a new topological technique for characterizing homoclinic tangles we have developed a topological approach called homotopic lobe dynamics for describing the qualitative structure of homoclinic tangles this approach begins from an efficient and accurate description of the initial development of a tangle up to some finite number of iterates j where the value of j indicates the amount of information that one puts into the theory our approach can then compute the topologically forced structure of the tangle at all iterates after j this allows one for example to predict a minimal set of homoclinic intersections this technique places few assumptions on the homoclinic tangles considered in fact one main advantage is its ability to describe the wide variety of behaviour seen in physically significant tangles

estimating the errors on measured entropy and mutual information abstract information entropy and the related quantity mutual information are used extensively as measures of complexity and to identify nonlinearity in dynamical systems expressions for the probability distribution of entropies and mutual informations calculated from finite amounts of data exist in the literature but the expressions have seldom been used in the field of nonlinear dynamics in this paper formulae for estimating the errors on observed information entropies and mutual informations are derived using the standard error analysis familiar to physicists their validity is demonstrated by numerical experiment for illustration the formulae are then used to evaluate the errors on the time lagged mutual information of the logistic map

on the symbolic dynamics of the henon map the authors study a method for finding all periodic orbits of the dissipative henon map for the canonical values a 1 4 b 0 3 this method is found to give indeed the good symbolic dynamics proposed earlier by two of them but it does not work as proposed for larger values of b this is seen in two ways although the method is thus not universally valid it can be used to study problems related to periodic orbits they find that for the usual values a 1 4 b 0 3 expressions for average quantities are not easily obtained precisely from periodic orbits in agreement with recent claims

braid analysis of a bouncing ball i identify the template organizing the chaotic dynamics of a bounding ball system i further show how to estimate the topological parameter values of the system directly from a time series a process i call topological time series analysis two distinct methods to determine the topological parameters are illustrated and compared the pruning front procedure and a braid analysis both procedures lead to compatible results

consistent distribution free k sample and independence tests for univariate random variables a popular approach for testing if two univariate random variables are statistically independent consists of partitioning the sample space into bins and evaluating a test statistic on the binned data the partition size matters and the optimal partition size is data dependent while for detecting simple relationships coarse partitions may be best for detecting complex relationships a great gain in power can be achieved by considering finer partitions we suggest novel consistent distribution free tests that are based on summation or maximization aggregation of scores over all partitions of a fixed size we show that our test statistics based on summation can serve as good estimators of the mutual information moreover we suggest regularized tests that aggregate over all partition sizes and prove those are consistent too we provide polynomial time algorithms which are critical for computing the suggested test statistics efficiently we show that the power of the regularized tests is excellent compared to existing tests and almost as powerful as the tests based on the optimal yet unknown in practice partition size in simulations as well as on a real data example

measuring and testing dependence by correlation of distances distance correlation is a new measure of dependence between random vectors distance covariance and distance correlation are analogous to product moment covariance and correlation but unlike the classical definition of correlation distance correlation is zero only if the random vectors are independent the empirical distance dependence measures are based on certain euclidean distances between sample elements rather than sample moments yet have a compact representation analogous to the classical covariance and correlation asymptotic properties and applications in testing independence are discussed implementation of the test and monte carlo results are also presented

scaling properties of a scattering system with an incomplete horseshoe we investigate a parameter dependent map describing a chaotic scattering system in parameter ranges leading to an incomplete horseshoe we construct an approximate symbolic dynamics which describes quite well the hyperbolic component of the invariant set its grammatical rules are correlated with the convergence properties of the thermodynamical formalism for the measures characterizing the invariant set

dehn twists and pseudo anosov diffeomorphisms we show that it is possible to obtain many pseudo anosov diffeomorphisms from dehn twists in particular we generalize a theorem of long and morton to obtain that iff is a pseudo anosov diffeomorphism of an oriented surface andt is the dehn twist around the simple closed curve then the isotopy class oft n f contains a pseudo anosov diffeomorphism except for at most 7 consecutive values ofn

determining thurston classes using nielsen types in previous work 3 we showed how the thurston or bers classifications of diffeomorphisms of surfaces could be obtained using the nielsen types of the lifts of the diffeomorphism to the unit disc in this paper we find improved conditions on the nielsen types for the thurston and bers classes we use them to verify that an example studied by nielsen is a pseudo anosov diffeomorphism with stretching factor of degree 4 this example is of interest in its own right but it also serves to illustrate exactly how the nielsen types are used for verifying examples we discuss the general usefulness of this method

new proofs of some results of nielsen given a compact oriented surface m with negative euler characteristic and a homeomorphism z m2 m2 what is the best representative cp m2 m2 in the isotopy class of s thurston s answered this question by defining pseudo anosov homeomorphisms a generalization of anosov diffeomorphisms on t2 which may be characterized as follows the homeomorphism rp is pseudo anosov iff there is a pair of transverse measured geodesic laminations i and au which intersect each closed nonperipheral geodesic in h4 which are topologically preserved by cp and which have their respective transverse measures multiplied under the action of cp by l n and d for some l 1 thurston showed that cp m m2 can be pieced together from homeomorphisms of subsurfaces which are either periodic or pseudoanosov more precisely thurston proved theorem 0 1 the preprint s provides an introduction to this theorem describing many of the useful properties of pseudo anosov diffeomorphisms the proof outlined in s is contained in 111

fixed points of pseudo anosov diffeomorphisms of surfaces this article supplements richard miller s paper 6 by establishing a result on the minimum number of fixed points in the homotopy class of a pseudo anosov surface transformation we thus verify one of the many farreaching results that william thurston announced in 1977 in a preprint s dealing with the classification up to homotopy of diffeomorphisms of compact surfaces thurston stated his theorems without proofs but with many hints at techniques fathi laudenbach poenaru and other workers at an intensive seminar at the university of orsay gave detailed proofs of some but not all of thurston s assertions and published their findings in a monograph 7 more recently miller 6 and gilman 2 have verified thurston s claim that his theorem 4 s p 81 follows from the classical investigations of jakob nielsen 4 using miller s exposition of the nielsenthurston theory we establish

symbolic dynamics and periodic orbits for the cardioid billiard the periodic orbits of the strongly chaotic cardioid billiard are studied by introducing a binary symbolic dynamics the corresponding partition is mapped to a topologically well ordered symbol plane in the symbol plane the pruning front is obtained from orbits running either into or through the cusp we show that all periodic orbits correspond to maxima of the lagrangian and give a complete list up to code length 15 the symmetry reduction is done on the level of the symbol sequences and the periodic orbits are classified using symmetry lines we show that there exists an infinite number of families of periodic orbits accumulating in length and that all other families of geometrically short periodic orbits eventually get pruned all these orbits are related to finite orbits starting and ending in the cusp we obtain an analytical estimate of the kolmogorov sinai entropy and find a good agreement with the numerically calculated value and the one obtained by averaging periodic orbits furthermore the statistical properties of periodic orbits are investigated

a kernel two sample test we propose a framework for analyzing and comparing distributions which we use to construct statistical tests to determine if two samples are drawn from different distributions our test statistic is the largest difference in expectations over functions in the unit ball of a reproducing kernel hilbert space rkhs and is called the maximum mean discrepancy mmd we present two distribution free tests based on large deviation bounds for the mmd and a third test based on the asymptotic distribution of this statistic the mmd can be computed in quadratic time although efficient linear time approximations are available our statistic is an instance of an integral probability metric and various classical metrics on distributions are obtained when alternative function classes are used in place of an rkhs we apply our two sample tests to a variety of problems including attribute matching for databases using the hungarian marriage method where they perform strongly excellent performance is also obtained when comparing distributions over graphs for which these are the first such tests

on the nielsen type and the classification for the mapping class group let s be a compact surface of genus g with n boundary components with 2g 2 n 0 and let m s be its mapping class group also known as the teichmiiller modular group using measurd foliations thurston 15 has shown that m s contains three different types of elements bers has shown this decomposition can be obtained using teichmiiller distance 3 subdividing one of the classes further he obtains four classes nielsen has a series of papers 7 8 parts i iii in which he develops an elaborate theory of fixed points of mapping classes if t is any homeomorphism of s and h a lift of t or of a power of 1 to the upper half plane nielsen s theory involves the assignment of a pair of integers to h called the nielsen type of h the purpose of this paper is to show how the thurston classes and the bers classes can be defined in terms of the nielsen types of the lifts of

nonparametric k sample tests via dynamic slicing k sample testing problems arise in many scientific applications and have attracted statisticians attention for many years we propose an omnibus nonparametric method based on an optimal discretization aka slicing of continuous random variables in the test the novelty of our approach lies in the inclusion of a term penalizing the number of slices i e the resolution of the discretization so as to regularize the corresponding likelihood ratio test statistic an efficient dynamic programming algorithm is developed to determine the optimal slicing scheme asymptotic and finite sample properties such as power and null distribution of the resulting test statistic are studied we compare the proposed testing method with some existing well known methods and demonstrate its statistical power through extensive simulation studies as well as a real data example a dynamic slicing method for the one sample testing problem is further developed and studied under the same framework supplementary materials inclu

estimating mutual information we present two classes of improved estimators for mutual information m x y from samples of random points distributed according to some joint probability density mu x y in contrast to conventional estimators based on binnings they are based on entropy estimates from k nearest neighbor distances this means that they are data efficient with k 1 we resolve structures down to the smallest possible scales adaptive the resolution is higher where data are more numerous and have minimal bias indeed the bias of the underlying entropy estimates is mainly due to nonuniformity of the density at the smallest resolved scale giving typically systematic errors which scale as functions of k n for n points numerically we find that both families become exact for independent distributions i e the estimator m x y vanishes up to statistical fluctuations if mu x y mu x mu y this holds for all tested marginal distributions and for all dimensions of x and y in addition we give estimators for redundancies between more than two random variables we compare our algorithms in detail with existing algorithms finally we demonstrate the usefulness of our estimators for assessing the actual independence of components obtained from independent component analysis ica for improving ica and for estimating the reliability of blind source separation

stability of the homology of the mapping class groups of orientable surfaces the mapping class group of f fgs r is f rgs wo a where a is the topological group of orientation preserving diffeomorphisms of f which are the identity on df and fix the s punctures when r 0 and 2g s 3 an important feature of f is that it acts properly discontinuously on the teichmiller space us the quotient is the moduli space f of isometry classes of complete hyperbolic metrics of finite area on f since us is homeomorphic to euclidean space and f is virtually torsion free we have

horseshoe templates with global torsion in a driven laser dimensional system with many longitudinal modes oscillating simultaneously hence in sec ii we describe the experimental system and check that topological analysis is applicable by estimating the lyapunov dimension of the attractors then sec iii is devoted to the extraction of upo from experimental data and the computation of their topological invariants we show there how careful signal processing can greatly help in extracting topological information from noisy time series of finite length and precision finally in sec iv we determine the templates of the different attractors compare them and relate them to the values of the modulation frequency in particular we show that these templates have a nonzero global torsion and that this torsion increases by one full turn when the period of modulation increases by 2p v r this observation is in close agreement with the theoretical studies of modulated nonlinear oscillators by gilmore and mccallum 10

markov shifts and topological entropy of families of homoclinic tangles abstract the existence of a homoclinic orbit in dynamical systems implies chaotic behaviour with positive entropy in this work we determine explicitly the markov shifts associated to certain smale horseshoe homoclinic orbits which allow us to compute a lower bound for the topological entropy that such a system can have it is done associating a heteroclinic orbit which belongs to the same isotopy class and then determining the markov partition of the dynamical core of an end periodic mapping

measuring dependence powerfully and equitably given a high dimensional data set we often wish to find the strongest relationships within it a common strategy is to evaluate a measure of dependence on every variable pair and retain the highest scoring pairs for follow up this strategy works well if the statistic used a has good power to detect non trivial relationships and b is equitable meaning that for some measure of noise it assigns similar scores to equally noisy relationships regardless of relationship type e g linear exponential periodic in this paper we define and theoretically characterize two new statistics that together yield an efficient approach for obtaining both power and equitability to do this we first introduce a new population measure of dependence and show three equivalent ways that it can be viewed including as a canonical smoothing of mutual information we then introduce an efficiently computable consistent estimator of our population measure of dependence and we empirically establish its equitability on a large class of noisy functional relationships this new statistic has better bias variance properties and better runtime complexity than a previous heuristic approach next we derive a second related statistic whose computation is a trivial side product of our algorithm and whose goal is powerful independence testing rather than equitability we prove that this statistic yields a consistent independence test and show in simulations that the test has good power against independence taken together our results suggest that these two statistics are a valuable pair of tools for exploratory data analysis

trellises formed by stable and unstable manifolds in the plane a trellis is the figure formed by the stable and unstable manifolds of a hyperbolic periodic point of a diseomorphism of a 2 manifold this paper describes and classifies some trellises the set of homoclinic points is linearly ordered as a subset of the stable manifold and again as a subset of the unstable manifold each homoclinic point is assigned a type number which is constant on its orbit combinatorial properties of trellises are studied using type numbers and the pair of linear orderings trellises are important because their closures in some cases are strange attractors and in other cases are ergodic zones introduction in chapter 33 of new methods of celestial mechanics poincare describes the figure formed by the stable and unstable manifolds of a hyperbolic fixed point of a transformation of the upper half plane as follows when we try to represent the figure formed by these two curves and their intersections each of which corresponds to a doubly asymptotic solution these intersections form a type of trellis tissue or grid with infinitely serrated mesh neither of the two curves must ever 6ut across itself again but must bend back upon itself in a very complex manner in order to cut across all of the meshes in the grid an infinite number of times the complexity of this figure will be striking and i shall not even try to draw it nothing is more suitable for providing us with an idea of the complex nature of the three body problem and of all the problems of dynamics in general the aim of this paper is to understand figures of the type poincare described so well if p is a hyperbolic periodic point of a diffeomorphism of a 2 manifold then the figure formed by its stable and unstable manifolds will be called the trellis of p whenever these manifolds have nonempty intersection figure 1 illustrates the beginning development of such a trellis similarly the figure formed by the stable and unstable manifolds of a cycle of hyperbolic periodic points is called the trellis of the cycle a cycle of hyperbolic periodic points of a diffeomorphism is a finite collection p of periodic points having a cyclic permutation p p such that the unstable manifold of p intersects the stable manifold of p for each p g p it is natural to conjecture that in some cases ergodic zones are the closures of trellises anosov transformations of the torus provide examples where this is the case 1 one may also conjecture that strange attractors are sometimes the closures of the unstable manifolds of cycles of hyperbolic periodic points for certain values of the parameters a and b the henon map of the plane defined by t x y 1 y ax by may provide examples received by the editors may 7 1985 1980 mathematics subject classification primary 58f13 secondary 58f05 58f11 1986 american mathematical society 0002 9947 86 1 00 25 per page

minimal topological chaos coexisting with a finite set of homoclinic and periodic orbits abstract in this note we explain how to find the minimal topological chaos relative to finite set of homoclinic and periodic orbits the main tool is the pruning method which is used for finding a hyperbolic map obtained uncrossing pieces of the invariant manifolds whose basic set contains all orbits forced by the finite set under consideration then we will show applications related to transport phenomena and to the problem of determining the orbits structure coexisting with a finite number of periodic orbits arising from the bouncing ball model

general curvature estimates for stable h surfaces in 3 manifolds applications we obtain an estimate for the norm of the second fundamental form of stable h surfaces in riemannian 3 manifolds with bounded sectional curvature our estimate depends on the distance to the boundary of the surface and on the bound on the sectional curvature but not on the manifold itself we give some applications in particular we obtain an interior gradient estimate for h sections in killing submersions

functional associations at global brain level during perception of an auditory illusion by applying maximal information coefficient abstract maximal information coefficient mic is a recently introduced information theoretic measure of functional association with a promising potential of application to high dimensional complex data sets here we applied mic to reveal the nature of the functional associations between different brain regions during the perception of binaural beat bb bb is an auditory illusion occurring when two sinusoidal tones of slightly different frequency are presented separately to each ear and an illusory beat at the different frequency is perceived we recorded sixty four channels eeg from two groups of participants musicians and non musicians during the presentation of bb and systematically varied the frequency difference from 1 hz to 48 hz participants were also presented non binuaral beat nbb stimuli in which same frequencies were presented to both ears across groups as compared to nbb i bb conditions produced the most robust changes in the mic values at the whole brain level when the frequency differences were in the classical alpha range 8 12 hz and ii the number of electrode pairs showing nonlinear associations decreased gradually with increasing frequency difference between groups significant effects were found for bbs in the broad gamma frequency range 34 48 hz but such effects were not observed between groups during nbb altogether these results revealed the nature of functional associations at the whole brain level during the binaural beat perception and demonstrated the usefulness of mic in characterizing interregional neural dependencies

degenerations of hyperbolic structures ii measured laminations in 3 manifolds that paper concerned the general theory of groups acting on r trees and the relationship of these actions to representations into sl2 c the purpose of the present paper is to develop the foundations of a general theory of measured laminations in 3 manifolds in the third paper 8 the results of this paper will be applied to study actions of 3 manifold groups on trees the starting point of 8 is the idea that dual to an action of qg m on an r tree is a codimension 1 measured lamination in m combining the latter results with 7 gives information about how hyperbolic structures on 3 manifolds can degenerate we view measured laminations in 3 manifolds as generalizations of both geodesic laminations on surfaces and surfaces in 3 manifolds most of this paper consists in extending thurston s theory of geodesic laminations and the hakenstallings waldhausen theory of incompressible surfaces to our context the paper is organized along the following lines in chapter i we develop the basic notions of codimension 1 measured laminations we prove one new and quite useful result theorem i 3 2 concerning a decomposition of these objects we also give a definition of the euler characteristic of a measured lamination and show that it has all the usual properties in chapter ii we introduce the branched surfaces of williams 16 and hatcher thurston we establish analogues for measured laminations carried by branched surfaces of some results of thurston s for train tracks and geodesic laminations by imitating an argument of plante s 10 we show that the leaves of a measured lamination have polynomial growth this is used to show that if a branched surface n carries some lamination all of whose leaves have virtually abelian fundamental groups and carries no disks or spheres then all laminations carried by n have zero euler characteristic we express the latter property by saying that n is flat in chapter iii generalizing haken s notion of normal surfaces we develop the theory of normal measured laminations and of normal branched surfaces in a

locally weighted regression an approach to regression analysis by local fitting abstract locally weighted regression or loess is a way of estimating a regression surface through a multivariate smoothing procedure fitting a function of the independent variables locally and in a moving fashion analogous to how a moving average is computed for a time series with local fitting we can estimate a much wider class of regression surfaces than with the usual classes of parametric functions such as polynomials the goal of this article is to show through applications how loess can be used for three purposes data exploration diagnostic checking of parametric models and providing a nonparametric regression surface along the way the following methodology is introduced a a multivariate smoothing procedure that is an extension of univariate locally weighted regression b statistical procedures that are analogous to those used in the least squares fitting of parametric functions c several graphical methods that are useful tools for understanding loess estimates and checking the a

periodic orbits as the skeleton classical and quantum chaos a description of a low dimensional deterministic chaotic system in terms of unstable periodic orbits cycles is a powerful tool for theoretical and experimental analysis of both classical and quantum deterministic chaos comparable to the familiar perturbation expansions for nearly integrable systems the infinity of orbits characteristic of a chaotic dynamical system can be resummed and brought to a selberg product form dominated by the short cycles and the eigenvalue spectrum of operators associated with the dynamical flow can then be evaluated in terms of unstable periodic orbits methods for implementing this computation for finite subshift dynamics are introduced

ergodicity of billiard flows and quadratic differentials on considere un systeme billard de 2 objets de masses m 1 et m 2 on montre que pour un ensemble dense de paires m 1 m 2 ce systeme est ergodique

continuity properties of entropy on donne une borne superieure sur le defaut en semicontinuite superieure des entropies topologiques et metriques d une auto application c r d une variete c r r 1

when strictly locally convex hypersurfaces are embedded in this paper we will prove hadamard stoker type theorems in the following ambient spaces mathcal m n times mathbb r where mathcal m n is a 1 4 pinched manifold and certain killing submersions e g berger spheres and heisenberg spaces that is under the condition that the principal curvatures of an immersed hypersurface are greater than some non negative constant depending on the ambient space we prove that such a hypersurface is embedded and we also study its topology

a consistent multivariate test of association based on ranks of distances summary we consider the problem of detecting associations between random vectors of any dimension few tests of independence exist that are consistent against all dependent alternatives we propose a powerful test that is applicable in all dimensions and consistent against all alternatives the test has a simple form is easy to implement and has good power

homoclinic tangencies generating partitions and curvature of invariant manifolds a method to compute the curvature of the unstable manifold is introduced and applied to henon map and duffing attractor showing that it allows the authors to locate the homoclinic tangencies and in turn to construct a generating partition the probability distribution of curvature values is investigated showing a power law decay finally the shape of the multifractal spectrum of effective liapunov exponents in nonhyperbolic systems is discussed

transport rates of a class of two dimensional maps and flows abstract a method is developed for estimating the transport rates of phase space areas for a class of two dimensional diffeomorphisms and flows the class of diffeomorphisms we considered are defined by the topological structure of their stable and unstable manifolds and hence are universal we show how to estimate the transport rates for a class of diffeomorphisms found by easton and for an extension of this class of diffeomorphisms which is found via a perturbation of the topology of the stable and unstable manifolds this is done by introducing symbolic dynamics and transfer matrices which in turn relate transport phenomena in phase space to markov processes in a precise manner in addition to the transport rates we use the transfer matrices to obtain estimates for the topological entropy averaged stretching rates and the elongation rate of the unstable manifold the flows we consider are two dimensional time periodic flows which can be reduced via poincare section to the extended family of maps we develop an analytical method based on chirikov s whisker map to classify a given flow according to the structure of its manifolds in its poincare section this allows the techniques developed here for maps to be directly applied to time periodic flows

cleaning up the record on the maximal information coefficient and equitability although we appreciate kinney and atwal s interest in equitability and maximal information coefficient mic we believe they misrepresent our work we highlight a few of our main objections below fig 1 equitability of mic and mutual information under a range of noise models the equitability of mic and mutual information across a subset of noise models analyzed in refs 1 and 4 for each noise model the relationships tested are as in ref 4 in each regarding our original paper 1 kinney and atwal 2 state mic is said to satisfy not just the heuristic notion of equitability but also the mathematical criterion of r2 equitability the latter being their formalization of the heuristic notion that we introduced this statement is simply false we were explicit in our paper that our claims regarding mic s performance were based on large scale simulations we tested mic s equitability through simulations these show that for a large collection of test functions with varied sample sizes noise levels and noise models mic roughly equals the coefficient of determination r2 relative to each respective noiseless function although we mathematically proved several things about mic none of our claims imply that it satisfies kinney and atwal s r2 equitability which would require that mic exactly equal r2 in the infinite data limit thus their proof that no dependence measure can satisfy r2 equitability although interesting does not uncover any error in our work and their suggestion that it does is a gross misrepresentation kinney and atwal seem ready to toss out equitability as a useful criterion based on their theoretical result we argue however that regardless of whether perfect equitability is possible approximate notions of equitability remain the right goal for many data exploration settings just as the theory of np completeness does not suggest we stop thinking about np complete problems but instead that we look for approximations and solutions in restricted cases an impossibility result about perfect equitability provides focus for further research but does not mean that useful solutions are unattainable similarly as others have noted 3 kinney and atwal s proof requires a highly permissive noise model and so the attainability of r2 equitability under more limited noise models such as those in our work remains an open question finally the authors argue that mutual information is more equitable than mic however they provide as justification only a single noise model only at limiting sample sizes n 5 000 as we ve shown in follow up work 4 which they themselves cite but fail to address mic is more equitable than mutual information estimation under many other realistic noise models even at a sample size of 5 000 kinney and atwal have stated it matters how one defines noise 5 and a useful statistic must indeed be robust to a wide range of noise models equally importantly we ve established in both our original and follow up work that at sample size regimes less than 5 000 mic is more equitable than mutual information estimates across all noise models tested mic s superior equitability in these settings is not an artifact we neglected as kinney and atwal suggest but rather a weakness of mutual information estimation and an important consideration for practitioners we expect that the understanding of equitability and mic will improve over time and that better methods may arise however accurate representations of the work thus far will allow researchers in the area to most productively and collectively move forward

equitability mutual information and the maximal information coefficient how should one quantify the strength of association between two random variables without bias for relationships of a specific form despite its conceptual simplicity this notion of statistical equitability has yet to receive a definitive mathematical formalization here we argue that equitability is properly formalized by a self consistency condition closely related to data processing inequality mutual information a fundamental quantity in information theory is shown to satisfy this equitability criterion these findings are at odds with the recent work of reshef et al reshef dn et al 2011 science 334 6062 1518 1524 which proposed an alternative definition of equitability and introduced a new statistic the maximal information coefficient mic said to satisfy equitability in contradistinction to mutual information these conclusions however were supported only with limited simulation evidence not with mathematical arguments upon revisiting these claims we prove that the mathematical definition of equitability proposed by reshef et al cannot be satisfied by any nontrivial dependence measure we also identify artifacts in the reported simulation evidence when these artifacts are removed estimates of mutual information are found to be more equitable than estimates of mic mutual information is also observed to have consistently higher statistical power than mic we conclude that estimating mutual information provides a natural and often practical way to equitably quantify statistical associations in large datasets

local convexity and nonnegative curvature gromov s proof of the sphere theorem an immersed hypersurface s in a riemannian manifold m will be called e convex for some e 0 if all principal curvatures have the same sign and absolute value at least e can one characterize all compact e convex immersed hypersurfaces of a complete manifold m if m is euclidean n space n 3 this problem is solved by a theorem of hadamard 14 generalized by hopf 15 s is embedded and bounds a convex n disk see 6 6 6 3 if m is a flat space form s is no longer necessarily embedded but it still bounds an immersed convex n disk the aim of this paper is to show the same fact for a manifold m of curvature k 0

r2 equitability is satisfiable kinney and atwal 1 make excellent points about mutual information the maximal information coefficient 2 3 and equitability one of their central claims however is that no nontrivial dependence measure can satisfy r 2 r2 equitability we argue that this is the result of a poorly constructed definition which we quote a dependence measure d x y d x y is r 2 r2 equitable if and only if when evaluated on a joint probability distribution p x y p x y that corresponds to a noisy functional relationship between two real random variables x and y the following relation holds d x y g r 2 f x y d x y g r2 f x y here g is a function that does not depend on p x y p x y and f is the function defining the noisy functional relationship i e y f x y f x

estimation of entropy and mutual information we present some new results on the nonparametric estimation of entropy and mutual information first we use an exact local expansion of the entropy function to prove almost sure consistency and central limit theorems for three of the most commonly used discretized information estimators the setup is related to grenander s method of sieves and places no assumptions on the underlying probability measure generating the data second we prove a converse to these consistency theorems demonstrating that a misapplication of the most common estimation techniques leads to an arbitrarily poor estimate of the true information even given unlimited data this inconsistency theorem leads to an analytical approximation of the bias valid in surprisingly small sample regimes and more accurate than the usual 1 n formula of miller and madow over a large region of parameter space the two most practical implications of these results are negative 1 information estimates in a certain data regime are likely contaminated by bias even if bias corrected estimators are used and 2 confidence intervals calculated by standard techniques drastically underestimate the error of the most common estimation methods finally we note a very useful connection between the bias of entropy estimators and a certain polynomial approximation problem by casting bias calculation problems in this approximation theory framework we obtain the best possible generalization of known asymptotic bias results more interesting this framework leads to an estimator with some nice properties the estimator comes equipped with rigorous bounds on the maximum error over all possible underlying probability distributions and this maximum error turns out to be surprisingly small we demonstrate the application of this new estimator on both real and simulated data