a finite element approximation for a class of caputo time fractional diffusion equations abstract we develop a fully discrete scheme for time fractional diffusion equations by using a finite difference method in time and a finite element method in space the fractional derivatives are used in caputo sense stability and error estimates are derived the accuracy and efficiency of the presented method is shown by conducting two numerical examples

stability and convergence of the difference methods for the space time fractional advection diffusion equation abstract in this paper we consider a space time fractional advection dispersion equation stfade on a finite domain the stfade is obtained from the standard advection dispersion equation by replacing the first order time derivative by the caputo fractional derivative of order 0 1 and the first order and second order space derivatives by the riemman liouville fractional derivatives of order 0 1 and of order 1 2 respectively for the space fractional derivatives d x u x t and d x u x t we adopted the grunwald formula and the shift grunwald formula respectively we propose an implicit difference method idm and an explicit difference method edm to solve this equation stability and convergence of these methods are discussed using mathematical induction we prove that the idm is unconditionally stable and convergent but the edm is conditionally stable and convergent numerical results are in good agreement with theoretical analysis

femoral strain during walking predicted with muscle forces from static and dynamic optimization mechanical strain plays an important role in skeletal health and the ability to accurately and noninvasively quantify bone strain in vivo may be used to develop preventive measures that improve bone quality and decrease fracture risk a non invasive estimation of bone strain requires combined musculoskeletal finite element modeling for which the applied muscle forces are usually obtained from static optimization so methods in this study we compared finite element predicted femoral strains in walking using muscle forces obtained from so to those obtained from forward dynamics fd simulation the general trends in strain distributions were similar between fd and so derived conditions and both agreed well with previously reported in vivo strain gage measurements on the other hand differences in peak maximum emax and minimum emin principal strain magnitudes were as high as 32 between fd emax emin 945 1271 e and so emax emin 752 859 e these large differences in strain magnitudes were observed during the first half of stance where so predicted lower gluteal muscle forces and virtually no co contraction of the hip adductors compared to fd the importance of these results will likely depend on the purpose application of the modeling procedure if the goal is to obtain a generalized strain distribution for adaptive bone remodeling algorithms then traditional so is likely sufficient in cases were strain magnitudes are critical as is the case with fracture risk assessment bone strain estimation may benefit by including muscle activation and contractile dynamics in so or by using fd when practical

adjustments to zatsiorsky seluyanov s segment inertia parameters zatsiorsky et al in contemporary problems in biomechanics pp 272 291 crc press massachusetts 1990a obtained by means of a gamma ray scanning technique the relative body segment masses center of mass cm positions and radii of gyration for samples of college aged caucasian males and females although these data are the only available and comprehensive set of inertial parameters regarding young adult caucasians they have been rarely utilized for biomechanical analyses of subjects belonging to the same or a similar population the main reason is probably that zatsiorsky et al used bony landmarks as reference points for locating segment cms and defining segment lengths some of these landmarks were markedly distant from the joint centers currently used by most researchers as reference points the purpose of this study was to adjust the mean relative cm positions and radii of gyration reported by zatsiorsky et al in order to reference them to the joint centers or other commonly used landmarks rather than the original landmarks the adjustments were based on a number of carefully selected sources of anthropometric data

adaptive discretization of fractional order viscoelasticity using sparse time history an efficient numerical method to integrate the constitutive response of fractional order viscoelasticity is developed the method can handle variable time steps to overcome the problem of the growing amount of data that has to be stored and used in each time step we introduce sparse quadrature we use an internal variable formulation of the viscoelastic equations where the internal variable is of stress type the rate equation that governs the evolution of the internal variable involves a fractional integral and can be identified as a volterra integral equation of the second kind with a weakly singular kernel for the numerical integration of the rate equation we adopt the finite element method in time in particular the discontinuous galerkin method with piecewise constant basis functions is used a priori and a posteriori error estimates are proved an adaptive strategy based on the a posteriori error estimate is developed finally the precision and effectiveness of the method are demonstrated by comparing the numerical solutions with analytical solutions

the fundamental problem of myoskeletal inverse dynamics and its implications the validity of current inverse dynamics models utilized for motion analysis is investigated it is shown that observables generated by the real biosystem such as ground reaction forces are incompatible with comparable responses of skeletodynamical inverse models currently in use this implies that results obtained with such models are erroneous to varying degrees while a quantification of these errors is difficult or impossible this phenomenon is termed the fundamental myoskeletal inverse dynamics problem a model fidelity indicator is proposed which for a specific inverse dynamics model applied to a particular motion provides a dimensionless numerical measure for the replicative validity of that model and the fidelity of its input data a practical example demonstrates the usefulness of this indicator it is suggested that the development of structurally sufficiently complex and biologically more realistic skeletomechanical models as well as substantial error reductions in data measuring and processing procedures will be necessary to improve the accuracy of inverse dynamics model computations

some new classes of complex symmetric operators we say that an operator t e b h is complex symmetric if there exists a conjugate linear isometric involution c so that t ct c we prove that binormal operators operators that are algebraic of degree two including all idempotents and large classes of rank one perturbations of normal operators are complex symmetric from an abstract viewpoint these results explain why the compressed shift and volterra integration operator are complex symmetric finally we attempt to describe all complex symmetric partial isometries obtaining the sharpest possible statement given only the data dim ker t dim ker t

center of mass velocity position predictions for balance control the purposes of this analysis were to predict the feasible movements during which balance can be maintained based on environmental contact force anatomical foot geometry and physiological muscle strength constraints and to identify the role of each constraint in limiting movement an inverted pendulum model with a foot segment was used with an optimization algorithm to determine the set of feasible center of mass cm velocity position combinations for movement termination the upper boundary of the resulting feasible region ran from a velocity of 1 1 s 1 normalized to body height at 2 4 foot lengths behind the heel to 0 45 s 1 over the heel to zero over the toe and the lower boundary from a velocity of 0 9 s 1 at 2 7 foot lengths behind the heel to zero over the heel forward falls would be initiated if states exceeded the upper boundary and backward falls would be initiated if the states fell below the lower boundary under normal conditions the constraint on the size of the base of support bos determined the upper and lower boundaries of the feasible region however friction and strength did limit the feasible region when friction levels were less than 0 82 when dorsiflexion was reduced more than 51 or when plantar flexion strength was reduced more than 35 these findings expand the long held concept that balance is based on cm position limits i e the horizontal cm position has to be confined within the bos to guarantee stable standing to a concept based on cm velocity position limits

an actuated dissipative spring mass walking model predicting human like ground reaction forces and the effects of model parameters abstract simple models are widely used to understand the mechanics of human walking the optimization based minimal biped model and spring loaded inverted pendulum slip model are two popular models that can achieve human like walking patterns however ground reaction forces grf from these two models still deviate from experimental data in this paper we proposed an actuated dissipative spring mass model by integrating these two models to realize more human like grf patterns we first explored the function of stiffness damping and weights of both energy cost and force cost in the objective function and found that these parameters have distinctly different influences on the optimized gait and grf profiles the stiffness and objective weight affect the number and size of peaks in the vertical grf and stance time the damping changes the relative size of the peaks but has little influence on stance time based on these observations these parameters were manually tuned at three different speeds to approach experimentally measured vertical grf and the highest correlation coefficient can reach 0 983 these results indicate that the stiffness damping and proper objective functions are all important factors in achieving human like motion for this simple walking model these findings can facilitate the understanding of human walking dynamics and may be applied in future biped models

wright functions as scale invariant solutions of the diffusion wave equation the time fractional diffusion wave equation is obtained from the classical diffusion or wave equation by replacing the first or second order time derivative by a fractional derivative of order 0 2 using the similarity method and the method of the laplace transform it is shown that the scale invariant solutions of the mixed problem of signalling type for the time fractional diffusion wave equation are given in terms of the wright function in the case 0 1 and in terms of the generalized wright function in the case 1 2 the reduced equation for the scale invariant solutions is given in terms of the caputo type modification of the

the discriminant capabilities of stability measures trunk kinematics and step kinematics in classifying successful and failed compensatory stepping responses by young adults abstract this study evaluated the discriminant capability of stability measures trunk kinematics and step kinematics to classify successful and failed compensatory stepping responses in addition the shared variance between stability measures step kinematics and trunk kinematics is reported the stability measures included the anteroposterior distance d between the body center of mass and the stepping limb toe the margin of stability mos as well as time to boundary considering velocity ttb v velocity and acceleration ttb a and mos ttb mos kinematic measures included trunk flexion angle and angular velocity step length and the time after disturbance onset of recovery step completion fourteen young adults stood on a treadmill that delivered surface accelerations necessitating multiple forward compensatory steps thirteen subjects fell from an initial disturbance but recovered from a second identical disturbance trunk flexion velocity at completion of the first recovery step and trunk flexion angle at completion of the second step had the greatest overall classification of all measures 92 3 ttb v and ttb a at completion of both steps had the greatest classification accuracy of all stability measures 80 8 the length of the first recovery step r 0 70 and trunk flexion angle at completion of the second recovery step r 0 54 had the largest correlations with stability measures although ttb v and ttb a demonstrated somewhat smaller discriminant capabilities than trunk kinematics the small correlations between these stability measures and trunk kinematics r 0 52 suggest that they reflect two important yet different aspects of a compensatory stepping response

finite difference approximations for fractional advection dispersion flow equations fractional advection dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium in this paper we develop practical numerical methods to solve one dimensional fractional advection dispersion equations with variable coefficients on a finite domain the practical application of these results is illustrated by modeling a radial flow problem use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site

complex symmetric weighted composition operators in several variables abstract in this paper we study the complex symmetric weighted composition operators on the hilbert space h s of analytic functions over the open unit ball with reproducing kernels k w s z 1 z w s where s n we completely characterize complex symmetric weighted composition operators with respect to two important conjugations j v and j which include hermitian weighted composition operators and normal weighted composition operators appeared in 17 surprisingly they are essential representatives of general conjugations of the form c j where c is unitary and j symmetric indeed these complex symmetric weighted composition operators can only be divided into two classes the size estimates and some other properties of these complex symmetric weighted composition operators are obtained moreover involutive weighted composition operators are completely characterized too as an application we construct concrete conjugations for the involutive composition operators which solves a problem raised by garcia and hammond in a general case

maximum voluntary joint torque as a function of joint angle and angular velocity model development and application to the lower limb measurements of human strength can be important during analyses of physical activities such measurements have often taken the form of the maximum voluntary torque at a single joint angle and angular velocity however the available strength varies substantially with joint position and velocity when examining dynamic activities strength measurements should account for these variations a model is presented of maximum voluntary joint torque as a function of joint angle and angular velocity the model is based on well known physiological relationships between muscle force and length and between muscle force and velocity and was tested by fitting it to maximum voluntary joint torque data from six different exertions in the lower limb isometric concentric and eccentric maximum voluntary contractions were collected during hip extension hip flexion knee extension knee flexion ankle plantar flexion and dorsiflexion model parameters are reported for each of these exertion directions by gender and age group this model provides an efficient method by which strength variations with joint angle and angular velocity may be incorporated into comparisons between joint torques calculated by inverse dynamics and the maximum available joint torques

an algorithm for determining gravity line location from posturographic recordings abstract in posturographic recordings the center of pressure cop displacement does not accurately reflect migration of the gravity line gl the vertical line passing through the body s center of gravity cog since the horizontal ground reaction force f x is proportional to the horizontal acceleration of the cog its second integral does represent the horizontal position of the gravity line glp however the initial constants of integration initial velocity and position are not known in this note a technique for estimating these integration constants is suggested zero point to zero point integration this method is based on a postulation that when f x 0 the glp and cop coincide by integrating f x from one zero point to another zero point both the instantaneous glp and its velocity are determined a validation of the suggested algorithm was performed using optical methods to determine the glp during a one legged standing tasks cross correlation values of glp determined via videography and the suggested algorithm ranged from 0 79 to 0 96 these results suggest that the zero point to zero point integration is an acceptable technique for determining glp from posturographic recordings

unitary equivalence to a complex symmetric matrix low dimensions abstract a matrix t m n c is uecsm if it is unitarily equivalent to a complex symmetric i e self transpose matrix we develop several techniques for studying this property in dimensions three and four among other things we completely characterize 4 4 nilpotent matrices which are uecsm and we settle an open problem which has lingered in the 3 3 case we conclude with a discussion concerning a crucial difference which makes dimension three so different from dimensions four and above

poincare series of some pure and mixed trace algebras of two generic matrices we work over a field k of characteristic zero the poincare series for the algebra cn 2 of gln invariants and the algebra tn 2 of gln concomitants of two generic n n matrices x and y are computed for n 6 both simply graded and bigraded cases are included the cases n 4 were known previously for c4 2 and c5 2 we construct a minimal set of generators and give an application to specht s theorem on unitary similarity of matrices by identifying the space mn2 of pairs of n n matrices with mn k2 we extend the action of gln to gln gl2 for n 5 we compute the poincare series for the polynomial invariants of this action when restricted to the subgroups gln sl2 and gln 1 where 1 is the maximal torus of sl2 consisting of diagonal matrices five conjectures are proposed concerning the numerators and denominators of various poincare series mentioned above some heuristic formulas and open problems are stated

finite element method for the space and time fractional fokker planck equation we develop the finite element method for the numerical resolution of the space and time fractional fokker planck equation which is an effective tool for describing a process with both traps and flights the time fractional derivative of the equation is used to characterize the traps and the flights are depicted by the space fractional derivative the stability and error estimates are rigorously established and we prove that the convergent order is o k 2 alpha h mu where k is the time step size and h the space step size numerical computations are presented which demonstrate the effectiveness of the method and confirm the theoretical claims

crank nicolson method for the fractional diffusion equation with the riesz fractional derivative we examine a numerical method to approximate to a fractional diffusion equation with the riesz fractional derivative in a finite domain which has second order accuracy in time and space level in order to approximate the riesz fractional derivative we use the fractional centered derivative approach we determine the error of the riesz fractional derivative to the fractional centered difference we apply the crank nicolson method to a fractional diffusion equation which has the riesz fractional derivative and obtain that the method is unconditionally stable and convergent numerical results are given to demonstrate the accuracy of the crank nicolson method for the fractional diffusion equation with using fractional centered difference approach

the variational iteration method for solving riesz fractional partial differential equations in this paper the variational iteration method is applied to obtain the solution for space fractional partial differential equations where the space fractional derivative is in the riesz sense on the basis of the properties and definition of the fractional derivative the iterative technique is carried out in a straightforward manner without the need for transforms or numerical approximations examples demonstrate that the series solution obtained shows agreement with the exact solutions of the problems solved

center of pressure control for balance maintenance during lateral waist pull perturbations in older adults when balance is disturbed location of the center of pressure cop contributes to a person s ability to recover from a perturbation this study investigated cop control prior to first step lift off fslo during lateral perturbations in older non fallers and fallers 38 non fallers and 16 fallers received lateral waist pulls at 5 different intensities crossover stepping responses at the intensity level where the largest number of subjects responded with crossover steps were analyzed whole body center of mass com and cop positions in the medio lateral ml direction with respect to the base of support bos and cop velocity were calculated an inverted pendulum model was used to define the bos stability boundary at fslo which was also adjusted using the cop position at fslo functional boundary no significant differences were found in the cop velocities between fallers and non fallers p 093 however the cop positions for fallers were located significantly more medial at fslo p 01 resulting in a significantly reduced functional boundary although the stability margins measures of stability based on the bos were significantly larger than zero for fallers p 004 they were not significantly different from zero for the functional boundary i e reaching the functional stability limit fallers had reduced functional limits of stability in the ml direction which would predispose them to more precarious stability conditions than non fallers this could be a cause for taking more steps than non fallers for balance recovery as we observed the functional boundary estimation may be a more sensitive marker of balance instability than the bos boundary

muscle recruitment by the min max criterion a comparative numerical study this paper introduces the min max criterion for simulation of muscle recruitment in multiple muscle systems the criterion is introduced and justified by comparison to two known criterion types the polynomial criterion and the soft saturation criterion the comparison is performed on a planar three muscle elbow model performing a dumbbell curl a generalized form of the soft saturation criterion is introduced and it is shown that the min max criterion can be interpreted as the limit of the classical criteria when the exponents in their mathematical expressions approach infinity we finally show how the min max criterion can be cast into a form that allows for efficient and robust numerical solution by linear programming it is concluded that the min max criterion possesses a number of attractive physiological as well as algorithmic advantages

finite difference spectral approximations for the time fractional diffusion equation in this paper we consider the numerical resolution of a time fractional diffusion equation which is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order a with 0 a 1 the main purpose of this work is to construct and analyze stable and high order scheme to efficiently solve the time fractional diffusion equation the proposed method is based on a finite difference scheme in time and legendre spectral methods in space stability and convergence of the method are rigourously established we prove that the full discretization is unconditionally stable and the numerical solution converges to the exact one with order o dt 2 a n m where dt n and m are the time step size polynomial degree and regularity of the exact solution respectively numerical experiments are carried out to support the theoretical claims

a comparison of muscle energy models for simulating human walking in three dimensions the popular hill model for muscle activation and contractile dynamics has been extended with several different formulations for predicting the metabolic energy expenditure of human muscle actions these extended models differ considerably in their approach to computing energy expenditure particularly in their treatment of active lengthening and eccentric work but their predictive abilities have never been compared in this study we compared the predictions of five different hill based muscle energy models in 3d forward dynamics simulations of normal human walking in a data tracking simulation that minimized muscle fatigue the energy models predicted metabolic costs that varied over a three fold range 2 45 7 15 j m kg with the distinction arising from whether or not eccentric work was subtracted from the net heat rate in the calculation of the muscle metabolic rate in predictive simulations that optimized neuromuscular control to minimize the metabolic cost all five models predicted similar speeds step lengths and stance phase durations however some of the models predicted a hip circumduction strategy to minimize metabolic cost while others did not and the accuracy of the predicted knee and ankle angles and ground reaction forces also depended on the energy model used the results highlights the need to clarify how eccentric work should be treated when calculating muscle energy expenditure the difficulty in predicting realistic metabolic costs in simulated walking even with a detailed 3d musculoskeletal model the potential for using such models to predict energetically optimal gait modifications and the room for improvement in existing muscle energy models and locomotion simulation frameworks

galerkin spectral method for the fractional nonlocal thermistor problem we develop and analyse a numerical method for the time fractional nonlocal thermistor problem by rigorous proofs some error estimates in different contexts are derived showing that the combination of the backward differentiation in time and the galerkin spectral method in space leads for an enough smooth solution to an approximation of exponential convergence in space

complex symmetric operators and applications ii a bounded linear operator t on a complex hilbert space h is called complex symmetric if t ct c where c is a conjugation an isometric antilinear involution of h we prove that t cj t where j is an auxiliary conjugation commuting with t t t we consider numerous examples including the poincare neumann singular integral bounded operator and the jordan model operator compressed shift the decomposition t cj t also extends to the class of unbounded c selfadjoint operators originally introduced by glazman in this context it provides a method for estimating the norms of the resolvents of certain unbounded operators

predicting dynamic postural instability using center of mass time to contact information our purpose was to determine whether spatiotemporal measures of center of mass motion relative to the base of support boundary could predict stepping strategies after upper body postural perturbations in humans we expected that inclusion of center of mass acceleration in such time to contact ttc calculations would give better predictions and more advanced warning of perturbation severity ttc measures were compared with traditional postural variables which do not consider support boundaries and with an inverted pendulum model of dynamic stability developed by hof et al 2005 the condition for dynamic stability journal of biomechanics 38 1 8 a pendulum was used to deliver sequentially increasing perturbations to 10 young adults who were strapped to a wooden backboard that constrained motion to sagittal plane rotation about the ankle joint subjects were instructed to resist the perturbations stepping only if necessary to prevent a fall peak center of mass and center of pressure velocity and acceleration demonstrated linear increases with postural challenge in contrast boundary relevant minimum ttc values decreased nonlinearly with postural challenge enabling prediction of stepping responses using quadratic equations when ttc calculations incorporated center of mass acceleration the quadratic fits were better and gave more accurate predictions of the ttc values that would trigger stepping responses in addition ttc minima occurred earlier with acceleration inclusion giving more advanced warning of perturbation severity our results were in agreement with ttc predictions based on hof s model and suggest that ttc may function as a control parameter influencing the postural control system s decision to transition from a stationary base of support to a stepping strategy

stepping boundary of external force controlled perturbations of varying durations comparison of experimental data and model simulations abstract this study investigated the stepping boundary the force that can be resisted without stepping for force controlled perturbations of different durations twenty two healthy young adults 19 37 years old were instructed to try not to step in response to 86 different force time combinations of forward waist pulls the forces at which 50 of subjects stepped f50 were identified for each tested perturbation durations results showed that f50 decreased hyperbolically when the perturbation s duration increased and converged toward a constant value about 10 bw for longer perturbations over 1500 ms the effect of perturbation duration was critical for the shortest perturbations less than 1 s in parallel a simple function was proposed to estimate this stepping boundary considering the dynamics of a linear inverted pendulum foot model and simple balance recovery reactions we could express the maximum pulling force that can be withstood without stepping as a simple function of the perturbation duration when used with values of the main model parameters determined experimentally this function replicated adequately the experimental results this study demonstrates for the first time that perturbation duration has a major influence on the outcomes of compliant perturbations such as force controlled pulls the stepping boundary corresponds to a constant perturbation force duration product and is largely explained by only two parameters the reaction time and the displacement of the center of pressure within the functional base of support future work should investigate pathological populations and additional parameters characterizing the perturbation time profile such as the time derivative of the perturbation

the condition for dynamic stability the well known condition for standing stability in static situations is that the vertical projection of the centre of mass com should be within the base of support bos on the basis of a simple inverted pendulum model an extension of this rule is proposed for dynamical situations the position of the vertical projection of the com plus its velocity times a factor rootl g should be within the bos l being leg length and g the acceleration of gravity it is proposed to name this vector quantity extrapolated centre of mass position xcom the definition suggests as a measure of stability the margin of stability b the minimum distance from xcom to the boundaries of the bos an alternative measure is the temporal stability margin tau the time in which the boundary of the bos would be reached without intervention some experimental data of subjects standing on one or two feet flatfoot and tiptoe are presented to give an idea of the usual ranges of these margins of stability example data on walking are also presented c 2004 elsevier ltd all rights reserved

adaptational responses in dynamic stability during disturbed walking in the elderly the purpose of this study was to examine the age related predictive and feedback adaptive locomotor improvements in the components of dynamic stability control during disturbed walking thirteen old 62 74 yrs and ten young 23 30 yrs male subjects performed a gait protocol on a gangway which included one covered element by exchanging this element the subjects walked either solely over hard surface or experienced a perturbation of the gait on the soft surface element the gait protocol consisted of a baseline on hard surface and an adaptation phase with 19 trials on soft or hard 2nd 8th and 19th surface the investigation of dynamic stability was made by using the margin of stability ms which was calculated as the difference between the base of support and the extrapolated center of mass cm horizontal velocity of cm and its vertical projection in anterior posterior direction as well as the eigenfrequency of an inverted pendulum generate the extrapolated cm as a result of the first unexpected disturbance ms was decreased in the step following the perturbation compared to baselines in both age groups this decrease was higher for the old participants compared to the young ones indicating a more unstable position in the step after the perturbation for the elderly in the following adaptation phase ms returned to baseline values in both age groups in the hard trial after the first unexpected perturbation both age groups increased ms at touchdown of the disturbed leg compared to baseline reflecting fast predictive adjustments our findings show that the well known age related biological impairments did not inhibit the adaptive improvements in the components of dynamic stability in the elderly however the feedback corrections after the first unexpected perturbation were less effective for the elderly this may increase the risk of falling

global optimization of statistical functions with simulated annealing abstract many statistical methods rely on numerical optimization to estimate a model s parameters unfortunately conventional algorithms sometimes fail even when they do converge there is no assurance that they have found the global rather than a local optimum we test a new optimization algorithm simulated annealing on four econometric problems and compare it to three common conventional algorithms not only can simulated annealing find the global optimum it is also less likely to fail on difficult functions because it is a very robust algorithm the promise of simulated annealing is demonstrated on the four econometric problems

complex symmetric partial isometries an operator t b h is complex symmetric if there exists a conjugate linear isometric involution c h h so that t ct c we provide a concrete description of all complex symmetric partial isometries in particular we prove that any partial isometry on a hilbert space of dimension 4 is complex symmetric

bounded symbols and reproducing kernel thesis for truncated toeplitz operators compressions of toeplitz operators to coinvariant subspaces of h2 are called truncated toeplitz operators we study two questions related to these operators the first raised by sarason is whether boundedness of the operator implies the existence of a bounded symbol the second is the reproducing kernel thesis we show that in general the answer to the first question is negative and we exhibit some classes of spaces for which the answers to both questions are positive

muscle oxygen consumption determined by nirs in relation to external force and emg local oxygen consumption in a muscle vo 2 can be determined by near infrared spectroscopy nirs in principle it should be possible to use this measure to validate musculoskeletal models however the relationship between vo 2 and external force or between vo 2 and surface emg as a measure for muscle activity is hardly known the aim of this study was 1 to evaluate the characteristics of the relationship between vo 2 and external moments and 2 to determine whether differences exist between the emg moment relationship and the vo 2 moment relationship subjects n 5 were asked to perform isometric contractions exerting combinations of elbow flexion and pro supination moments at force levels up to 70 of their maximum simultaneous surface emg and nirs measurements were performed on the m biceps breve bb and the m brachioradialis br a linear relationship was found between emg and vo 2 for the bb vo 2 and emg were linearly related to both the flexion moment and the pro supination moment however for the br only a linear relationship with flexion moment was found as expected based on the findings above the relationship between vo 2 and elbow flexion moment can be described by a linear equation under the conditions of this study isometric and force levels up to 70 these findings suggest that load sharing is independent of force level and that next to emg vo 2 can be used for the validation of musculoskeletal models

an approximate solution of a nonlinear fractional differential equation by adomian decomposition method the aim of the present analysis is to apply adomian decomposition method for the solution of a nonlinear fractional differential equation finally the solution obtained by the decomposition method has been numerically evaluated and presented in the form of tables and then compared with those obtained by truncated series method a good agreement of the results is observed

a survey on fuzzy fractional differential and optimal control nonlocal evolution equations abstract we survey some representative results on fuzzy fractional differential equations controllability approximate controllability optimal control and optimal feedback control for several different kinds of fractional evolution equations optimality and relaxation of multiple control problems described by nonlinear fractional differential equations with nonlocal control conditions in banach spaces are considered

effects of walking speed on the step by step control of step width abstract young healthy adults walking at typical preferred speeds use step by step adjustments of step width to appropriately redirect their center of mass motion and ensure mediolateral stability however it is presently unclear whether this control strategy is retained when walking at the slower speeds preferred by many clinical populations we investigated whether the typical stabilization strategy is influenced by walking speed twelve young neurologically intact participants walked on a treadmill at a range of prescribed speeds 0 2 1 2 m s the mediolateral stabilization strategy was quantified as the proportion of step width variance predicted by the mechanical state of the pelvis throughout a step calculated as r2 magnitude from a multiple linear regression our ability to accurately predict the upcoming step width increased over the course of a step the strength of the relationship between step width and pelvis mechanics at the start of a step was reduced at slower speeds however these speed dependent differences largely disappeared by the end of a step other than at the slowest walking speed 0 2 m s these results suggest that mechanics dependent adjustments in step width are a consistent component of healthy gait across speeds and contexts however slower walking speeds may ease this control by allowing mediolateral repositioning of the swing leg to occur later in a step thus encouraging slower walking among clinical populations with limited sensorimotor control

can stability really predict an impending slip related fall among older adults the primary purpose of this study was to systematically evaluate and compare the predictive power of falls for a battery of stability indices obtained during normal walking among community dwelling older adults one hundred and eighty seven community dwelling older adults participated in the study after walking regularly for 20 strides on a walkway participants were subjected to an unannounced slip during gait under the protection of a safety harness full body kinematics and kinetics were monitored during walking using a motion capture system synchronized with force plates stability variables including feasible stability region measurement margin of stability the maximum floquet multiplier the lyapunov exponents short and long term and the variability of gait parameters including the step length step width and step time were calculated for each subject sensitivity of predicting slip outcome fall vs recovery was examined for each stability variable using logistic regression results showed that the feasible stability region measurement predicted fall incidence among these subjects with the highest sensitivity 68 4 except for the step width with an sensitivity of 60 2 no other stability variables could differentiate fallers from those who did not fall for the sample included in this study the findings from the present study could provide guidance to identify individuals at increased risk of falling using the feasible stability region measurement or variability of the step width

a new analysis for the convergence of the gradient discretization method for multidimensional time fractional diffusion and diffusion wave equations abstract we consider as discretization in space the gdm gradient discretization method developed recently in droniou et al 2018 to approximate multidimensional time fractional diffusion and diffusion wave equations where the fractional order of the time derivative is respectively satisfying 0 1 and 1 2 the time fractional derivative is given in the caputo sense the time discretization is performed using a uniform mesh for the time fractional diffusion equation we derive an implicit scheme in addition to an l l 2 a priori estimate which can be derived from a reasoning in bradji 2018 we present and prove a new l 2 h 0 1 a priori estimate for the discrete problem under the assumption that the exact solution is sufficiently smooth these a priori estimates allow to prove error estimates in discrete norms of l 2 h 0 1 and l l 2 the convergence order in these estimates is optimal in the sense of two points of view the first one is that the order in space is the same one proved in droniou et al 2018 for elliptic equation whereas the second point of view is that the particular case of this order when 1 is the same one obtained in droniou et al 2018 for the standard heat equation for the time fractional diffusion wave equation we derive two implicit schemes a full convergence analysis is carried out for both schemes in particular we develop new a priori estimates which yield error estimates in several discrete norms for each scheme the convergence is proved to be unconditional and optimal the convergence results are obtained thanks to a comparison with appropriately chosen auxiliary gradient schemes and to the stated a priori estimates these results improve the ones of bradji 2017 in which a conditional convergence is proved for a sushi scheme using stabilization and hybrid interfaces approximating a time fractional diffusion wave equation for both cases of time fractional diffusion and diffusion wave equations we show the well posedness in the sense that the discrete solutions depend continuously on the data of the considered problems the stated results can be extended to multi term time fractional diffusion and diffusion wave equations one of the main features when using gdm is that their results hold for all the schemes within the framework of gdm conforming and nonconforming finite element mixed finite element hybrid mixed mimetic family some multi point flux approximation finite volume schemes and some discontinuous galerkin schemes some examples of schemes recovered by the gss gradient schemes presented in this work are sketched among these examples we quote the one presented in jin et al 2015 to approximate time fractional diffusion equations this work extends and improves some results presented in brief or stated without proof in the notes bradji 2018 we present some numerical tests using sushi introduced in eymard et al 2010 to support the theoretical results

unitary equivalence to a complex symmetric matrix an algorithm abstract we present a necessary and sufficient condition for a 3 3 matrix to be unitarily equivalent to a symmetric matrix with complex entries and an algorithm whereby an arbitrary 3 3 matrix can be tested this test generalizes to a necessary and sufficient condition that applies to almost every n n matrix the test is constructive in that it explicitly exhibits the unitary equivalence to a complex symmetric matrix

moment arms and musculotendon lengths estimation for a three dimensional lower limb model this paper presents a set of polynomial expressions that can be used as regression equations to estimate length and three dimensional moment arms of 43 lower limb musculotendon actuators these equations allow one to find at a low computational cost the musculotendon geometric parameters required for numerical simulation of large musculoskeletal models nominal values for these biomechanical parameters were established using a public domain musculoskeletal model of the lower limb ieee trans biomed eng 37 1990 757 to fit these nominal values regression equations with different levels of complexity were generated based on the number of generalized coordinates of the joints spanned by each musculotendon actuator least squares fitting was used to identify regression equation coefficients the goodness of the fit and confidence intervals were assessed and the best fitting equations selected

single and multiple step balance recovery responses can be different at first step lift off following lateral waist pull perturbations in older adults abstract an inability to recover lateral balance with a single step is predictive of future falls in older adults this study investigated if balance stability at first step lift off fslo would be different between multiple and single stepping responses to lateral perturbations 54 healthy older adults received left and right waist pulls at 5 different intensities levels 1 5 crossover stepping responses at and above intensity level 3 that induced both single and multiple steps were analyzed whole body center of mass com and center of pressure cop positions in the medio lateral direction with respect to the base of support were calculated an inverted pendulum model was used to define the lateral stability boundary which was also adjusted using the cop position at fslo functional boundary no significant differences were detected in the cop positions between the responses at fslo p 0 075 indicating no difference in the functional boundaries between the responses significantly smaller stability margins were observed at first step landing for multiple steps at all levels p 0 024 while stability margins were also significantly smaller at fslo for level 3 and 4 p 0 048 these findings indicate that although reduced stability at first foot contact would be associated with taking additional steps stepping responses could also be attributable to the com motion state as early as first step lift off preceding foot contact perturbation based training interventions aimed at improving the reactive control of stability would reduce initial balance instability at first step lift off and possibly the consequent need for multiple steps in response to balance perturbations

the relationship between two different mechanical cost functions and muscle oxygen consumption abstract inverse dynamic models often use cost functions to solve the load sharing problem although it is often assumed that energy is minimised most cost functions are based on mechanically related measures like muscle force or stress the aim of this study was to analyse the relationships of two cost functions with experimentally determined data on muscle energy consumption four subjects performed isometric contractions generating combinations of elbow flexion extension and pro supination moments muscle oxygen consumption v o 2 of the m biceps breve m biceps longum m brachioradialis and m triceps laterale was measured with near infrared spectroscopy both cost functions were implemented into an existing inverse dynamic shoulder and elbow model and the individual cost values per muscle were calculated normalised and subsequently compared to experimental v o 2 values the minimum stress cost function led to a good correspondence between v o 2 and cost for the m triceps laterale but for the flexor muscles cost was significantly lower a newly proposed energy related cost function showed however a far better correspondence the inclusion of a linear term and muscle mass in the new criterion led model results to correspond better to experimental results the energy related cost function appeared to be a better measure for muscle energy consumption than the stress cost function and led to more realistic predictions of muscle activation

simultaneous positive and negative external mechanical work in human walking in human walking the center of mass motion is similar to an inverted pendulum viewing double support as a transition from one inverted pendulum to the next we hypothesized that the leading leg performs negative work to redirect the center of mass velocity while simultaneously the trailing leg performs positive work to replace the lost energy to test this hypothesis we developed a method to quantify the external mechanical work performed by each limb individual limbs method traditional measures of external mechanical work use the sum of the ground reaction forces acting on the limbs combined limbs method allowing for the mathematical cancellation of simultaneous positive and negative work during multiple support periods we expected to find that the traditional combined limbs method underestimates external mechanical work by a substantial amount we used both methods to measure the external mechanical work performed by humans walking over a range of speeds we found that during double support the legs perform a substantial amount of positive and negative external work simultaneously the combined limbs measures of positive and negative external work were approximately 33 less than those calculated using the individual limbs method at all speeds the trailing leg performs greater than 97 of the double support positive work while the leading leg performs greater than 94 of the double support negative work

simulation of quadrupedal locomotion using a rigid body model locomotion of the horse is simulated using a mathematical model based on rigid body dynamics a general method to generate the equations of motion for a two dimensional rigid body model with an arbitrary number of hinge joints is presented and a numerical solution method restricted to tree structured models is described joint movements originating from muscular forces or moments are simulated but the method also allows that parts of the model follow strictly the pattern of kinematic data moment generators with first order linear feedback were used as a rotational muscle equivalent ground hoof interaction forces are approximated by a viscoelastic model and pseudo coulomb friction in vertical and horizontal directions respectively results of model simulations are compared to experimentally recorded data subsequently adjustments are made to improve the agreement between simulation and experimental results