global behavior of positive solutions of nonlinear three point boundary value problems we investigate the structure of the positive solution set for nonlinear three point boundary value problems of the form u h t f u 0 u 0 0 u 1 lambdau eta where eta epsilon 0 1 is given lambda epsilon 0 1 n is a parameter f epsilon c 0 infinity 0 infinity satisfies f s 0 for s 0 and h epsilon c 0 1 0 infinity is not identically zero on any subinterval of 0 1 our main results demonstrate the existence of continua of positive solutions of the above problem c 2004 elsevier ltd all rights reserved

nonlocal problems involving spatial structure for coupled reaction diffusion systems the purpose of this work is to investigate the uniqueness and existence of solutions for a system of nonlinear parabolic equations weakly coupled with ordinary differential equations in bounded domains subject to nonlocal boundary conditions that involve values of the unknown functions inside of the spatial domain the solutions of such problems allow for considering not only physical chemical or biological properties but also spatial structure of the modelled system the uniqueness of classical solutions of the nonlocal boundary problems for coupled reaction diffusion systems is proved by means of comparison principles for differential inequalities the existence of the unique solution is obtained via a monotone iterative method applications are given to some model problems in epidemiology and ecology with the nonlocal boundary conditions based on the inverse distance weighting interpolation and cokriging

a note on the bitsadze samarskii type nonlocal boundary value problem in a banach space abstract the bitsadze samarskii type nonlocal boundary value problem u t a u t f t 0 t 1 u 0 u 1 u 0 1 in an arbitrary banach space e with the positive operator a is considered the well posedness of this boundary value problem in the spaces of smooth functions is established the new exact schauder s estimates of solutions of the boundary value problems for elliptic equations are obtained

the blow up rate for a degenerate parabolic equation with a non local source abstract in this paper we establish the local existence of the solution and the finite time blow up result for the equation v v m xx a l l v q dx x l l 0 moreover it is proved that the solution has global blow up and there exist constants c 1 c 2 such that c 1 t 1 1 q 1 max x l l v x c 2 t 1 1 q 1 where t 1 is the blow up time

reaction diffusion equations with nonlocal boundary and nonlocal initial conditions in this paper we investigate the existence multiplicity and asymptotic behavior of solutions for a reaction diffusion equation under nonlocal boundary and nonlocal initial conditions a distinctive feature of this type of boundary and initial conditions is that the parabolic equation may possess multiple solutions even if the reaction function is smooth including linear functions of special concern is the asymptotic behavior of these multiple solutions in relation to the solutions of the corresponding steady state problem which may also possess multiple solutions in some special cases both the time dependent and the steady state problems possess an infinite number of solutions and each of the time dependent solutions converges to one of the steady state solutions

a quasilinear reaction diffusion system coupled via nonlocal sources in this paper we establish the critical exponent and the blow up rate for a quasilinear reaction diffusion system with nonlocal nonlinear sources and inner absorptions subject to homogeneous dirichlet conditions and nonnegative initial data it is found that the critical exponent is determined by the interaction among all the six nonlinear exponents from all the three kinds of the nonlinearities while the blow up rate is independent of the nonlinear diffusion exponents due to the effects of the coupled nonlocal sources two kinds of characteristic algebraic systems are introduced to get simple descriptions for the critical exponent and the blow up rate respectively we prove moreover that the blow up could be global due to the nonlocality of the nonlinear sources

operator semigroups flows and their invariant sets abstract let b h be the bounded operators on a hubert space h an operator semi group is an absolutely convex unital subsemigroup of the ball of b h such a induces a flow on h1 the unit ball of h namely s x sx let l be the lattice of closed invariant sets of this flow both the weak and norm topologies on h1 induce hausdorff metrics on the hyperspace of closed convex subsets of h1 we show that various subsets of i are closed subspaces of these hyperspaces there exist nontrivial operator semigroups which have the same invariant sets as the ball of b h i e the transitive semigroup problem has a negative solution however we can characterize the ball of b h in terms of certain transitivity properties we obtain an analogue of lomonosov s theorem for operator semigroups which are transitive on the unit sphere of h let a be a weakly closed unital subalgebra of b h and let a1 be its unit ball we show that the existence of minimal invariant sets for a yields hyperinvariant subspaces we characterize various operator algebras for which a1 is strongly precompact including von neumann algebras triangular algebras csl algebras and the standard function algebras acting by multiplication on a l2 space this yields structural results for certain algebras generated by a subnormal operator

asymptotic behavior of solutions of reaction diffusion equations with nonlocal boundary conditions abstract this paper is concerned with some dynamical property of a reaction diffusion equation with nonlocal boundary condition under some conditions on the kernel in the boundary condition and suitable conditions on the reaction function the asymptotic behavior of the time dependent solution is characterized in relation to a finite or an infinite set of constant steady state solutions this characterization is determined solely by the initial function and it leads to the stability and instability of the various steady state solutions in the case of finite constant steady state solutions the time dependent solution blows up in finite time when the initial function in greater than the largest constant solution also discussed is the decay property of the solution when the kernel function in the boundary condition prossesses alternating sign in its domain

the operational matrices of bernstein polynomials for solving the parabolic equation subject to specification of the mass abstract some physical problems in science and engineering are modelled by the parabolic partial differential equations with nonlocal boundary specifications in this paper a numerical method which employs the bernstein polynomials basis is implemented to give the approximate solution of a parabolic partial differential equation with boundary integral conditions the properties of bernstein polynomials and the operational matrices for integration differentiation and the product are introduced and are utilized to reduce the solution of the given parabolic partial differential equation to the solution of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the new technique

the utilization of total mass to determine the switching points in the symmetric boundary control problem with a linear reaction term abstract the authors study the problem u t u x x u 0 x 1 t 0 u x 0 0 and u 0 t u 1 t t where t u 0 for t 2 k t t 2 k 1 and t 0 for t 2 k 1 t t 2 k 2 k 0 1 2 with t 0 0 and the sequence t k is determined by the equations 0 1 u x t k d x m for k 1 3 5 and 0 1 u x t k d x m for k 2 4 6 and where 0 m m note that the switching points t k k 1 2 3 are unknown existence and uniqueness are demonstrated theoretical estimates of the t k and t k 1 t k are obtained and numerical verifications of the estimates are presented the case of u x 0 t u x 1 t t is also considered and analyzed

the two level finite difference schemes for the heat equation with nonlocal initial condition abstract in this paper the two level finite difference schemes for the one dimensional heat equation with a nonlocal initial condition are analyzed as the main result we obtain conditions for the numerical stability of the schemes in addition we revise the stability conditions obtained in 21 for the crank nicolson scheme we present several numerical examples that confirm the theoretical results within linear as well as nonlinear problems in some particular cases it is shown that for small regions of the time step size values the explicit ftcs scheme is stable while certain implicit methods such as crank nicolson scheme are not

function algebras with a strongly precompact unit ball abstract let be a finite positive borel measure with compact support k c and regard l as an algebra of multiplication operators on the hilbert space l 2 then consider the subalgebra a k of all continuous functions on k that are analytic on the interior of k and the subalgebra r k defined as the uniform closure of the rational functions with poles outside k froelich and marsalli showed that if the restriction of the measure to the boundary of k is discrete then the unit ball of a k is strongly precompact and that if the unit ball of r k is strongly precompact then the restriction of the measure to the boundary of each component of c k is discrete the aim of this paper is to provide three examples that go to clarify the results of froelich and marsalli in particular it is shown that the converses to both statements are false

nonlocal in time problem for anisotropic parabolic equations with variable exponents of nonlinearities abstract in this study we consider nonlinear parabolic equations with variable exponents of the nonlinearities the nonlocal problem for these equations is investigated and the existence and uniqueness theorems are proved for the problem

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

on a class of parabolic equations with nonlocal boundary conditions in this paper we study a class of parabolic equations subject to a nonlocal boundary condition the problem is a generalized model for a theory of ion diffusion in channels by using energy method we first derive some a priori estimates for solutions and then prove that the problem has a unique global solution moreover under some assumptions on the nonlinear boundary condition it is shown that the solution blows up in finite time finally the long time behavior of solution to a linear problem is also studied in the paper

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

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

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

nonlocal solutions of parabolic equations with strongly elliptic differential operators abstract the paper deals with second order parabolic equations on bounded domains with dirichlet conditions in arbitrary euclidean spaces their interest comes from being models for describing reaction diffusion processes in several frameworks a linear diffusion term in divergence form is included which generates a strongly elliptic differential operator a further linear part of integral type is present which accounts of nonlocal diffusion behaviours the main result provides a unifying method for studying the existence and localization of solutions satisfying nonlocal associated boundary conditions the cauchy multipoint and the mean value conditions are included in this investigation the problem is transformed into its abstract setting and the proofs are based on the homotopic invariance of the leray schauder topological degree a bounding function i e lyapunov like function theory is developed which is new in this infinite dimensional context it allows that the associated vector fields have no fixed points on the boundary of their domains and then it makes possible the use of a degree argument

revisiting the stability of 2d passive biped walking local behavior models of biped walking have demonstrated that stable walking motions are possible without active control stability of these motions has typically been quantified by studying the stability of an associated poincare map orbital stability however additional insight may be obtained by examining how perturbations evolve over the short term local stability for example there may be regions where small perturbations actually diverge from the unperturbed trajectory even if over the entire cycle small but perhaps not large perturbations are dissipated we present techniques to calculate local stability and demonstrate the utility of these techniques by examining the local stability of the 2d compass biped these techniques are relevant to the design of controllers to maintain stability in robots and in understanding how the neuromuscular system maintains stability in humans

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

composition operators with a minimal commutant abstract let c be a composition operator on the hardy space h 2 induced by a linear fractional self map of the unit disk we consider the question whether the commutant of c is minimal in the sense that it reduces to the weak closure of the unital algebra generated by c we show that this happens in exactly three cases when is either a non periodic elliptic automorphism or a parabolic non automorphism or a loxodromic self map of the unit disk also we consider the case of a composition operator induced by a univalent analytic self map of the unit disk that fixes the origin and that is not necessarily a linear fractional map but in exchange its konigs s domain is bounded and strictly starlike with respect to the origin and we show that the operator c has a minimal commutant furthermore we provide two examples of univalent analytic self maps of the unit disk such that c is compact but it fails to have a minimal commutant

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