lipschitz p integral operators and lipschitz p nuclear operators national natural science foundation of china 10526034 10701063 nsf dms 0800061 dms 1068838 dms 1200370

lipschitz operator ideals and the approximation property abstract we establish the basics of the theory of lipschitz operator ideals with the aim of recovering several classes of lipschitz maps related to absolute summability that have been introduced in the literature in the last years as an application we extend the notion and main results on the approximation property for banach spaces to the case of metric spaces

leg stiffness primarily depends on ankle stiffness during human hopping abstract when humans hop in place or run forward they adjust leg stiffness to accommodate changes in stride frequency or surface stiffness the goal of the present study was to determine the mechanisms by which humans adjust leg stiffness during hopping in place five subjects hopped in place at 2 2 hz while we collected force platform and kinematic data each subject completed trials in which they hopped to whatever height they chose preferred height hopping and trials in which they hopped as high as possible maximum height hopping leg stiffness was approximately twice as great for maximum height hopping as for preferred height hopping ankle torsional stiffness was 1 9 times greater while knee torsional stiffness was 1 7 times greater in maximum height hopping than in preferred height hopping we used a computer simulation to examine the sensitivity of leg stiffness to the observed changes in ankle and knee stiffness our model consisted of four segments foot shank thigh head arms trunk interconnected by three torsional springs ankle knee hip in the model increasing ankle stiffness by 1 9 fold as observed in the subjects caused leg stiffness to increase by 2 0 fold increasing knee stiffness by 1 7 fold had virtually no effect on leg stiffness thus we conclude that the primary mechanism for leg stiffness adjustment is the adjustment of ankle stiffness

calf muscle moment work and efficiency in level walking role of series elasticity abstract moment and work of the human calf muscles in level walking were determined by means of an emg to force processor based on a muscle analogue hof and van den berg 1981 j biomechanics 14 747 758 759 770 771 785 787 792 nine subjects four women five men walked on a level treadmill at speeds between 0 5 and 2 5 m s 1 in their self chosen pace and at forced pace with steplengths between 0 3 and 1 1 m the calf muscles are normally only active in the stance phase the moment increases with a variable course to a peak just before push off this peak moment increases with the walking speed from the reference moment the value in standing on the toes with one leg at zero speed to 1 5 2 1 times this value at a speed of 2 m s 1 and decreases at still greater speeds during the roll over phase work is done on the calf muscles negative work followed by positive work in push off the negative work is constant 0 20 0 36 j kg 1 depending on the subject the positive work increases linearly with steplength not with speed from zero at ca 0 35 m to 0 50 j kg 1 at a steplength of 1 1 m the interaction between the contractile and the series elastic component in the muscle could be studied by means of the analogue a great part of the work done on the muscle and of the positive work done by the contractile component are stored in the series elastic component the stored energy is released at a high rate in push off this mechanism ideally requires a concerted contraction i e a contraction in which the activation is matched to the load to the effect that the length of the contractile component remains constant the muscle then behaves like a spring consequences are a only little of the negative work gets lost b the length of the contractile component remains close to the optimum of the force length relation c the shortening speed of the contractile component is now in the range where the muscle works at a high efficiency and d high power peaks can be delivered due to the catapult action

lipschitz p r s integral operators and lipschitz p r s nuclear operators abstract we introduce the notions of strongly lipschitz p r s nuclear operators and strongly lipschitz p r s integral operators we develop a theory of strongly lipschitz p r s nuclear operators and strongly lipschitz p r s integral operators which closely parallels the theory for linear operators their close relationships with other lipschitz operator ideals are studied

comparison between emg to force processing and kinetic analysis for the calf muscle moment in walking and stepping abstract in emg to force processing the force of a muscle or the corresponding muscle moment with respect to the joint is determined from emg and muscle length by means of a model of the muscle s contractile and elastic properties in kinetic analysis the joint moment is calculated from the ground reaction force and the positions and accelerations of the body segments by means of newtonian equations based on a rigid body segment model these two fundamentally different methods have been compared by determining simultaneously the calf muscle moment and the total ankle moment in walking at slow moderate and fast speed and in stepping up and down a low bench as long as there is no activity of other muscles the moments obtained by either method should be identical the analysis was restricted to such periods as assessed from the emg of tibialis anterior in each experimental condition three steps were analyzed in five subjects of 21 22 yr in comparison with the results of kinetic analysis emg processing shows differences between 7 and 54 nm r m s with an average of 22 nm r m s normalized with respect to the r m s value of the moment itself the differences amount to 0 13 0 51 with an average of 0 22 in the majority of cases there was no systematic deviation between emg processing and kinetic analysis and the r m s difference was of the same order of magnitude as determined previously in ergometer experiments

mechanical output from individual muscles during explosive leg extensions the role of biarticular muscles the main result of this study is that biarticular leg muscles contribute significantly to the work done at joints due to transfer of power during explosive leg extensions in particular a net power transfer was shown from hip to knee joint during jumping and sprinting seven elite athletes performed explosive one legged jump and sprint push offs kinematics ground reaction forces and el tromyography emg of leg muscles were recorded the mechanical output of six individu muscle groups was estimated by using hill based muscle models the emg and kinematics served as input to these models for jumping as well as for sprinting the model estimated similar results for the relative work contribution done about a joint due to transfer of power by the biarticular muscles rectus femoris showed a power transfer from hip to knee joint while in contrast hamstrings showed a power transfer from knee to hip joint regardless of these opposite directions of power transfer a net transfer occurred from the hip to the knee joint the relative work contribution of hamstrings done in hip extension was 7 in jumping and 11 in sprinting for rectus femoris the relative work contribution done in knee extension was 21 in jumping and 31 in sprinting power transferring actions by gastrocnemius from knee to ankle contributed 25 in jumping and 28 in sprinting to the work done in piantar flexion these results support the hypothesis that the action of biarticular muscles contributes to a net transfer of power from proximal to distal joints during explosive leg extensions this action of the biarticular muscles causes an efficient conversion of body segment rotations into the desired translation of the body centre of gravity

the mechanics of running how does stiffness couple with speed a mathematical model for terrestrial running is presented based on a leg with the properties of a simple spring experimental force platform evidence is reviewed justifying the formulation of the model the governing differential equations are given in dimensionless form to make the results representative of animals of all body sizes the dimensionless input parameters are u a horizontal froude number based on forward speed and leg length v a vertical froude number based on vertical landing velocity and leg length and kleg a dimensionless stiffness for the leg spring results show that at high forward speed kleg is a nearly linear function of both u and v while the effective vertical stiffness is a quadratic function of u for each u v pair the simulation shows that the vertical force at mid step may be minimized by the choice of a particular step length a particularly useful specification of the theory occurs when both kleg and v are assumed fixed when kleg 15 and v 0 18 the model makes predictions of relative stride length s and initial leg angle theta o that are in good agreement with experimental data obtained from the literature

integration of the soliton hierarchy with self consistent sources in contrast with the soliton equations the evolution of the eigenfunctions in the lax representation of soliton equation with self consistent sources sescs possesses singularity we present a general method to treat the singularity to determine the evolution of scattering data the akns hierarchy with self consistent sources the mkdv hierarchy with self consistent sources the nonlinear schrodinger equation hierarchy with self consistent sources the kaup newell hierarchy with self consistent sources and the derivative nonlinear schrodinger equation hierarchy with self consistent sources are integrated directly by using the inverse scattering method the n soliton solutions for some sescs are presented it is shown that the insertion of a source may cause the variation of the velocity of soliton this approach can be applied to all other 1 1 dimensional soliton hierarchies

a motion analysis marker based method of determining centre of pressure during two legged hopping the fixed position of force plates has led researchers to pursue alternative methods of determining centre of pressure cop location to date errors reported using alternative methods to the force plate during dynamic tasks have been high the aim of this study was to investigate the accuracy of a motion analysis markerbased system to determine cop during a two legged hopping task five markers were attached to the left and right feet of eight healthy adults 5 females 3 males age 25 072 8 years height 1 7570 07 m mass 71 3711 3 kg multivariate forward stepwise and forced entry linear regression was used with data from five participants to determine cop position during quiet standing and hopping at various frequencies maximum standard error of the estimate of cop position was 12 mm in the anteroposterior direction and 8 mm in the mediolateral cross validation was performed using the remaining 3 participants maximum root mean square difference between the force plate and marker method was 14 mm for mediolateral cop and 20 mm for anteroposterior cop during 1 5 hz hopping differences reduced to a maximum of 7 mm mediolateral and 14 mm anteroposterior for the other frequencies the smallest difference in calculated sagittal plane ankle moment and timing of maximum moment was during 3 0 hz hopping and largest at 1 5 hz results indicate the marker based method of determining cop may be a suitable alternative to a force plate to determine cop position during a two legged hopping task at frequencies greater than 1 5 hz 2014 elsevier ltd all rights reserved

leg stiffness in human running comparison of estimates derived from previously published models to direct kinematic kinetic measures abstract it is not presently clear whether mathematical models used to estimate leg stiffness during human running are valid therefore leg stiffness during the braking phase of ground contact of running was calculated directly using synchronous kinematic high speed motion analysis and kinetic force platform analysis and compared to stiffness calculated using four previously published kinetic models nineteen well trained male middle distance runners age 21 1 4 1 yr vo2max 69 5 7 5 mlo2 kg 1 min 1 completed a series of runs of increasing speed from 2 5 to 6 5 m s 1 leg stiffness was calculated directly from kinetic kinematic analysis using both vertical and horizontal forces to obtain the resultant force in the line of leg compression model 1 values were also estimated using four previously published mathematical models where only force platform derived and anthropometric measures were required models 2 5 morin et al 2005 morin et al 2011 blum et al 2009 farley et al 1993 respectively the greatest statistical similarity between leg stiffness values occurred with models 1 and 2 the poorest similarity occurred when values from model 4 were compared with model 1 analyses suggest that the poor correlation between model 1 other models may have resulted from errors in the estimation in change in leg length during the braking phase previously published mathematical models did not provide accurate leg stiffness estimates although model 2 used by morin et al 2005 provided reasonable estimates that could be further improved by the removal of systematic error using a correction factor k 1 0496 kmodel2

pfaffianization of the two dimensional toda lattice pfaffianization procedure due to hirota and ohta is applied to the two dimensional toda lattice as a result a pfaffianized version of the two dimensional toda lattice is found

leg stiffness measures depend on computational method leg stiffness is often computed from ground reaction force grf registrations of vertical hops to estimate the force resisting capacity of the lower extremity during ground contact with leg stiffness values incorporated in a spring mass model to describe human motion individual biomechanical characteristics including leg stiffness were investigated in 40 healthy males our aim is to report and discuss the use of 13 different computational methods for evaluating leg stiffness from a double legged repetitive hopping task using only grf registrations four approximations for the velocity integration constant were combined with three mathematical expressions giving 12 methods for computing stiffness using double integrations one frequency based method that considered ground contact times was also trialled the 13 methods thus defined were used to compute stiffness in four extreme cases which were the stiffest and most compliant consistent and variable subjects all methods provided different stiffness measures for a given individual but the between method variations in stiffness were consistent across the four atypical subjects the frequency based method apparently overestimated the actual stiffness values whereas double integrations measures were more consistent in double integrations the choice of the integration constant and mathematical expression considerably affected stiffness values as variations during hopping were more or less emphasized stating a zero centre of mass position at take off gave more consistent results and taking a weighted average of the force or displacement curve was more forgiving to variations in performance in any case stiffness values should always be accompanied by a detailed description of their evaluation methods as our results demonstrated that computational methods affect calculated stiffness

new non isospectral integrable couplings of the akns system abstract by means of semi direct sums of lie algebras and the compatibility condition of lax pair an efficient way to construct non isospectral integrable couplings is proposed as its applications new non isospectral integrable couplings of the akns system are obtained the method can be devoted to constructing non isospectral integrable couplings of other nonlinear evolution equations

dynamic similarity during human running about froude and strouhal dimensionless numbers abstract dynamic similarity is a widely used concept in the fluid mechanics field and consists in placing two different sized systems in equivalent experimental conditions this enables removal of the effects of size and prediction of the behavior of a full size system from a scale model the aim of this study was to test whether the froude number nfr or the strouhal number str could be used as a criterion for dynamic similarity during running fifteen male subjects ran barefoot on a runway in three experimental conditions i all subjects ran at the same speed v 3 5 m s 1 ii the speed was determined from nfr iii the stride frequency was determined from str antero posterior fy and vertical fz ground reaction force components were assessed the similarity between the subjects was analysed from scale factor sets computed from anthropometric and kinetic data the use of str implied strong inter subject similarity for temporal parameters mean r 0 96 time to fz peak time to fy braking peak fy zero fore aft shear time to fy propulsive peak while nfr induced fewer and lower similarities mean r 0 75 fy zero fore aft shear time to fy propulsive peak fy braking impulse that only concerned antero posterior parameters this study brought experimental evidence that neither nfr nor str were sufficient for dynamic similarity during running but that each of them made its own contribution these findings suggested that the concomitant use of nfr and str should be assessed to induce inter subject dynamic similarity during running

optimal muscular coordination strategies for jumping this paper presents a detailed analysis of an optimal control solution to a maximum height squat jump based upon how muscles accelerate and contribute power to the body segments during the ground contact phase of jumping quantitative comparisons of model and experimental results expose a proximal to distal sequence of muscle activation i e from hip to knee to ankle we found that the contribution of muscles dominates both the angular acceleration and the instantaneous power of the segments however the contributions of gravity and segmental motion are insignificant except the latter become important during the final 10 of the jump vasti and gluteus maximus muscles are the major energy producers of the lower extremity these muscles are the prime movers of the lower extremity because they dominate the angular acceleration of the hip toward extension and the instantaneous power of the trunk in contrast the ankle plantarflexors soleus gastrocnemius and the other plantarflexors dominate the total energy of the thigh though these muscles also contribute appreciably to trunk power during the final 20 of the jump therefore the contribution of these muscles to overall jumping performance cannot be neglected we found that the biarticular gastrocnemius increases jump height i e the net vertical displacement of the center of mass of the body from standing by as much as 25 however this increase is not due to any unique biarticular action e g proximal to distal power transfer from the knee to the ankle since jumping performance is similar when gastrocnemius is replaced with a uniarticular ankle plantarflexor

the n soliton solutions of the sine gordon equation with self consistent sources abstract the hierarchy of the sine gordon equation with self consistent sources is derived by using the eigenfunctions of recursion operator the bilinear form of the sine gordon equation with self consistent sources is given and the n soliton solutions are obtained through hirota method and wronskian technique respectively some novel determinantal identities are presented to treat the nonlinear term in the time evolution and finish the wronskian verifications

effects of independently altering body weight and mass on the energetic cost of a human running model the mechanisms underlying the metabolic cost of running and legged locomotion in general remain to be well understood prior experimental studies show that the metabolic cost of human running correlates well with the vertical force generated to support body weight the mechanical work done and changes in the effective leg stiffness further previous work shows that the metabolic cost of running decreases with decreasing body weight increases with increasing body weight and mass and does not significantly change with changing body mass alone in the present study we seek to uncover the basic mechanism underlying this existing experimental data we find that an actuated spring mass mechanism representing the effective mechanics of human running provides a mechanistic explanation for the previously reported changes in the metabolic cost of human running if the dimensionless relative leg stiffness effective stiffness normalized by body weight and leg length is regulated to be constant the model presented in this paper provides a mechanical explanation for the changes in metabolic cost due to changing body weight and mass which have been previously measured experimentally and highlights the importance of active leg stiffness regulation during human running

comparison of whole body vertical stiffness and leg stiffness during single leg hopping in place in children and adults in the hopping literature whole body vertical stiffness and leg stiffness are used interchangeably due to most of the movement occurring in the vertical direction however there is some anterior posterior movement of the center of mass and displacements of the foot during hopping in place in both children and adults further it is not understood if leg stiffness show a similar pattern as whole body vertical stiffness when increasing hopping frequency the purpose of this study was to test if whole body vertical stiffness and leg stiffness are different during single leg hopping in place in children and adults across a range of frequencies seventeen children aged 5 11years and 16 young adults participated in this study the subjects hopped at their preferred frequency as well as 20 below 20 above and 40 above preferred frequency our results demonstrate that both whole body vertical stiffness and leg stiffness increase when increasing hopping frequency for children and adults however whole body vertical stiffness consistently overestimates leg stiffness due to a similar peak force but a greater leg length change compared to vertical com displacement this suggests a considerable horizontal com movement from landing to mid stance during hopping children aged 5 11years old showed lower absolute values but higher normalized values of two stiffness measures than adults this suggests somewhat adult like stiffness control in children but a reduced ability to manipulate the horizontal movement during single leg hopping in place when compared to adults

duality for lipschitz p summing operators abstract building upon the ideas of r arens and j eells 1956 1 we introduce the concept of spaces of banach space valued molecules whose duals can be naturally identified with spaces of operators between a metric space and a banach space on these spaces we define analogues of the tensor norms of chevet 1969 3 and saphar 1970 14 whose duals are spaces of lipschitz p summing operators in particular we identify the dual of the space of lipschitz p summing operators from a finite metric space to a banach space answering a question of j farmer and w b johnson 2009 6 and use it to give a new characterization of the non linear concept of lipschitz p summing operator between metric spaces in terms of linear operators between certain banach spaces more generally we define analogues of the norms of j t lapreste 1976 11 whose duals are analogues of a pietsch s p r s summing operators a pietsch 1980 12 as a special case we get a lipschitz version of q p dominated operators

intralimb compensation strategy depends on the nature of joint perturbation in human hopping due to the well described spring mass dynamics of bouncing gaits human hopping is a tractable model for elucidating basic neuromuscular compensation principles we tested whether subjects would employ a multi joint or single joint response to stabilize leg stiffness while wearing a spring loaded ankle foot orthosis afo that applied localized resistive and assistive torques to the ankle we analyzed kinematics and kinetics data from nine subjects hopping in place on one leg at three frequencies 2 2 2 4 and 2 8hz and three orthosis conditions freely articulating afo afo with plantarflexion resistance and afo with plantarflexion assistance leg stiffness was invariant across afo conditions however compensation strategy depended upon the nature of the applied load biological ankle stiffness increased in response to a resistive load at twice the rate that it decreased with an assitive load ankle adjustments alone fully compensated for an assistive load with no net change in combined biological plus applied total ankle stiffness px0 133 in contrast a resistive load resulted in a 7 4 9 0 increase in total ankle stiffness across frequencies and a concomitant 10 15 increase in knee joint stiffness at each frequency pp0 037 the increased knee joint stiffness in response to resistive ankle load allowed subjects to maintain a more flexed knee at mid stance which attenuated the effect of the increased total ankle joint stiffness to preserve leg stiffness and whole limb biomechanical performance our findings suggest humans maintain invariant leg stiffness in bouncing gaits through different intralimb compensation strategies that are specific to the nature of the joint loading r2008 elsevier ltd all rights reserved

numerical bifurcation analysis of the bipedal spring mass model abstract the spring mass model and its numerous extensions are currently one of the best candidates for templates of human and animal locomotion however with increasing complexity their applications can become very time consuming in this paper we present an approach that is based on the calculation of bifurcations in the bipedal spring mass model for walking since the bifurcations limit the region of stable walking locomotion can be studied by computing the corresponding boundaries originally the model was implemented as a hybrid dynamical system our new approach consists of the transformation of the series of initial value problems on different intervals into a single boundary value problem using this technique discontinuities can be avoided and sophisticated numerical methods for studying parametrized nonlinear boundary value problems can be applied thus appropriate extended systems are used to compute transcritical and period doubling bifurcation points as well as turning points we show that the resulting boundary value problems can be solved by the simple shooting method with sufficient accuracy making the application of the more extensive multiple shooting superfluous the proposed approach is fast robust to numerical perturbations and allows determining complete manifolds of periodic solutions of the original problem

comparing different approaches for determining joint torque parameters from isovelocity dynamometer measurements strength or maximum joint torque is a fundamental factor governing human movement and is regularly assessed for clinical and rehabilitative purposes as well as for research into human performance this study aimed to identify the most appropriate protocol for fitting a maximum voluntary torque function to experimental joint torque data three participants performed maximum isometric and concentric eccentric knee extension trials on an isovelocity dynamometer and a separate experimental protocol was used to estimate maximum knee extension angular velocity a nine parameter maximum voluntary torque function which included angle angular velocity and neural inhibition effects was fitted to the experimental torque data and three aspects of this fitting protocol were investigated using an independent experimental estimate of maximum knee extension angular velocity gave lower variability in the high concentric velocity region of the maximum torque function compared to using dynamometer measurements alone a weighted root mean square difference rmsd score function that forced the majority 73 92 of experimental data beneath the maximum torque function was found to best account for the one sided noise in experimental torques resulting from sub maximal effort by the participants the suggested protocol an appropriately weighted rmsd score function and an independent estimate of maximum knee extension angular velocity gave a weighted rmsd of between 11 and 13 nm 4 5 of maximum isometric torque it is recommended that this protocol be used in generating maximum voluntary joint torque functions in all torque based modelling of dynamic human movement

two binary darboux transformations for the kdv hierarchy with self consistent sources two binary integral type darboux transformations for the kdv hierarchy with self consistent sources are proposed in contrast with the darboux transformation for the kdv hierarchy one of the two binary darboux transformations provides non auto backlund transformation between two nth kdv equations with self consistent sources with different degrees the formula for the m times repeated binary darboux transformations are presented this enables us to construct the n soliton solution for the kdv hierarchy with self consistent sources

bilinear backlund transformation for the kdv equation with a source a bilinear backlund transformation for the kdv equation with a source is obtained by using the bilinear transformation method superposition formulae for solutions are then constructed which satisfy bianchi s exchange property and it is shown that they have a simple structure when they are written in the bilinear forms the inverse scattering transform for the equation is also derived from the backlund transformation

footfall dynamics for racewalkers and runners barefoot on compliant surfaces the main purpose of this study is to investigate the role of footfall surface compliance on the physical parameters affecting barefoot racewalkers and runners these parameters are identified using a new inverted pendulum body model with a forward moving foot pivot model correlations of footfall loads measured for four compliant surface mats showed leg foot compression stiffness for both gaits were in the range of 10 8 12 9 kn m with the initial stiffness spikes in the range of 6 5 52 kn m the average leg damping factor was about 0 6 for racewalkers and 6 for runners for both gaits there was negative leg damping just prior to foot lift off compared to the peak reactions for the rigid surface a mat of intermediate compliance 1020 kn m was effective in reducing the runners peak reaction spikes by as much as 17

a simply stabilized running model the spring loaded inverted pendulum slip or monopedal hopper is an archetypal model for running in numerous animal species although locomotion is generally considered a complex task requiring sophisticated control strategies to account for coordination and stability we show that stable gaits can be found in the slip with both linear and air springs controlled by a simple fixed leg reset policy we first derive touchdown to touchdown poincare maps under the common assumption of negligible gravitational effects during the stance phase we subsequently include and assess these effects and briefly consider coupling to pitching motions we investigate the domains of attraction of symmetric periodic gaits and bifurcations from the branches of stable gaits in terms of nondimensional parameters

a comparison and update of direct kinematic kinetic models of leg stiffness in human running abstract direct kinematic kinetic modelling currently represents the gold standard in leg stiffness quantification during three dimensional 3d motion capture experiments however the medial lateral components of ground reaction force and leg length have been neglected in current leg stiffness formulations it is unknown if accounting for all 3d would alter healthy biologic estimates of leg stiffness compared to present direct modelling methods this study compared running leg stiffness derived from a new method multiplanar method which includes all three cartesian axes against current methods which either only include the vertical axis line method or only the plane of progression uniplanar method twenty healthy female runners performed shod overground running at 5 0 m s three dimensional motion capture and synchronised in ground force plates were used to track the change in length of the leg vector hip joint centre to centre of pressure and resultant projected ground reaction force leg stiffness was expressed as dimensionless units as a percentage of an individual s bodyweight divided by standing leg length bw ll leg stiffness using the line method was larger than the uniplanar method by 15 6 bw ll p

integrability of the semi discrete toda equation with self consistent sources a new procedure to construct and solve soliton equations with self consistent sources sescss is applied to the semi discrete toda equation based on its bilinear from bilinear backlund transformation bt for the semi discrete toda escs is presented starting from the bt a lax pair is derived for the semi discrete toda escs

leg stiffness and stride frequency in human running when humans and other mammals run the body s complex system of muscle tendon and ligament springs behaves like a single linear spring leg spring a simple spring mass model consisting of a single linear leg spring and a mass equivalent to the animal s mass has been shown to describe the mechanics of running remarkably well force platform measurements from running animals including humans have shown that the stiffness of the leg spring remains nearly the same at all speeds and that the spring mass system is adjusted for higher speeds by increasing the angle swept by the leg spring the goal of the present study is to determine the relative importance of changes to the leg spring stiffness and the angle swept by the leg spring when humans alter their stride frequency at a given running speed human subjects ran on treadmill mounted force platform at 2 5ms 1 while using a range of stride frequencies from 26 below to 36 above the preferred stride frequency force platform measurements revealed that the stiffness of the leg spring increased by 2 3 fold from 7 0 to 16 3 knm 1 between the lowest and highest stride frequencies the angle swept by the leg spring decreased at higher stride frequencies partially offsetting the effect of the increased leg spring stiffness on the mechanical behavior of the spring mass system we conclude that the most important adjustment to the body s spring system to accommodate higher stride frequencies is that leg spring becomes stiffer

predicting maximum eccentric strength from surface emg measurements the origin of the well documented discrepancy between maximum voluntary and in vitro tetanic eccentric strength has yet to be fully understood this study aimed to determine whether surface emg measurements can be used to reproduce the in vitro tetanic force velocity relationship from maximum voluntary contractions five subjects performed maximal knee extensions over a range of eccentric and concentric velocities on an isovelocity dynamometer whilst emg from the quadriceps were recorded maximum voluntary mvc force length velocity data were estimated from the dynamometer measurements and a muscle model normalised amplitude length velocity data were obtained from the emg signals dividing the mvc forces by the normalised amplitudes generated emg corrected force length velocity data the goodness of fit of the in vitro tetanic force velocity function to the mvc and emg corrected forces was assessed based on a number of comparative scores the in vitro tetanic force velocity function provided a significantly better fit to the emg corrected forces compared to the mvc forces p or 0 05 furthermore the emg corrected forces generated realistic in vitro tetanic force velocity profiles a 58 19 increase in maximum eccentric strength is theoretically achievable through eliminating neural factors in conclusion emg amplitude can be used to estimate in vitro tetanic forces from maximal in vivo force measurements supporting neural factors as the major contributor to the difference between in vitro and in vivo maximal force

suspending loads decreases load stability but may slightly improve body stability here we seek to determine how compliantly suspended loads could affect the dynamic stability of legged locomotion we theoretically model the dynamic stability of a human carrying a load using a coupled spring mass damper model and an actuated spring loaded inverted pendulum model as these models have demonstrated the ability to correctly predict other aspects of locomotion with a load in prior work such as body forces and energetic cost we report that minimizing the load suspension natural frequency and damping ratio significantly reduces the stability of the load mass but may slightly improve the body stability of locomotion when compared to a rigidly attached load these results imply that a highly compliant load suspension could help stabilize body motion during human animal or robot load carriage but at the cost of a more awkward less stable load

effects of altered stride frequency and contact time on leg spring behavior in human running abstract many studies have demonstrated that contact time is a key factor affecting both the energetics and mechanics of running the purpose of the present study was to further explore the relationships between contact time t c step frequency f and leg stiffness k leg in human running since f is a compound parameter depending on both contact and aerial time the specific goal of this study was to independently vary f and t c and to investigate their respective effects on spring mass characteristics during running seeking to determine if the changes in k leg observed when running at different f are mainly due to inherent changes in t c we compared three types of constant 3 33 m s 1 running conditions in 10 male subjects normal running at the subject s freely chosen f running with decreased and increased f and decreased and increased t c at the imposed freely chosen f the data from the varied f trials showed that the variation of t c was strongly correlated to that of k leg r 2 0 90 and the variation of f was also significantly correlated to that of k leg r 2 0 47 further changes in t c obtained in various t c conditions were significantly correlated to changes in k leg r 2 0 96 these results confirm that leg stiffness was significantly influenced by step frequency variations during constant speed running as earlier demonstrated but our more novel finding is that compared to step frequency the effect of contact time variations appears to be a stronger and more direct determinant of k leg indeed 90 96 of the variance in k leg can be explained by contact time whether this latter parameter is directly controlled or indirectly controlled through its close relationship with step frequency in conclusion from the comparison of two experimental procedures i e imposing various step frequency conditions vs asking subjects to intentionally vary contact time at their freely chosen step frequency it appears that changes in leg stiffness are mainly related to changes in contact time rather than to those in step frequency step frequency appears to be an indirect factor influencing leg stiffness through its effect on contact time which could be considered a major determinant of this spring mass characteristic of human running

on the pfaffianized kp equation with self consistent sources abstract a new procedure called source generation is applied to the pfaffianized kp equation as a result the pfaffianized kp equation with self consistent sources escs is obtained this coupled system cannot only be reduced to the pfaffianized kp equation but also reduced to the kp equation with self consistent sources kpescs so the pfaffianized kp escs can be viewed as a pfaffian version of the kpescs which indicates the commutativity of the source generation procedure and pfaffianization

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

reducing ground reaction forces in gymnastics landings may increase internal loading the aim of this study was to use a subject specific seven link wobbling mass model of a gymnast and a multi layer model of a landing mat to determine landing strategies that minimise ground reaction forces grf and internal forces subject specific strength parameters were determined that defined the maximum voluntary torque angle angular velocity relationship at each joint these relationships were used to produce subject specific lumped linear muscle models for each joint muscle activation histories were optimised using a simplex algorithm to minimise grf or bone bending moments for forward and backward rotating vault landings optimising the landing strategy to minimise each of the grf reduced the peak vertical and horizontal grf by 9 for the backward rotating vault and by 8 and 48 for the forward rotating vault compared to a matching simulation however most internal loading measures bone bending moments joint reaction forces and muscle forces increased compared to the matching simulation optimising the landing strategy to minimise the peak bone bending moments resulted in reduced internal loading measures and in most cases reduced grf bone bending moments were reduced by 27 during the forward rotating vault and by 2 during the backward rotating vault landings when compared to the matching simulations it is possible for a gymnast to modify their landing strategy in order to minimise internal forces and lower grf however using a reduction in grf due to a change in landing strategy as a basis for a reduction in injury potential in vaulting movements may not be appropriate since internal loading can increase

soldier relevant body borne loads increase knee joint contact force during a run to stop maneuver the purpose of this study was to understand the effects of load carriage on human performance specifically during a run to stop rts task using opensim analysis tools knee joint contact force grounds reaction force leg stiffness and lower extremity joint angles and moments were determined for nine male military personnel performing a rts under three load configurations light 6kg medium 20kg and heavy 40kg subject based means for each biomechanical variable were submitted to repeated measures anova to test the effects of load during the rts body borne load significantly increased peak knee joint contact force by 1 2 bw p 0 001 and peak vertical p 0 001 and anterior posterior p 0 002 ground reaction forces by 0 6 bw and 0 3 bw respectively body borne load also had a significant effect on hip p 0 026 posture with the medium load and knee p 0 046 posture with the heavy load with the heavy load participants exhibited a substantial albeit non significant increase in leg stiffness p 0 073 and d 0 615 increases in joint contact force exhibited during the rts were primarily due to greater grfs that impact the soldier with each incremental addition of body borne load the stiff leg extended knee and large braking force the soldiers exhibited with the heavy load suggests their injury risk may be greatest with that specific load configuration further work is needed to determine if the biomechanical profile exhibited with the heavy load configuration translates to unsafe shear forces at the knee joint and consequently a higher likelihood of injury

modela r as a froude and strouhal dimensionless numbers combination for dynamic similarity in running the aim of this study was to test the hypothesis that running at fixed fractions of froude nfr and strouhal str dimensionless numbers combinations induce dynamic similarity between humans of different sizes nineteen subjects ran in three experimental conditions i constant speed ii similar speed nfr and iii similar speed and similar step frequency nfr and str combination in addition to anthropometric data temporal kinematic and kinetic parameters were assessed at each stage to measure dynamic similarity informed by dimensional scale factors and by the decrease of dimensionless mechanical parameter variability over a total of 54 dynamic parameters dynamic similarity from scale factors was met for 16 mean r 0 51 32 mean r 0 49 and 52 mean r 0 60 parameters in the first the second and the third experimental conditions respectively the variability of the dimensionless preceding parameters was lower in the third condition than in the others this study shows that the combination of nfr and str computed from the dimensionless energy ratio at the center of gravity modela r ensures dynamic similarity between different sized subjects the relevance of using similar experimental conditions to compare mechanical dimensionless parameters is also proved and will highlight the study of running techniques or equipment and will allow the identification of abnormal and pathogenic running patterns modela r may be adapted to study other abilities requiring bounces in human or animal locomotion or to conduct investigations in comparative biomechanics

lipschitz compact operators abstract we introduce the notion of lipschitz compact weakly compact finite rank approximable operators from a pointed metric space x into a banach space e we prove that every strongly lipschitz p nuclear operator is lipschitz compact and every strongly lipschitz p integral operator is lipschitz weakly compact a theory of lipschitz compact weakly compact finite rank operators which closely parallels the theory for linear operators is developed in terms of the lipschitz transpose map of a lipschitz operator we state lipschitz versions of schauder type theorems on the weak compactness of the adjoint of a weakly compact linear operator

the non isospectral modified kadomtsev petviashvili equation with self consistent sources and its coupled system abstract a new algebraic method called source generation procedure is applied to construct non isospectral soliton equations with self consistent sources as results the non isospectral modified kadomtsev petviashvili equation with self consistent sources mkpescs and its gramm type determinant solutions are obtained by using the source generation procedure furthermore a new coupled system of the non isospectral mkpescs and its pfaffian solutions are constructed

conservation laws and self consistent sources for a super kn hierarchy in the present paper a super extension of the kaup newell kn hierarchy is proposed by matrix lie super algebras and the super kn hierarchy with self consistent sources is established furthermore infinitely many conservation laws of the super integrable hierarchy are presented the methods derived by us can be generalized to other nonlinear equation hierarchies

binary symmetry constraints of n wave interaction equations in 1 1 and 2 1 dimensions binary symmetry constraints of the n wave interaction equations in 1 1 and 2 1 dimensions are proposed to reduce the n wave interaction equations into finite dimensional liouville integrable systems a new involutive and functionally independent system of polynomial functions is generated from an arbitrary order square matrix lax operator and used to show the liouville integrability of the constrained flows of the n wave interaction equations the constraints on the potentials resulting from the symmetry constraints give rise to involutive solutions to the n wave interaction equations and thus the integrability by quadratures are shown for the n wave interaction equations by the constrained flows

pfaffianization of the discrete kp equation we use the procedure of ohta and hirota to generate an integrable coupled system of discrete equations from the discrete kp equation

on the mechanical power output required for human running insight from an analytical model in this paper the dynamics of human running on flat terrain and the required mechanical power output with its dependency on various parameters is investigated knowing the required mechanical power output is of relevance due to its relationship with the metabolic power for example a better understanding of the dependencies of required mechanical power output on weight running and wind speed step frequency ground contact time etc is very valuable for the assessment analysis and optimization of running performance therefore a mathematical model based on very few assumptions is devised the purpose of the proposed model is to relate running speed and required mechanical power output as an algebraic function of the runner s mass height step rate ground contact time and wind speed this is relevant in order to better understand the mechanical energy cost of locomotion and how much it depends on which parameters the first of the main energy dissipation mechanisms is due to vertical oscillation i e during each step some of the potential energy difference gets transformed into heat the second mechanism is due to the anterior ground reaction force during the first part of stance and the third is due to aerodynamic drag with the approximations of constant running speed and a sinusoidal vertical ground reaction force profile one obtains closed algebraic expressions for the center of mass trajectory and the required mechanical power output comparisons of model predictions and reported performance data suggest that approximately a quarter of the ground impact energy is stored during the first part of ground contact and then released during the remaining stance phase further one can conclude from the model that less mechanical power output is required when running with higher step rates and a higher center of mass non intuitive is the result that a shorter ground contact time is beneficial for fast runs while the opposite holds for slow runs an important advantage of the devised model compared to others is that it leads to closed algebraic expressions for the center of mass trajectory and mechanical power output which are functions of measurable quantities i e of step rate ground contact time running speed runner s mass center of mass height aerodynamic drag at some given speed wind speed and heart rate moreover the model relies on very few assumptions which have been verified and the only tuning parameter is the ratio of recovered elastic energy

simple method for measuring center of mass work during field running abstract the purpose of this study was to propose and validate a new simple method for calculation of center of mass work during field running in order to avoid the use of costly and inconvenient measurement devices this method relies on spring mass model and measurements of average horizontal velocity and contact and flight times during running ten male recreational subjects ran on a dynamometer treadmill at different velocities ranging from 2 22 to 4 44 m s 1 during 4 min 30 s for each velocity twenty consecutive steps were analyzed after 3 min 30 s the potential w pot forward kinetic w kinf and the total center of mass w ext work data obtained with this new method were compared with the reference data calculated from ground reaction force measurements w ext w pot and w kinf values calculated with the proposed method were respectively 3 39 0 77 higher 4 14 0 72 lower and 7 34 1 08 higher than values obtained by the reference method furthermore significant linear regressions close to the identity line were obtained between the reference and the proposed method values of works r 0 99 p w ext r 0 98 p w pot r 0 98 p w kinf it was concluded that this new method could provide a good estimate of center of mass work in field running thanks to a few simple mechanical parameters

human ankle joint stiffness over the full range of muscle activation levels abstract system identification techniques have been used to track changes in dynamic stiffness of the human ankle joint over a wide range of muscle contraction levels subjects lay supine on an experimental table with their left foot encased in a rigid low inertia cast which was fixed to an electro hydraulic actuator operating as a position servo subjects generated tonic plantarflexor or dorsiflexor torques of different magnitudes ranging from rest to maximum voluntary contractions mvc during repeated presentations of a stochastic ankle angular position perturbation compliance impulse response functions irf were determined from every 2 5 s perturbation sequence the gain g natural freqency n and damping parameters of the second order model providing the best fit to each irf were determined and used to compute the corresponding inertial i viscous b and elastic k stiffness parameters the behaviour of these parameters with mean torque was found to follow two simple rules first the elastic parameter k increased in proportion to mean ankle torque as it was varied from rest to mvc these changes were considerable involving increases of more than an order of magnitude second the damping parameter remained almost invariant over the entire range of contractions despite the dramatic changes in k

maximising somersault rotation in tumbling performing complex somersaulting skills during the flight phase of tumbling requires the generation of linear and angular momenta during the approach and takeoff phases this paper investigates how approach characteristics and takeoff technique affect performance with a view to maximising somersault rotation in tumbling a five segment planar simulation model customised to an elite gymnast was used to produce a simulation which closely matched a recorded performance of a double layout somersault by the elite gymnast three optimisations were carried out to maximise somersault rotation with different sets of initial conditions using the same initial linear and angular momentum as the double layout somersault and varying the joint torque activation timings allowed a double straight somersault to be performed with 19 more rotation potential than the actual performance increasing the approach velocity to a realistic maximum of 7 ms 1 resulted in a 42 reduction in rotation potential when the activation timings were unchanged but allowed a triple layout somersault to be performed with an increase of 31 in rotation potential when activation timings were re optimised increasing also the initial angular momentum to a realistic maximum resulted in a 4 reduction in rotation potential when the activation timings were unchanged but allowed a triple straight somersault to be performed with a further increase of 9 in rotation potential when activation timings were re optimised it is concluded that the limiting factor to maximising somersault rotation is the ability to generate high linear and angular velocities during the approach phase coupled with the ability to adopt consonant activation timings during the takeoff phase