existence of self shrinkers to the degree one curvature flow with a rotationally symmetric conical end abstract given a smooth symmetric homogeneous of degree one function f 1 n satisfying i f 0 for all i 1 n and a rotationally symmetric cone c in r n 1 we show that there is a f self shrinker i e a hypersurface in r n 1 which satisfies f 1 n 1 2 x n 0 where x is the position vector n is the unit normal vector and 1 n are principal curvatures of that is asymptotic to c at infinity

universality in mean curvature flow neckpinches we study noncompact surfaces evolving by mean curvature flow without any symmetry assumptions we prove that any solution that is c 3 close at some time to a standard neck will develop a neckpinch singularity in finite time will become asymptotically rotationally symmetric in a space time neighborhood of its singular set and will have a unique tangent flow

rigidity of self shrinkers and translating solitons of mean curvature flows abstract in this paper we prove that any complete m dimensional spacelike self shrinkers in pseudo euclidean spaces r n m n must be affine planes and there exists no complete m dimensional spacelike translating soliton in r n m n these results are proved by using a new omori yau maximal principle we also derive a rigidity theorem of self shrinking hypersurfaces in euclidean space with gauss image lies in a regular ball

generic mean curvature flow i generic singularities it has long been conjectured that starting at a generic smooth closed embedded surface in r 3 the mean curvature ow remains smooth until it arrives at a singularity in a neighborhood of which the ow looks like concentric spheres or cylinders that is the only singularities of a generic ow are spherical or cylindrical we will address this conjecture here and in a sequel the higher dimensional case will be addressed elsewhere the key to showing this conjecture is to show that shrinking spheres cylinders and planes are the only stable self shrinkers under the mean curvature ow we prove this here in all dimensions an easy consequence of this is that every singularity other than spheres and cylinders can be perturbed away

embedded minimal surfaces of finite topology in this paper we prove that any complete embedded minimal surface m in r 3 with finite topology and compact boundary possibly empty is conformally a compact riemann surface m with boundary punctured in a finite number of interior points and that m can be represented in terms of meromorphic data on its conformal completion m in particular we demonstrate that m is a minimal surface of finite type and describe how this property permits a classification of the asymptotic behavior ofm

kinetic analysis of the lower limbs during walking what information can be gained from a three dimensional model kinetic analyses joint moments powers and work of the lower limbs were performed during normal walking to determine what further information can be gained from a three dimensional model over planar models it was to be determined whether characteristic moment and power profiles exist in the frontal and transverse planes across subjects and how much work was performed in these planes kinetic profiles from nine subjects were derived using a three dimensional inverse dynamics model of the lower limbs and power profiles were then calculated by a dot product of the angular velocities and joint moments resolved in a global reference system characteristic joint moment profiles across subjects were found for the hip knee and ankle joints in all planes except for the ankle frontal moment as expected the major portion of work was performed in the plane of progression since the goal of locomotion is to support the body against gravity while generating movements which propel the body forward however the results also showed that substantial work was done in the frontal plane by the hip during walking 23 of the total work at that joint the characteristic joint profiles suggest defined motor patterns and functional roles in the frontal and transverse planes kinetic analysis in three dimensions is necessary particularly if the hip joint is being examined as a substantial amount of work was done in the frontal plane of the hip to control the pelvis and trunk against gravitational forces

morphological texture based maximum likelihood pixel classification based on local granulometric moments abstract morphological granulometries are one parameter filter sequences that monotonically decrease image area a size distribution is generated by measuring the residual area after each iteration of the filter sequence normalization yields a probability distribution function whose moments can be employed as image signatures by measuring residual area locally over a window about each point of an image instead of over the entire image local texture features are generated at each pixel and these features can be employed for pixel classification by using several granulometric structuring element generating sequences numerous moment sets result each carrying different textural information a detailed analysis of this pixel classification methodology using a gaussian maximum likelihood classifier is provided included is a statistical study of classification accuracy feature optimization and robustness with respect to various relevant noise models

carleman inequalities for the dirac and laplace operators and unique continuation on demontre les meilleures inegalites de type carleman possibles dans le cas de l operateur de dirac on obtient la propriete de prolongement unique pour d v ou v l loc r n 3n 2 2 n 3

morphological granulometric estimation of random patterns in the context of parameterized random sets abstract morphological features are used to estimate the state of a random pattern set governed by a multivariate probability distribution the feature vector is composed of granulometric moments and pattern estimation involves feature based estimation of the parameter vector governing the random set under such circumstances the joint density of the features and parameters is a generalized function concentrated on a solution manifold and estimation is determined by the conditional density of the parameters given an observed feature vector the paper explains the manner in which the joint probability mass of the parameters and features is distributed and the way the conditional densities give rise to optimal estimators according to the distribution of probability mass whether constrained or not to the solution manifold the estimation theory is applied using analytic representation of linear granulometric moments the effects of random perturbations in the shape parameter vector is discussed and the theory is applied to random sets composed of disjoint random shapes the generalized density framework provides a proper mathematical context for pattern estimation and gives insight via the distribution of mass on solution manifolds to the manner in which morphological probes discriminate random sets relative to their distributions and the manner in which the use of additional probes can be beneficial for better estimation

validation of net joint loads calculated by inverse dynamics in case of complex movements application to balance recovery movements the joint forces and moments driving the motion of a human subject are classically computed by an inverse dynamic calculation however even if this process is theoretically simple many sources of errors may lead to huge inaccuracies in the results moreover a direct comparison with in vivo measured loads or with gold standard values from literature is only possible for very specific studies therefore assessing the inaccuracy of inverse dynamic results is not a trivial problem and a simple method is still required this paper presents a simple method to evaluate both 1 the consistency of the results obtained by inverse dynamics 2 the influence of possible modifications in the inverse dynamic hypotheses this technique concerns recursive calculation performed on full kinematic chains and consists in evaluating the loads obtained by two different recursive strategies it has been applied to complex 3d whole body movements of balance recovery a recursive newton euler procedure was used to compute the net joint loads two models were used to represent the subject bodies considering or not the upper body as a unique rigid segment the inertial parameters of the body segments were estimated from two different sets of scaling equations de leva p 1996 adjustments to zatsiorsky suleyanov s segment inertia parameters journal of biomechanics 29 1223 1230 dumas r cheze l verriest j p 2006b adjustments to mcconville et al and young et al body segment inertial parameters journal of biomechanics in press using this comparison technique it has been shown that for the balance recovery motions investigated 1 the use of the scaling equations proposed by dumas et al instead of those proposed by de leva improves the consistency of the results average relative influence up to 30 for the transversal moment 2 the arm motions dynamically influence the recovery motion in a non negligible way average relative influence up to 15 and 30 for the longitudinal force and the transversal moment respectively

whole body inverse dynamics over a complete gait cycle based only on measured kinematics this paper presents a three dimensional 3d whole body multi segment model for inverse dynamics analysis over a complete gait cycle based only on measured kinematic data the sequence of inverse dynamics calculations differs significantly from the conventional application of inverse dynamics using force plate data a new validated smooth transition assumption was used to solve the indeterminacy problem in the double support phase kinematic data is required for all major body segments and hence a whole body gait measurement protocol is presented finally sensitivity analyses were conducted to evaluate the effects of digital filtering and body segment parameters on the accuracy of the prediction results the model gave reasonably good estimates of sagittal plane ground forces and moment however the estimates in the other planes were less good which we believe is largely due to their small magnitudes in comparison to the sagittal forces and moment the errors observed are most likely caused by errors in the kinematic data resulting from skin movement artefact and by errors in the estimated body segment parameters a digital filtering cut off frequency of 4 5 hz was found to produce the best results it was also shown that errors in the mass properties of body segments can play a crucial role with changes in properties sometimes having a disproportionate effect on the calculated ground reactions the implication of these results is that even when force plate data is available the estimated joint forces are likely to suffer from similar errors

asymptotic behavior and stability of mean curvature flow with a conical end abstract if the initial hypersurface of an immortal mean curvature flow is asymptotic to a regular cone whose entropy is small the flow will become asymptotically self expanding moreover the expander that gives rise to the limiting flow is asymptotically stable as an equilibrium solution of the normalized mean curvature flow

heterogeneous morphological granulometries abstract the most basic class of binary granulometries is composed of unions of openings by structuring elements that are homogeneously scaled by a single parameter these univariate granulometries have previously been extended to multivariate granulometries in which each structuring element is scaled by an individual parameter this paper introduces the more general class of filters in which each structuring element is scaled by a function of its sizing parameter the result being multivariate heterogeneous granulometries owing to computational considerations of particular importance are the univariate heterogeneous granulometries for which scaling is by functions of a single variable the basic morphological properties of heterogeneous granulometries are given analytic and geometric relationships between multivariate and univariate heterogeneous pattern spectra are explored and application to texture classification is discussed the homogeneous granulometric mixing theory both the representation of granulometric moments and the asymptotic theory concerning the distributions of granulometric moments is extended to heterogeneous scaling

fast algorithm for sequential machine to compute pattern spectrum via city block distance transform abstract computing pattern spectrum is one of the key methods of determining shape size distribution of objects in the image morphological algorithms for computing the pattern spectrum by successive opening and pixel counting are computationally very costly on sequential machines in this paper a fast algorithm via city block distance transform with morphological concept for the purpose is proposed

the dirichlet problem for the minimal surface system in arbitrary dimensions and codimensions let be a bounded c2 domain in n and m be a continuous map the dirichlet problem for the minimal surface system asks whether there exists a lipschitz map f m with f and with the graph of f a minimal submanifold in n m for m 1 the dirichlet problem was solved more than 30 years ago by jenkins and serrin 12 for any mean convex domains and the solutions are all smooth this paper considers the dirichlet problem for convex domains in arbitrary codimension m we prove that if m satisfies 8n sup d2 2 sup d 1 then the dirichlet problem for is solvable in smooth maps here is the diameter of such a condition is necessary in view of an example of lawson and osserman 13 in order to prove this result we study the associated parabolic system and solve the cauchy dirichlet problem with as initial data 2003 wiley periodicals inc

minimal cones and self expanding solutions for mean curvature flows in this paper we study self expanding solutions for mean curvature flows and their relationship to minimal cones in euclidean space ilmanen lectures on mean curvature flow and related equations trieste notes 1995 proved the existence of self expanding hypersurfaces with prescribed tangent cones at infinity if the cone is c 3 alpha regular and mean convex but not area minimizing we can prove that the corresponding self expanding hypersurfaces are smooth embedded and have positive mean curvature everywhere see theorem 1 1 as a result for regular minimal but not area minimizing cones we can give an affirmative answer to a problem arisen by lawson brothers in proc sympos pure math 44 441 464 1986

compactness of self shrinkers in r3 with fixed genus abstract we prove the compactness of self shrinkers in r 3 with bounded entropy and fixed genus as a corollary we show that numbers of ends of such surfaces are uniformly bounded by the entropy and genus

construction of complete embedded self similar surfaces under mean curvature flow part i we carry out the first main step towards the construc tion of new examples of complete embedded self similar surfaces under mean curvature flow an approximate solution is obtained by taking two known examples of self similar surfaces and desin gularizing the intersection circle using an appropriately modified singly periodic scherk surface called the core using an inverse function theorem we show that for small boundary conditions on the core there is an embedded surface close to the core that is a solution of the equation for self similar surfaces this provides us with an adequate central piece to substitute for the intersection

phase dependent changes in local dynamic stability during walking in elderly with and without knee osteoarthritis previously we reported reduced time averaged knee local stability in the unaffected but not the affected leg of elderly with knee osteoarthritis oa compared to controls since stability may show phase related changes we reanalyzed the dataset reported previously using time dependent local stability t and also calculated time averaged local stability s for comparison we studied treadmill walking at increasing speeds focusing on sagittal plane knee movements 16 patients 12 healthy peers and 15 young subjects were measured we found a clear maximum in t i e minimum in stability at around 60 of the stride cycle stancemax t a second clear maximum swingmax t at around 95 followed by a minimum between 70 and 100 swingmin t stancemax t of both legs was significantly higher in the oa than the young control group values for healthy elderly fell between those of the other groups were significantly higher than in young adults but there was only a trend towards a significant difference with the stancemax t of the oa group s affected side time averaged and time dependent stability measures within one leg were uncorrelated while time dependent stability measures at the affected side were inversely correlated with s at the unaffected side the results indicate that time dependent local dynamic stability might provide a more detailed insight into the problems of gait stability in oa than conventional averaged local dynamic stability measures and support the notion that the paradoxical decline in unaffected side time averaged local stability may be caused by a trade off between affected and unaffected side stability

existence and non existence of area minimizing hypersurfaces in manifolds of non negative ricci curvature we study minimal hypersurfaces in manifolds of non negative ricci curvature euclidean volume growth and quadratic curvature decay at infinity by comparison with capped spherical cones we identify a precise borderline for the ricci curvature decay above this value no complete area minimizing hypersurfaces exist below this value in contrast we construct examples

pattern spectrum and multiscale shape representation the results of a study on multiscale shape description smoothing and representation are reported multiscale nonlinear smoothing filters are first developed using morphological opening and closings g matheron 1975 used openings and closings to obtain probabilistic size distributions of euclidean space sets continuous binary images these distributions are used to develop a concept of pattern spectrum a shape size descriptor a pattern spectrum is introduced for continuous graytone images and arbitrary multilevel signals as well as for discrete images by developing a discrete size family of patterns large jumps in the pattern spectrum at a certain scale indicate the existence of major protruding or intruding substructures of the signal at the scale an entropy like shape size complexity measure is also developed based on the pattern spectrum for shape representation a reduced morphological skeleton transform is introduced for discrete binary and graytone images this transform is a sequence of skeleton components sparse images which represent the original shape at various scales it is shown that the partially reconstructed images from the inverse transform on subsequences of skeleton components are the openings of the image at a scale determined by the number of eliminated components in addition two way correspondences are established among the degree of shape smoothing via multiscale openings or closings the pattern spectrum zero values and the elimination or nonexistence of skeleton components at certain scales

shape decomposition and representation using a recursive morphological operation abstract this paper presents a recursive morphological operation developed in order to perform efficient shape representation this operation uses a structuring element as a geometrical primitive to evaluate the shape of an object it results in a set of loci of the translated structuring elements that are included in the object but which do not overlap the analysis of its computational complexity shows that it is usually less time consuming than morphological erosion by using this operation an object decomposition algorithm is then developed for shape representation it decomposes an object into a union of simple and non overlapping object components the object is represented by the sizes and loci of its object components this representation is information preserving shift and scale invariant and non rebundant it has been compared with skeletons morphological decomposition chain codes and quatrees in terms of compression ability and image processing facility experimental results shows that it is very compact especially if information loss is allowed because of the non overlap between object components many image processing tasks can be easily performed by directly using this shape representation

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures

the three dimensional determination of internal loads in the lower extremity a three dimensional model of the lower limb containing 47 muscles was developed to study the differences between a two and three dimensional approach for determining internal loads the role of the dynamic joint representation and the behavior of different load bearing criteria in walking and running the problem of redundancy of the musculo skeletal system was resolved by applying inverse dynamics and static optimization methods different hypothetical load bearing capabilities of hinge spherical and intermediate joint types for the knee and the ankle joints were tested it was found that even almost planar movements such as walking and running are associated with significant three dimensional intersegment moments especially in the frontal plane thus a two dimensional approach may underestimate internal loads up to 60 it is shown that pure hinge joints are inappropriate for modeling the dynamical joint function of the knee and ankle joints a more flexible joint representation in combination with a squared muscle stress minimization criterion predicted a lot of synergistic as well as antagonistic muscle activation which was also found in the emg patterns the results indicate the importance of muscular joint stabilization in natural human movements compared to in vivo measurements it is speculated that the predicted force magnitudes are considerably overestimated due to error propagation and still insufficient anatomical models thus increased efforts to improve further the reliability of internal load calculations should be made in the future

a general regularity theory for stable codimension 1 integral varifolds we give a necessary and sufficient geometric structural condition which we call the a structural hypothesis for a stable codimension 1 integral varifold on a smooth riemannian manifold to correspond to an embedded smooth hypersurface away from a small set of generally unavoidable singularities the a structural hypothesis says that no point of the support of the varifold has a neighborhood in which the support is the union of three or more embedded c 1 a hypersurfaces with boundary meeting only along their common boundary we establish that whenever a stable integral n varifold on a smooth n 1 dimensional riemannian manifold satisfies the a structural hypothesis for some a 0 1 2 its singular set is empty if n 6 discrete if n 7 and has hausdorff dimension n 7 if n 8 in view of well known examples this is the best possible general dimension estimate on the singular set of a varifold satisfying our hypotheses we also establish compactness of mass bounded subsets of the class of stable codimension 1 integral varifolds satisfying the a structural hypothesis for some a 0 1 2 the a structural hypothesis on an n varifold for any a 0 1 2 is readily implied by either of the following two hypotheses i the varifold corresponds to an absolutely area minimizing rectifiable current with no boundary ii the singular set of the varifold has vanishing n 1 dimensional hausdorff measure thus our theory subsumes the well known regularity theory for codimension 1 area minimizing rectifiable currents and settles the long standing question as to which weakest size hypothesis on the singular set of a stable minimal hypersurface guarantees the validity of the above regularity conclusions an optimal strong maximum principle for stationary codimension 1 integral varifolds follows from our regularity and compactness theorems

uniqueness of self similar shrinkers with asymptotically conical ends in this paper we show the uniqueness of smooth embedded selfshrinkers asymptotic to generalized cylinders of infinite order also we construct non rotationally symmetric self shrinking ends asymptotic to generalized cylinders with rate as fast as any given polynomial

unique continuation and absence of positive eigenvalues for schrodinger operators on considere l operateur de laplace sur r n et une fonction v x sur un sous ensemble connexe ouvert de r n on montre que si n 3 une propriete de prolongement unique est vraie pour v l loc n 2 r n dans l espace de sobolev h loc 2 q r n avec q 2n n 2

further results on map optimality and strong consistency of certain classes of morphological filters morphological openings and closings can be viewed as consistent map estimators of smooth random binary image signals immersed in i i d clutter or suffering from i i d random dropouts we revisit this viewpoint under much more general assumptions and show that quite surprisingly the above interpretation is still valid

positive solutions to schrodinger equations and geometric applications a variant of li tam theory which associates to each end of a complete riemannian manifold a positive solution of a given schrodinger equation on the manifold is developed it is demonstrated that such positive solutions must be of polynomial growth of fixed order under a suitable scaling invariant sobolev inequality consequently a finiteness result for the number of ends follows in the case when the sobolev inequality is of particular type the finiteness result is proven directly as an application an estimate on the number of ends for shrinking gradient ricci solitons and submanifolds of euclidean space is obtained

sensitivity of the knee joint response muscle forces and stability to variations in gait kinematics kinetics abstract sensitivity analysis of the knee joint response to variations in gait kinematics kinetics as reported in the literature is crucial for improved understanding and more effective prevention treatment strategies using our validated finite element musculoskeletal fe ms model of lower extremity we investigate the sensitivity of knee anterior cruciate ligament acl muscle and contact forces plus stability to the reported scatter in asymptomatic gait data three highly loaded stance instants 25 50 and 75 with five levels mean 0 5sd and sd for each of six knee joint angles moments are used employing taguchi approach 25 experiments and regression equations acl force drops significantly at larger flexion angles all periods and smaller internal moment at 75 only but increases with the flexion moment tibiofemoral tf medial lateral contact force partitioning is found contrary to the common claim most sensitive to changes in the adduction angle and not in the adduction moment total tf contact force increases significantly at greater moments but not angles especially in the sagittal plane forces in lateral hamstrings are significantly influenced by changes in adduction angles moments larger flexion moments at 25 and 50 significantly increase forces in quadriceps and on patellofemoral pf contact sagittal moment adduction moment at 75 and flexion angle at 25 contribute most to the joint stability a strong inverse correlation exists between the joint stability and the total tf compression force these findings can be exploited to adapt and modify intact injured and reconstructed knee joint responses during gait

non homothetic granulometric mixing theory with application to blood cell counting a granulometry is a family of morphological openings by scaled structuring elements as the scale increases increasing image area is removed normalizing removed area by the total area yields the pattern spectrum of the image the pattern spectrum is a probability distribution function and its moments are known as granulometric moments modeling the image as a random set the pattern spectrum is a random function and its moments are random variables the original granulometric mixing theory provides closed form representation of the granulometric moments shows that the distributions of the moments are asymptotically normal and gives asymptotic expressions for the means and variances of the granulometric moments the theory applies to random set models formed as disjoint unions of randomly scaled image primitives grains the scaled grains are known as homothetics the theory can be used in a method of moments fashion to estimate the parameters governing the random scaling factors and mixture proportions application is limited by the homothetic assumption this paper drops the homothetic requirement and provides a mixing theory for a disjoint union of fully randomized primitives whereas the original asymptotic theory gives expressions for the moments themselves and is applied by taking expectations afterwards the non homothetic theory involves the granulometric size density which is the mean of the original size distribution prior to normalization hence the representation concerns expectations not random variables nonetheless a similar method of moments approach can be used to estimate mixture proportions a large part of the paper is devoted to estimate blood cell proportions that correspond to cell age categories each cell class is represented by a random grain and the problem is to estimate the proportions of cells occurring in the various age categories the random behavior of the cells in each category makes the non homothetic theory appropriate because the estimation strategy leads to a system of nonlinear equations whose solution presents computational difficulties the estimation is accomplished via a divide and conquer strategy in which the full mixture problem is partitioned into smaller problems and the solutions of these problems are joined to solve the full problem

granulometric moments and corneal endothelium status abstract specular microscopy is a common practice in ophthalmology the corneal endothelium status is usually evaluated by means of the density the hexagonality the mean the standard deviation and the coefficient of variation of cell areas we propose to replace the cell area moments by the corresponding moments of a different probability distribution the granulometric size distribution associated to a disc all cells touching the frame are ignored by the area moments but used by the granulometric moments twenty images have been analyzed when the size of the focused region is reduced the area moments show a greater variation than the corresponding granulometric moments

morphological pattern spectrum classification of noisy shapes exterior granulometries abstract moments of morphological granulometric pattern spectra have been used for classification of both texture and patterns the present paper concerns shape classification in the presence of edge noise when using granulometries generated by linear structuring elements although these have been shown to provide excellent texture classification it is shown here that they cannot successfully discriminate between close convex shapes when there is even modest edge noise by redefining a linear granulometry so that it operates on the linear convex hull in the direction of the generating structuring element excellent shape recognition is achieved even for edge noise exceeding that typically encountered in practice these newly proposed granulometries are called exterior granulometries and for a given shape class optimal feature sets from amongst their first three pattern spectrum moments are obtained their superior performance is demonstrated by a probabilistic comparison between the stability of the pattern spectrum means for both ordinary and exterior linear granulometries

unique continuation for elliptic operators a geometric variational approach dans un sous ensemble connexe ouvert de r n n 3 on considere l equation d ordre 2 lu div a x u b x u v x u 0 ou a x a id j x est une matrice symetrique reelle on etudie les proprietes d unicite de l operateur

off line signature verification by local granulometric size distributions a fundamental problem in the field of off line signature verification is the lack of a signature representation based on shape descriptors and pertinent features the main difficulty lies in the local variability of the writing trace of the signature which is closely related to the identity of human beings in this paper we propose a new formalism for signature representation based on visual perception a signature image consists of 512 spl times 128 pixels and is centered on a grid of rectangular retinas which are excited by local portions of the signature granulometric size distributions are used for the definition of local shape descriptors in an attempt to characterize the amount of signal activity exciting each retina on the focus of the attention grid experimental evaluation of this scheme is made using a signature database of 800 genuine signatures from 20 individuals two types of classifiers a nearest neighbor and a threshold classifier show a total error rate below 0 02 percent and 1 0 percent respectively in the context of random forgeries

a three dimensional kinematic and dynamic model of the lower limb a model describing the kinematics and dynamics of the lower limb is presented the lower limb is modeled as a sequence of four rigid links connected by three universal rotary joints representing the hip knee and ankle joints each joint is modeled as a sequence of three single axis rotational joints thus ascribing to the lower limb a total of 12 degrees of freedom a method is described to measure the gait variables so that all nine angles can be computed based on the positions of nine markers placed on the subject during a gait study the gait variables are then used in an iterative newton euler formulation to compute the moments exerted about the axes of each joint during gait

a practical method for calculating largest lyapunov exponents from small data sets detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest lyapunov exponent lyapunov exponents quantify the exponential divergence of initially close state space trajectories and estimate the amount of chaos in a system we present a new method for calculating the largest lyapunov exponent from an experimental time series the method follows directly from the definition of the largest lyapunov exponent and is accurate because it takes advantage of all the available data we show that the algorithm is fast easy to implement and robust to changes in the following quantities embedding dimension size of data set reconstruction delay and noise level furthermore one may use the algorithm to calculate simultaneously the correlation dimension thus one sequence of computations will yield an estimate of both the level of chaos and the system complexity

a local regularity theorem for mean curvature flow this paper proves curvature bounds for mean curvature flows and other related flows in regions of spacetime where the gaussian densities are close to 1

spring like gait mechanics observed during walking in both young and older adults a spring loaded inverted pendulum model successfully demonstrated the oscillatory behavior of the center of mass com and corresponding ground reaction forces grfs of young healthy subjects this study questioned whether spring like leg walking dynamics are consistently observed in the walking of older adults that exhibit different gait characteristics such as slower gait speed from the young eight young and eight older adult subjects participated in overground walking experiments performed at four different gait speeds ranging from their self selected speed to a maximum walking speed to calculate the effective leg stiffness the damped compliant leg model with a curved foot was used the model parameters of leg stiffness and damping constant were optimized to achieve the best fit between model and human grfs data we observed that the grfs data from both age groups were reasonably well fitted by spring like leg dynamics throughout the broad range of gait speeds the leg stiffness and damping constant consistently increased as a function of the walking speed in both age groups but slightly greater variations of the model parameters were observed for the older adults trials the results imply that human walking dynamics and the variation with respect to age can be well captured by spring like leg dynamics

backward uniqueness for parabolic equations it is shown that a function u satisfying t u m u u u x t mem x 2 in n br 0 t and u x 0 0 for x n br must vanish identically in n br 0 t

quaternions and joint angles in an analysis of local stability of gait for different variants of walking speed and treadmill slope the authors describe an example of application of a nonlinear time series analysis directed at identifying the presence of deterministic chaos in human gait kinematic data by means of the largest lyapunov exponent lle a positive lle value is interpreted as an indicator of local instability the research was aimed at assessment of the influence of both walking speed and ground slope on the resilience of gait control to infinitesimally small perturbations that occur naturally during walking the analysis of treadmill gait data was carried out twofold 1 for the time series representing the following joint angles hip flexion extension knee flexion extension and dorsiflexion plantarflexion of the ankle and 2 for the time series representing rotations of foot tibia and femur segments through euler angles converted to a quaternion representation a comparison between both approaches as well as a dependency between treadmill inclination and lle values constitute the original aspects of this study the lle value was estimated threefold for every time series as the short term lle for both the first step and the first stride and as the long term lle over a fixed interval between the fourth and the tenth stride it was confirmed that all considered movements are characterized by positive lle values which quantify a local instability moreover a tendency to attenuate the perturbation consequences is evident in all variants of walking speed and treadmill slope

the effect of walking speed on local dynamic stability is sensitive to calculation methods local dynamic stability has been assessed by the short term local divergence exponent s which quantifies the average rate of logarithmic divergence of infinitesimally close trajectories in state space both increased and decreased local dynamic stability at faster walking speeds have been reported this might pertain to methodological differences in calculating s therefore the aim was to test if different calculation methods would induce different effects of walking speed on local dynamic stability ten young healthy participants walked on a treadmill at five speeds 60 80 100 120 and 140 of preferred walking speed for 3min each while upper body accelerations in three directions were sampled from these time series s was calculated by three different methods using a a fixed time interval and expressed as logarithmic divergence per stride time s a b a fixed number of strides and expressed as logarithmic divergence per time s b and c a fixed number of strides and expressed as logarithmic divergence per stride time s c mean preferred walking speed was 1 16 0 09m s there was only a minor effect of walking speed on s a s b increased with increasing walking speed indicating decreased local dynamic stability at faster walking speeds whereas s c decreased with increasing walking speed indicating increased local dynamic stability at faster walking speeds thus the effect of walking speed on calculated local dynamic stability was significantly different between methods used to calculate local dynamic stability therefore inferences and comparisons of studies employing s should be made with careful consideration of the calculation method language en

embedded self similar shrinkers of genus 0 we conrm a well known conjecture that the round sphere is the only compact embedded self similar shrinking solution of mean curvature ow in r 3 with genus 0 more generally we show that the only properly embedded self similar shrinkers in r 3 with vanishing intersection form are the sphere the cylinder and the plane this answers two questions posed by t ilmanen

effect of low pass filtering on joint moments from inverse dynamics implications for injury prevention analyses of joint moments are important in the study of human motion and are crucial for our understanding of e g how and why acl injuries occur such analyses may be affected by artifacts due to inconsistencies in the equations of motion when force and movement data are filtered with different cut off frequencies the purpose of this study was to quantify the effect of these artifacts and compare joint moments calculated with the same or different cut off frequency for the filtering of force and movement data 123 elite handball players performed sidestep cutting while the movement was recorded by eight 240 hz cameras and the ground reaction forces were recorded by a 960 hz force plate knee and hip joint moments were calculated through inverse dynamics with four different combinations of cut off frequencies for signal filtering movement 10 hz force 10 hz 10 10 movement 15 hz force 15 hz movement 10 hz force 50 hz 10 50 movement 15 hz force 50 hz the results revealed significant differences especially between conditions with different filtering of force and movement mean sd peak knee abduction moment for the 10 10 and 10 50 condition were 1 27 0 53 and 1 64 0 68 nm kg respectively ranking of players based on knee abduction moments were affected by filtering condition out of 20 players with peak knee abduction moment higher than mean 1 sd with the 10 50 condition only 11 were still above mean 1 sd when the 10 10 condition was applied hip moments were very sensitive to filtering cut off mean sd peak hip flexion moment was 3 64 0 75 and 5 92 1 80 under the 10 10 and 10 50 conditions respectively based on these findings force and movement data should be processed with the same filter conclusions from previous inverse dynamics studies where this was not the case should be treated with caution

revisiting the stability of 2d passive biped walking local behavior models of biped walking have demonstrated that stable walking motions are possible without active control stability of these motions has typically been quantified by studying the stability of an associated poincare map orbital stability however additional insight may be obtained by examining how perturbations evolve over the short term local stability for example there may be regions where small perturbations actually diverge from the unperturbed trajectory even if over the entire cycle small but perhaps not large perturbations are dissipated we present techniques to calculate local stability and demonstrate the utility of these techniques by examining the local stability of the 2d compass biped these techniques are relevant to the design of controllers to maintain stability in robots and in understanding how the neuromuscular system maintains stability in humans

rigidity of complete spacelike translating solitons in pseudo euclidean space abstract in this paper we investigate the parametric version and non parametric version of rigidity theorem of spacelike translating solitons in pseudo euclidean space r n m n firstly we classify m dimensional complete spacelike translating solitons in r n m n and prove the only complete spacelike translating solitons in r n m n are the spacelike m planes secondly we generalize the rigidity theorem of entire spacelike lagrangian translating solitons to spacelike translating solitons with general codimensions

kinematic variability and local dynamic stability of upper body motions when walking at different speeds a ubiquitous characteristic of elderly and patients with gait disabilities is that they walk slower than healthy controls many clinicians assume these patients walk slower to improve their stability just as healthy people slow down when walking across ice however walking slower also leads to greater variability which is often assumed to imply deteriorated stability if this were true then slowing down would be completely antithetical to the goal of maintaining stability this study sought to resolve this paradox by directly quantifying the sensitivity of the locomotor system to local perturbations that are manifested as natural kinematic variability eleven young healthy subjects walked on a motorized treadmill at five different speeds three dimensional movements of a single marker placed over the first thoracic vertebra were recorded during continuous walking mean stride to stride standard deviations and maximum finite time lyapunov exponents were computed for each time series to quantify the variability and local dynamic stability respectively of these movements quadratic regression analyses of the dependent measures vs walking speed revealed highly significant u shaped trends for all three mean standard deviations but highly significant linear trends with significant or nearly significant quadratic terms for five of the six finite time lyapunov exponents subjects exhibited consistently better local dynamic stability at slower speeds for these five measures these results support the clinically based intuition that people who are at increased risk of falling walk slower to improve their stability even at the cost of increased variability

spatial size distributions applications to shape and texture analysis this paper proposes new descriptors for binary and gray scale images based on newly defined spatial size distributions ssd the main idea consists of combining a granulometric analysis of the image with a comparison between the geometric covariograms for binary images or the auto correlation function for gray scale images of the original image and its granulometric transformation the usual granulometric size distribution then arises as a particular case of this formulation examples are given to show that in those cases in which a finer description of the image is required the more complex descriptors generated from the ssd could be advantageously used it is also shown that the new descriptors are probability distributions so their intuitive interpretation and properties can be appropriately studied from the probabilistic point of view the usefulness of these descriptors in shape analysis is illustrated by some synthetic examples and their use in texture analysis is studied various cases of ssd and several former methods for texture classification are compared

granolds a novel texture representation in this paper we describe a new image texture representation using two different parameterised monotonic mappings this new technique transforms the input image into a function on two dimensions that may be regarded as a surface that we have named the granold the nature of this surface is such that corners appear at positions where there are simultaneously large changes in the response of the monotonic mappings to the input image the shape and position of these corners is then analysed to provide information about the texture of the input image the use of grey level thresholds and morphological granulometries in the formation of the granold is presented then these mappings are generalised and a mathematical analysis of the new technique including constraints for the parameterised monotonic mappings is presented a number of experiments are described that use grey level thresholds and morphological granulometries to fundamentally test and demonstrate the new technique are presented the results from these experiments are presented and discussed including a qualitative discussion of the behaviour of the new technique in response to changes in the input image the conclusion is that the new technique performs in an expected repeatable and predictable manner in accordance with its mathematical analysis c 2000 pattern recognition society published by elsevier science ltd all rights reserved

dynamic knee loads during gait predict proximal tibial bone distribution abstract this study tested the validity of the prediction of dynamic knee loads based on gait measurements the relationship between the predicted loads at the knee and the distribution of bone between the medial and lateral sides of the tibia was examined the motion and external forces and moments at the knee were measured during gait and a statically determinate muscle model was used to predict the corresponding forces on the medial and lateral tibial plateaus in particular the relationship between the knee adduction moment during gait and the ratio or distribution of medial to lateral tibial bone mineral content was studied bone mineral content was measured with dual energy x ray absorptiometry in four regions two proximal regions 20 mm in height one medial and one lateral and two distal regions 20 mm in height one medial and one lateral the best single predictor of the medial lateral ratio of proximal bone mineral content bone distribution was the adduction moment r 2 0 31 p 0 003 adding weight negative coefficient p 0 0004 and the ratio of the average predicted peak force on the medial plateau to the predicted peak force on the lateral plateau positive coefficient p 0 0033 to the regression model significantly increased the ability to predict the proximal medial lateral bone distribution r 2 0 72 p 0 0001 distally neither the subject characteristics nor the gait moments and predicted forces were significant predictors of the bone distribution the lack of a correlation distally may be reflective of the forces being more evenly distributed further from the tibial plateau while it has long been suggested that the adduction moment is the primary determinate of the distribution of load between the medial and lateral plateaus this is the first evidence of its relationship to the underlying bone distribution

a parametric approach to numerical modeling of tkr contact forces in vivo knee contact forces are difficult to determine using numerical methods because there are more unknown forces than equilibrium equations available we developed parametric methods for computing contact forces across the knee joint during the stance phase of level walking three dimensional contact forces were calculated at two points of contact between the tibia and the femur one on the lateral aspect of the tibial plateau and one on the medial side muscle activations were parametrically varied over their physiologic range resulting in a solution space of contact forces the obtained solution space was reasonably small and the resulting force pattern compared well to a previous model from the literature for kinematics and external kinetics from the same patient peak forces of the parametric model and the previous model were similar for the first half of the stance phase but differed for the second half the previous model did not take into account the transverse external moment about the knee and could not calculate muscle activation levels ultimately the parametric model will result in more accurate contact force inputs for total knee simulators as current inputs are not generally based on kinematics and kinetics inputs from tkr patients