algebraic signal processing theory cooley tukey type algorithms for dcts and dsts in this paper we systematically derive a large class of fast general radix algorithms for various types of real discrete fourier transforms real dfts including the discrete hartley transform dht based on the algebraic signal processing theory this means that instead of manipulating the transform definition we derive algorithms by manipulating the polynomial algebras underlying the transforms using one general method the same method yields the well known cooley tukey fast fourier transform fft as well as general radix discrete cosine and sine transform algorithms the algebraic approach makes the derivation concise unifies and classifies many existing algorithms yields new variants enables structural optimization and naturally produces a human readable structural algorithm representation based on the kronecker product formalism we show for the first time that the general radix cooley tukey and the lesser known bruun algorithms are instances of the same generic algorithm further we show that this generic algorithm can be instantiated for all four types of the real dft and the dht

algebraic signal processing theory sampling for infinite and finite 1 d space we derive a signal processing framework called space signal processing that parallels time signal processing as such it comes in four versions continuous discrete infinite finite each with its own notion of convolution and fourier transform as in time these versions are connected by sampling theorems that we derive in contrast to time however space signal processing is based on a different notion of shift called space shift which operates symmetrically our work rigorously connects known and novel concepts into a coherent framework most importantly it shows that the sixteen discrete cosine and sine transforms are the space equivalent of the discrete fourier transform and hence can be derived by sampling the platform for our work is the algebraic signal processing theory an axiomatic approach and generalization of linear signal processing that we recently introduced

sensitivity analysis of discrete markov chains via matrix calculus abstract markov chains are often used as mathematical models of natural phenomena with transition probabilities defined in terms of parameters that are of interest in the scientific question at hand sensitivity analysis is an important way to quantify the effects of changes in these parameters on the behavior of the chain many properties of markov chains can be written as simple matrix expressions and hence matrix calculus is a powerful approach to sensitivity analysis using matrix calculus we derive the sensitivity and elasticity of a variety of properties of absorbing and ergodic finite state chains for absorbing chains we present the sensitivities of the moments of the number of visits to each transient state the moments of the time to absorption the mean number of states visited before absorption the quasistationary distribution and the probabilities of absorption in each of several absorbing states for ergodic chains we present the sensitivity of the stationary distribution the mean first passage time matrix the fundamental matrix and the kemeny constant we include two examples of application of the results to demographic and ecological problems

hex splines a novel spline family for hexagonal lattices this paper proposes a new family of bivariate nonseparable splines called hex splines especially designed for hexagonal lattices the starting point of the construction is the indicator function of the voronoi cell which is used to define in a natural way the first order hex spline higher order hex splines are obtained by successive convolutions a mathematical analysis of this new bivariate spline family is presented in particular we derive a closed form for a hex spline of arbitrary order we also discuss important properties such as their fourier transform and the fact they form a riesz basis we also highlight the approximation order for conventional rectangular lattices hex splines revert to classical separable tensor product b splines finally some prototypical applications and experimental results demonstrate the usefulness of hex splines for handling hexagonally sampled data

markov chain sensitivity measured by mean first passage times abstract the purpose of this article is to present results concerning the sensitivity of the stationary probabilities for an n state time homogeneous irreducible markov chain in terms of the mean first passage times in the chain

recommendation for the minimum number of steps to analyze when performing the uncontrolled manifold analysis on walking data abstract the uncontrolled manifold ucm analysis quantifies the extent to which co variation among a set of variables facilitates consistent performance by partitioning variance in those variables into two components then calculating their normalized difference i e the synergy index although ucm derived measures are thought to depend on the number of data points analyzed the minimum number needed to reasonably approximate true values of these measures is unknown for each of two performance variables related to mechanical stability of gait we evaluated changes in ucm derived measures when increasing the number of analyzed points here steps fourteen older adults walked on a treadmill while motion capture tracked movement for each subject n steps where n 2 99 were randomly sampled from the first 100 then used to calculate ucm derived variables for each subject variables were expressed as a percent of the subject specific value with n 100 and averaged across 50 simulations for each n 95 confidence intervals cis were calculated from group data the minimum number of steps to reasonably approximate a variables was defined as the value of n for which the lower ci was 90 of the value with n 100 regardless of performance variable reasonable approximations of the synergy index were attained with n 16 steps whereas n 50 steps were needed for each of the variance components however the differences between using 16 steps and 50 steps were small collecting 15 20 steps is recommended for a reasonable approximation of the synergy indices considered herein particularly when data collection is constrained to a limited number of steps

discrete signal processing on graphs frequency analysis signals and datasets that arise in physical and engineering applications as well as social genetics biomolecular and many other domains are becoming increasingly larger and more complex in contrast to traditional time and image signals data in these domains are supported by arbitrary graphs signal processing on graphs extends concepts and techniques from traditional signal processing to data indexed by generic graphs this paper studies the concepts of low and high frequencies on graphs and low high and band pass graph signals and graph filters in traditional signal processing these concepts are easily defined because of a natural frequency ordering that has a physical interpretation for signals residing on graphs in general there is no obvious frequency ordering we propose a definition of total variation for graph signals that naturally leads to a frequency ordering on graphs and defines low high and band pass graph signals and filters we study the design of graph filters with specified frequency response and illustrate our approach with applications to sensor malfunction detection and data classification

on single and double soules matrices abstract until now the concept of a soules basis matrix of sign pattern n consisted of an orthogonal matrix r r n n generated in a certain way from a positive n vector which has the property that for any diagonal matrix diag 1 n with 1 n 0 the symmetric matrix a r r t has nonnegative entries only in the present paper we introduce the notion of a pair of double soules basis matrices of sign pattern n which is a pair of matrices p q each in r n n which are not necessarily orthogonal and which are generated in a certain way from two positive vectors but such that pq t i and such that for any of the aforementioned diagonal matrices the matrix a p q t also has nonnegative entries only we investigate the interesting properties which such matrices a have as a preamble to the above investigation we show that the iterates a k r k r k t generated in the course of the qr algorithm when it is applied to a r r t where r is a soules basis matrix of sign pattern n are again symmetric matrices generated by the soules basis matrices r k of sign pattern n which are themselves modified as the algorithm progresses our work here extends earlier works by soules and elsner et al

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

local dynamic stability in turning and straight line gait successful community and household ambulation require the ability to navigate corners and maneuver around obstacles posing unique challenges compared to straight line walking the challenges associated with turning may contribute to an increased incidence of falling and the occurrence of fall related injuries a measure of stability applied to turning gait may be able to quantify a system s response to naturally occurring disturbances associated with turning and identify subjects at greater risk of falling an index of stability has been used previously to assess the rate of kinematic separation local dynamic stability during straight line gait the purpose of this study was to determine if local dynamic stability during constant speed turning is reduced compared to straight line treadmill walking maximum finite time lyapunov exponents lambda were used to estimate the local stability of able bodied subjects n 19 sagittal plane hip knee and ankle trajectories for turning compared to straight line walking at two different walking speeds turning lambda was greater than straight lambda for the hip right knee and ankle p 0 05 turning lambda for the left knee angle was similar to straight lambda there were no differences in lambda between left and right limbs for the hip and ankle and also no differences between the inside and outside limbs during turning for all joints these findings indicate able bodied subjects hip right knee and ankle kinematics are less locally stable while turning than walking in a straight line and may be used as a comparative tool for determining the efficacy of therapeutic interventions for mobility impaired populations

new perturbation bounds for denumerable markov chains this paper is devoted to perturbation analysis of denumerable markov chains bounds are provided for the deviation between the stationary distribution of the perturbed and nominal chain where the bounds are given by the weighted supremum norm in addition bounds for the perturbed stationary probabilities are established furthermore bounds on the norm of the asymptotic decomposition of the perturbed stationary distribution are provided where the bounds are expressed in terms of the norm of the ergodicity coefficient or the norm of a special residual matrix refinements of our bounds for doeblin markov chains are considered as well our results are illustrated with a number of examples

on explicit forms for ergodicity coefficients abstract explicit forms for ergodicity coefficients are known only when the l1 norm or the l norm is used 11 15 the purpose of this paper is to give a new approach to ergodicity coefficients and their functional forms realizations of this general procedure provide further ergodicity coefficients their explicit forms and new eigenvalue bounds which are compared with the deutsch zenger numerical radius 2 8

algebraic signal processing theory 1 d space in our paper titled ldquoalgebraic signal processing theory foundation and 1 d timerdquo appearing in this issue of the ieee transactions on signal processing we presented the algebraic signal processing theory an axiomatic and general framework for linear signal processing the basic concept in this theory is the signal model defined as the triple a m phi where a is a chosen algebra of filters m an associated a module of signals and phi is a generalization of the z transform each signal model has its own associated set of basic sp concepts including filtering spectrum and fourier transform examples include infinite and finite discrete time where these notions take their well known forms in this paper we use the algebraic theory to develop infinite and finite space signal models these models are based on a symmetric space shift operator which is distinct from the standard time shift we present the space signal processing concepts of filtering or convolution ldquoz transform rdquo spectrum and fourier transform for finite length space signals we obtain 16 variants of space models which have the 16 discrete cosine and sine transforms dcts dsts as fourier transforms using this novel derivation we provide missing signal processing concepts associated with the dcts dsts establish them as precise analogs to the dft get deep insight into their origin and enable the easy derivation of many of their properties including their fast algorithms

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

transition matrices for well conditioned markov chains abstract let t r n n be an irreducible stochastic matrix with stationary distribution vector set a i t and define the quantity 3 t 1 2 max j 1 n j a j 1 where aj j 1 n are the n 1 n 1 principal submatrices of a obtained by deleting the jth row and column of a results of cho and meyer and of kirkland show that 3 provides a sensitive measure of the conditioning of under perturbation of t moreover it is known that 3 t n 1 2 n in this paper we investigate the class of irreducible stochastic matrices t of order n such that 3 t n 1 2 n for such matrices correspond to markov chains with desirable conditioning properties we identify some restrictions on the zero nonzero patterns of such matrices and construct several infinite classes of matrices for which 3 is as small as possible

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

perturbation bounds for quantum markov processes and their fixed points we investigate the stability of quantum markov processes with respect to perturbations of their transition maps in the first part we introduce a condition number that measures the sensitivity of fixed points of a quantum channel to perturbations we establish upper and lower bounds on this condition number in terms of subdominant eigenvalues of the transition map in the second part we consider quantum markov processes that converge to a unique stationary state and we analyze the stability of the evolution at finite times in this way we obtain a linear relation between the mixing time of a quantum markov process and the sensitivity of its fixed point with respect to perturbations of the transition map

algebraic signal processing theory foundation and 1 d time this paper introduces a general and axiomatic approach to linear signal processing sp that we refer to as the algebraic signal processing theory asp basic to asp is the linear signal model defined as a triple a m phi where familiar concepts like the filter space and the signal space are cast as an algebra a and a module m respectively the mapping phi generalizes the concept of a z transform to bijective linear mappings from a vector space of signal samples into the module m common concepts like filtering spectrum or fourier transform have their equivalent counterparts in asp once these concepts and their properties are defined and understood in the context of asp they remain true and apply to specific instantiations of the asp signal model for example to develop signal processing theories for infinite and finite discrete time signals for infinite or finite discrete space signals or for multidimensional signals we need only to instantiate the signal model to one that makes sense for that specific class of signals filtering spectrum fourier transform and other notions follow then from the corresponding asp concepts similarly common assumptions in sp translate into requirements on the asp signal model for example shift invariance is equivalent to a being commutative for finite duration signals shift invariance then restricts a to polynomial algebras we explain how to design signal models from the specification of a special filter the shift the paper illustrates the general asp theory with the standard time shift presenting a unique signal model for infinite time and several signal models for finite time the latter models illustrate the role played by boundary conditions and recover the discrete fourier transform dft and its variants as associated fourier transforms finally asp provides a systematic methodology to derive fast algorithms for linear transforms this topic and the application of asp to space dependent signals and to multidimensional signals are pursued in companion papers

gait stability variability and complexity on inclined surfaces this study evaluated the gait stability variability and complexity of healthy young adults on inclined surfaces a total of 49 individuals walked on a treadmill at their preferred speed for 4min at inclinations of 6 8 and 10 in upward up and downward down conditions and in horizontal 0 condition gait variability was assessed using average standard deviation trunk acceleration between strides var gait stability was assessed using margin of stability mos and maximum lyapunov exponent s and gait complexity was assessed using sample entropy sen trunk variability var increased in the medial lateral ml anterior posterior and vertical directions for all inclined conditions the sen values indicated that movement complexity decreased almost linearly from down to up conditions reflecting changes in gait pattern with longer and slower steps as inclination increased the down conditions were associated with the highest variability and lowest stability in the mos ml but not in s stability was lower in up conditions which exhibited the largest s values the overall results support the hypothesis that inclined surfaces decrease gait stability and alter gait variability particularly in up conditions

efficient algorithms for computing the 2 d hexagonal fourier transforms in this paper representations of the two dimensional 2 d signals are presented that reduce the computation of 2 d discrete hexagonal fourier transforms 2 d dhfts the representations are based on the concept of the covering that reveals the mathematical structure of the transforms specifically a set of unitary paired transforms is derived that splits the 2 d dhft into a number of smaller one dimensional 1 d dfts examples for the 8 spl times 4 and 16 spl times 8 hexagonal lattices are described in detail the number of multiplications required for computing the 8 spl times 4 and 16 spl times 8 point dhfts are equal 20 and 136 respectively in the general n spl ges 8 case the number of multiplications required to compute the 2n spl times n point dhft by the paired transforms equals n sup 2 log n 1 n

sensitivity of the stationary distribution vector for an ergodic markov chain stationary distribution vectors p sup infinity for markov chains with associated transition matrices t are important in the analysis of many models in the mathematical sciences such as queuing networks input output economic models and compartmental tracer analysis models the purpose of this paper is to provide insight into the sensitivity of p sup infinity to perturbations in the transition probabilities of t and to understand some of the difficulties in computing an accurate p sup infinity the group inverse a sup number of i t is shown to be of fundamental importance in understanding sensitivity or conditioning of p sup infinity the main result shows that if there is a state that is accessible from every other state and the corresponding column of t has no small off diagonal elements then p sup infinity cannot be sensitive to small perturbations in t ecological examples are given a new stable algorithm for calculating a sup number is described

geometric transformations on the hexagonal grid the hexagonal grid has long been known to be superior to the more traditional rectangular grid system in many aspects in image processing and machine vision related fields however systematic developments of the mathematical backgrounds for the hexagonal grid are conspicuously lacking the purpose of this paper is to study geometric transformations on the hexagonal grid formulations of the transformation matrices are carried out in a symmetrical hexagonal coordinate frame a trio of new trigonometric functions are defined in this paper to facilitate the rotation transformations a fast algorithm for rounding an arbitrary point to the nearest hexagonal grid point is also presented

explicit forms for ergodicity coefficients and spectrum localization explicit forms for orgodicity coefficients which bound the non unit eigenvalues of finite stochastic matrices are known in the cases p 1 when the lp norm is used 3 6 7 the purpose of this report is to generalize these results to arbitrary real matrices in particular square matrices with a real valued eigenvalue by giving a generalized and to some extent simplified synthesis of methods used hitherto alternative procedures are mentioned in the light of the framework established the results are of particular relevance for a primitive nonnegative matrix for which a bound results for the non perron frobenius eigenvalues

upper bounds on the maximum modulus of subdominant eigenvalues of nonnegative matrices abstract upper bounds for the maximum modulus of the subdominant roots of square nonnegative matrices are obtained we provide a unified approach that yields or improves upon most of the bounds that have been obtained so far

algebraic signal processing theory 1 d nearest neighbor models we present a signal processing framework for the analysis of discrete signals represented as linear combinations of orthogonal polynomials we demonstrate that this representation implicitly changes the associated shift operation from the standard time shift to the nearest neighbor shift introduced in this paper using the algebraic signal processing theory we construct signal models based on this shift and derive their corresponding signal processing concepts including the proper notions of signal and filter spaces z transform convolution spectrum and fourier transform the presented results extend the algebraic signal processing theory and provide a general theoretical framework for signal analysis using orthogonal polynomials

generalized inverses of markovian kernels in terms of properties of the markov chain abstract all one condition generalized inverses of the markovian kernel i p where p is the transition matrix of a finite irreducible markov chain can be uniquely specified in terms of the stationary probabilities and the mean first passage times of the underlying markov chain special sub families include the group inverse of i p kemeny and snell s fundamental matrix of the markov chain and the moore penrose g inverse the elements of some sub families of the generalized inverses can also be re expressed involving the second moments of the recurrence time variables some applications to kemeny s constant and perturbations of markov chains are also considered

improved bounds for a condition number for markov chains abstract let p be the transition matrix for an n state homogeneous ergodic markov chain set q i p and let q q i j be the group generalized inverse of q a well known condition number due to funderlic and meyer which is used in the error analysis for the computation of the stationary distribution vector 1 n t of the chain is 4 max 1 i j n q i j in this paper we refine two upper estimates on 4 due to meyer in the course of proving one of our results we show that q i j j 1 i q 1 j where q j is the n 1 n 1 principal submatrix of q obtained from deleting its j th row and column and we characterize the case of equality the fact that we have a tight upper bound on the individual entries of the group inverse allows us to apply it in other contexts in which the group inverse arises for an irreducible nonnegative matrix such applications include for instance bounds on the second order partial derivative of the perron root with respect to any entry of the matrix and on the elasticity of the perron root with respect to any entry of the matrix

derivatives and perturbations of eigenvectors for a matrix bf a z whose entries are complex valued functions of a complex variable z results are presented concerning derivatives of an eigenvector bf x z of bf a z associated with a simple eigenvalue lambda z when bf x z is restricted to satisfy a constraint of the form sigma bf x z 1 where sigma is a rather arbitrary scaling function the differentiation formulas lead to a new approach for analyzing the sensitivity of an eigenvector under small perturbations in the underlying matrix finally an application is given which concerns the analysis of a finite markov chain subject to perturbations in the transition probabilities

applications of paz s inequality to perturbation bounds for markov chains abstract in several papers meyer singly and with coauthors established the usefulness of the group generalized inverse in the study and computations of various aspects of markov chains here we are interested in those results which concern bounds on the condition number of the chain and on the error in the computation of the stationary distribution vector we show that a lemma due to paz can be used to improve sometimes by a factor of 2 some of the constants in the bounds obtained in the aforementioned papers

on rayleigh ritz ratios of a generalized laplacian matrix of directed graphs abstract in c w wu algebraic connectivity of directed graphs linear and multilinear algebra in press a generalization of fiedler s notion of algebraic connectivity to directed graphs was presented that inherits many properties of the algebraic connectivity of undirected graphs and has applications to the synchronization of coupled dynamical systems with both constant and time varying coupling however it did not inherit nonnegativity and some of its bounds for combinatorial properties such as maximum directed cut and isoperimetric number are less strict that their undirected counterparts in particular these bounds do not have the corresponding bounds for undirected graphs as limiting cases the purpose of this paper is to present a refinement of this algebraic connectivity which preserve nonnegativity for strongly connected graphs and have bounds which contain the undirected graphs as special cases in particular we study quantities related to a generalized laplacian matrix of directed graphs and obtain bounds on combinatorial properties such as diameter bandwidth and bisection width for general directed graphs finally we give an application to the synchronization of coupled dynamical systems with constant coupling in particular we show that strong enough cooperative coupling will synchronize a network of coupled systems if the underlying directed graph is strongly connected

the discrete cosine transform each discrete cosine transform dct uses n real basis vectors whose components are cosines in the dct 4 for example the j th component of boldv k is cos j frac 1 2 k frac 1 2 frac pi n these basis vectors are orthogonal and the transform is extremely useful in image processing if the vector boldx gives the intensities along a row of pixels its cosine series sum c k boldv k has the coefficients c k boldx boldv k n they are quickly computed from a fast fourier transform but a direct proof of orthogonality by calculating inner products does not reveal how natural these cosine vectors are we prove orthogonality in a different way each dct basis contains the eigenvectors of a symmetric second difference matrix by varying the boundary conditions we get the established transforms dct 1 through dct 4 other combinations lead to four additional cosine transforms the type of boundary condition dirichlet or neumann centered at a meshpoint or a midpoint determines the applications that are appropriate for each transform the centering also determines the period n 1 or n in the established transforms n frac 1 2 or n frac 1 2 in the other four the key point is that all these eigenvectors of cosines come from simple and familiar matrices

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

efficient computation of the bounds of continuous time imprecise markov chains abstract when the initial distribution and transition rates for a continuous time markov chain are not known precisely robust methods are needed to study the evolution of the process in time to avoid judgements based on unwarranted precision we follow the ideas successfully applied in the study of discrete time model to build a framework of imprecise markov chains in continuous time the imprecision in the distributions over the set of states is modelled with upper and lower expectation functionals which equivalently represent sets of probability distributions uncertainty in transitions is modelled with sets of transition rates compatible with available information the kolmogorov s backward equation is then generalised into the form of a generalised differential equation with generalised derivatives and set valued maps the upper and lower expectation functionals corresponding to imprecise distributions at given times are determined by the maximal and minimal solutions of these equations the second part of the paper is devoted to numerical methods for approximating the boundary solutions the methods are based on discretisation of the time interval a uniform and adaptive grid discretisations are examined the latter is computationally much more efficient than the former one but is not applicable on every interval therefore to achieve maximal efficiency a combination of the methods is used

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

energy cost of balance control during walking decreases with external stabilizer stiffness independent of walking speed human walking requires active neuromuscular control to ensure stability in the lateral direction which inflicts a certain metabolic load the magnitude of this metabolic load has previously been investigated by means of passive external lateral stabilization via spring like cords in the present study we applied this method to test two hypotheses 1 the effect of external stabilization on energy cost depends on the stiffness of the stabilizing springs and 2 the energy cost for balance control and consequently the effect of external stabilization on energy cost depends on walking speed fourteen healthy young adults walked on a motor driven treadmill without stabilization and with stabilization with four different spring stiffnesses between 760 and 1820 nm 1 at three walking speeds 70 100 and 130 of preferred speed energy cost was calculated from breath by breath oxygen consumption gait parameters mean and variability of step width and stride length and variability of trunk accelerations were calculated from kinematic data on average external stabilization led to a decrease in energy cost of 6 p 0 005 as well as a decrease in step width 24 p 0 001 step width variability 41 p 0 001 and variability of medio lateral trunk acceleration 12 5 p 0 005 increasing stabilizer stiffness increased the effects on both energy cost and medio lateral gait parameters up to a stiffness of 1260 nm 1 contrary to expectations the effect of stabilization was independent of walking speed p 0 111 these results show that active lateral stabilization during walking involves an energetic cost which is independent of walking speed

the effects of gait strategy on metabolic rate and indicators of stability during downhill walking when walking at a given speed humans often appear to prefer gait patterns that minimize metabolic rate thereby maximizing metabolic economy however recent experiments have demonstrated that humans do not maximize economy when walking downhill the purpose of this study was to investigate whether this non metabolically optimal behavior is the result of a trade off between metabolic economy and gait stability we hypothesized that humans have the ability to modulate their gait strategy to increase either metabolic economy or stability but that increase in one measure will be accompanied by decrease in the other subjects walked downhill using gait strategies ranging from risky to conservative which were either prescribed by verbal instructions or induced by the threat of perturbations we quantified spatiotemporal gait characteristics metabolic rate and several indicators of stability previously associated with fall risk stride period variability step width variability lyapunov exponents floquet multipliers and stride period fractal index when subjects walked using conservative gait strategies stride periods and lengths decreased metabolic rate increased and anteroposterior maximum lyapunov exponents increased which has previously been interpreted as an indicator of decreased stability these results do not provide clear support for the proposed trade off between economy and stability particularly when stability is approximated using complex metrics however several gait pattern changes previously linked to increased fall risk were observed when our healthy subjects walked with a conservative strategy suggesting that these changes may be a response to rather than a cause of increased fall risk

a unified perturbation analysis framework for countable markov chains abstract in this paper we are devoted to singular perturbation analysis for discrete time or continuous time markov chains we modify and extend the drift condition method well known for regular perturbation to develop a new framework for singular perturbation analysis our results extend and improve the corresponding ones in 2 for singularly perturbed markov chains by allowing a general perturbation form less restrictive conditions and more computable bounds our analysis covers the regular perturbation analysis and hence unifies singular and regular perturbation analysis furthermore our results are illustrated by two two dimensional markov chains including a discrete time queue and a continuous time level dependent quasi birth death process

active arm swing and asymmetric walking leads to increased variability in trunk kinematics in young adults abstract fall induced injuries are a leading cause for occupational injuries with the majority originating from challenging same level walking surfaces despite current perturbation and fall prevention paradigms occupational fall prevalence remains stable typically these paradigms do not account for arm swing which has been demonstrated to affect the center of mass movement during walking this study examined the effect of different arm swing on postural control during symmetric and asymmetric walking fifteen healthy young adults age 23 4 2 8 walked symmetrically and asymmetrically with three arm motions normal held and active on a split belt treadmill caren extended system motek medical amsterdam nl mean standard deviation and maximal values of trunk linear and angular velocity and whole body angular momentum were calculated in all three axes additionally step length time and width mean and coefficient of variation margin of stability and harmonic ratios were calculated compared to normal and held conditions active arm increased trunk linear and angular velocity standard deviation max velocity values mean step length and time as well as the coefficient of variation for step length time and width furthermore whole body angular momentum increased as a function of arm swing amplitude active arm swing further reduced harmonic ratios in the mediolateral and anteroposterior directions asymmetric walking increased average step time and width as well as increased the coefficient of variation for step length and time but reduced left average step length and step width coefficient of variation further asymmetric walking increased mediolateral margin of stability and reduced anteroposterior and mediolateral harmonic ratios finally results demonstrated that actively increasing arm swing increases trunk linear and angular velocity variability in healthy young adults during symmetric and asymmetric treadmill walking findings may be due to active arm swing and asymmetric walking causing a disproportional contribution to trunk and center of mass movement causing participants to modify their base of support to maintain stability

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

effects of perturbation magnitude on dynamic stability when walking in destabilizing environments external perturbations applied to the walking surface or visual field can challenge an individual s ability to maintain stability during walking accurately quantifying and predicting changes in stability during walking will further our understanding of how individuals respond to challenges encountered during daily life and guide the development of assessments and rehabilitation interventions for individuals at increased risk of falling this study is the first to determine how orbital and local dynamic stability metrics including maximum floquet multipliers and local divergence exponents change in response to continuous mediolateral visual and surface perturbations of different amplitudes eleven healthy individuals walked in a fully immersive virtual environment participants completed two 3 min walking trials each under the following nine conditions no perturbations surface perturbations at each of 3 amplitudes and visual perturbations at each of 5 amplitudes all perturbations were applied as continuous pseudo random oscillations during both surface and visual perturbations individuals were significantly more orbitally and locally unstable compared to un perturbed walking as walking surface perturbation amplitudes increased individuals were more orbitally but not locally unstable as visual perturbation amplitudes increased individuals were more locally but not orbitally unstable between lower and higher amplitudes overall these dynamic stability metrics were much less sensitive to changes in perturbation amplitudes than to differences between un perturbed and perturbed walking or to differences between mechanical and visual perturbations this suggests that the type of perturbation s applied has a far greater impact than the magnitude of those perturbations in determining the response that will be elicited

algebraic signal processing theory 2 d spatial hexagonal lattice we develop the framework for signal processing on a spatial or undirected 2 d hexagonal lattice for both an infinite and a finite array of signal samples this framework includes the proper notions of z transform boundary conditions filtering or convolution spectrum frequency response and fourier transform in the finite case the fourier transform is called discrete triangle transform like the hexagonal lattice this transform is nonseparable the derivation of the framework makes it a natural extension of the algebraic signal processing theory that we recently introduced namely we construct the proper signal models given by polynomial algebras bottom up from a suitable definition of hexagonal space shifts using a procedure provided by the algebraic theory these signal models in turn then provide all the basic signal processing concepts the framework developed in this paper is related to mersereau s early work on hexagonal lattices in the same way as the discrete cosine and sine transforms are related to the discrete fourier transform a fact that will be made rigorous in this paper

stationary distributions and mean first passage times of perturbed markov chains stationary distributions of perturbed finite irreducible discrete time markov chains are intimately connected with the behaviour of associated mean first passage times this interconnection is explored through the use of generalized matrix inverses some interesting qualitative results regarding the nature of the relative and absolute changes to the stationary probabilities are obtained together with some improved bounds ams classification 15a51 60j10 60j20 65f20 65f35

limits of recovery against slip induced falls while walking abstract slip induced falls in gait often have devastating consequences the purposes of this study were 1 to select the determinants that can best discriminate the outcomes recoveries or falls of an unannounced slip induced in gait and to find their corresponding threshold i e the limits of recovery which can clearly separate these two outcomes and 2 to verify these results in a subset of repeated slip trials based on the data collected from 69 young subjects during a slip induced in gait nine different ways of combining the center of mass com stability the hip height and its vertical velocity were investigated with the aid of logistic regression the results revealed that the com stability s and limb support represented by the quotient of hip vertical velocity to hip height s hip recorded at the instant immediately prior to the recovery step touchdown were sufficiently sensitive to account for all 100 variance in falls and specific enough to account for nearly all 98 3 variability in recoveries this boundary s hip 0 22 s 0 25 which quantifies the risk of falls in the stability limb support quotient s s hip domain was fully verified using second slip and third slip trials n 76 with classification of falls at 100 and recoveries at 98 6 the severity of an actual fall is likely to be greater further below the boundary while the likelihood of a fall diminishes above it finally the slope of the boundary also indicates the tradeoff between the stability and limb support whereby high stability can compensate for the insufficiency in limb support or vice versa

strong stability and perturbation bounds for discrete markov chains we consider the stability problem of discrete markov chains when their transition matrices are perturbed for finite irreducible markov chains many perturbation bounds for the stationary vector are available in this paper we identify a condition under which these bounds are carried over to discrete irreducible markov chains which are not finite

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

on optimal condition numbers for markov chains let t and tilde t t e be arbitrary nonnegative irreducible stochastic matrices corresponding to two ergodic markov chains on n states a function is called a condition number for markov chains with respect to the norm pair if pi tilde pi alpha leq kappa t e beta here and tilde pi are the stationary distribution vectors of the two chains respectively various condition numbers particularly with respect to the 1 and norm pairs have been suggested in the literature they were ranked according to their size by cho and meyer in a paper from 2001 in this paper we first of all show that what we call the generalized ergodicity coefficient tau p a rm sup y t e 0 frac y ta p y 1 where e is the n vector of all 1 s and a is the group generalized inverse of a i t is the smallest condition number of markov chains with respect to the p norm pair we use this result to identify the smallest condition number of markov chains among the and 1 norm pairs these are respectively 3 and 6 in the cho meyer list of 8 condition numbers kirkland has studied 3 t he has shown that kappa 3 t geq frac n 1 2n and he has characterized transition matrices for which equality holds we prove here again that 2 3 t 6 which appears in the cho meyer paper and we characterize the transition matrices t for which kappa 6 t frac n 1 n there is actually only one such matrix t j n i n 1 where j n is the n n matrix of all 1 s

spectral theorem for convex monotone homogeneous maps and ergodic control we consider convex maps f rn rn that are monotone i e that preserve the product ordering of rn and nonexpansive for the sup norm this includes convex monotone maps that are additively homogeneous i e that commute with the addition of constants we show that the fixed point set of f when it is nonempty is isomorphic to a convex inf subsemilattice of rn whose dimension is at most equal to the number of strongly connected components of a critical graph defined from the tangent affine maps of f this yields in particular an uniqueness result for the bias vector of ergodic control problems this generalizes results obtained previously by lanery romanovsky and schweitzer and federgruen for ergodic control problems with finite state and action spaces which correspond to the special case of piecewise affine maps f we also show that the length of periodic orbits of f is bounded by the cyclicity of its critical graph which implies that the possible orbit lengths of f are exactly the orders of elements of the symmetric group on n letters

synergies in the ground reaction forces and moments during double support in curb negotiation in young and older adults abstract falls that occur while negotiating steps are a leading cause of death in older adults although recent efforts have improved understanding of the mechanics and control of stepping behaviors the double support phase during stepping is understudied therefore we quantified the stability of the resultant forces and moments acting on the body during this phase these quantities determine the movement of the body and therefore their stability is essential for successful stepping behavior we measured the ground reaction variables grvs under both feet as healthy young n 10 and older adults n 10 stepped up and down a curb we employed the uncontrolled manifold method to evaluate the hypotheses that the grvs covary to stabilize the resultant force and moment in the three coordinate directions robust stabilization of the resultant forces and moments was observed while stepping up however while stepping down the stability of the resultant moment was prioritized over that of the resultant forces in the vertical and the anterior posterior directions and the stability of the resultant medio lateral force was prioritized over that of the resultant anterior posterior force the salience of stabilizing whole body angular momentum and medio lateral motion during locomotion is well known but their prioritization during adaptive gait is a novel result and is possibly related to the higher likelihood of falling during descent versus ascent finally contrary to our expectations we observed no age differences in our stability indices indicating that healthy aging does not diminish the stability of the resultant forces and moments

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

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

acute perturbation of the group inverse abstract for any n n complex matrix a let a g be the group inverse of a when a is singular a matrix b a e is said to be an acute perturbation of a if e is small and the spectral radius b g b a g a 1 the acute perturbation coincides with the stable perturbation of the group inverse if the matrix b satisfies condition r b n a 0 n b r a 0 which was introduced by castro gonzalez et al 2008 8 furthermore several examples are provided to illustrate the acute perturbation of the group inverse

treadmill belt width but not feedback from the lower visual field influences the noise characteristics of step width time series abstract step kinematic variability which has been associated with gait related fall risk is thought to be attributed to neuromotor noise altered neuromotor control of step kinematics would be expected to manifest as changes in the noise related characteristics of the step kinematic time series this study determined the effects of eliminating feedback from the lower visual field and reducing treadmill belt width on the noise characteristics of step width time series and statistical measures of step width variability during treadmill walking we hypothesized that eliminating feedback from the lower visual field and reducing treadmill belt width would both alter the noise characteristics of step width time series reflected by decreased fractal scaling and increase statistical measures of step width variability eighteen young adults performed four randomly ordered walking trials during which we manipulated visual feedback from the lower visual field normal and obstructed and treadmill belt width wide and narrow reducing treadmill belt width but not eliminating feedback from the lower visual field significantly reduced the fractal scaling of step width time series indicating a shift towards white uncorrelated noise these results suggest that accounting for the influence of treadmill belt width on step width time series may be an important consideration in both laboratory and clinical settings further work is needed to clarify the effects of vision on measures of step width identify the mechanism s underlying the observed shift towards white uncorrelated noise associated with reduced treadmill belt width and to assess the potential relationship between the noise characteristics of step width time series and fall risk

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

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

a classification of invariant distributions and convergence of imprecise markov chains abstract we analyse the structure of imprecise markov chains and study their convergence by means of accessibility relations we first identify the sets of states so called minimal permanent classes that are the minimal sets capable of containing and preserving the whole probability mass of the chain these classes generalise the essential classes known from the classical theory we then define a class of extremal imprecise invariant distributions and show that they are uniquely determined by the values of the upper probability on minimal permanent classes moreover we give conditions for unique convergence to these extremal invariant distributions

stability and criticality analysis for integer linear programs with markovian problem data this paper presents the stability and criticality analysis of integer linear programs with respect to perturbations in stochastic data given as markov chains these perturbations affect the initial distribution the transition matrix or the stationary distribution of markov chains stability analysis is concerned with obtaining the set of all perturbations for which a solution remains optimal this paper gives expressions for stability regions for perturbations in the initial distribution the transition matrix the stationary distribution and the product of elements of the transition matrix and the stationary distribution furthermore criticality measures that describe the sensitivity of the objective function with respect to an element of the problem data are derived stability regions that preserve the stochasticity of the problem data are given finally stability regions for perturbations of elements of the transition matrix given that the problem is not linear in the initial distribution or the transition matrix are obtained using a small perturbation analysis the results are applied to sensor placement problems and numerical examples are given

measure valued differentiation for stationary markov chains we study general state space markov chains that depend on a parameter say sufficient conditions are established for the stationary performance of such a markov chain to be differentiable with respect to specifically we study the case of unbounded performance functions and thereby extend the result on weak differentiability of stationary distributions of markov chains to unbounded mappings first a closed form formula for the derivative of the stationary performance of a general state space markov chain is given using an operator theoretic approach in a second step we translate the derivative formula into unbiased gradient estimators specifically we establish phantom type estimators and score function estimators we illustrate our results with examples from queueing theory

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

mixing times with applications to perturbed markov chains abstract a measure of the mixing time or time to stationarity in a finite irreducible discrete time markov chain is considered the statistic i j 1 m m ij j where j is the stationary distribution and mij is the mean first passage time from state i to state j of the markov chain is shown to be independent of the initial state i so that i for all i is minimal in the case of a periodic chain yet can be arbitrarily large in a variety of situations an application considering the effects perturbations of the transition probabilities have on the stationary distributions of markov chains leads to a new bound involving for the 1 norm of the difference between the stationary probability vectors of the original and the perturbed chain when is large the stationary distribution of the markov chain is very sensitive to perturbations of the transition probabilities

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

comparison of perturbation bounds for the stationary distribution of a markov chain abstract the purpose of this paper is to review and compare the existing perturbation bounds for the stationary distribution of a finite irreducible homogeneous markov chain

the effects of step width and arm swing on energetic cost and lateral balance during running in walking humans prefer a moderate step width that minimizes energetic cost and vary step width from step to step to maintain lateral balance arm swing also reduces energetic cost and improves lateral balance in running humans prefer a narrow step width that may present a challenge for maintaining lateral balance however arm swing in running may improve lateral balance and help reduce energetic cost to understand the roles of step width and arm swing we hypothesized that net metabolic power would be greater at step widths greater or less than preferred and when running without arm swing we further hypothesized that step width variability indicator of lateral balance would be greater at step widths greater or less than preferred and when running without arm swing ten subjects ran 3 m s at four target step widths 0 15 20 and 25 leg length ll with arm swing at their preferred step width with arm swing and at their preferred step width without arm swing we measured metabolic power step width and step width variability when subjects ran at target step widths less 0 ll or greater 15 20 and 25 ll than preferred both net metabolic power demand by 3 9 12 and 15 and step width variability by 7 33 46 and 69 increased when running without arm swing both net metabolic power demand by 8 and step width variability by 9 increased compared to running with arm swing it appears that humans prefer to run with a narrow step width and swing their arms so as to minimize energetic cost and improve lateral balance

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

digraph based conditioning for markov chains for an irreducible stochastic matrix t we consider a certain condition number c t w hich measures the stability of the corresponding stationary distribution when t is perturbed we characterize the strongly connected directed graphs d such that c t is bounded as t ranges over sd the set of stochastic matrices whose directed graph is contained in d for those digraphs d for which c t is bounded we find the maximum value of c t as t ranges over sd

effect of function perturbation on the steady state distribution of genetic regulatory networks optimal structural intervention the dynamics of a rule based gene regulatory network are determined by the regulatory functions in conjunction with whatever probability distributions are involved in network transitions in the case of boolean networks bns and more generally probabilistic boolean networks pbns there has been a significant amount of investigation into the effect of perturbing gene states in particular the design of intervention strategies based on finite or infinite horizon control polices this paper considers the less investigated issue of function perturbations a single function perturbation affects network dynamics and alters the long run distribution whereas any individual gene perturbation has only transient effects and does not change the long run distribution we derive analytic results for changes in the steady state distributions of pbns resulting from modifications to the underlying regulatory rules and apply the derived results to find optimal structural interventions to avoid undesirable states the results are applied to a wnt5a network and a mammalian cell cycle related network respectively to achieve more favorable steady state distributions and reduce the risk of getting into aberrant phenotypes

series expansions for continuous time markov processes we present update formulas that allow us to express the stationary distribution of a continuous time markov process with denumerable state space having generator matrix q through a continuous time markov process with generator matrix q under suitable stability conditions numerical approximations can be derived from the update formulas and we show that the algorithms converge at a geometric rate applications to sensitivity analysis and bounds on perturbations are discussed as well numerical examples are presented to illustrate the efficiency of the proposed algorithm

a model of foot placement during gait foot placement is known to affect balance during gait however how foot placements are chosen is unknown objectives of this research were to analyze swing trajectories of the foot during gait with respect to the pelvis and to propose a model of foot placement control which provides a stable base of support effects of gait speed and vision on this model of foot placement were then examined foot trajectories were analyzed using spherical coordinates referenced to the pelvis termed the pelvic spherical coordinates pscs a model was developed based on this coordinate system which predicts foot placement in terms of position and velocity with respect to the pelvis it is proposed that foot placements are chosen to minimize the sum of psc coordinates of the stance and swing feet foot velocity at heel contact is proposed to minimize the sum of the psc stance and swing angular velocities experimental data were collected to test this model during walking trials of different speeds both with and without vision results showed that sums of stance and swing feet pscs were very close to zero at heel contact supporting the positional control hypothesis sums of psc velocities however were not zero at heel contact rather individual swing leg psc velocities were zero at heel contact suggesting independent swing leg velocity control vision did not have an effect on position or velocity variables at heel contact gait speed did have an effect particularly on psc velocities at heel contact abstract truncated at 250 words

predicted threshold against backward balance loss in gait the purpose of this study was to determine the minimum forward center of mass com velocity required to prevent backward loss of balance in gait as function of the initial com position we hypothesized that these threshold values would be different from those previously published for standing because of the postural differences between gait and standing to investigate this issue we constructed a seven link nine degree of freedom biomechanical model and employed dynamic optimization to estimate these threshold values under two initial postural conditions 1 the posture at the beginning of swing phase i e at toe off and 2 symmetric bipedal standing in particular for a range of possible com positions posterior to the base of support bos simulated annealing was used to search for the minimum velocity that could carry the com into the bos and avoid backward loss of balance we found that the stability boundary against backward balance loss in walking had a similar overall trend as that previously published for standing in general the minimal com velocity necessary to prevent a backward loss of balance in walking was greater than that in symmetric bipedal standing and the difference could approach 30 or more when the com started 0 5 and 1 0 foot lengths behind the bos these discrepancies suggest that simpler biomechanical models while being more efficient and easier to employ may not always be adequate for exploring stability limits of humans

conditioning of the entries in the stationary vector of a google type matrix abstract using combinatorial and analytic techniques we give conditioning bounds for the stationary vector t of a stochastic matrix of the form ca 1 c b where c 0 1 is a scalar and a and b are stochastic matrices the latter being rank one such matrices and their stationary vectors arise as a key component in google s pagerank algorithm the conditioning bounds considered include normwise absolute componentwise and relative componentwise and the bounds depend on c and on quantities such as the number of dangling nodes which correspond to rows of a having all entries equal or the lengths of certain cycles in the directed graph associated with a it is shown that if vertex j is on only long cycles in that directed graph then the corresponding entry in t exhibits better conditioning properties and that for dangling nodes the sensitivity of the corresponding entries in t decreases as the number of dangling nodes increases conditions are given that are sufficient to ensure that an iterate of the power method accurately reflects the relative ordering of two entries in t

sensitivity to perturbation of the stationary distribution some refinements abstract the sensitivity of the unique stationary distribution of a finite markov chain which has a state accessible in one step from every other state is examined using ordinary matrix inversion involving an arbitrary vector a result of funderlic and meyer is sharpened the presence of the arbitrary vector allows extension of the result to stochastic matrices with other structures the examples given by funderlic and meyer are reconsidered

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

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

perturbation bounds and degree of imprecision for uniquely convergent imprecise markov chains abstract the effect of perturbations of parameters for uniquely convergent imprecise markov chains is studied we provide the maximal distance between the distributions of original and perturbed chain and maximal degree of imprecision given the imprecision of the initial distribution the bounds on the errors and degrees of imprecision are found for the distributions at finite time steps and for the stationary distributions as well

predicted threshold against backward balance loss following a slip in gait the purpose of this study was to use a 7 link moment actuated human model to predict at liftoff of the trailing foot in gait the threshold of the center of mass com velocity relative to the base of support bos required to prevent backward balance loss during single stance recovery from a slip five dynamic optimization problems were solved to find the minimum com velocities that would allow the simulation to terminate with the com above the bos when the com started 0 25 0 5 0 75 1 0 and 1 25 foot lengths behind the heel of the stance foot i e behind the bos the initial joint angles of the model were based on averaged data from experimental trials foot ground contact was modeled using 16 visco elastic springs distributed under the stance foot slipping was modeled by setting the sliding coefficient of friction of these springs to 0 02 the forward velocity of the com necessary to avoid a backward balance loss is nearly two times larger under slip conditions under non slip conditions the predicted threshold for backward balance loss following a slip agreed well with experimental data collected from 99 young adults in response to 927 slips during walking in all trials in which a subject s com had a velocity below the predicted threshold the subject s recovery foot landed posterior to the slipping foot as predicted finally combining experimental data with optimization we verified that the 7 link model could more accurately predict gait stability than a 2 link model