positive dependence in markov chains abstract a general notion of positive dependence among successive observations in a finite state stationary process is studied with particular attention to the case of a stationary ergodic markov chain this dependence condition can be expressed as a positivity condition on the joint probability matrices of pairs of observations some useful conditions equivalent to positive dependence are obtained for reversible chains but shown not to be equivalent for nonreversible chains

multivariate smoothing spline functions given data z i g t i varepsilon i 1 leqq i leqq n where g is the unknown function the t i are known d dimensional variables in a domain omega and the varepsilon i are i

testing for stochastic independence application to blind source separation in this paper we address the issue of testing for stochastic independence and its application as a guide to selecting the standard independent component analysis ica algorithms when solving blind source separation bss problems our investigation focuses on the problem of establishing tests for the quality of separation among recovered sources obtained by ica algorithms in an unsupervised environment we review existing tests and propose two contingency table based algorithms the first procedure is based on the measure of goodness of fit of the observed signals to the model of independence provided by the power divergence pd family of test statistics we provide conditions that guarantee the validity of the independence test when the individual sources are nonstationary when the sources exhibit significant time dependence we show how to adopt hotelling s t sup 2 test statistic for zero mean to create an accurate test of independence experimental results obtained from a variety of synthetic and real life benchmark data sets confirm the success of the pd based test when the individual source samples preserve the so called constant cell probability assumption as well as the validity of the t sup 2 based test for sources with significant time dependence

the effect of dependence on chi squared tests of fit tests of fit that assume i i d observations may be affected by dependence among the observations this effect is studied for the pearson chi squared test when the data form a stationary stochastic process the general form of asymptotic distribution theory under the null hypothesis is outlined and examples are given for testing fit to a specified normal law it is shown that when observations come from a quite general class of gaussian stationary processes positive correlation among the observations is confounded with lack of normality

irrotational or divergence free interpolation this paper deals with interpolation of vectorial data in we build specific vectorial splines which interpolate the data and meet some required vectorial properties we so derive irrotational or divergence free interpolating splines numerical examples compare the so obtained interpolating functions with the ones obtained by independent interpolation on each variable

regularity of c 1 smooth surfaces with prescribed p mean curvature in the heisenberg group we consider a c1 smooth surface with prescribed p or h mean curvature in the 3 dimensional heisenberg group assuming only the prescribed p mean curvature h in c 0 we show that any characteristic curve is c2 smooth and its line curvature equals h in the nonsingular domain by introducing characteristic coordinates and invoking the jump formulas along characteristic curves we can prove that the legendrian or horizontal normal gains one more derivative therefore the seed curves are c2 smooth we also obtain the uniqueness of characteristic and seed curves passing through a common point under some mild conditions respectively these results can be applied to more general situations

a codazzi like equation and the singular set for c1 smooth surfaces in the heisenberg group in this paper we study the structure of the singular set for a c1 smooth surface in the 3 dimensional heisenberg group 1 we discover a codazzi like equation for the p area element along the characteristic curves on the surface information obtained from this ordinary differential equation helps us to analyze the local configuration of the singular set and the characteristic curves in particular we can estimate the size and obtain the regularity of the singular set we understand the global structure of the singular set through a hopf type index theorem we also justify the codazzi like equation by proving a fundamental theorem for local surfaces in 1

area stationary surfaces in the heisenberg group h1 we consider area stationary surfaces perhaps with a volume constraint in the heisenberg group h1 endowed with its carnot caratheodory distance by analyzing the first variation of area we characterize c2 area stationary surfaces as those with mean curvature zero or constant if a volume preserving condition is assumed and such that the characteristic curves meet orthogonally the singular curves moreover a minkowski type formula relating the area the mean curvature and the volume is obtained for volume preserving area stationary surfaces enclosing a given region as a consequence of the characterization of area stationary surfaces we refine the bernstein type theorem given in jih hsin cheng jenn fang hwang andrea malchiodi paul yang minimal surfaces in pseudohermitian geometry ann sc norm super pisa cl sci 5 4 1 2005 129 177 mr 2165405 and nicola garofalo scott pauls the bernstein problem in the heisenberg group arxiv math dg 0209065 to describe all c2 entire area stationary graphs over the xy plane in a calibration argument shows that these graphs are globally area minimizing finally by using the description of the singular set in jih hsin cheng jenn fang hwang andrea malchiodi paul yang minimal surfaces in pseudohermitian geometry ann sc norm super pisa cl sci 5 4 1 2005 129 177 mr 2165405 the characterization of area stationary surfaces and the ruling property of constant mean curvature surfaces we prove our main results where we classify area stationary surfaces in with or without a volume constraint and non empty singular set in particular we deduce the following counterpart to alexandrov uniqueness theorem in euclidean space any compact connected c2 surface in area stationary under a volume constraint must be congruent to a rotationally symmetric sphere obtained as the union of all the geodesics of the same curvature joining two points as a consequence we solve the isoperimetric problem in assuming c2 smoothness of the solutions 13 completely reducible hypersurfaces in a pencil pages 672 688 j v pereira s yuzvinsky preview pdf 324 k related articles 14 modules over motivic cohomology pages 689 727 oliver rondigs paul arne ostvaer preview pdf 303 k related articles 15 the fundamental group of period domains over finite fields pages 728 741 sascha orlik preview pdf 154 k related articles articles 1 15

isoperimetric and sobolev inequalities for carnot caratheodory spaces and the existence of minimal surfaces after hormander s fundamental paper on hypoellipticity 54 the study of partial differential equations arising from families of noncommuting vector fields has developed significantly in this paper we study some basic functional and geometric properties of general families of vector fields that include the hormander type as a special case similar to their classical counterparts such properties play an important role in the analysis of the relevant differential operators both linear and nonlinear to motivate our results we recall some classical inequalities let e c r be a caccioppoli set a measurable set having a locally finite perimeter then one has the isoperimetric inequality

strongly stable surfaces in sub riemannian 3 space forms abstract a surface of constant mean curvature cmc equal to h in a sub riemannian 3 manifold is strongly stable if it minimizes the functional area 2 h volume up to second order in this paper we obtain some criteria ensuring strong stability of surfaces in sasakian 3 manifolds we also produce new examples of c 1 complete cmc surfaces with empty singular set in the sub riemannian 3 space forms by studying those ones containing a vertical line as a consequence we are able to find complete strongly stable non vertical surfaces with empty singular set in the sub riemannian hyperbolic 3 space m 1 in relation to the bernstein problem in m 1 we discover strongly stable c entire minimal graphs in m 1 different from vertical planes these examples are in clear contrast with the situation in the first heisenberg group where complete strongly stable surfaces with empty singular set are vertical planes finally we analyze the strong stability of cmc surfaces of class c 2 and non empty singular set in the sub riemannian 3 space forms when these surfaces have isolated singular points we deduce their strong stability even for variations moving the singular set

umbilicity and characterization of pansu spheres in the heisenberg group for n 2 we define a notion of umbilicity for hypersurfaces in the heisenberg group hn we classify umbilic hypersurfaces in some cases and prove that pansu spheres are the only umbilic spheres with positive constant p or horizontal mean curvature in hn up to heisenberg translations

scattered noisy hermite data fitting using an extension of the weighted least squares method given a set of scattered hermite data with noise we use an extension of the weighted least squares method to find the solution based on the bivariate spline this method can adjust some weights according to different noise sizes to get a better approximation we show that our method produces a unique spline to fit the data also we give the error bound for the method in addition we give some probability analysis for our method finally we present some numerical experiments to demonstrate the performance of our method

minimal surfaces and harmonic functions in the heisenberg group abstract we study the blow up of h perimeter minimizing sets in the heisenberg group h n n 2 we show that the lipschitz approximations rescaled by the square root of excess converge to a limit function assuming a stronger notion of local minimality we prove that this limit function is harmonic for the kohn laplacian in a lower dimensional heisenberg group

almost sure convergence of smoothingdm splines for noisy data the purpose of this paper is to study the convergence of smoothingd m splines relative to sets of data perturbed by a random noise conditions of almost sure convergence and error estimates are given

slowly decaying correlations testing normality nuisance parameters abstract slowly decaying serial correlations can cause goodness of fit tests for a distribution to reject the null hypothesis with probability tending to one with increasing sample size when the null distribution is completely specified simple null hypothesis the problem is actually worse than for the case where there are nuisance parameters to be estimated composite null hypothesis in particular we consider here testing normality we discuss limit theorems and propose correction terms to be incorporated in certain goodness of fit statistics to improve the rates of convergence for the simple hypothesis case we show how the problem is solved automatically for the composite null hypothesis case when the nuisance parameters are estimated from the data simulations illustrate the results

rectifiability and perimeter in the heisenberg group in this paper we fully extend to the heisenberg group endowed with its intrinsic carnot caratheodory metric and perimeter the classical de giorgi s rectifiability divergence theorems

regularity of mean curvature flow of graphs on lie groups free up to step 2 abstract we consider smooth solutions of the mean curvature flow of graphs over bounded domains in a lie group free up to step two and not necessarily nilpotent endowed with a one parameter family of riemannian metrics collapsing to a subriemannian metric 0 as 0 we establish c k estimates for this flow that are uniform as 0 and as a consequence prove long time existence for the subriemannian mean curvature flow of the graph our proof extend to the setting of every step two carnot group not necessarily free and can be adapted following our previous work in capogna et al 2013 to the total variation flow

a critical set with nonnull image has large hausdorff dimension the question of how complicated a critical set must be to have a nonnull image is answered by relating its hausdorff dimension to the holder differentiability of the map this leads to a new extension of the morse sard theorem the main tool is an extended version of morse s lemma

the regularity of euclidean lipschitz boundaries with prescribed mean curvature in three dimensional contact sub riemannian manifolds abstract in this paper we consider a set e with prescribed mean curvature f c and euclidean lipschitz boundary e inside a three dimensional contact sub riemannian manifold m we prove that if is locally a regular intrinsic graph the characteristic curves are of class c 2 the result is shape and improves the ones contained in capogna et al 2009 and galli and ritore 2015

vectorial interpolation using radial basis like functions this paper deals with vector field interpolation i e the data are r3 values located in scattered r3 points while the interpolating function is a function from r3 into r3 in order to take into account possible connections between the components of the interpolant we derive it by solving a variational spline problem involving the rotational and the divergence of the interpolant and depending on a parameter significative of the balance of the rotational part and of the divergence part and on the order m of derivatives of the rotational and divergence involved in the minimized seminorm we so obtain interpolants whose expression is x ni 1 x xi ai pm 1 x where is some 3 3 matrix function pm 1 is a degree m 1 vectorial polynomial and where the ai are r3 vectors besides the ai meet a relation generalizing the usual orthogonality to all polynomials of degree at most m 1 for 1 we find the usual m harmonic splines in each component of numerical examples show the interest of the method and we compare the so obtained functions with the ones obtained by matlab s procedures

computer experiments for the analysis of extreme value phenomena the analysis of extreme values occupies an important role in the study of many physical phenomena in many instances measurements of the phenomenon of interest form a time series in which successive observations are statistically dependent i e autocorrelated and frequently the probabilities of occurence of extreme values of such sequences are of interest however the presence of autocorrelation very often does not permit analytical approaches for obtaining such probabilities and related statistical properties this study proposes the use of computer experiments in the analysis of extreme value phenomena in such cases and in particular the use of computer experiments to quantify the effect of dependence in the underlying sequence on the statistical properties of the extreme values computer experiments are described for the investigation of the statistical behavior of the maximum value of an underlying sequence exhibiting an exponential type correlation structure statistical descriptors of the maximum such as its distribution functions and selected moments e g mean and variance are determined and the effects of varying degrees of autocorrelation are quantified effects of other variables such as the size of the underlying sequences also are examined and discussed

approximation by smoothing variational vector splines for noisy data this paper addresses the problem of constructing some free form curves and surfaces from given to different types of data exact and noisy data we extend the theory of d m splines over a bounded domain for noisy data to the smoothing variational vector splines both results of convergence for respectively the exact and noisy data are established as soon as some estimations of errors are given

area stationary and stable surfaces of class c1 in the sub riemannian heisenberg group h1 abstract we consider surfaces of class c 1 in the 3 dimensional sub riemannian heisenberg group h 1 assuming the surface is area stationary i e a critical point of the sub riemannian perimeter under compactly supported variations we show that its regular part is foliated by horizontal straight lines in case the surface is complete and oriented without singular points and stable i e a second order minimum of perimeter we prove that the surface must be a vertical plane this implies the following bernstein type result a complete locally area minimizing intrinsic graph of a c 1 function in h 1 is a vertical plane

a bayesian independence test for small datasets we propose a bayesian test for independence among signals where only a small dataset is available traditional frequentist approaches often fail in this case due to inaccurate estimation of either the source statistical models or the threshold used by the test statistics in addition these frequentist methods cannot incorporate prior information into the computation of the test statistics our procedure renders parametric the nonparametric problem of testing for independence by quantizing the observed data samples into a table of cell counts the test statistic is based on the likelihood of the observed cell counts under the independence hypothesis where the marginal cell probabilities are modeled by independent symmetric dirichlet priors we apply our bayesian test to validate the solutions to the problem of blind source separation with small datasets using both synthetic and real life benchmark data the experimental results indicate that our approach can overcome the scarcity of data samples and significantly outperform the standard frequentist parametric methods with a proper selection of the prior parameters

testing normality in autoregressive models summary it is shown that any test of normality computed from autoregressive residuals has the same limiting null distribution as for the standard case of independent identically distributed observations with estimated parameters some numerical results are given to indicate that this approximation is acceptable for sample size 20 in first and second order models limited numerical results are also given to explore the effect of incorrectly specifying the order it was discovered independently by mukantseva 1977 pierce kopecky 1979 and loynes 1980 that any conventional test of normality computed from regression residuals has the same limiting null distribution as for the case of independent identically distributed observations where only the mean and variance are estimated it is shown here that the same result holds for such tests when computed from residuals in autoregressive models of any given order in both the regression and autoregression cases it is necessary for this result that the fitting includes estimation of the mean i e a constant term in the model the results here could be extended as by pierce kopecky 1979 to tests of distributional form other than normal but this seems of limited interest for the autoregressive situation only asymptotic distribution theory under the null hypothesis of normality is considered here some limited numerical results are given to investigate the rate of convergence these results indicate that for first and second order autoregressive models the asymptotic approximation derived here may be adequate for sample sizes as small as 20 there is the practical difficulty that lack of fit indicated by a test of normality of errors for an autoregressive model of specified order may be due to incorrect specification of the order rather than to nonnormality of errors some very limited numerical results pertaining to this problem are also given here some related theoretical work is given by moore 1982 and moore gleser 1983 the argument is presented in terms of first order autoregression no essential changes being needed for extension to higher order models write the model as

smoothness of lipschitz minimal intrinsic graphs in heisenberg groups hn n 1 we prove that lipschitz intrinsic graphs in the heisenberg groups h n with n 1 which are vanishing viscosity solutions of the minimal surface equation are smooth

approximation by discrete variational splines in this paper we present a numerical approximation of curves and surfaces from a given scattered data set an approximating curve or surface problem is obtained by minimizing a quadratic functional in a parametric finite element space its solution is called a discrete smoothing variational spline the existence and uniqueness of this problem are shown as long as the convergence of the method is established finally some particular examples are given

existence and uniqueness for p area minimizers in the heisenberg group in 3 we studied p mean curvature and the associated p minimal surfaces in the heisenberg group from the viewpoint of pde and differential geometry in this paper we look into the problem through the variational formulation we study a generalized p area and associated p minimizers in general dimensions we prove the existence and investigate the uniqueness of minimizers since this is reduced to solving a degenerate elliptic equation we need to consider the effect of the singular set and this requires a careful study we define the notion of weak solution and prove that in a certain sobolev space a weak solution is a minimizer and vice versa we also give many interesting examples in dimension 2 an intriguing point is that in dimension 2 a c 2 smooth solution from the pde viewpoint may not be a minimizer however this statement is true for higher dimensions due to the relative smallness of the size of the singular set