stability of peakons the peakons are peaked solitary wave solutions of a certain nonlinear dispersive equation that is a model in shallow water theory and the theory of hyperelastic rods we give a very simple proof of the orbital stability of the peakons in the h1 norm 2000 john wiley sons inc

optimal digital filtering requires a different cut off frequency strategy for the determination of the higher derivatives the present study investigated four different filtering and differentiation sequences for the calculation of the higher derivatives from noisy displacement data when using a second order butterworth filter and first order finite differences these were 1 the conventional sequence i e filtering the displacement data and then differentiating 2 filtering the displacement with a different cut off frequency depending upon optimal 0th 1st and 2nd derivatives 3 double filtering and differentiation only for acceleration and 4 differentiation and then filtering separately in each derivative domain i e treating the noisy higher derivatives as individual signals thirty levels of time domain and 30 levels of frequency domain computer generated pure noise signals were superimposed on 24 reference signals which simulated the medial lateral anterior posterior and vertical displacement patterns of eight markers attached to the lower extremity segments during walking the optimum cut off frequency for the displacement velocity and acceleration data was calculated as the one that produced the minimum root mean square error between the reference and noisy data in each derivative domain the results indicated that the conventional strategy has to be reconsidered and modified as the best results were obtained by the second strategy the optimum cut off frequency for acceleration was lower than that required for the velocity which in turn was lower than the optimum cut off frequency for displacement the findings of the present study will contribute to the development of existing and future automatic filtering techniques based on digital filtering

discrete wavelet transform a tool in smoothing kinematic data abstract motion analysis systems typically introduce noise to the displacement data recorded butterworth digital filters have been used to smooth the displacement data in order to obtain smoothed velocities and accelerations however this technique does not yield satisfactory results especially when dealing with complex kinematic motions that occupy the low and high frequency bands the use of the discrete wavelet transform as an alternative to digital filters is presented in this paper the transform passes the original signal through two complementary low and high pass fir filters and decomposes the signal into an approximation function and a detail function further decomposition of the signal results in transforming the signal into a hierarchy set of orthogonal approximation and detail functions a reverse process is employed to perfectly reconstruct the signal inverse transform back from its approximation and detail functions the discrete wavelet transform was applied to the displacement data recorded by pezzack et al 1977 the smoothed displacement data were twice differentiated and compared to pezzack et al s acceleration data in order to choose the most appropriate filter coefficients and decomposition level on the basis of maximizing the percentage of retained energy pre and minimizing the root mean square error rmse daubechies wavelet of the fourth order db4 at the second decomposition level showed better results than both the biorthogonal and coiflet wavelets pre 97 5 rmse 4 7 rad s 2 the db4 wavelet was then used to compress complex displacement data obtained from a noisy mathematically generated function results clearly indicate superiority of this new smoothing approach over traditional filters

endpoint error in smoothing and differentiating raw kinematic data an evaluation of four popular methods endpoint error describes the erratic behavior at the beginning and end of the computed acceleration data which is commonly observed after smoothing and differentiating raw displacement data to evaluate endpoint error produced by four popular smoothing and differentiating techniques lanshammar s 1982 j biomechanics 15 99 105 modification of the pezzack et al 1977 j biomechanics 10 377 382 raw angular displacement data set was truncated at three different locations corresponding to the major peaks in the criterion acceleration curve also for each data subset three padding conditions were applied each data subset was smoothed and differentiated using the butterworth digital filter cubic spline quintic spline and fourier series to obtain acceleration values rms residual errors were calculated between the computed and criterion accelerations in the endpoint regions although no method completely eliminated endpoint error the results demonstrated clear superiority of the quintic spline over the other three methods in producing accurate acceleration values close to the endpoints of the modified pezzack et al 1977 data set in fact the quintic spline performed best with non padded data cumulative error 48 0 rad s 2 conversely when applied to non padded data the butterworth digital filter produced wildly deviating values beginning more than the 10 points from the terminal data point cumulative error 226 6 rad s 2 each of the four methods performed better when applied to data subsets padded by linear extrapolation average cumulative error 68 8 rad s 2 than when applied to analogous subsets padded by reflection average cumulative error 86 1 rad s 2

the camassa holm equation as a geodesic flow on the diffeomorphism group misiolek j geom phys 24 203 208 1998 has shown that the camassa holm equation is a geodesic flow on the bott virasoro group in this paper it is shown that the camassa holm equation for the case 0 is the geodesic spray of the weak riemannian metric on the diffeomorphism group of the line or the circle obtained by right translating the h1 inner product over the entire group this paper uses the right trivialization technique to rigorously verify that the euler poincare theory for lie groups can be applied to diffeomorphism groups the observation made in this paper has led to physically meaningful generalizations of the ch equation to higher dimensional manifolds

interacting fisher wright diffusions in a catalytic medium we study the longtime behaviour of interacting systems in a randomly fluctuating space time medium and focus on models from population genetics there are two prototypes of spatial models in population genetics spatial branching processes and interacting fisher wright diffusions quite a bit is known on spatial branching processes where the local branching rate is proportional to a random environment catalytic medium

on the integrable perturbations of the camassa holm equation we present an investigation of the nonlinear partial differential equations pde which are asymptotically representable as a linear combination of the equations from the camassa holm hierarchy for this purpose we use the infinitesimal transformations of dependent and independent variables of the original pde this approach is helpful for the analysis of the systems of the pde which can be asymptotically represented as the evolution equations of polynomial structure we present an investigation of the nonlinear partial differential equations pde which are asymptotically representable as a linear combination of the equations from the camassa holm hierarchy for this purpose we use the infinitesimal transformations of dependent and independent variables of the original pde this approach is helpful for the analysis of the systems of the pde which can be asymptotically represented as the evolution equations of polynomial structure

coalescing random walks and voter model consensus times on the torus in mathbb z d voter models on z d and on the torus of site n in z d are considered the second ones get trapped with probability 1 as t tends to infinity at all 0 s or all 1 s the asymptotic growth of these trapping times are examined as n tends to infinity this is done with limit thewems for coalescing random walk systems on the torus of side n in z d

global existence for a new periodic integrable equation we establish the local well posedness for a new periodic integrable equation we show that the equation has classical solutions that blowup in finite time as well as classical solutions which exist globally in time

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

geometry and curvature of diffeomorphism groups withh1metric and mean hydrodynamics holm marsden and ratiu adv in math 137 1998 1 81 derived a new model for the mean motion of an ideal fluid in euclidean space given by the equationv t u t v t 2 u t t u t grad p t where div u 0 andv 1 2 u in this model the momentumvis transported by the velocityu with the effect that nonlinear interaction between modes corresponding to length scales smaller than is negligible we generalize this equation to the setting of ann dimensional compact riemannian manifold the resulting equation is the euler poincare equation associated with the geodesic flow of theh1right invariant metric on ds the group of volume preserving hilbert diffeomorphisms of classhs we prove that the geodesic spray is continuously differentiable fromtds m intottds m so that a standard picard iteration argument proves existence and uniqueness on a finite time interval our goal in this paper is to establish the foundations for lagrangian stability analysis following arnold ann inst grenoble16 1966 319 361 to do so we use submanifold geometry and prove that the weak curvature tensor of the right invarianth1metric on ds is a bounded trilinear map in thehstopology from which it follows that solutions to jacobi s equation exist using such solutions we are able to study the infinitesimal stability behavior of geodesics

low regularity solutions of the periodic fornberg whitham equation this paper studies low regularity solutions of the periodic fornberg whitham equation with initial value the existence and the uniqueness of solutions are proved the results are illustrated by considering the periodic peakons of the periodic fornberg whitham equation

diffusive clustering in an infinite system of hierarchically interacting diffusions we study a countable system of interacting diffusions on the interval 0 1 indexed by a hierarchical group a particular choice of the interaction guaranties we are in the diffusive clustering regime this means clusters of components with values either close to 0 or close to 1 grow on various different scales however single components oscillate infinitely often between values close to 0 and close to 1 in such a way that they spend fraction one of their time together and close to the boundary the processes in the whole class considered and starting with a shift ergodic initial law have the same qualitative properties universality

global weak solutions for a new periodic integrable equation with peakon solutions abstract we prove the existence of global weak solutions for a new periodic integrable equation with peakon solutions

well posedness and blow up solutions for an integrable nonlinearly dispersive model wave equation abstract we establish local well posedness in the sobolev space hs with any s 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the camassa holm equation however unlike the more familiar korteweg devries model we demonstrate conditions on the initial data that lead to finite time blow up of certain solutions

on the well posedness problem and the scattering problem for the dullin gottwald holm equation in this paper we study the well posedness of the cauchy problem and the scattering problem for a new nonlinear dispersive shallow water wave equation the so called dgh equation which was derived by dullin gottwald and holm the issue of passing to the limit as the dispersive parameter tends to zero for the solution of the dgh equation is investigated and the convergence of solutions to the dgh equation as 2 0 is studied and the scattering data of the scattering problem for the equation can be explicitly expressed the new exact peaked solitary wave solutions are obtained in the dgh equation after giving the condition of existing peakon in the dgh equation it turns out to be nonlinearly stable for the peakon in the dgh equation

application of singular spectrum analysis to the smoothing of raw kinematic signals motion capture systems currently used in biomechanical analysis introduce systematic measurement errors that appear in the form of noise in recorded displacement signals the noise is unacceptably amplified when differentiating displacements to obtain velocities and accelerations to avoid this phenomenon it is necessary to smooth the displacement signal prior to differentiation in order to eliminate the noise introduced by the experimental system the use of singular spectrum analysis ssa is presented in this paper as an alternative to traditional digital filtering methods ssa is a novel non parametric technique based on principles of multivariate statistics the original time series is decomposed into a number of additive time series each of which can be easily identified as being part of the modulated signal or as being part of the random noise several examples that demonstrate the superiority of this technique over other methods used in biomechanical analysis are presented in this paper

local well posedness and stability of peakons for a generalized dullin gottwald holm equation abstract we establish the local well posedness for a generalized dullin gottwald holm equation by using kato s theory furthermore the orbital stability of the peaked solitary waves is also proved

global existence and blow up solutions for a nonlinear shallow water equation considered herein are the problems of the existence of global solutions and the formation of singularities for a new nonlinear shallow water wave equation derived by dullin gottward and holm blow up can occur only in the form of wave breaking a wave breaking mechanism for solutions with certain initial profiles is described in detail and the exact blow up rate is established the blow up set for a class of initial profiles and lower bounds of the existence time of the solution are also determined

on precision limits for derivatives numerically calculated from noisy data abstract this paper has three purposes 1 1 to verify an error formula from which the maximal precision in derivatives obtained from noisy measurement data can be calculated the formula is verified by comparison with the resulting noise in derivatives obtained by local least squares polynomial fitting it is also verified that differentiation by fourier series expansion gives noise exceeding the minimum value according to the error formula 2 2 an index called the relative noise amplification rna is introduced for an arbitrary differentiating filter it is defined as the noise transmission of the filter divided by the minimal noise transmission according to the above mentioned error formula when the filter produces unbiased estimates the value of the rna always exceeds one the rna can be used as a quality index for differentiating filters the filter with the smallest value on the rna also has the smallest amount of noise superimposed on the calculated derivatives provided the input noise is white and additive 3 3 the bandwidth and the relative noise amplification of differentiating filters obtained by local least squares polynomial fitting are presented the 0th 1st and 2nd order derivatives have been investigated for polynomials up to 8th order these results can be used for the determination of suitable filter parameters in various applications

existence and uniqueness of low regularity solutions for the dullin gottwald holm equation we establish the local existence and uniqueness of solutions for the dullin gottwald holm equation with continuously differentiable periodic initial data the regularity conditions needed for the cauchy problem via the semigroup approach of quasilinear hyperbolic equations of evolution or the viscosity method are significantly lowered

influence of body segments parameters estimation models on inverse dynamics solutions during gait the purpose of the present study was to examine the in uence of anthropometric data on joint kinetics during gait we particularly focused on the sensitivity of inverse dynamics solutions to the use of models for body segment parameters bsp estimation six often used estimation models were selected to provide bsp values for the three segments of the lower limb kinematics and dynamics were sampled from seven subjects performing barefoot gait at three different speeds joint kinetics were estimated with the bottom up method using bsp values derived from each estimation model as anthropometric inputs the bsp estimates were highly sensitive to the model used with deviations ranging from at least 9 73 up to 60 maximal variations of peak values for the hip joint exion extension moment during the swing phase were 20 11 hence our ndings suggest that the in uence of bsp cannot be neglected observed deviations are especially due to the effect of varying simultaneously the mass moments of inertia and the center of mass location values according to the underlying relationship of interdependency linking each component considering both the differences found in joint kinetics and the level of accuracy of bsp models evidence is provided that using multiple regression bsp estimation functions derived from zatsiorsky and seluyanov biomechanics viii b 1983 pp 1152 1159 should be recommended to assess joint kinetics

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

influence of body segment parameter estimation on calculated ground reaction forces in highly dynamic movements the effect of body segment parameter estimation bsp on the inverse dynamics modelling results has not yet been demonstrated in specific groups during athletic movements with high segment accelerations therefore the purpose of this study was to analyse this effect in ski jumpers as representatives of a specific group i e low body mass index by comparing calculated and measured ground reaction forces during ski jumping imitation jumps full body kinematics and vertical ground reaction forces were recorded of 9 ski jumpers performing three imitation jumps each bsp were estimated using three previously published one individually optimized and one ski jumper group specific model vertical ground reaction forces were calculated using the vertical acceleration of the segments as well as the bsp of the single models in a top down approach statistical analysis revealed a main model effect concerning the root mean square error between the calculated and the measured ground reaction force with deviations between the models of 53 individual optimization and the application of the ski jumper group specific model increased the accuracy of the calculated ground reaction forces by 11 and 7 respectively compared to the best performing published model the results of inverse dynamics modelling are very sensitive to the bsp estimation for specific groups like ski jumpers during movements incorporating high segment accelerations this emphasizes the importance of selecting adequate bsp estimation models or methods when analysing specific groups in highly dynamic movements in order to increase the accuracy of the inverse dynamics analyses results

low regularity solutions of the periodic general degasperis procesi equation abstract this paper studies low regularity solutions of the periodic general degasperis procesi equation with an initial value the existence and the uniqueness of solutions are proved the results are illustrated by considering the periodic peakons of the periodic general degasperis procesi equation

euler equations on homogeneous spaces and virasoro orbits abstract we show that the following three systems related to various hydrodynamical approximations the korteweg de vries equation the camassa holm equation and the hunter saxton equation have the same symmetry group and similar bihamiltonian structures it turns out that their configuration space is the virasoro group and all three dynamical systems can be regarded as equations of the geodesic flow associated to different right invariant metrics on this group or on appropriate homogeneous spaces in particular we describe how arnold s approach to the euler equations as geodesic flows of one sided invariant metrics extends from lie groups to homogeneous spaces we also show that the above three cases describe all generic bihamiltonian systems which are related to the virasoro group and can be integrated by the translation argument principle they correspond precisely to the three different types of generic virasoro orbits finally we discuss interrelation between the above metrics and kahler structures on virasoro orbits as well as open questions regarding integrable systems corresponding to a finer classification of the orbits

on the weak solutions to a shallow water equation we obtain the existence of global in time weak solutions to the cauchy problem for a one dimensional shallow water equation that is formally integrable and can be obtained by approximating directly the hamiltonian for euler s equation in the shallow water regime the solution is obtained as a limit of viscous approximation the key elements in our analysis are some new a priori one sided supernorm and space time higher norm estimates on the first order derivatives 2000 john wiley sons inc

on the geometric approach to the motion of inertial mechanical systems according to the principle of least action the spatially periodic motions of one dimensional mechanical systems with no external forces are described in the lagrangian formalism by geodesics on a manifold configuration space the group d of smooth orientation preserving diffeomorphisms of the circle the periodic inviscid burgers equation is the geodesic equation on d with the l 2 right invariant metric however the exponential map for this right invariant metric is not a c 1 local diffeomorphism and the geometric structure is therefore deficient on the other hand the geodesic equation on d for the h 1 rightinvariant metric is also a re expression of a model in mathematical physics we show that in this case the exponential map is a c 1 local diffeomorphism and that if two diffeomorphisms are sufficiently close on d t heycan be joined by a unique length minimizing geodesic a state of the system is transformed to another nearby state by going through a uniquely determined flow that minimizes the energy we also analyse for both metrics the breakdown of the geodesic flow

blow up phenomena for the weakly dissipative dullin gottwald holm equation revisited in this paper we investigate the cauchy problem for the weakly dissipative dullin gottwald holm equation we first establish the local well posedness result by kato s semigroup theory then we obtain the precise blow up scenario and present some blow up results for strong solutions to the equation finally we discuss the blow up rate of the wave breaking solutions this result complements the early one in the literature such as novruzov j math phys 54 092703 2013

the use of optimally regularized fourier series for estimating higher order derivatives of noisy biomechanical data abstract optimally regularized filtered fourier series can be used most effectively for estimating higher order derivatives of noisy data sequences such as occur in biomechanical investigations as the optimal filtering process takes place in the derivative domain use being made of the statistical information contained in the periodogram of the data sequence no numerical differentiation as such takes place and all the problems known to be associated with such a procedure are avoided the optimally filtered approximation to the function itself is obtained by the same technique the algorithm is based on known results from time series and approximation theory and offers several advantages when compared with other techniques the optimally filtered approximations of the function and all its required derivatives are obtained in a single run compact analytical fourier representations of the filtered function and the derivatives are available allowing interpolation between data points and further analytical treatment of the function all the desirable properties of fourier approximations such as periodicity end point continuity etc can be utilized if fast fourier transforms are used the algorithm is computationally far more efficient than other techniques

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

traveling wave solutions of the degasperis procesi equation we classify all weak traveling wave solutions of the degasperis procesi equation in addition to smooth and peaked solutions the equation is shown to admit more exotic traveling waves such as cuspons stumpons and composite waves

a comparison of automatic filtering techniques applied to biomechanical walking data abstract the purpose of this study was to compare and evaluate six automatic filtering techniques commonly used in biomechanics for filtering gait analysis kinematic signals namely 1 power spectrum signal to noise ratio assessment 2 generalised cross validation spline 3 least squares cubic splines 4 regularisation of fourier series 5 regression model and 6 residual analysis a battery of 1440 signals representing the displacements of seven markers attached upon the surface of the right lower limbs and one marker attached upon the surface of the sacrum during walking were used their original signal and added noise characteristics were known a priori the signals were filtered with every technique and the root mean square error between the filtered and reference signal was calculated for each derivative domain results indicated that among the investigated techniques there is not one that performs best in all the cases studied generally the techniques of power spectrum estimation least squares cubic splines and generalised cross validation produced the most acceptable results

a type of bounded traveling wave solutions for the fornberg whitham equation abstract in this paper by using bifurcation method we successfully find the fornberg whitham equation u t u x x t u x u u x x x u u x 3 u x u x x has a type of traveling wave solutions called kink like wave solutions and antikink like wave solutions they are defined on some semifinal bounded domains and possess properties of kink waves and anti kink waves their implicit expressions are obtained for some concrete data the graphs of the implicit functions are displayed and the numerical simulation is made the results show that our theoretical analysis agrees with the numerical simulation