a mass transportation approach to quantitative isoperimetric inequalities a sharp quantitative version of the anisotropic isoperimetric inequality is established corresponding to a stability estimate for the wulff shape of a given surface tension energy this is achieved by exploiting mass transportation theory especially gromov s proof of the isoperimetric inequality and the brenier mccann theorem a sharp quantitative version of the brunn minkowski inequality for convex sets is proved as a corollary

longest increasing subsequences in sliding windows we consider the problem of finding the longest increasing subsequence in a sliding window over a given sequence lisw we propose an output sensitive data structure that solves this problem in time o n log log n output for a sequence of n elements this data structure substantially improves over the naive generalization of the longest increasing subsequence algorithm and in fact produces an output sensitive optimal solution

local dynamic stability of treadmill walking intrasession and week to week repeatability abstract repetitive falls degrade the quality of life of elderly people and of patients suffering of various neurological disorders in order to prevent falls while walking one should rely on relevant early indicators of impaired dynamic balance the local dynamic stability lds represents the sensitivity of gait to small perturbations divergence exponents maximal lyapunov exponents assess how fast a dynamical system diverges from neighbor points although numerous findings attest the validity of lds as a fall risk index reliability results are still sparse the present study explores the intrasession and intersession repeatability of gait lds using intraclass correlation coefficients icc and standard error of measurement sem ninety five healthy individuals performed 5 min treadmill walking in two sessions separated by 9 days trunk acceleration was measured with a 3d accelerometer three time scales were used to estimate lds over 4 10 strides 4 10 over one stride 1 and over one step 0 5 the intrasession repeatability was assessed from three repetitions of either 35 strides or 70 strides taken within the 5 min tests the intersession repeatability compared the two sessions which totalized 210 strides the intrasession iccs 70 strides estimates 35 strides estimates were 0 52 0 18 for 4 10 and 0 84 0 77 for 1 and 0 5 the intersession iccs were around 0 60 the sem results revealed that 0 5 measured in medio lateral direction exhibited the best reliability sufficient to detect moderate changes at individual level 20 however due to the low intersession repeatability one should average several measurements taken on different days in order to better approximate the true lds

the sharp quantitative isoperimetric inequality a quantitative sharp form of the classical isoperimetric inequality is proved thus giving a positive answer to a conjecture by hall

when human walking becomes random walking fractal analysis and modeling of gait rhythm fluctuations abstract we present a random walk fractal analysis of the stride to stride fluctuations in the human gait rhythm the gait of healthy young adults is scale free with long range correlations extending over hundreds of strides this fractal scaling changes characteristically with maturation in children and older adults and becomes almost completely uncorrelated with certain neurologic diseases stochastic modeling of the gait rhythm dynamics based on transitions between different neural centers reproduces distinctive statistical properties of the gait pattern by tuning one model parameter the hopping transition range the model can describe alterations in gait dynamics from childhood to adulthood including a decrease in the correlation and volatility exponents with maturation

step width variability but not step length variability or step time variability discriminates gait of healthy young and older adults during treadmill locomotion the variability of spatial and temporal step kinematics have separately been shown to prospectively predict falls by older adults however the published literature has not addressed the relative importance of the information related to locomotion control contributed by variability of spatial variables temporal variables or both we conducted a post hoc analysis to determine the extent to which the variability of spatial and temporal step kinematics are independent descriptors of locomotion control in healthy young and older adults a second purpose of the analysis was to establish the extent to which treadmill walking mimics overground walking using a benchmark for step kinematics eighteen young adults and 12 older adults walked at a self selected velocity on an instrumented treadmill from which step length variability step width variability and step time variability were computed stepwise discriminant analysis correctly classified group membership of 70 percent of the 30 subjects i e young or older adult using only step width variability p 0 037 wilk s lamda 0 854 step width variability was 70 57 percent larger than step length variability p 0 001 this relationship was similar to that of the benchmark established for overground locomotion the results suggest that for healthy younger and older adults step width variability is a more meaningful descriptor of locomotion control than step length variability and step time variability given the potential clinical impact further systematic study and improvement of the methods and technology for acquiring step kinematic data are warranted

effect of noise and filtering on largest lyapunov exponent of time series associated with human walking this study aimed to determine the effect of added noise filtering and time series length on the largest lyapunov exponent lye value calculated for time series obtained from a passive dynamic walker the simplest passive dynamic walker model comprising of two massless legs connected by a frictionless hinge joint at the hip was adopted to generate walking time series the generated time series was used to construct a state space with the embedding dimension of 3 and time delay of 100 samples the lye was calculated as the exponential rate of divergence of neighboring trajectories of the state space using rosenstein s algorithm to determine the effect of noise on lye values seven levels of gaussian white noise snr 55 25db with 5db steps were added to the time series in addition the filtering was performed using a range of cutoff frequencies from 3hz to 19hz with 2hz steps the lye was calculated for both noise free and noisy time series with different lengths of 6 50 100 and 150 strides results demonstrated a high percent error in the presence of noise for lye therefore these observations suggest that rosenstein s algorithm might not perform well in the presence of added experimental noise furthermore findings indicated that at least 50 walking strides are required to calculate lye to account for the effect of noise finally observations support that a conservative filtering of the time series with a high cutoff frequency might be more appropriate prior to calculating lye

on line scheduling with monotone subsequence constraints abstract in this paper we study an on line scheduling problem that is motivated by applications such as carpooling the ride requests arrive on line over list and specify the service locations among m locations the goal is to determine a schedule that maximizes the number of satisfied requests using k servers we consider two variants of the problem with respect to constraints on the service location in the monotone service direction variant the service locations of the requests assigned to a server must be monotonic in the strict monotone service direction variant the service locations must be strictly monotonic we present lower bounds on the competitive ratio for both variants for the monotone service direction variant we give an optimal straightforward algorithm if k m and we prove that no deterministic on line algorithm can achieve a constant competitive ratio if k m for the strict monotone service direction variant we give a lower bound max 2 m 1 m 1 on the competitive ratio of barely deterministic algorithms if k 2 and a lower bound of max m 2 1 if k 2 we propose a balanced interval algorithm for the strict monotone service direction variant if k 2 and report some numerical experiments to evaluate the efficiency of this algorithm

response time is more important than walking speed for the ability of older adults to avoid a fall after a trip abstract we previously reported that the probability of an older adult recovering from a forward trip and using a lowering strategy increases with decreased walking velocity and faster response time to determine the within subject interaction of these variables we asked three questions 1 is the body orientation at the time that the recovery foot is lowered to the ground tilt angle critical for successful recovery 2 can a simple inverted pendulum model using subject specific walking velocity and response time as input variables predict this body orientation and thus success of recovery 3 is slower walking velocity or faster response time more effective in preventing a fall after a trip tilt angle was a perfect predictor of a successful recovery step indicating that the recovery foot placement must occur before the tilt angle exceeds a critical value of between 23 and 26 from vertical the inverted pendulum model predicted the tilt angle from walking velocity and response time with an error of 0 4 2 2 and a correlation coefficient of 0 93 the model predicted that faster response time was more important than slower walking velocity for successful recovery in a typical individual who is at risk for falling we predicted that a reduction of response time to a normal value allows a 77 increase in safe walking velocity the mathematical model produced patient specific recommendations for fall prevention and suggested the importance of directing therapeutic interventions toward improving the response time of older adults

possible biomechanical origins of the long range correlations in stride intervals of walking when humans walk the time duration of each stride varies from one stride to the next these temporal fluctuations exhibit long range correlations it has been suggested that these correlations stem from higher nervous system centers in the brain that control gait cycle timing existing proposed models of this phenomenon have focused on neurophysiological mechanisms that might give rise to these long range correlations and generally ignored potential alternative mechanical explanations we hypothesized that a simple mechanical system could also generate similar long range correlations in stride times we modified a very simple passive dynamic model of bipedal walking to incorporate forward propulsion through an impulsive force applied to the trailing leg at each push off push off forces were varied from step to step by incorporating both sensory and motor noise terms that were regulated by a simple proportional feedback controller we generated 400 simulations of walking with different combinations of sensory noise motor noise and feedback gain the stride time data from each simulation were analyzed using detrended fluctuation analysis to compute a scaling exponent a this exponent quantified how each stride interval was correlated with previous and subsequent stride intervals over different time scales for different variations of the noise terms and feedback gain we obtained short range correlations 1 0 our results indicate that a simple biomechanical model of walking can generate long range correlations and thus perhaps these correlations are not a complex result of higher level neuronal control as has been previously suggested

longest increasing subsequences from patience sorting to the baik deift johansson theorem we describe a simple one person card game patience sorting its analysis leads to a broad circle of ideas linking young tableaux with the longest increasing subsequence of a random permutation via the schensted correspondence a recent highlight of this area is the work of baik deift johansson which yields limiting probability laws via hard analysis of toeplitz determinants

walking variability during continuous pseudo random oscillations of the support surface and visual field abstract walking on uneven surfaces or while undergoing perturbations has been associated with increased gait variability in both modeling and human studies previous gait research involving continuous perturbations has focused on sinusoidal oscillations which can result in individuals predicting the perturbation and or entraining to it therefore we examined the effects of continuous pseudo random support surface and visual field oscillations on 12 healthy young participants participants walked in a virtual reality environment under no perturbation nop anterior posterior ap walking surface and visual oscillation and mediolateral ml walking surface and visual oscillation conditions participants exhibited shorter p 0 005 wider p

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

fast computation of a longest increasing subsequence and application we consider the complexity of computing a longest increasing subsequence lis parameterised by the length of the output namely we show that the maximal length k of an increasing subsequence of a permutation of the set of integers 1 2 n can be computed in time o nloglogk in the ram model improving the previous 30 year bound of o nlogk the algorithm also improves on the previous o nloglogn bound the optimality of the new bound is an open question reducing the computation of a longest common subsequence lcs between two strings to an lis computation leads to a simple o rloglogk time algorithm for two sequences having r pairs of matching symbols and an lcs of length k

active control of lateral balance in human walking we measured variability of foot placement during gait to test whether lateral balance must be actively controlled against dynamic instability the hypothesis was developed using a simple dynamical model that can walk down a slight incline with a periodic gait resembling that of humans this gait is entirely passive except that it requires active control for a single unstable mode con ned mainly to lateral motion an especially e cient means of controlling this instability is to adjust lateral foot placement we hypothesized that similar active feedback control is performed by humans with fore aft dynamics stabilized either passively or by very low level control the model predicts that uncertainty within the active feedback loop should result in variability in foot placement that is larger laterally than fore aft in addition loss of sensory information such as by closing the eyes should result in larger increases in lateral variability the control model also predicts a slight coupling between step width and length we tested 15 young normal human subjects and found that lateral variability was 79 larger than fore aft variability with eyes open and a larger increase in lateral variability 53 vs 21 with eyes closed consistent with the model s predictions we also found that the coupling between lateral and fore aft foot placements was consistent with a value of 0 13 predicted by the control model our results imply that humans may harness passive dynamic properties of the limbs in the sagittal plane but must provide signi cant active control in order to stabilize lateral motion 2000 elsevier science ltd all rights reserved

assessing gait stability the influence of state space reconstruction on inter and intra day reliability of local dynamic stability during over ground walking estimating local dynamic stability is considered a powerful approach to identify persons with balance impairments its validity has been studied extensively and provides evidence that short term local dynamic stability is related to balance impairments and the risk of falling thus far however this relation has only been proven on group level for clinical use differences on the individual level should also be detectable requiring reliability to be high in the current study reliability of short term local dynamic stability was investigated within and between days participants walked 500 m back and forth on a straight outdoor footpath on 2 non consecutive days and 3d linear accelerations were measured using an accelerometer dynaport minimod the state space was reconstructed using 4 common approaches all based on delay embedding within session intra class correlation coefficients were good 0 70 however between session intra class correlation coefficients were poor to moderate 0 63 and influenced by the reconstruction method the same holds for the smallest detectable difference which ranged from 17 to 46 depending on the state space reconstruction method the best within and between session intra class correlation coefficients and smallest detectable differences were achieved with a state space reconstruction with a fixed time delay and number of embedding dimensions overall due to the influence of biological variation and measurement error the short term local dynamic stability can only be used to detect substantial differences on the individual level

tight bounds for permutation flow shop scheduling in flow shop scheduling there are m machines and n jobs such that every job has to be processed on the machines in the fixed order 1 m in the permutation flow shop problem it is also required that each machine process the set of all jobs in the same order formally given n jobs along with their processing times on each machine the goal is to compute a single permutation of the jobs n n that minimizes the maximum job completion time makespan of the schedule resulting from the previously best known approximation guarantee for this problem was o m log m 1 2 sviridenko m 2004 a note on permutation flow shop problem ann oper res 129 247 252 in this paper we obtain an improved o min m1 2 n1 2 approximation algorithm for the permutation flow shop scheduling problem by finding a connection between the scheduling problem and the longest increasing subsequence problem our approximation ratio is relative to the lower bounds of maximum job length and maximum machine load and is the best possible such result this also resolves an open question from potts et al potts c d shmoys d williamson 1991 permutation vs nonpermutation flow shop schedules oper res lett 10 281 284 by algorithmically matching the gap between permutation and nonpermutation schedules we also consider the weighted completion time objective for the permutation flow shop scheduling problem using a natural linear programming relaxation and our algorithm for the makespan objective we obtain an o min m1 2 n1 2 approximation algorithm for minimizing the total weighted completion time improving on the previously best known guarantee of em for any constant e 0 smutnicki c 1998 some results of the worst case analysis for flow shop scheduling eur j oper res 109 66 87 we give a matching lower bound on the integrality gap of our linear programming relaxation

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

the validity of stability measures a modelling approach measures calculated from unperturbed walking patterns such as variability measures and maximum floquet multipliers are often used to study the stability of walking however it is unknown if and to what extent these measures correlate to the probability of falling we studied whether in a simple model of human walking i e a passive dynamic walker the probability of falling could be predicted from maximum floquet multipliers kinematic state variability and step time variability we used an extended version of the basic passive dynamic walker with arced feet and a hip spring the probability of falling was manipulated by varying the foot radius and hip spring stiffness or varying these factors while co varying the slope to keep step length constant the simulation data indicated that floquet multipliers and kinematic state variability correlated inconsistently with probability of falling step time variability correlated well with probability of falling but a more consistent correlation with the probability of falling was found by calculating the variability of the log transform of the step time our findings speak against the use of maximum floquet multipliers and suggest instead that variability of critical variables may be a good predictor of the probability to fall

measuring step kinematic variability on an instrumented treadmill how many steps are enough variability of step kinematics has been associated with falls by older adults however between study differences with regard to the number of steps used to compute variability have varied by an order of magnitude if the number of steps used to compute variability is too low there is the potential for a statistically spurious outcome on the other hand for subjects with mobility impairments a protocol requiring too many steps to estimate variability imposes an unnecessary burden on the subjects we have determined the minimum number of steps needed to estimate the variability of spatial and temporal step kinematics more than 700 steps were collected during level walking on an instrumented treadmill accurate estimation of step kinematic variability required at least 400 steps the increased error in estimating the mean and standard deviations of the step kinematic variables with too few steps can impose an experimental cost with regard to statistical design considerations the extent to which translation of these results can be made to the variability of spatial and temporal step kinematics collected during over ground walking awaits further research

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 selection principle for the sharp quantitative isoperimetric inequality we introduce a new variational method for the study of isoperimetric inequalities with quantitative terms the method is general as it relies on a penalization technique combined with the regularity theory for quasiminimizers of the perimeter two notable applications are presented first we give a new proof of the sharp quantitative isoperimetric inequality in mathbb r n second we positively answer a conjecture by hall concerning the best constant for the quantitative isoperimetric inequality in mathbb r 2 in the small asymmetry regime

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

dangers of geometric filtering abstract we argue that there are unrecognised dangers in the popular and powerful geometric filtering method carelessly applied geometric filtering can produce apparent structure in data which is pure noise or can cause severe distortions in clean deterministic data the explanations for these effects are straightforward and the dangers are easily avoided by taking simple precautions in the filtering process

dynamic stability of human walking in visually and mechanically destabilizing environments abstract understanding how humans remain stable during challenging locomotor activities is critical to developing effective tests to diagnose patients with increased fall risk this study determined if different continuous low amplitude perturbations would induce specific measureable changes in measures of dynamic stability during walking we applied continuous pseudo random oscillations of either the visual scene or support surface in either the anterior posterior or mediolateral directions to subjects walking in a virtual environment with speed matched optic flow floquet multipliers and short term local divergence exponents both increased indicating greater instability during perturbed walking these responses were generally much stronger for body movements occurring in the same directions as the applied perturbations likewise subjects were more sensitive to both visual and mechanical perturbations applied in the mediolateral direction than to those applied in the anterior posterior direction consistent with previous experiments and theoretical predictions these responses were likewise consistent with subjects anecdotal perceptions of which perturbation conditions were most challenging contrary to the floquet multipliers and short term local divergence exponents which both increased long term local divergence exponenets decreased during perturbed walking however this was consistent with specific changes in the mean log divergence curves which indicated that subjects movements reached their maximum local divergence limits more quickly during perturbed walking overall the floquet multipliers were less sensitive but reflected greater specificity in their responses to the different perturbation conditions conversely the short term local divergence exponents exhibited less specificity in their responses but were more sensitive measures of instability in general

beyond competitive analysis the competitive analysis of online algorithms has been criticized as being too crude and unrealistic we propose refinements of competitive analysis in two directions the first restricts the power of the adversary by allowing only certain input distributions while the other allows for comparisons between information regimes for online decision making we illustrate the first with an application to the paging problem as a byproduct we characterize completely the work functions of this important special case of the k server problem we use the second refinement to explore the power of lookahead in server and task systems

influence of simulated neuromuscular noise on movement variability and fall risk in a 3d dynamic walking model people at risk of falling exhibit increased gait variability which may predict future falls however the causal mechanisms underlying these correlations are not well known increased neuronal noise associated with aging likely leads to increased gait variability which could in turn lead to increased fall risk this paper presents a model of how changes in neuromuscular noise independently affect gait variability and probability of falling and aims to determine the extent to which changes in gait variability directly predict fall risk we used a dynamic walking model that incorporates a lateral step controller to maintain lateral stability noise was applied to this controller to approximate neuromuscular noise in humans noise amplitude was varied between low amplitudes that did not induce falls and high amplitudes for which the model always fell with increases in noise amplitude the model fell more often and after fewer steps gait variability increased with noise amplitude and predicted increased probability of falling importantly these relationships were not linear at either low gait variability or very high gait variability small increases in noise and variability affected probability of falling very little conversely at intermediate noise and or variability levels the same small increases resulted in large increases in probability of falling our results validate the idea that age related increases in neuromuscular noise likely play a direct contributing role in increasing fall risk however neuromuscular noise remains only one of many important factors that need to be considered these findings have important implications for fall prevention research and practice