metriques de carnot cartheodory et quasiisometries des espaces symetriques de rang un we exhibit a rigidity property of the simple groups sp n 1 and f7 20 which implies mostow rigidity this property does not extend to o n 1 and u n 1 the proof relies on quasiconformal theory applied in the cr setting extensions are given to a class of solvable lie groups as a byproduct a result on quasiisometries of infinite nilpotent groups is obtained dans cet article on etablit une propriete de rigidite des groupes simples de rang un sp n 1 n 2 2 et fqui entraine la rigidite de mostow 1 theoreme toute quasiisornwtrie de 1 espace hyperbolique quaternionien hhn n 2 resp du plan hyperbolique de cayley cah2 est d distance bornee d une isometrie i e differe d une isomnetrie par une application qui diplace les points d une distance bornee une application f entre espaces metriques est une quasiisometrie s il existe des constantes l et c telles que l image de f soit c dense et que pour tous x y c d x y d fx fy ld x y c l une quasiisometrie entre des g et g est une sorte d isomorphisme virtuel dans la categorie topologique en effet cela correspond a une action de g sur un fibre principal c de groupe g sur une base compacte cf ra un isomorphisme entre sous groupes cocompacts covolume fini suffit pour les groupes simples de rang un de groupes de lie s etend en une quasiisometrie des espaces symetriques ou des groupes de lie si celle ci est proche d une isometrie des espaces symetriques resp un isomorphisme des groupes de lie les sous groupes sont conjugues c est la rigidite de mostow m2 la propriete ci dessus ne s etend pas aux groupes o n 1 et u n 1 en effet paragraphe 11 7 ceux ci ont beaucoup de quasiisometries au moins this content downloaded from 207 46 13 92 on sun 20 nov 2016 04 26 46 utc all use subject to http about jstor org terms

gradient estimate for solutions to poisson equations in metric measure spaces abstract let x d be a complete pathwise connected metric measure space with a locally ahlfors q regular measure where q 1 suppose that x d supports a local 1 2 poincare inequality and a suitable curvature lower bound for the poisson equation u f on x d moser trudinger and sobolev inequalities are established for the gradient of u the local holder continuity with optimal exponent of solutions is obtained

efficient computation of ihara coefficients using the bell polynomial recursion abstract the ihara zeta function has proved to be a powerful tool in the analysis of graph structures it is determined by the prime cycles of a finite graph g v e and can be characterized in terms of a quasi characteristic polynomial of the adjacency matrix t of the oriented line graph associated to g the coefficients of this polynomial referred to as ihara coefficients have been used to characterize graphs in a permutation invariant manner and allow for an efficient evaluation of the ihara zeta function in this paper we present a novel method for computing the ihara coefficients we first establish a characterization of the ihara coefficients in terms of complete bell polynomials and by exploiting a recursive relation for the latter we show how the ihara coefficients can be efficiently computed in o e 2 provided that the eigenvalues of t are known

an adaptive timestepping algorithm for stochastic differential equations we introduce a variable timestepping procedure using local error control for the pathwise strong numerical integration of a system of stochastic differential equations forced by a single wiener process the milstein method is used to advance the numerical solution and the stepsizes are determined via two local error estimates that roughly correspond to leading order deterministic and stochastic local error components one advantage of using two error controls is an increased flexibility that allows for the treatment of both drift and diffusion dominated regimes in a consistent manner numerical results are presented and the generalization of this approach to wider classes of problems and methods is discussed

detecting the number of clusters in n way probabilistic clustering recently there has been a growing interest in multiway probabilistic clustering some efficient algorithms have been developed for this problem however not much attention has been paid on how to detect the number of clusters for the general n way clustering n 2 to fill this gap this problem is investigated based on n way algebraic theory in this paper a simple yet efficient detection method is proposed by eigenvalue decomposition evd which is easy to implement we justify this method in addition its effectiveness is demonstrated by the experiments on both simulated and real world data sets

on quotients of spaces with ricci curvature bounded below abstract let m g be a smooth riemannian manifold and g a compact lie group acting on m effectively and by isometries it is well known that a lower bound of the sectional curvature of m g is again a bound for the curvature of the quotient space which is an alexandrov space of curvature bounded below moreover the analogous stability property holds for metric foliations and submersions the goal of the paper is to prove the corresponding stability properties for synthetic ricci curvature lower bounds specifically we show that such stability holds for quotients of rcd k n spaces under isomorphic compact group actions and more generally under metric measure foliations and submetries an rcd k n space is a metric measure space with an upper dimension bound n and weighted ricci curvature bounded below by k in a generalized sense in particular this shows that if m g has ricci curvature bounded below by k r and dimension n then the quotient space is an rcd k n space additionally we tackle the same problem for the cd cd and mcp curvature dimension conditions we provide as well geometric applications which include a generalization of kobayashi s classification theorem of homogeneous manifolds to rcd k n spaces with essential minimal dimension n n a structure theorem for rcd k n spaces admitting actions by large compact groups and geometric rigidity results for orbifolds such as cheng s maximal diameter and maximal volume rigidity theorems finally in two appendices we apply the methods of the paper to study quotients by isometric group actions of discrete spaces and of super ricci flows

pointwise lipschitz functions on metric spaces abstract for a metric space x we study the space d x of bounded functions on x whose pointwise lipschitz constant is uniformly bounded d x is compared with the space lip x of bounded lipschitz functions on x in terms of different properties regarding the geometry of x we also obtain a banach stone theorem in this context in the case of a metric measure space we also compare d x with the newtonian sobolev space n 1 x in particular if x supports a doubling measure and satisfies a local poincare inequality we obtain that d x n 1 x

the poincare inequality is an open ended condition let p 1 and let x d be a complete metric measure space with borel and doubling that admits a 1 p poincare inequality then there exists e 0 such that x d admits a 1 q poincare inequality for every q p e quantitatively

lipschitz continuity of cheeger harmonic functions in metric measure spaces abstract we use the heat equation to establish the lipschitz continuity of cheeger harmonic functions in certain metric spaces the metric spaces under consideration are those that are endowed with a doubling measure supporting a 1 2 poincare inequality and in addition supporting a corresponding sobolev poincare type inequality for the modification of the measure obtained via the heat kernel examples are given to illustrate the necessity of our assumptions on these spaces we also provide an example to show that in the general setting the best possible regularity for the cheeger harmonic functions is lipschitz continuity

geometry and analysis of dirichlet forms let e be a regular strongly local dirichlet form on l2 x m and d the associated intrinsic distance assume that the topology induced by d coincides with the original topology on x and that x is compact satisfies a doubling property and supports a weak 1 2 poincare inequality we first discuss the non coincidence of the intrinsic length structure and the gradient structure under the further assumption that the ricci curvature of x is bounded from below in the sense of lott sturm villani the following are shown to be equivalent i the heat flow of e gives the unique gradient flow of u ii e satisfies the newtonian property iii the intrinsic length structure coincides with the gradient structure moreover for the standard resistance dirichlet form on the sierpinski gasket equipped with the kusuoka measure we identify the intrinsic length structure with the measurable riemannian and the gradient structures we also apply the above results to the coarse ricci curvatures and asymptotics of the gradient of the heat kernel

derivations and alberti representations abstract we relate generalized lebesgue decompositions of measures in terms of curve fragments alberti representations and weaver derivations this correspondence leads to a geometric characterization of the local norm on the weaver cotangent bundle of a metric measure space x the local norm of a form df sees how fast f grows on curve fragments seen by this implies a new characterization of differentiability spaces in terms of the a e equality of the local norm of df and the local lipschitz constant of f as a consequence the lip lip inequality of keith must be an equality we also provide dimensional bounds for the module of derivations in terms of the assouad dimension of x

a posteriori error analysis for higher order dissipative methods for evolution problems we prove a posteriori error estimates for time discretizations by the discontinuous galerkin method dg q and the corresponding implicit runge kutta radau method irk r q of arbitrary order q 0 for both linear and nonlinear evolution problems of the form u prime mathfrak f u f with 2 angle bounded operator mathfrak f the key ingredient is a novel higher order reconstruction widehat u of the discrete solution u which restores continuity and leads to the differential equation widehat u prime pi mathfrak f u f for a suitable interpolation operator and piecewise polynomial approximation f of f we discuss applications to linear pde such as the convection diffusion equation 1 2 and the wave equation formally and nonlinear pde corresponding to subgradient operators 1 such as the p laplacian as well as lipschitz operators 1 2 we also derive conditional a posteriori error estimates for the time dependent minimal surface problem

p poincare inequality versus poincare inequality some counterexamples we point out some of the differences between the consequences of p poincare inequality and that of poincare inequality in the setting of doubling metric measure spaces based on the geometric characterization of poincare inequality given in durand cartagena et al mich math j 60 2011 we obtain a geometric property implied by the support of a p poincare inequality and demonstrate by examples that an analogous geometric characterization for finite p is not possible the examples we give are metric measure spaces which are doubling and support an poincare inequality but support no finite p poincare inequality in particular these examples show that one cannot expect a self improving property for poincare inequality in the spirit of keith zhong ann math 167 2 575 599 2008 we also show that the persistence of poincare inequality under measured gromov hausdorff limits fails for poincare inequality

does error control suppress spuriosity in the numerical solution of initial value ordinary differential equations to what extent does local error control confer global properties this work concentrates on global steady states or fixed points it is shown that for systems of equations spurious fixed points generally cease to exist when local error control is used for scalar problems on the other hand locally adaptive algorithms generally avoid spurious fixed points by an indirect method the stepsize selection process causes spurious fixed points to be unstable however problem classes exist where for arbitrarily small tolerances stable spurious fixed points persist with significant basins of attraction a technique is derived for generating such examples

a study of graph spectra for comparing graphs and trees the spectrum of a graph has been widely used in graph theory to characterise the properties of a graph and extract information from its structure it has also been employed as a graph representation for pattern matching since it is invariant to the labelling of the graph there are however a number of potential drawbacks in using the spectrum as a representation of a graph firstly more than one graph may share the same spectrum it is well known for example that very few trees can be uniquely specified by their spectrum secondly the spectrum may change dramatically with a small change structure there are a wide variety of graph matrix representations from which the spectrum can be extracted among these are the adjacency matrix combinatorial laplacian normalised laplacian and unsigned laplacian spectra can also be derived from the heat kernel matrix and path length distribution matrix the choice of matrix representation clearly has a large effect on the suitability of spectrum in a number of pattern recognition tasks in this paper we investigate the performance of the spectra as a graph representation in a variety of situations firstly we investigate the cospectrality of the various matrix representations over large graph and tree sets extending the work of previous authors we then show that the euclidean distance between spectra tracks the edit distance between graphs over a wide range of edit costs and we analyse the accuracy of this relationship we then use the spectra to both cluster and classify the graphs and demonstrate the effect of the graph matrix formulation on error rates these results are produced using both synthetic graphs and trees and graphs derived from shape and image data

backtrackless walks on a graph the aim of this paper is to explore the use of backtrackless walks and prime cycles for characterizing both labeled and unlabeled graphs the reason for using backtrackless walks and prime cycles is that they avoid tottering and can increase the discriminative power of the resulting graph representation however the use of such methods is limited in practice because of their computational cost in this paper we present efficient methods for computing graph kernels which are based on backtrackless walks in a labeled graph and whose worst case running time is the same as that of kernels based on random walks for clustering unlabeled graphs we construct feature vectors using ihara coefficients since these coefficients are related to the frequencies of prime cycles in the graph to efficiently compute the low order coefficients we present an o v 3 algorithm which is better than the o v 6 worst case running time of previously known algorithms in the experimental evaluation we apply the proposed method to clustering both labeled and unlabeled graphs the results show that using backtrackless walks and prime cycles instead of random walks can increase the accuracy of recognition

mean square stability properties of an adaptive time stepping sde solver we consider stability properties of a class of adaptive time stepping schemes based upon the milstein method for stochastic differential equations with a single scalar forcing in particular we focus upon mean square stability for a class of linear test problems with multiplicative noise we demonstrate that desirable stability properties can be induced in the numerical solution by the use of two realistic local error controls one for the drift term and one for the diffusion

model problems in numerical stability theory for initial value problems in the past numerical stability theory for initial value problems in ordinary differential equations has been dominated by the study of problems with essentially trivial dynamics whilst this has resulted in a coherent and self contained body of knowledge it has not thoroughly addressed the problems of real interest in applications recently there have been a number of studies of numerical stability for wider classes of problems admitting more complicated dynamics this on going work is unified and possible directions for future work are outlined in particular striking similarities between this new developing stability theory and the classical non linear stability theory are emphasised the classical theories of a b and algebraic stability for runge kutta methods are briefly reviewed and it is emphasised that the classes of equations to which these theories apply linear decay and contractive problems only admit trivial dynamics four other categories of equations gradient dissipative conservative and hamiltonian systems are considered relationships and differences between the possible dynamics in each category which range from multiple competing equilibria to fully chaotic solutions are highlighted and it is stressed that the wide range of possible behaviour allows a large variety of applications runge kutta schemes which preserve the dynamical structure of the underlying problem are sought and indications of a strong relationship between the developing stability theory for these new categories and the classical existing stability theory for the older problems are given algebraic stability in particular is seen to play a central role the effects of error control are considered and multi step methods are discussed briefly finally various open problems are described

phase space error control for dynamical systems variable time stepping algorithms for initial value ordinary differential equations are traditionally designed to solve a problem for a fixed initial condition and over a finite time it can be shown that these algorithms may perform poorly for long time computations with initial conditions that lie in a small neighborhood of a fixed point in this regime there are orbits that are bounded in space but unbounded in time and the classical error per step or error per unit step philosophy may be improved upon a new error criterion is introduced that essentially bounds the truncation error at each step by a fraction of the solution arc length over the corresponding time interval this new control can be incorporated within a standard algorithm as an additional constraint at negligible additional computational cost it is shown that this new criterion has a positive effect on the linear stability properties and hence improves behavior in the neighborhood of stable fixed points furthermore spurious fixed points and period two solutions are prevented the new criterion is shown to be admissible in the sense that it can always be satisfied with nonzero stepsizes implementation details and numerical results are given

depth based hypergraph complexity traces from directed line graphs in this paper we aim to characterize the structure of hypergraphs in terms of structural complexity measure measuring the complexity of a hypergraph in a straightforward way tends to be elusive since the hyperedges of a hypergraph may exhibit varying relational orders we thus transform a hypergraph into a line graph which not only accurately reflects the multiple relationships exhibited by the hyperedges but is also easier to manipulate for complexity analysis to locate the dominant substructure within a line graph we identify a centroid vertex by computing the minimum variance of its shortest path lengths a family of centroid expansion subgraphs of the line graph is then derived from the centroid vertex we compute the depth based complexity traces for the hypergraph by measuring either the directed or undirected entropies of its centroid expansion subgraphs the resulting complexity traces provide a flexible framework that can be applied to both hypergraphs and graphs we perform hyper graph classification in the principal component space of the complexity trace vectors experiments on hyper graph datasets abstracted from bioinformatics and computer vision data demonstrate the effectiveness and efficiency of the complexity traces highlightswe aim to characterize hypergraphs in terms of structural complexity measures straightforwardly measuring the complexity of a hypergraph tends to be elusive we transform a hypergraph into a line graph for measuring the complexities we compute depth based complexity traces for the hypergraph on line graphs the complexity traces provide a flexible framework for hypergraphs and graphs

dirichlet problem at infinity on gromov hyperbolic metric measure spaces we study the asymptotic dirichlet problem for p harmonic functions in a very general setting of gromov hyperbolic metric measure spaces

modulus and the poincare inequality on metric measure spaces the purpose of this paper is to develop the understanding of modulus and the poincare inequality as defined on metric measure spaces various definitions for modulus and capacity are shown to coincide for general collections of metric measure spaces consequently modulus is shown to be upper semi continuous with respect to the limit of a sequence of curve families contained in a converging sequence of metric measure spaces moreover several competing definitions for the poincare inequality are shown to coincide if the underlying measure is doubling one such characterization considers only continuous functions and their continuous upper gradients and extends work of heinonen and koskela applications include showing that the p poincare inequality with a doubling measure for p 1 persists through to the limit of a sequence of converging pointed metric measure spaces this extends results of cheeger a further application is the construction of new doubling measures in euclidean space which admit a 1 poincare inequality

spectral curvature clustering scc this paper presents novel techniques for improving the performance of a multi way spectral clustering framework govindu in proceedings of the 2005 ieee computer society conference on computer vision and pattern recognition cvpr 05 vol 1 pp 1150 1157 2005 chen and lerman 2007 preprint in the supplementary webpage for segmenting affine subspaces specifically it suggests an iterative sampling procedure to improve the uniform sampling strategy an automatic scheme of inferring the tuning parameter from the data a precise initialization procedure for k means as well as a simple strategy for isolating outliers the resulting algorithm spectral curvature clustering scc requires only linear storage and takes linear running time in the size of the data it is supported by theory which both justifies its successful performance and guides our practical choices we compare it with other existing methods on a few artificial instances of affine subspaces application of the algorithm to several real world problems is also discussed

definitions of quasiconformality summary we establish that the infinitesimal h definition for quasiconforreal mappings on carnot groups implies global quasisymmetry and hence the absolute continuity on almost all lines our method is new even in r where we obtain that the limsup condition in the h definition can be replaced by a liminf condition this leads to a new removability result for quasi conformal mappings in euclidean spaces an application to parametrizations of chord arc surfaces is also given

modeling the shape of the scene a holistic representation of the spatial envelope in this paper we propose a computational model of the recognition of real world scenes that bypasses the segmentation and the processing of individual objects or regions the procedure is based on a very low dimensional representation of the scene that we term the spatial envelope we propose a set of perceptual dimensions naturalness openness roughness expansion ruggedness that represent the dominant spatial structure of a scene then we show that these dimensions may be reliably estimated using spectral and coarsely localized information the model generates a multidimensional space in which scenes sharing membership in semantic categories e g streets highways coasts are projected closed together the performance of the spatial envelope model shows that specific information about object shape or identity is not a requirement for scene categorization and that modeling a holistic representation of the scene informs about its probable semantic category

hypergraph imaging an overview hypergraph theory as originally developed by berge hypergraphe dunod paris 1987 is a theory of finite combinatorial sets modeling lot of problems of operational research and combinatorial optimization this framework turns out to be very interesting for many other applications in particular for computer vision in this paper we are going to survey the relationship between combinatorial sets and image processing more precisely we propose an overview of different applications from image hypergraph models to image analysis it mainly focuses on the combinatorial representation of an image and shows the effectiveness of this approach to low level image processing in particular to segmentation edge detection and noise cancellation

space filling with metric measure spaces we show a sharp relationship between the existence of space filling mappings with an upper gradient in a lorentz space and the poincare inequality in a general metric setting as key examples we consider these phenomena in cantor diamond spaces and the heisenberg groups

the essential stability of local error control for dynamical systems although most adaptive software for initial value problems is designed with an accuracy requirement control of the local error it is frequently observed that stability is imparted by the adaptation this relationship between local error control and numerical stability is given a firm theoretical underpinning the dynamics of numerical methods with local error control are studied for three classes of ordinary differential equations dissipative contractive and gradient systems dissipative dynamical systems are characterised by having a bounded absorbing set mathcal b which all trajectories eventually enter and remain inside the exponentially contractive problems studied have a unique globally exponentially attracting equilibrium point and thus they are also dissipative since the absorbing set mathcal b may be chosen to be a ball of arbitrarily small radius around the equilibrium point the gradient systems studied are those for which the set of equilibria comprises isolated points and all traje

from local to global in quasiconformal structures abstract we exhibit a large class of metric spaces whose infinitesimal quasiconformal structure is strong enough to capture the global quasiconformal structure a sufficient condition for this to happen is described in terms of a poincare type inequality

differentiability and poincare type inequalities in metric measure spaces abstract we demonstrate the necessity of a poincare type inequality for those metric measure spaces that satisfy cheeger s generalization of rademacher s theorem for all lipschitz functions taking values in a banach space with the radon nikodym property this is done by showing the existence of a rich structure of curve fragments that connect nearby points similar in nature to semmes s pencil of curves for the standard poincare inequality using techniques similar to cheeger kleiner 12 we show that our conditions are also sufficient we also develop another characterization of rnp lipschitz differentiability spaces by connecting points by curves that form a rich structure of partial derivatives that were first discussed in 5

differentiability of lipschitz functions on banach spaces abstract the main result of this note says that if the norm of a banach space e is differentiable frechet gateaux or in some intermediate sense away from the origin then every locally lipschitz function defined on an open subset g of e is differentiable in the same sense at every point of some subset of g dense in g as the main corollary we prove that every lipschitz function on an asplund space is frechet differentiable on a dense set moreover in all these results the mean value theorem for the points of differentiability which is equivalent to the possibility of computing the clarke subdifferential from the points of differentiability is shown to hold

geometry and analysis of dirichlet forms ii abstract given a regular strongly local dirichlet form e under assumption that the lower bound of the ricci curvature of bakry emery the local doubling and local poincare inequalities are satisfied we obtain that i the intrinsic differential and distance structures of e coincide ii the cheeger energy functional ch d e is a quadratic norm this shows that ii is necessary for the riemannian ricci curvature defined by ambrosio gigli savare to be bounded from below this together with some recent results of ambrosio gigli savare yields that the heat flow gives a gradient flow of boltzman shannon entropy under the above assumptions we also obtain an improvement on kuwada s duality theorem for dirichlet forms under the assumptions of doubling and poincare inequalities finally dirichlet forms are constructed to show that doubling and poincare inequalities are not enough to obtain either i or ii above that is the lower bound of the bakry emery curvature condition is essential

weak and strong convergence of derivations and stability of flows with respect to mgh convergence abstract this paper is devoted to the study of weak and strong convergence of derivations and of the flows associated to them when dealing with a sequence of metric measure structures x d m n m n weakly convergent to m in particular under curvature assumptions either only on the limit metric structure x d m or on the whole sequence of metric measure spaces we provide several stability results

lipschitz algebras and derivations ii exterior differentiation abstract basic aspects of differential geometry can be extended to various non classical settings lipschitz manifolds rectifiable sets sub riemannian manifolds banach manifolds wiener space etc although the constructions differ in each of these cases one can define a module of measurable 1 forms and a first order exterior derivative we give a general construction which applies to any metric space equipped with a finite measure and produces the desired result in all of the above cases it agrees with cheeger s construction when the latter is defined it also applies to an important class of dirichlet spaces where however the known first order differential calculus in general differs from ours although the two are related

a differentiable structure for metric measure spaces abstract the main result of this paper is the provision of conditions under which a metric measure space admits a differentiable structure this differentiable structure gives rise to a finite dimensional l cotangent bundle over the given metric measure space and then to a sobolev space h 1 p over the given metric measure space the latter which is reflexive for p 1 this extends results of cheeger geom funct anal 9 1999 3 428 to a wider collection of metric measure spaces

quasiconformal maps in metric spaces with controlled geometry this paper develops the foundations of the theory of quasiconformal maps in metric spaces that satisfy certain bounds on their mass and geometry the principal message is that such a theory is both relevant and viable the first main issue is the problem of definition which we next describe quasiconformal maps are commonly understood as homeomorphisms that distort the shape of infinitesimal balls by a uniformly bounded amount this requirement makes sense in every metric space given a homeomorphism f from a metric space x to a metric space y then for x x and r 0 set

error estimates and adaptive time step control for a class of one step methods for stiff ordinary differential equations we prove new optimal a priori error estimates for a class of implicit one step methods for stiff ordinary differential equations obtained by using the discontinuous galerkin method with piecewise polynomials of degree zero and one starting from these estimates we propose a new algorithm for automatic time step control and we discuss the relation between this algorithm and earlier algorithms implemented in packages for the numerical solution of stiff ordinary differential equations

trust region algorithms and timestep selection unconstrained optimization problems are closely related to systems of ordinary differential equations odes with gradient structure in this work we prove results that apply to both areas we analyze the convergence properties of a trust region or levenberg marquardt algorithm for optimization the algorithm may also be regarded as a linearized implicit euler method with adaptive timestep for gradient odes from the optimization viewpoint the algorithm is driven directly by the levenberg marquardt parameter rather than the trust region radius this approach is discussed for example in r fletcher practical methods of optimization 2nd ed john wiley new york 1987 but no convergence theory is developed we give a rigorous error analysis for the algorithm establishing global convergence and an unusual extremely rapid type of superlinear convergence the precise form of superlinear convergence is exhibited the ratio of successive displacements from the limit point is bounded above and below by geometrically decreasing sequences we also show how an inexpensive change to the algorithm leads to quadratic convergence from the ode viewpoint this work contributes to the theory of gradient stability by presenting an algorithm that reproduces the correct global dynamics and gives very rapid local convergence to a stable steady state

structure of measures in lipschitz differentiability spaces we prove the equivalence of two seemingly very di erent ways of generalising rademacher s theorem to metric measure spaces the rst was introduced by cheeger and is based upon di erentiation with respect to another xed chart func tion the second approach is new for this generality and originates in some ideas of alberti it is based upon forming partial derivatives along a very rich structure of lipschitz curves analogous to the di erentiability theory of euclidean spaces by examining this structure further we naturally arrive to several descriptions of lipschitz di erentiability spaces

differentiability of lipschitz functions in lebesgue null sets we show that if n 1 then there exists a lebesgue null set in mathbb r n containing a point of differentiability of each lipschitz function f mathbb r n rightarrow mathbb r n 1 in combination with the work of others this completes the investigation of when the classical rademacher theorem admits a converse avoidance of sigma porous sets arising as irregular points of lipschitz functions plays a key role in the proof

quasisymmetric parametrizations of two dimensional metric spheres according to the classical uniformization theorem every smooth riemannian surface z homeomorphic to the 2 sphere is conformally diffeomorphic to s2 the unit sphere inr3 equipped with the riemannian metric induced by the ambient euclidean metric the availability of a similar uniformization procedure for spheres with a generalized conformal structure is highly desirable in particular in connection with thurston s hyperbolization conjecture this was addressed by cannon in his combinatorial riemann mapping theorem 7 he considers topological surfaces equipped with a sequence of shinglings a combinatorial structure that leads to a notion of approximate conformal moduli of rings he then finds conditions that imply the existence of coordinate systems on the surface that relate these combinatorial moduli to classical analytic moduli in the plane in this paper we develop a uniformization theory for a different type of generalized conformal structure we start with a metric space z homeomorphic to s2 and ask for conditions under which z can be mapped onto s2 by a quasisymmetric homeomorphism the class of quasisymmetries is an appropriate analog of conformal1 mappings in a metric space context quasisymmetric homeomorphisms also arise in the theory of gromov hyperbolic metric spaces quasi isometries between gromov hyperbolic spaces induce quasisymmetric boundary homeomorphisms our setup has the advantage that we can exploit recent notions and methods from analysis

a polynomial characterization of hypergraphs using the ihara zeta function the aim of this paper is to seek a compact characterization of irregular unweighted hypergraphs for the purposes of clustering to this end we develop a polynomial characterization for hypergraphs based on the ihara zeta function we investigate the flexibility of the polynomial coefficients for learning relational structures with different relational orders furthermore we develop an efficient method for computing the coefficient set our representation for hypergraphs takes into account not only the vertex connections but also the hyperedge cardinalities and thus can distinguish different relational orders which is prone to ambiguity if the hypergraph laplacian is used in our experimental evaluation we demonstrate the effectiveness of the proposed characterization for clustering irregular unweighted hypergraphs and its advantages over the spectral characterization of the hypergraph laplacian