s integers

word problems of groups formal languages characterizations and decidability abstract let g be a group and let g be a monoid homomorphism from the set of all strings over some finite alphabet onto the group g the set is then called a generating set for g and the language 1 1 is called the word problem of g with respect to the generating set via the homomorphism and is denoted by w g we consider nine conditions that hold in each such language of the form w g and determine which combinations of these conditions are equivalent to the property of the language in question being the word problem of a group we show that each of these nine conditions is decidable for the family of regular languages but that each is undecidable for the family of one counter languages the languages accepted by one counter pushdown automata we also show that the property of a language being the word problem of a group is undecidable for the family of one counter languages but is decidable for the family of deterministic context free languages the languages accepted by deterministic pushdown automata

non linear relaxation time for stochastic processes driven by non gaussian noises decay of unstable states abstract the non linear relaxation time nlrt technique is developed for langevin equations with non gaussian noises exact results are obtained for the cases of dichotomous markov noise and poisson white shot noise as an explicit application we study the decay of unstable states under non gaussian fluctuations

on the definition of word hyperbolic groups formal languages based on multiplication tables of finitely generated groups are investigated and used to give a linguistic characterization of word hyperbolic groups

on the rational subset problem for groups we use language theory to study the rational subset problem for groups and monoids we show that the decidability of this problem is preserved under graph of groups constructions with finite edge groups in particular it passes through free products amalgamated over finite subgroups and hnn extensions with finite associated subgroups we provide a simple proof of a result of grunschlag showing that the decidability of this problem is a virtual property we prove further that the problem is decidable for a direct product of a group g with a monoid m if and only if membership is uniformly decidable for g automaton subsets of m it follows that a direct product of a free group with any abelian group or commutative monoid has decidable rational subset membership

analogues of cayley graphs for topological groups we define for a compactly generated totally disconnected locally compact group a graph called a rough cayley graph that is a quasi isometry invariant of the group this graph carries information about the group structure in an analogous way to the ordinary cayley graph for a finitely generated group with this construction the machinery of geometric group theory can be applied to topological groups this is illustrated by a study of groups where the rough cayley graph has more than one end and a study of groups where the rough cayley graph has polynomial growth

nlrt formalism for asymmetric dichotomous markov noise the authors present a general theory of nonlinear relaxation time nlrt for stochastic transient dynamics of systems driven by an asymmetric dichotomous markov noise dmn two limiting cases of this general result are studied the poissonian white shot noise wsn and the gaussian white noise gwn

master equation description of external poisson white noise in finite systems we consider finite systems with random control parameters a theory for a unified description of internal fluctuations and external noise is presented internal fluctuations are modeled by a one step markovian master equation external noise is introduced by random parameters in the master equation it is modeled by a poisson white noise the unified description of fluctuations features a markovian master equation with nonvanishing transition probabilities for all steps in the state space alternative formulations are given in terms of the generating function poisson representation and the equations for the factorial moments an expansion around the thermodynamic limit is considered the theory permits the calculation of finite size effects it predicts the existence of a coupling of the two types of fluctuations leading to crossed fluctuation contributions two examples are considered i a poisson counting process with fluctuating parameter ii a creation and annihilation process with source terms and fluctuations in each of the creation annihilation and source parameters in the second example a complete analysis is given for the stationary distribution and associated moments for a finite system and also in the thermodynamic limit the different role of the fluctuations of the three parameters is discussed explicit crossed fluctuations contributions are found the effect of the system size on the type of transitions induced by external noise in the thermodynamic limit is discussed

the complexity of grigorchuk groups with application to cryptography abstract the turing complexity of the word problems of a class of groups introduced by grigorchuk 1985 is examined in particular it is shown that such problems of permutation groups of the infinite complete binary tree yield natural complete sets that separate time and space complexity classes if they are distinct a refinement of savitch s translation theorem as well as a similar result restricted for time complexity follow new families of nonfinitely presented groups are shown to have word problems uniformly solvable in simultaneous logspace and quadratic time a new family of public key cryptosystems based on these word problems is constructed