on hamiltonian stationary lagrangian spheres in non einstein kahler surfaces hamiltonian stationary lagrangian spheres in kahler einstein surfaces are minimal we prove that in the family of non einstein kahler surfaces given by the product 1 2 of two complete orientable riemannian surfaces of different constant gauss curvatures there is only a non minimal hamiltonian stationary lagrangian sphere this example defined when the surfaces 1 and 2 are spheres is unstable

unbounded solutions of the equation asymptotic behavior of solutions of first order differential equation with deviating arguments in the email protected t email protected email protected i t y email protected i y email protected i is discussed for t a criterion for representing solutions in exponential form is proved inequalities for solution estimation are given sufficient conditions for the existence of unbounded solutions are derived a relevant illustrative example is given as well known results are discussed and compared

brief paper preservation of exponential stability for linear non autonomous functional differential systems we consider preservation of exponential stability for a system of linear equations with a distributed delay under the addition of new terms and a delay perturbation as particular cases the system includes models with concentrated delays and systems of integrodifferential equations our method is based on bohl perron type theorems

algebraic relation of two meromorphic mappings on a kahler manifold having the same inverse images of hyperplanes abstract let m be a complete kahler manifold whose universal covering is biholomorphic to a ball b m r 0 0 r 0 in c m in this paper we will show that if two meromorphic mappings f and g from m into p n c have the same inverse images for 2 n 2 hyperplanes h i i 1 2 n 2 with multiplicities counted to level l 0 and satisfy the condition c then the map f g is algebraically degenerate over c where l 0 is a positive integer and is a non negative number explicitly estimated our result generalizes the previous result of h fujimoto for the case of mappings on c m to the case of mappings on a complete kahler manifold as above m and also improves his result by giving an explicit estimate for the number l 0

asymptotic equivalence of neutral and infinite retarded differential equations staffan a demontre qu une equation differentielle fonctionnelle fde neutre avec d operateur stable peut souvent etre traitee comme une fde a retard avec retardement infini le but principal de l article est d illustrer comment les techniques de staffans peuvent etre employees pour etudier le comportement asymptotique de solutions de certaines fde neutres non lineaires et non autonomes en particulier on considere une equation neutre scalaire de la forme d dt x t cx t r f t x t x t r

fractional subpixel diffusion and fuzzy logic approach for ultrasound speckle reduction speckle is the dominant source of noise in ultrasound imaging and is a kind of multiplicative noise it is difficult to design a filter to remove speckle effectively in this paper a novel fuzzy subpixel fractional partial difference fsfpd for ultrasound speckle reduction is proposed euler lagrange equation acts as an increasing function of the fractional derivative s absolute value of the image intensity function the fractional order partial difference is computed in the frequency and fuzzy domain with subpixel precision we test the proposed method on both synthetic and real breast ultrasound bus images the comparisons of the experimental results show that the proposed method can preserve edges and structural details of ultrasound images well while removing speckle noise in addition the filtered images are assessed and evaluated by radiologists using double blind method the results demonstrate that the discrimination rate of breast cancers has been highly improved after employing the proposed method

determining the optimal level of smoothing in cortical thickness analysis a hierarchical approach based on sequential statistical thresholding article i nfo the extent of smoothing applied to cortical thickness maps critically influences sensitivity anatomical precision and resolution of statistical change detection theoretically it could be optimized by increasing the trade off between vertex wise sensitivity and specificity across several levels of smoothing but to date neither parametric nor nonparametric methods are able to control the error at the vertex level if the null hypothesis is rejected after smoothing of cortical thickness maps to overcome these drawbacks we applied sequential statistical thresholding based on a simple hierarchical model this methodology aims at controlling erroneous detections firstly at the level of clusters over smoothed statistical maps and secondly at the vertex level over unsmoothed statistical maps by applying an adaptive false discovery rate fdr procedure to clusters previously detected the superior performance of the proposed methodology over other conventional procedures was demonstrated in simulation studies as expected only the hierarchical method yielded a predictable false discovery proportion near the predefined fdr q value for any smoothing level at the same time as being as sensitive as the others at the optimal setting it was therefore the only method able to approximate the optimal size of spatial smoothing when the true change was assumed unknown the hierarchical method was further validated in a cross sectional study comparing moderate alzheimer s disease ad patients with healthy elderly subjects results suggest that the extent of cortical thinning reported in previous ad studies might be artificially inflated by the choice of inadequate smoothing in these cases interpretation should be based on the location of local maxima of suprathreshold regions rather than on the spatial extent of the detected signal in the statistical parametric map

denoising natural images based on a modified sparse coding algorithm this paper proposes a novel image reconstruction method for natural images using a modified sparse coding sc algorithm proposed by us this sc algorithm exploited the maximum kurtosis as the maximizing sparse measure criterion at one time a fixed variance term of sparse coefficients is used to yield a fixed information capacity on the other hand in order to improve the convergence speed we use a determinative basis function which is obtained by a fast fixed point independent component analysis fastica algorithm as the initialization feature basis function of our sparse coding algorithm instead of using a random initialization matrix the experimental results show that by using our sc algorithm the feature basis vectors of natural images can be successfully extracted then exploiting these features the original images can be reconstructed easily furthermore compared with the standard ica method the experimental results show that our sc algorithm is indeed efficient and effective in performing image reconstruction task

wavelet threshold estimators for data with correlated noise wavelet threshold estimators for data with stationary correlated noise are constructed by applying a level dependent soft threshold to the coefficients in the wavelet transform a variety of threshold choices is proposed including one based on an unbiased estimate of mean squared error the practical performance of the method is demonstrated on examples including data from a neurophysiological context the theoretical properties of the estimators are investigated by comparing them with an ideal but unattainable bench mark that can be considered in the wavelet context as the risk obtained by ideal spatial adaptivity and more generally is obtained by the use of an oracle that provides information that is not actually available in the data it is shown that the level dependent threshold estimator performs well relative to the bench mark risk and that its minimax behaviour cannot be improved on in order of magnitude by any other estimator the wavelet domain structure of both short and long range dependent noise is considered and in both cases it is shown that the estimators have near optimal behaviour simultaneously in a wide range of function classes adapting automatically to the regularity properties of the underlying model the proofs of the main results are obtained by considering a more general multivariate normal decision theoretic problem

harmonic maps and constant mean curvature surfaces in h 2 x r we introduce a it hyperbolic gauss map into the poincar e disk for any surface in bbb h 2 times bbb r with regular vertical projection and prove that if the surface has constant mean curvature h 1 2 this hyperbolic gauss map is harmonic conversely we show that every nowhere conformal harmonic map from an open simply connected riemann surface sigma into the poincar e disk is the hyperbolic gauss map of a two parameter family of such surfaces as an application we obtain that any holomorphic quadratic differential on sigma can be realized as the abresch rosenberg holomorphic differential of some and generically infinitely many complete surfaces with h 1 2 in bbb h 2 times bbb r a similar result applies to minimal surfaces in the heisenberg group rm nil 3 finally we classify all complete minimal vertical graphs in bbb h 2 times bbb r

frequency domain volume rendering by the wavelet x ray transform we describe a wavelet based x ray rendering method in the frequency domain with a smaller time complexity than wavelet splatting standard fourier volume rendering is summarized and interpolation and accuracy issues are briefly discussed we review the implementation of the fast wavelet transform in the frequency domain the wavelet x ray transform is derived and the corresponding fourier wavelet volume rendering algorithm fwvr is introduced fwvr uses haar or b spline wavelets and linear or cubic spline interpolation various combinations are tested and compared with wavelet splatting ws we use medical mr and ct scan data as well as a 3 d analytical phantom to assess the accuracy time complexity and memory cost of both fwvr and ws the differences between both methods are enumerated

empirical bayes approach to improve wavelet thresholding for image noise reduction wavelet threshold algorithms replace small magnitude wavelet coefficients with zero and keep or shrink the other coefficients this is basically a local procedure because wavelet coefficients characterize the local regularity of a function although a wavelet transform has decorrelating properties structures in images like edges are never decorrelated completely and these structures appear in the wavelet coefficients a classification based on a local criterion like coefficient magnitude is not the perfect method to distinguish important uncorrupted coefficients from coefficients dominated by noise we therefore introduce a geometrical prior model for configurations of important wavelet coefficients and combine this with local characterization of a classical threshold procedure into a bayesian framework the local characterization is incorporated into the conditional model whereas the prior model describes only configurations not coefficient values more precisely local characterization favors co

biorthogonal bases of compactly supported wavelets orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide we show here that under fairly general conditions exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual riesz bases of compactly supported wavelets we give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions and we present a sufficient condition for the decay of their fourier transforms we study the regularity of these biorthogonal bases we provide several families of examples all symmetric corresponding to linear phase filters in particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity we also show how to construct symmetric biorthogonal wavelet bases close to a nonsymmetric orthonormal basis

a bayesian spatiotemporal model for very large data sets functional mri provides a unique perspective of neuronal organization however these data include many complex sources of spatiotemporal variability which require spatial preprocessing and statistical analysis for the latter bayesian models provide a promising alternative to classical inference which uses results from gaussian random field theory to assess the significance of spatially correlated statistic images a bayesian approach generalizes the application of these ideas in that 1 random fields are used to model all spatial parameters not solely observation error 2 their smoothness is optimized and 3 a broader class of models can be compared the main problem however is computational due to the large number of voxels in a brain volume sampling methods are time consuming however approximate inference using variational bayes vb offers a principled and transparent way to specify assumptions necessary for computational tractability penny et al 2005b described such a scheme using a joint spatial prior and approximated the joint posterior density with one that factorized over voxels however a further computational bottleneck is encountered when evaluating the log model evidence used to compare models this has lead to dividing a brain volume into slices and treating each independently this amounts to approximating the spatial prior over a full volume with stacked 2d priors that is smoothness along the z axis is not included in the model here we describe a vb scheme that approximates the zero mean joint spatial prior with a non zero mean empirical prior that factors over voxels thereby overcoming this problem we do this by modifying the original vb algorithm of penny et al using the conditional form of a so called conditional autoregressive car prior to update a marginal prior over voxels we refer to this as a spatially informed voxel wise prior svp and use them to spatially regularise general linear model glm and autoregressive ar coefficients over time to model serial correlations this algorithm scales more favourably with the number of voxels providing a truly 3d spatiotemporal model over volumes containing tens of thousands of voxels we compare the scaling of compute times with the number of voxels and performance with a joint prior applied to synthetic and single subject data

preservation of stability in delay equations under delay perturbations we consider a class of linear delay equations with perturbed time lags and present conditions which guarantee that the asymptotic stability of the trivial solution of the equation at hand is preserved under these perturbations as an example we show how our results can be used to obtain an estimate on the maximum allowable sampling interval in the stabilization of a hybrid system with feedback delays we also present applications of our perturbation theorem to obtain stability conditions for delay equations with multiple delays

asymptotic convergence criteria of solutions of delayed functional differential equations abstract the convergence of all solutions of the scalar equation x f t x t is studied under the main assumption that every constant is its solution the criterion and the sufficient conditions for convergence are proved for the considered classes of equations it is among others proved that all solutions are convergent if and only if there exists at least one increasing and convergent solution as applications criteria for some classes of linear equations as well as for nonlinear equations are obtained some comparisons with already known results are given too

robust h sub spl infin performance for linear delay differential systems with time varying uncertainties the present paper concerns robust h spl infin performance for linear delay differential systems which involve an uncertain time delay and time varying norm bounded parameter uncertainties based on the lyapunov functional a simple criterion is proposed which assures the pseudo quadratic stability as a well as an h sub spl infin norm bound the criterion is given in the form of a linear matrix inequality which is affine or convex in additional scalar parameters a simple criterion is presented to evaluate the extent of the performance robustness

a hopf differential for constant mean curvature surfaces in s2 r and h2 r a basic tool in the theory of constant mean curvature cmc surfaces in space forms is the holomorphic quadratic differential discovered by h hopf in this paper we generalize this differential to immersed cmc surfaces in the product spaces s r and h r and prove a corresponding result about the geometry of immersed cmc spheres s in these target spaces

theoretical comparison of fourier and wavelet encoding in magnetic resonance imaging a theoretical comparison of the wavelet and fourier encoding methods is made with respect to resolution sensitivity to artifact and signal to noise ratios snr s a general mathematical description is developed in which magnetic resonance mr image encoding is represented by a projection of the function representing the mr signal density onto an approximation subspace of the finite energy functions characteristics of the subspace are used to define a generalized point spread function for space variant systems using the formal model of mr image encoding it is shown that wavelet encoding approaches the resolution limit defined by fourier encoding artifact is treated according to whether or not the source of the variation in the measured data is stationary nonstationary imperfections perturb the projection operation and result in encoding method dependent effects which can be modeled by a distortion matrix suitable for treating shift variant systems a framework is developed in which to derive expressions for snr s applicable to a general class of mr encoding methods

a nonlocal maximum likelihood estimation method for rician noise reduction in mr images postacquisition denoising of magnetic resonance mr images is of importance for clinical diagnosis and computerized analysis such as tissue classification and segmentation it has been shown that the noise in mr magnitude images follows a rician distribution which is signal dependent when signal to noise ratio snr is low it is particularly difficult to remove the random fluctuations and bias introduced by rician noise the objective of this paper is to estimate the noise free signal from mr magnitude images we model images as random fields and assume that pixels which have similar neighborhoods come from the same distribution we propose a nonlocal maximum likelihood nlml estimation method for rician noise reduction our method yields an optimal estimation result that is more accurate in recovering the true signal from rician noise than nl means algorithm in the sense of snr contrast and method error we demonstrate that nlml performs better than the conventional local maximum likelihood lml estimation method in preserving and defining sharp tissue boundaries in terms of a well defined sharpness metric while also having superior performance in method error

embedded minimal surfaces exotic spheres and manifolds with positive ricci curvature let n be a three dimensional riemannian manifold let e be a closed embedded surface in n then it is a question of basic interest to see whether one can deform in its isotopy class to some canonical embedded surface from the point of view of geometry a natural canonical surface will be the extremal surface of some functional defined on the space of embedded surfaces the simplest functional is the area functional the extremal surface of the area functional is called the minimal surface such minimal surfaces were used extensively by meeks yau my21 in studying group actions on three dimensional manifolds in my2 the theory of minimal surfaces was used to simplify and strengthen the classical dehn s lemma loop theorem and the sphere theorem in the setting there one minimizes area among all immersed surfaces and proves that the extremal object is embedded in this paper we minimize area among all embedded surfaces isotopic to a fixed embedded surface in the category of these surfaces we prove a general existence theorem theorem 1 a particular consequence of this theorem is that for irreducible manifolds an embedded incompressible surface is isotopic to an embedded incompressible surface with minimal area we also prove that there exists an embedded sphere of least area enclosing a fake cell provided the complementary volume is not a standard ball and provided there exists no embedded one sided rp2 by making use of the last result and a cutting and pasting argument we are able to settle a well known problem in the theory of three dimensional manifolds we prove that the covering space of any irreducible orientable three dimensional manifold is irreducible it is possible to exploit our existence theorem to study

anomaly detection in images with smooth background via smooth sparse decomposition in various manufacturing applications such as steel composites and textile production anomaly detection in noisy images is of special importance although there are several methods for image den

notes on submanifolds in warped products in this paper we study submanifolds in the warped product m fr where m is a riemannian manifold f is the warping function we investigate the graphs with positive gaussian curvature in m2 r the minimal graphs in m fr and the totally geodesic submanifolds in m fr

denoising functional mr images a comparison of wavelet denoising and gaussian smoothing we present a general wavelet based denoising scheme for functional magnetic resonance imaging fmri data and compare it to gaussian smoothing the traditional denoising method used in fmri analysis one dimensional wavelab thresholding routines were adapted to two dimensional 2 d images and applied to 2 d wavelet coefficients to test the effect of these methods on the signal to noise ratio snr we compared the snr of 2 d fmri images before and after denoising using both gaussian smoothing and wavelet based methods we simulated a fmri series with a time signal in an active spot and tested the methods on noisy copies of it the denoising methods were evaluated in two ways by the average temporal snr inside the original activated spot and by the shape of the spot detected by thresholding the temporal snr maps denoising methods that introduce much smoothness are better suited for low snrs but for images of reasonable quality they are not preferable because they introduce heavy deformations wavelet based denoising methods that introduce less smoothing preserve the sharpness of the images and retain the original shapes of active regions we also performed statistical parametric mapping on the denoised simulated time series as well as on a real fmri data set false discovery rate control was used to correct for multiple comparisons the results show that the methods that produce smooth images introduce more false positives the less smoothing wavelet based methods although generating more false negatives produce a smaller total number of errors than gaussian smoothing or wavelet based methods with a large smoothing effect

nonexistence of complete conformal metrics with prescribed scalar curvatures on a manifold of nonpositive curvature abstract let m g be a complete simply connected riemannian manifold with nonpositive curvature k its scalar curvature and k a smooth function on m we obtain a nonexistence result of complete metrics on m conformal to g and with k as their scalar curvature

estimation of false discovery rates for wavelet denoised statistical parametric maps correction for multiple comparisons in neuroimaging data is an important area of research recently wavelet based methods have gained popularity and have been reported to achieve better sensitivity compared to spatial domain methods however these techniques produce smoothed statistical maps which are difficult to interpret the generated maps have to be thresholded again in the spatial domain to delineate active from inactive regions the selection of a proper threshold satisfying the required error rate control is not straightforward in this paper a framework is proposed for thresholding wavelet denoised maps in which a rejection region is fixed and the achieved false discovery rate fdr is estimated this approach provides a meaningful strategy to choose thresholds for wavelet denoised statistical parametric maps spms two fdr estimation algorithms were used to assess the achieved error rate control when thresholding wavelet filtered spms at various rejection regions their performance was evaluated using both simulated and resting fmri data the proposed framework was also applied on in vivo data

differential inequalities on complete riemannian manifolds and applications this paper treats various aspects of the asymptotic behavior of solutions of certain elliptic equations of geometric interest on complete riemannian manifolds sharp results relating the rate of volume growth of a complete riemannian manifold and the growth of its harmonic and subharmonic functions can be found in e22 together with references to related results in sect 2 of this paper we consider solutions of the more restrictive inequality au e o where div o grad and show that these must be unbounded if m n has even quadratically exponential volume growth and that u must have faster than linear growth if t

constant mean curvature spacelike hypersurfaces in lorentzian warped products and calabi bernstein type problems abstract in this paper we provide several uniqueness and non existence results for complete parabolic constant mean curvature spacelike hypersurfaces in lorentzian warped products under appropriate geometric assumptions as a consequence of this parametric study we obtain very general uniqueness and non existence results for a large family of uniformly elliptic edp s so solving the calabi bernstein problem in a wide family of spacetimes

noise estimation and reduction in magnetic resonance imaging using a new multispectral nonlocal maximum likelihood filter denoising of magnetic resonance mr images enhances diagnostic accuracy the quality of image manipulations such as registration and segmentation and parameter estimation the first objective of this paper is to introduce a new high performance nonlocal filter for noise reduction in mr image sets consisting of progressively weighted that is multispectral images this filter is a multispectral extension of the nonlocal maximum likelihood filter nlml performance was evaluated on synthetic and textit in vivo t 2 and t 1 weighted brain imaging data and compared to the nonlocal means nlm and its multispectral version that is ms nlm and the nonlocal maximum likelihood nlml filters visual inspection of filtered images and quantitative analyses showed that all filters provided substantial reduction of noise further as expected the use of multispectral information improves filtering quality in addition numerical and experimental analyses indicated that the new multispectral nlml filter ms nlml demonstrated markedly less blurring and loss of image detail than seen with the other filters evaluated in addition since noise standard deviation sd is an important parameter for all of these nonlocal filters a multispectral extension of the method of maximum likelihood estimation mle of noise amplitude is presented and compared to both local and nonlocal mle methods numerical and experimental analyses indicated the superior performance of this multispectral method for estimation of noise sd

ends of hyperbolic 3 manifolds let n h3 f be a hyperbolic 3 manifold which is homeomorphic to the interior of a compact 3 manifold we prove that n is geometrically tame as a consequence we prove that f s limit set l is either the entire sphere at infinity or has measure zero we also prove that n s geodesic flow is ergodic if and only if lf is the entire sphere at infinity department of mathematics stanford university stanford california 94305 current address department of mathematics university of michigan ann arbor michigan 48109 e mail address dick canary math lsa umich edu this content downloaded from 40 77 167 14 on wed 15 jun 2016 05 36 06 utc all use subject to http about jstor org terms

prescribing curvature on open surfaces in this article we study the problem sometimes called the berger nirenberg problem of prescribing the curvature on a riemann surface that is on an oriented surface equipped with a conformal class of riemannian metrics

positive solutions of yamabe type equations on complete manifolds and applications we study the semilinear equation u a x u b x u 1 on a complete riemannian manifold we determine conditions on the coefficients that guarantee existence and nonexistence of positive solutions a very general uniqueness result is also established our main results are valid without explicit curvature assumptions and appear to be new even in rm

bayesian hierarchical multi subject multiscale analysis of functional mri data we develop a methodology for bayesian hierarchical multi subject multiscale analysis of functional magnetic resonance imaging fmri data we begin by modeling the brain images temporally with a standard general linear model after that we transform the resulting estimated standardized regression coefficient maps through a discrete wavelet transformation to obtain a sparse representation in the wavelet space subsequently we assign to the wavelet coefficients a prior that is a mixture of a point mass at zero and a gaussian white noise in this mixture prior for the wavelet coefficients the mixture probabilities are related to the pattern of brain activity across different resolutions to incorporate this information we assume that the mixture probabilities for wavelet coefficients at the same location and level are common across subjects furthermore we assign for the mixture probabilities a prior that depends on a few hyperparameters we develop an empirical bayes methodology to estimate the hyperparameters and as these hyperparameters are shared by all subjects we obtain precise estimated values then we carry out inference in the wavelet space and obtain smoothed images of the regression coefficients by applying the inverse wavelet transform to the posterior means of the wavelet coefficients an application to computer simulated synthetic data has shown that when compared to single subject analysis our multi subject methodology performs better in terms of mean squared error finally we illustrate the utility and flexibility of our multi subject methodology with an application to an event related fmri dataset generated by postle 2005 through a multi subject fmri study of working memory related brain activation

wavelets and functional magnetic resonance imaging of the human brain the discrete wavelet transform dwt is widely used for multiresolution analysis and decorrelation or whitening of nonstationary time series and spatial processes wavelets are naturally appropriate for analysis of biological data such as functional magnetic resonance images of the human brain which often demonstrate scale invariant or fractal properties we provide a brief formal introduction to key properties of the dwt and review the growing literature on its application to fmri we focus on three applications in particular i wavelet coefficient resampling or wavestrapping of 1 d time series 2 to 3 d spatial maps and 4 d spatiotemporal processes ii wavelet based estimators for signal and noise parameters of time series regression models assuming the errors are fractional gaussian noise fgn and iii wavelet shrinkage in frequentist and bayesian frameworks to support multiresolution hypothesis testing on spatially extended statistic maps we conclude that the wavelet domain is a rich source of new concepts and techniques to enhance the power of statistical analysis of human fmri data

the existence of embedded minimal surfaces and the problem of uniqueness it was in the early nineteen thirties that douglas and radd solved the plateau problem for a rectifiable jordan curve in euclidean space the minimal surfaces that they obtained are realized by conformal harmonic maps from the unit disk in 1948 morrey generalized the theorem of douglas and radd to minimal surfaces in a homogeneously regular riemannian manifold in both cases the solutions are defective as they may possess branch points it was not until 1968 that osserman 24 was able to prove that interior branch points do not exist on the douglas radd morrey solution osserman s theorem in this full generality was proved by alt 2 3 and gulliver 8 if the jordan curve is real analytic gulliver and lesley 9 showed that the solution surface is free of branch points even on its boundary hence at least in this case we know that the douglas radd morrey solution is an immersion and represents a classical surface however the douglas radd morrey solution may in general have selfintersections it was then a classical problem to find conditions on the jordan curve to guarantee that the solution is an embedded disk in 1932 radd was the first to produce a general condition of this type he proved that if the curve can be projected onto a convex curve in the plane then it bounds a unique embedded minimal disk which is a graph over this plane it was not until the seventies that there was further progress on the problem of embeddedness osserman had conjectured that if a jordan curve is extreme that is lies on the boundary of its convex hull then the douglas rad6 solution is embedded gulliver and spruck 10 proved this conjecture under the additional assumption that the absolute total curvature of the jordan curve is less than 4 z right after this work tomi and tromba 32 showed that an extreme curve must bound an embedded minimal disk which need not be stable in 1976 almgren and simon 1 established the existence for such a curve of an embedded stable minimal disk which need not be a douglas radd solution finally in the same year the authors 18 proved that any douglas rad6 solution must be embedded

a new approach for uniqueness of complete maximal hypersurfaces in spatially parabolic grw spacetimes abstract complete maximal hypersurfaces in generalized robertson walker spacetimes whose fiber has a parabolic universal riemannian covering are studied under natural boundedness assumptions a spacelike hypersurface is shown to be parabolic when it is endowed with a conformal metric this fact allows us to get for complete maximal hypersurfaces several uniqueness results as an application all the entire solutions to the maximal hypersurface equation under certain constrains are obtained solving new calabi bernstein problems

adapting to unknown smoothness via wavelet shrinkage abstract we attempt to recover a function of unknown smoothness from noisy sampled data we introduce a procedure sureshrink that suppresses noise by thresholding the empirical wavelet coefficients the thresholding is adaptive a threshold level is assigned to each dyadic resolution level by the principle of minimizing the stein unbiased estimate of risk sure for threshold estimates the computational effort of the overall procedure is order n log n as a function of the sample size n sureshrink is smoothness adaptive if the unknown function contains jumps then the reconstruction essentially does also if the unknown function has a smooth piece then the reconstruction is essentially as smooth as the mother wavelet will allow the procedure is in a sense optimally smoothness adaptive it is near minimax simultaneously over a whole interval of the besov scale the size of this interval depends on the choice of mother wavelet we know from a previous paper by the authors that traditional smoot

a schoen theorem for minimal surfaces in h 2 r abstract in this paper we prove that a complete minimal surface immersed in h 2 r with finite total curvature and two ends each one asymptotic to a vertical geodesic plane must be a horizontal catenoid moreover we give a geometric description of minimal ends of finite total curvature in h 2 r we also prove that a minimal complete end e with finite total curvature is properly immersed and that the gaussian curvature of e is locally bounded in terms of the geodesic distance to its boundary

wspm wavelet based statistical parametric mapping recently we have introduced an integrated framework that combines wavelet based processing with statistical testing in the spatial domain in this paper we propose two important enhancements of the framework first we revisit the underlying paradigm i e that the effect of the wavelet processing can be considered as an adaptive denoising step to improve the parameter map followed by a statistical detection procedure that takes into account the non linear processing of the data with an appropriate modification of the framework we show that it is possible to reduce the bias of the method with respect to the best linear estimate providing conservative results that are closer to the original data second we propose an extension of our earlier technique that compensates for the lack of shift invariance of the wavelet transform we demonstrate experimentally that both enhancements have a positive effect on performance in particular we present a reproducibility study for multi session data that compares wspm against spm with different amounts of smoothing the full approach is available as a toolbox named wspm for the spm2 software it takes advantage of multiple options and features of spm such as the general linear model

detection of focal changes in human cortical thickness spherical wavelets versus gaussian smoothing subtle but progressive variations in human cortical thickness have been associated with the initial phases of prevalent neurological and psychiatric conditions but slight changes in cortical thickness at preclinical stages are typically masked by effects of the gaussian kernel smoothing on the cortical surface shape descriptors here we present the first study aimed at detecting changes in human cortical thickness maps by applying soft thresholding to multiresolution spherical wavelet coefficients in order to make gaussian and wavelet smoothing methods comparable the trade off between sensitivity and specificity was optimized to detect simulated thickness changes in various cortical areas of healthy elderly subjects results revealed a better sensitivity specificity trade off when using wavelet based methods as compared to gaussian smoothing in both the whole neocortex p 10 7 and cortical region based statistical analyses p 10 9 which was mainly due to the higher specificity obtained with the wavelet approach the lower smoothing introduced by wavelets and their adaptive properties may account for the enhanced specificity and sensitivity when compared with gaussian spatial filters these results strongly support the use of spherical wavelet methods to detect subtle variations in cortical thickness maps which may be crucial in better understanding the course of neuronal loss in normal aging and in finding early markers of cortical degeneration

a non integrated defect relation and the uniqueness theorem for meromorphic maps of a complete kahler manifold into pn c abstract in this paper a non integrated defect relation is established for meromorphic maps of an m dimensional complete kahler manifold m into p n c intersecting q hypersurfaces d 1 d q in p n c located in k subgeneral position where d 1 d q are said to be in k subgeneral position if at most k k n of these hypersurfaces meet at one point in p n c the related uniqueness theorem is also given

fractional splines and wavelets we extend schoenberg s family of polynomial splines with uniform knots to all fractional degrees alpha 1 these splines which involve linear combinations of the one sided power functions x alpha max 0 x alpha are alpha holder continuous for alpha 0 we construct the corresponding b splines by taking fractional finite differences and provide an explicit characterization in both time and frequency domains we show that these functions satisfy most of the properties of the traditional b splines including the convolution property and a generalized fractional differentiation rule that involves finite differences only we characterize the decay of the b splines that are not compactly supported for nonintegral alpha s their most astonishing feature in reference to the strang fix theory is that they have a fractional order of approximation alpha 1 while they reproduce the polynomials of degree lceil alpha rceil for alpha frac 1 2 they satisfy all the requirements for a multiresolution analysis of ll 2 riesz bounds two scale relation and may therefore be used to build new families of wavelet bases with a continuously varying order parameter our construction also yields symmetrized fractional b splines which provide the connection with duchon s general theory of radial m s splines including thin plate splines in particular we show that the symmetric version of our splines can be obtained as the solution of a variational problem involving the norm of a fractional derivative

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

the control of the false discovery rate in multiple testing under dependency benjamini and hochberg suggest that the false discovery rate may be the appropriate error rate to control in many applied multiple testing problems a simple procedure was given there as an fdr controlling procedure for independent test statistics and was shown to be much more powerful than comparable procedures which control the traditional familywise error rate we prove that this same procedure also controls the false discovery rate when the test statistics have positive regression dependency on each of the test statistics corresponding to the true null hypotheses this condition for positive dependency is general enough to cover many problems of practical interest including the comparisons of many treatments with a single control multivariate normal test statistics with positive correlation matrix and multivariate t furthermore the test statistics may be discrete and the tested hypotheses composite without posing special difficulties for all other forms of dependency a simple conservative modification of the procedure controls the false discovery rate thus the range of problems for which a procedure with proven fdr control can be offered is greatly increased 1 1 simultaneous hypotheses testing the control of the increased type i error when testing simultaneously a family of hypotheses is a central issue in the area of multiple comparisons rarely are we interested only in whether all hypotheses are jointly true or not which is the test of the intersection null hypothesis in most applications we infer about the individual hypotheses realizing that some of the tested hypotheses are usually true we hope not all and some are not we wish to decide which ones are not true indicating statistical discoveries an important such problem is that of multiple endpoints in a clinical trial a new treatment is compared with an existing one in terms of a large number of potential benefits endpoints

entire radially symmetric graphs with prescribed mean curvature in warped product spaces abstract this study investigates entire radially symmetric graphs in the warped product m f r where m is a riemannian manifold with a pole and f m r is the warping function the main results demonstrate the existence and uniqueness of entire radially symmetric graphs with prescribed mean curvature

non integrated defect of meromorphic maps on kahler manifolds the purpose of this article is twofold the first is to establish a truncated non integrated defect relation for meromorphic mappings from a complete kahler manifold quotien of a ball into a projective variety intersecting hypersurfaces in subgeneral position we also apply it to the gauss mapping from a closed regular submanifold of mathbb c m the second aim is to establish an above type theorem with truncation level 1 for differentiably nondegenerate meromorphic mappings

which ambient spaces admit isoperimetric inequalities for submanifolds we give simple conditions on an ambient manifold that are necessary and sufficient for isoperimetric inequalities to hold

wavelet based fmri analysis 3 d denoising signal separation and validation metrics we present a novel integrated wavelet domain based framework w ica for 3 d denoising functional magnetic resonance imaging fmri data followed by source separation analysis using independent component analysis ica in the wavelet domain we propose the idea of a 3 d wavelet based multi directional denoising scheme where each volume in a 4 d fmri data set is sub sampled using the axial sagittal and coronal geometries to obtain three different slice by slice representations of the same data the filtered intensity value of an arbitrary voxel is computed as an expected value of the denoised wavelet coefficients corresponding to the three viewing geometries for each sub band this results in a robust set of denoised wavelet coefficients for each voxel given the de correlated nature of these denoised wavelet coefficients it is possible to obtain more accurate source estimates using ica in the wavelet domain the contributions of this work can be realized as two modules first in the analysis module we combine a new 3 d wavelet denoising approach with signal separation properties of ica in the wavelet domain this step helps obtain an activation component that corresponds closely to the true underlying signal which is maximally independent with respect to other components second we propose and describe two novel shape metrics for post ica comparisons between activation regions obtained through different frameworks we verified our method using simulated as well as real fmri data and compared our results against the conventional scheme gaussian smoothing spatial ica s ica the results show significant improvements based on two important features 1 preservation of shape of the activation region shape metrics and 2 receiver operating characteristic curves it was observed that the proposed framework was able to preserve the actual activation shape in a consistent manner even for very high noise levels in addition to significant reduction in false positive voxels

integrated wavelet processing and spatial statistical testing of fmri data we introduce an integrated framework for detecting brain activity from fmri data which is based on a spatial discrete wavelet transform unlike the standard wavelet based approach for fmri analysis we apply the suitable statistical test procedure in the spatial domain for a desired significance level this scheme has one remaining degree of freedom characterizing the wavelet processing which is optimized according to the principle of minimal approximation error this allows us to determine the threshold values in a way that does not depend on data while developing our framework we make only conservative assumptions consequently the detection of activation is based on strong evidence we have implemented this framework as a toolbox wspm for the spm2 software taking advantage of multiple options and functions of spm such as the setup of the linear model and the use of the hemodynamic response function we show by experimental results that our method is able to detect activation patterns the results are comparable to those obtained by spm even though statistical assumptions are more conservative

controlling the false discovery rate a practical and powerful approach to multiple testing summary the common approach to the multiplicity problem calls for controlling the familywise error rate fwer this approach though has faults and we point out a few a different approach to problems of multiple significance testing is presented it calls for controlling the expected proportion of falsely rejected hypotheses the false discovery rate this error rate is equivalent to the fwer when all hypotheses are true but is smaller otherwise therefore in problems where the control of the false discovery rate rather than that of the fwer is desired there is potential for a gain in power a simple sequential bonferronitype procedure is proved to control the false discovery rate for independent test statistics and a simulation study shows that the gain in power is substantial the use of the new procedure and the appropriateness of the criterion are illustrated with examples

the geometry of finite topology bryant surfaces we prove that in the three dimensional hyperbolic space h 3 the properly embedded constant mean curvature one surfaces with finite topology are of finite total curvature and each ends are regular this imply in particular that the hrosphere is the only properly embedded simply connected constant mean curvature one surface in h 3

statistical approach to segmentation of single channel cerebral mr images a statistical model is presented that represents the distributions of major tissue classes in single channel magnetic resonance mr cerebral images using the model cerebral images are segmented into gray matter white matter and cerebrospinal fluid csf the model accounts for random noise magnetic field inhomogeneities and biological variations of the tissues intensity measurements are modeled by a finite gaussian mixture smoothness and piecewise contiguous nature of the tissue regions are modeled by a three dimensional 3 d markov random field mrf a segmentation algorithm based on the statistical model approximately finds the maximum a posteriori map estimation of the segmentation and estimates the model parameters from the image data the proposed scheme for segmentation is based on the iterative conditional modes icm algorithm in which measurement model parameters are estimated using local information at each site and the prior model parameters are estimated using the segmentation after each cycle of iterations application of the algorithm to a sample of clinical mr brain scans comparisons of the algorithm with other statistical methods and a validation study with a phantom are presented the algorithm constitutes a significant step toward a complete data driven unsupervised approach to segmentation of mr images in the presence of the random noise and intensity inhomogeneities

a hopf theorem for non constant mean curvature and a conjecture of a d alexandrov we prove a uniqueness theorem for immersed spheres of prescribed non constant mean curvature in homogeneous three manifolds in particular this uniqueness theorem proves a conjecture by a d alexandrov about immersed spheres of prescribed weingarten curvature in mathbb r 3 for the special but important case of prescribed mean curvature as a consequence we extend the classical hopf uniqueness theorem for constant mean curvature spheres to the case of immersed spheres of prescribed antipodally symmetric mean curvature in mathbb r 3

non integrated defect relation for meromorphic maps from a kahler manifold intersecting hypersurfaces in subgeneral of pn c abstract in this article we establish a truncated non integrated defect relation for meromorphic mappings from an m dimensional complete kahler manifold into p n c intersecting q hypersurfaces q 1 q q in k subgeneral position of degree d i i e the intersection of any k 1 hypersurfaces is emptyset we will prove that i 1 q f u 1 q i k n 1 n 1 u u 1 d where u is explicitly estimated and d is the least common multiple of d i s our result generalizes and improves previous results in the last part of this paper we will apply this result to study the distribution of the gauss map of minimal surfaces

a costa hoffman meeks type surface in mathbb h 2 times mathbb r we show the existence in the space 2 x of a family of embedded minimal surfaces of genus 1 k and finite total extrinsic curvature with two catenoidal type ends and one middle planar end the proof is based on a gluing procedure

modeling brain activation in fmri using group mrf noise confounds present serious complications to functional magnetic resonance imaging fmri analysis the amount of discernible signals within a single dataset of a subject is often inadequate to obtain satisfactory intra subject activation detection to remedy this limitation we propose a novel group markov random field gmrf that extends each subject s neighborhood system to other subjects to enable information coalescing a distinct advantage of gmrf over standard fmri group analysis is that no stringent one to one voxel correspondence is required instead intra and inter subject neighboring voxels are jointly regularized to encourage spatially proximal voxels to be assigned similar labels across subjects our proposed group extended graph structure thus provides an effective means for handling inter subject variability also adopting a group wise approach by integrating group information into intra subject activation as opposed to estimating a single average group map permits inter subject differences to be characterized and studied gmrf can be elegantly implemented as a single mrf thus enabling all subjects activation maps to be simultaneously and collaboratively segmented with global optimality guaranteed in the case of binary labeling we validate our technique on synthetic and real fmri data and demonstrate gmrf s superior performance over standard fmri analysis

least area incompressible surfaces in 3 manifolds let m be a riemannian manifold and let f be a closed surface a map f f m is called least area if the area of f is less than the area of any homotopic map from f to m note that least area maps are always minimal surfaces but that in general minimal surfaces are not least area as they represent only local stationary points for the area function the existence of least area immersions in a homotopy class of maps has been established when the homotopy class satisfies certain injectivity conditions on the fundamental group 18 17 in this paper we shall consider the possible singularities of such immersions our results show that the general philosophy is that least area surfaces intersect least meaning that the intersections and self intersections of least area immersions are as small as their homotopy classes allow when measured correctly one should note that evidence supporting this view had been found by meeks yau in their embedding theorems for minimal disks and 2 spheres 13 143 our main result asserts that if a least area immersion is homotopic to an embedding then it has no self intersections which clearly exemplifies the above philosophy the precise result is the following

stable minimal surfaces in m ℝ in 4 we developed the theory of properly embedded minimal surfaces in m r where m is a compact riemannian surface one of the first results in 4 is that a properly embedded noncompact minimal surface in m r of bounded gaussian curvature is quasiperiodic in the following sense given any sequence of vertical translates n of a subsequence of the n converges on compact subsets of m r to another properly embedded minimal surface by the curvature estimates of schoen 8 every properly embedded stable minimal surface in m r has bounded curvature therefore every properly embedded noncompact stable minimal surface in m r is quasiperiodic this quasiperiodicity property will be essential in proving the next theorem throughout this paper m denotes a compact riemannian surface

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed