synchrony of neural oscillators induced by random telegraphic currents when a neuron receives a randomly fluctuating input current its reliability of spike generation improves compared with the case of a constant input current mainen and sejnowski science 268 1503 1995 this phenomenon can be interpreted as phase synchronization between uncoupled nonlinear oscillators subject to a common external input we analyze this phenomenon using dynamical models of neurons assuming the input current to be a simple random telegraphic signal that jumps between two values and the neuron to be always purely self oscillatory the internal state of the neuron randomly jumps between two limit cycles corresponding to the input values which can be described by random phase maps when the switching time of the input current is sufficiently long using such a random map description we discuss the synchrony of neural oscillators subject to fluctuating inputs especially when the phase maps are monotonic we can generally show that the lyapunov exponent is negative namely phase synchronization is stable and reproducibility of spike timing improves

proof of the angular momentum mass inequality for axisymmetric black holes we prove that for any vacuum maximal asymptotically flat axisymmetric initial data for einstein equations close to extreme kerr data the inequality j m is satisfied where m and j are the total mass and angular momentum of the data the proof consists in showing that extreme kerr is a local minimum of the mass

quantifying phase synchronization using instances of hilbert phase slips abstract we propose to quantify phase synchronization between two signals x t and y t by calculating variance in the hilbert phase of y t at instances of phase slips exhibited by x t the proposed approach is tested on numerically simulated coupled chaotic roessler systems and second order autoregressive processes furthermore we compare the performance of the proposed and original approaches using uterine electromyogram signals and show that both approaches yield consistent results a standard phase synchronization approach which involves unwrapping the hilbert phases 1 t and 2 t of the two signals and analyzing the variance in the n 1 t m 2 t mod 2 n and m are integers was used for comparison the synchronization indexes obtained from the proposed approach and the standard approach agree reasonably well in all of the systems studied in this work our results indicate that the proposed approach unlike the traditional approach does not require the non invertible transformations unwrapping of the phases and calculation of mod 2 and it can be used to reliably to quantify phase synchrony between two signals

lasalle type theorems for stochastic differential delay equations the main aim of this paper is to establish the lasalle type asymptotic convergence theorems for the solutions of stochastic differential delay equations these stochastic versions are then applied to establish sufficient criteria for the stochastically asymptotic stability of the delay equations several examples are also given for illustration

synchronization in a system of globally coupled oscillators with time delay we study the synchronization phenomena in a system of globally coupled oscillators with time delay in the coupling the self consistency equations for the order parameter are derived which depend explicitly on the amount of delay analysis of these equations reveals that the system in general exhibits discontinuous transitions in addition to the usual continuous transition between the incoherent state and a multitude of coherent states with different synchronization frequencies in particular the phase diagram is obtained on the plane of the coupling strength and the delay time and ubiquity of multistability as well as suppression of the synchronization frequency is manifested numerical simulations are also performed to give consistent results

potential symmetries and direct reduction methods of order two for partial differential equations written in conservative form a remarkable link between potential symmetries and direct reduction methods of order two is enlightened

spurious detection of phase synchronization in coupled nonlinear oscillators coupled nonlinear systems under certain conditions exhibit phase synchronization which may change for different frequency bands or with the presence of additive system noise in both cases fourier filtering is traditionally used to preprocess data we investigate to what extent the phase synchronization of two coupled r ossler oscillators depends on 1 the broadness of their power spectrum 2 the width of the bandpass filter and 3 the level of added noise we find that for identical coupling strengths oscillators with broader power spectra exhibit weaker synchronization further we find that within a broad bandwidth range bandpass filtering reduces the effect of noise but can lead to a spurious increase in the degree of phase synchronization with narrowing bandwidth even when the coupling between the two oscillators remains the same

hidden and contact symmetries of ordinary differential equations the lie algebra for the maximal contact symmetries of third order ordinary differential equations odes is examined for type i and ii hidden symmetries where the analysis of hidden symmetries for point symmetries is extended to contact symmetries ones invariant under the group associated with the ten dimensional maximal lie algebra may produce type i hidden symmetries for two parameter subgroups and type ii hidden symmetries for certain solvable non abelian three parameter subgroups in the third order odes when they are reduced in order a new class of type ii hidden symmetries is recognized in which contact symmetries transform to point symmetries in some reduction paths two examples of odes invariant under subgroups of the ten parameter group under which y 0 is invariant demonstrate the new class of type ii hidden symmetries

phase synchronization in temperature and precipitation records abstract we study phase synchronization between atmospheric variables such as daily mean temperature and daily precipitation records we find significant phase synchronization between records of oxford and vienna as well as between the records of precipitation and temperature in each city to find the time delay in the synchronization between the records we study the time lag phase synchronization when the records are shifted by a variable time interval of days we also compare the results of the method with the classical cross correlation method and find that in certain cases the phase synchronization yields more significant results

learning driver response relationships from synchronization patterns we test recent claims that causal driver response relationships can be deduced from interdependencies between simultaneously measured time series we apply two recently proposed interdependence measures that should give results similar to cross predictabilities used by previous authors the systems that we study are asymmetrically coupled simple models lorenz roessler and henon models the couplings being such that they lead to generalized synchronization if the data were perfect noise free infinitely long we should be able to detect at least in some cases which of the coupled systems is the driver and which the response this might no longer be true if the time series has finite length instead estimated interdependencies depend strongly on which of the systems has a higher effective dimension at the typical neighborhood sizes used to estimate them and causal relationships are more difficult to detect we also show that slightly different variants of the interdependence measure can have quite different sensitivities

a group classification of the general second order coupled diffusion system we classify second order systems of coupled heat diffusion equations the underlying system contains several unknown functions the forms of which are specified via the method of group classification in particular the canonical forms of the low dimensional lie algebras and the lie algebras of higher dimension provide a means to specify the functional forms of these arbitrary functions

a nine dimensional lorenz system to study high dimensional chaos we examine the dynamics of three dimensional cells with square planform in dissipative rayleigh benard convection by applying a triple fourier series ansatz up to second order we obtain a system of nine nonlinear ordinary differential equations from the governing hydrodynamic equations depending on two control parameters namely the rayleigh number and the prandtl number the asymptotic behaviour can be stationary periodic quasiperiodic or chaotic a period doubling cascade is identified as a route to chaos hereafter the asymptotic behaviour progressively evolves towards a hyperchaotic attractor for given values of control parameters beyond the accumulation point we observe a low dimensional chaotic attractor as is currently done for dissipative systems although the correlation dimension strongly suggests that this attractor could be embedded in a three dimensional space a topological characterization reveals that a higher dimensional space must be used thus we reconstruct a four dimensional model which is found to be in agreement with the properties of the original dynamics the nine dimensional lorenz model could therefore play a significant role in developing tools to characterize chaotic attractors embedded in phase space with a dimension greater than 3

synchronization of delay bursting neuron system with stochastic noise via linear controllers abstract synchronization of bursting neuron system with stochastic noise is investigated when parameters are known based on lyapunov stability theory via appropriate controllers synchronization of delay hindmarsh rose system with stochastic noise is obtained when parameters are unknown by adaptive control technique corresponding parameter update laws are proposed to achieve synchronization of hindmarsh rose system with stochastic noise finally numerical simulations are given to demonstrate the effectiveness of the proposed schemes

staggered polarization of vertex models with u_q widehat sl n symmetry in this paper we give an explicit formula for level 1 vertex operators related to u q widehat sl n as operators on the fock spaces we derive also their commutation relations as an application we calculate with the vector representation of u q widehat sl n thereby extending the recent work on the staggered polarization of the xxz model

on generalized synchronization of different order chaotic systems a submanifold approach regulation theory is used to address the synchronization phenomena of chaotic systems our results are based on the solution of the francis isidori byrnes equations to derive the synchronization submanifold thus conditions for complete or partial synchronization are depicted this analysis is not restrictive with respect to the master and the slave systems dimensions therefore it can be applied to strictly different systems with the same order or even different order systems finally workbench examples are presented to illustrate the results

synchronization of chaotic trajectories using control we demonstrate that two identical chaotic systems can be made to synchronize by applying small judiciously chosen temporal parameter perturbations to one of them this idea is illustrated with a numerical example other issues related to synchronization are also discussed

conditional lie backlund symmetry and reduction of evolution equations we suggest a generalization of the notion of invariance of a given partial differential equation with respect to a lie backlund vector field such a generalization proves to be effective and enables us to construct principally new ansatz reducing evolution type equations to several ordinary differential equations in the framework of the said generalization we obtain principally new reductions of a number of nonlinear heat conductivity equations ut uxx f u ux with poor lie symmetry and obtain their exact solutions it is shown that these solutions cannot be constructed by means of the symmetry reduction procedure

group classification of the general second order evolution equation semi simple invariance groups in this paper we consider the problem of group classification of the generic second order evolution equation in one spatial variable we construct all inequivalent evolution equations whose invariance groups are either semi simple or semi direct products of semi simple and solvable lie groups the obtained lists of invariant equations contain both already known equations and the broad classes of new evolution equations possessing nontrivial lie symmetry

understanding dynamics of the system using hilbert phases an application to study neonatal and fetal brain signals the hilbert phase phi t of a signal x t exhibits slips when the magnitude of their successive phase difference phi t i 1 phi t i exceeds pi by applying this approach to periodic uncorrelated and long range correlated data we show that the standard deviation of the time difference between the successive phase slips deltatau normalized by the percentage of slips in the data is characteristic of the correlation in the data we consider a 50x50 square lattice and model each lattice point by a second order autoregressive ar2 process further we model a subregion of the lattice using a different set of ar2 parameters compared to the rest by applying the proposed approach to the lattice model we show that the two distinct parameter regions introduced in the lattice are clearly distinguishable finally we demonstrate the application of this approach to spatiotemporal neonatal and fetal magnetoencephalography signals recorded using 151 superconducting quantum interference device sensors to identify the sensors containing the neonatal and fetal brain signals and discuss the improved performance of this approach over the traditionally used spectral approach

singular harmonic maps and applications to general relativity the study of axially symmetric stationary multi black hole configurations and the force between co axially rotating black holes involves as a first step an analysis on the boundary regularity of the so called reduced singular harmonic maps we carry out this analysis by considering those harmonic maps as solutions to some homogeneous divergence systems of partial differential equations with singular coefficients our results extend previous works by weinstein comm pure appl math 43 903 948 1990 comm pure appl math 45 1183 1203 1992 and by li and tian manu math 73 1 83 89 1991 commun math phys 149 1 30 1992 differential geometry pde on manifolds vol 54 pp 317 326 1993 this paper is based on the ph d thesis of the author singular harmonic maps into hyperbolic spaces and applications to general relativity phd thesis the state university of new jersey rutgers 2009

robust synchronization of chaotic systems the question of robustness of synchronization with respect to small arbitrary perturbations of the underlying dynamical systems is addressed we present examples of chaos synchronization demonstrating that normal hyperbolicity is a necessary and sufficient condition for the synchronization manifold to be smooth and persistent under small perturbations the same examples however show that in real applications normal hyperbolicity is not sufficient to give quantitative bounds for deformations of the synchronization manifold i e even in the case of normal hyperbolicity two almost identical systems may cause large synchronization errors

mitigating the effects of measurement noise on granger causality computing granger causal relations among bivariate experimentally observed time series has received increasing attention over the past few years such causal relations if correctly estimated can yield significant insights into the dynamical organization of the system being investigated since experimental measurements are inevitably contaminated by noise it is thus important to understand the effects of such noise on granger causality estimation the first goal of this paper is to provide an analytical and numerical analysis of this problem specifically we show that due to noise contamination 1 spurious causality between two measured variables can arise and 2 true causality can be suppressed the second goal of the paper is to provide a denoising strategy to mitigate this problem specifically we propose a denoising algorithm based on the combined use of the kalman filter theory and the expectation maximization algorithm numerical examples are used to demonstrate the effectiveness of the denoising approach

the existence of contact transformations for evolution type equations we investigate the existence of a one parameter group of contact transformations for evolution type equations ut f t x u ux uxx u n subscripts denote differentiation unless otherwise indicated where u n is the nth derivative of u with respect to x we prove that contact transformations of evolution equations where f is expandable as a power series in terms of all derivatives of order higher than one are just extended lie point transformations this result is extended to the case with m independent space variables as a consequence we obtain an ansatz for determining lie point transformations for nth order evolution equations with m independent space variables examples are given to verify the results obtained as well as to show how lie point transformations of these evolution type partial differential equations can be calculated from this ansatz

synchronization of chaotic structurally nonequivalent systems synchronization features are explored for a pair of chaotic high dimensional bidirectionally coupled structurally nonequivalent systems we find two regimes of synchronization in dependence on the coupling strength creation of a lower dimensional chaotic state and for larger coupling a transition toward a stable periodic motion we characterize this new state showing that it is associated with an abrupt transition in the lyapunov spectrum the robustness of this state against noise is discussed and the use of this dynamical property as a possible approach for the control of chaos is outlined

mean phase coherence as a measure for phase synchronization and its application to the eeg of epilepsy patients abstract we apply the concept of phase synchronization of chaotic and or noisy systems and the statistical distribution of the relative instantaneous phases to electroencephalograms eegs recorded from patients with temporal lobe epilepsy using the mean phase coherence as a statistical measure for phase synchronization we observe characteristic spatial and temporal shifts in synchronization that appear to be strongly related to pathological activity in particular we observe distinct differences in the degree of synchronization between recordings from seizure free intervals and those before an impending seizure indicating an altered state of brain dynamics prior to seizure activity

time scale synchronization of chaotic oscillators this paper presents the result of the investigation of chaotic oscillator synchronization a new approach to detect the synchronized behaviour of chaotic oscillators has been proposed this approach is based on the analysis of different time scales in the time series generated by the coupled chaotic oscillators this approach has been applied for the coupled rossler and lorenz systems

synchronization of chaos and hyperchaos using linear and nonlinear feedback functions using the method of variable feedback synchronization of chaotic and hyperchaotic systems is presented the robustness of the method based on the flexibility of choices of feedback functions is exemplified linear and nonlinear feedback functions and their linear superpositions are used for synchronization calculations with model systems indicate that functions constructed by linear superposition of feedback functions that independently synchronize a system are more efficient in achieving synchronization than the functions from which they are made we have not noticed any difference in the technique of synchronization based on the number of positive lyapunov exponents

assessing statistical significance of phase synchronization index an application to study baroreflex function in critically ill infants abstract phase differences of two signals in perfect synchrony exhibit a narrow band distribution whereas the phase differences of two asynchronous signals exhibit uniform distribution we assess the statistical significance of the phase synchronization between two signals by using a signed rank test to compare the distribution of their phase differences to the theoretically expected uniform distribution for two asynchronous signals using numerical simulation of a second order autoregressive ar2 process we show that the proposed approach correctly identifies the coupling between the ar2 process and the driving white noise we also identify the optimal p value that distinguishes coupled scenarios from uncoupled ones to identify the limiting cases we study the phase synchronization between two independent white noises as a function of bandwidth of the filter in a different second simulation we identify the frequency bandwidth below which the proposed approach fails and suggest using a data driven approach for those scenarios finally we demonstrate the application of this approach to study the coupling between beat to beat cardiac intervals and continuous blood pressure obtained from critically ill infants to characterize the baroreflex function

a note on the lasalle type theorems for stochastic differential delay equations the main aim of this note is to improve some results obtained in the author s earlier paper 1999 j math anal appl 236 350 369 from the improved result follow some useful criteria on the stochastic asymptotic stability and boundedness

detecting phase synchronization in noisy data from coupled chaotic oscillators two schemes are proposed to detect phase synchronization from chaotic data contaminated by noise the first is a neighborhood based method which links time delay embedding with instantaneous phase estimation the second adopts the local projection method as a preprocessing filter to noisy data both schemes utilize the state recurrence an important feature of chaotic data the proposed schemes are applied to data measured from two typical chaotic systems i e the coupled rossler systems and the coupled lorenz systems respectively the results show that phase synchronization which may be buried by noise is detected even when the noise level is high moreover the overestimation of the degree of phase synchronization which may be introduced by the hilbert transform combined with a traditional linear bandpass filter can be avoided when the data are contaminated by only measurement noise

chaos synchronization of general complex dynamical networks recently it has been demonstrated that many large scale complex dynamical networks display a collective synchronization motion here we introduce a time varying complex dynamical network model and further investigate its synchronization phenomenon based on this new complex network model two network chaos synchronization theorems are proved we show that the chaos synchronization of a time varying complex network is determined by means of the inner coupled link matrix the eigenvalues and the corresponding eigenvectors of the coupled configuration matrix rather than the conventional eigenvalues of the coupled configuration matrix for a uniform network especially we do not assume that the coupled configuration matrix is symmetric and its off diagonal elements are nonnegative which in a way generalizes the related results existing in the literature

vertex operators of admissible modules ofuq c 1 n using our recent bosonic realization ofuq sp2n we construct explicitly the vertex operators for the level 1 2 modules ofuq sp2n using bosonic fields our method contains a detailed analysis of all theq intertwining relations

synchronization of spectral components and its regularities in chaotic dynamical systems the chaotic synchronization regime in coupled dynamical systems is considered it has been shown that theonset of a synchronous regime is based on the appearance of a phase relation between the interacting chaoticoscillator frequency components of fourier spectra the criterion of synchronization of spectral components aswell as the measure of synchronization has been discussed the universal power law has been described themain results are illustrated by coupled rossler systems van der pol and van der pol duf ng oscillators doi 10 1103 physreve 71 056204 pacs numberssd 05 45 xt 05 45 tpi introduction

cross correlation of instantaneous phase increments in pressure flow fluctuations applications to cerebral autoregulation we investigate the relationship between the blood flow velocities bfv in the middle cerebral arteries and beat to beat blood pressure bp recorded from a finger in healthy and post stroke subjects during the quasisteady state after perturbation for four different physiologic conditions supine rest head up tilt hyperventilation and co2 rebreathing in upright position to evaluate whether instantaneous bp changes in the steady state are coupled with instantaneous changes in the bfv we compare dynamical patterns in the instantaneous phases of these signals obtained from the hilbert transform as a function of time we find that in post stroke subjects the instantaneous phase increments of bp and bfv exhibit well pronounced patterns that remain stable in time for all four physiologic conditions while in healthy subjects these patterns are different less pronounced and more variable we propose an approach based on the cross correlation of the instantaneous phase increments to quantify the coupling between bp and bfv signals we find that the maximum correlation strength is different for the two groups and for the different conditions for healthy subjects the amplitude of the cross correlation between the instantaneous phase increments of bp and bfv is small and attenuates within 3 5 heartbeats in contrast for post stroke subjects this amplitude is significantly larger and cross correlations persist up to 20 heartbeats further we show that the instantaneous phase increments of bp and bfv are cross correlated even within a single heartbeat cycle we compare the results of our approach with three complementary methods direct bp bfv cross correlation transfer function analysis and phase synchronization analysis our findings provide insight into the mechanism of cerebral vascular control in healthy subjects suggesting that this control mechanism may involve rapid adjustments within a heartbeat of the cerebral vessels so that bfv remains steady in response to changes in peripheral bp

group theoretical framework for potential symmetries of evolution equations we develop algebraic approach to the problem of classification of potential symmetries of nonlinear evolution equations it is essentially based on the recently discovered fact r zhdanov j math phys 50 053522 2009 that any such symmetry is mapped into a contact symmetry the approach enables using the classical results on classification of contact symmetries of nonlinear evolution equations by sokolov and magadeev to classify evolution equations admitting potential symmetries we construct several examples of new nonlinear fourth order evolution equations admitting potential symmetries since the symmetries obtained depend on nonlocal variables they cannot be derived by the infinitesimal lie approach

new classes of symmetries for partial differential equations new classes of symmetries for partial differential equations are introduced by writing a given partial differential equation s in a conserved form a related system t with potentials as additional dependent variables is obtained the lie group of point transformations admitted by t induces a symmetry group of s new symmetries may be obtained for s that are neither point nor lie backlund symmetries they are determined by a completely algorithmic procedure significant new symmetries are found for the wave equation with a variable wave speed and the nonlinear diffusion equation

adaptive synchronization between two different noise perturbed chaotic systems with fully unknown parameters abstract this paper investigates adaptive synchronization between two different noise perturbed chaotic systems with fully unknown parameters based on lyapunov stability theory the general expression of the suitable nonlinear adaptive synchronization controller is developed at the same time the noise perturbed update laws of parameters are proposed to guarantee the synchronization of two different unknown chaotic systems theoretical analysis and numerical simulations on adaptive synchronization between rossler system and noise perturbed chua s circuits lorenz system and noise perturbed chen system with fully unknown parameters show the efficiency of the proposed adaptive scheme

on relation between potential and contact symmetries of evolution equations we prove that any evolution equation that can be written in the form of a conservation law reduces to an evolution equation admitting the group of contact symmetries leaving the temporal variable invariant and vice versa one of the consequences of this fact is that any evolution equation admitting the potential symmetries can be transformed into another evolution equation so that the potential symmetries map into contact symmetries of the latter based on this fact is out group approach to classification of evolution equations possessing nonlocal symmetries we present several examples of classifications of second order evolution equations admitting potential symmetries

an algorithm based on variable feedback to synchronize chaotic and hyperchaotic systems in recent years the subject of how to synchronize chaotic and hyperchaotic systems has attracted much attention due to its theoretic interest and potential applications to secure communications various approaches have been proposed and pursued in this work an algorithm based on variable feedback is developed analytically and applied to a nine dimensional lorenz system numerically one finds that the proposed method can be applied to chaotic systems with one or more lyapunov exponents it is also demonstrated that the given method can be further simplified for concrete systems

regularity of harmonic maps with prescribed singularities in this paper we studied the regularity problem for harmonic maps into hyperbolic spaces with prescribed singularities along codimension two submanifolds this is motivated from one of hawking s conjectures on the uniqueness of kerr solutions among all axially symmetric asymptotically flat stationary solutions to the vacuum einstein equation in general relativity

impulsively synchronizing chaotic systems with delay and applications to secure communication in this paper the presence of transmission delay and sampling delay in chaos based secure communication systems by employing impulsive synchronization is studied a time delayed impulsive differential system with delayed impulses modeling the synchronization error between the driving and response schemes employed in such communication systems is presented the equi attractivity property of the error dynamics is investigated and the sufficient conditions leading to this property are obtained a set of upper bounds on the delay terms involved in the system are also obtained and a numerical example is given a communication security scheme employing hyperchaotic systems possessing continuous driving impulsive driving and delay is proposed and simulation results are given to demonstrate the performance of the scheme

cooperative behavior of a chain of synaptically coupled chaotic neurons abstract a coupled linear chain of hindmarsh rose model neurons with reciprocal inhibition between neighboring neurons exhibited synchronous oscillations in which neighboring neurons burst out of phase and next nearest neighbor neurons burst in phase the bifurcations observed inside this out of phase regime were qualitatively the same for all chains with an even number of neurons and were similar to those observed in a single isolated cell although the organization of the behavior of a chain of coupled neurons was more regular than that of an isolated cell when noise was added to the synaptic coupling strengths there was less hysteresis in the system and many of the bifurcations with smaller basins of attraction were eliminated making the system even more regular these results suggest that in populations of bursting neurons with reciprocal inhibition the chaotic behavior found in single cells is suppressed

bifurcation analysis of two coupled periodically driven duffing oscillators bifurcation diagrams and phase diagrams of two coupled periodically driven identical duffing oscillators are presented it is shown that the global pattern of bifurcation curves in parameter space consists of repeated subpatterns similar to the superstructure observed for single periodically driven strictly dissipative oscillators the subpattern itself however is different from that of a single duffing oscillator due in particular to hopf bifurcations that are newly added to the bifurcation scenario

contact symmetries and solutions by reduction of partial differential equations the correspondence between a contact symmetry of a second order pde e and a point symmetry of the equivalent first order system s is used to determine a class of solutions for e which may not always be invariant under contact transformation we name these solutions pseudo invariant because they are determined from a point symmetry of s via a reduction method invariant solutions under the related contact transformation are included in the family of pseudo invariant solutions

quantum r matrix and intertwiners for the kashiwara algebra we study the algebrab q g presented by kashiwara and introduce intertwiners similar toq vertex operators we show that a matrix determined by 2 point functions of the intertwiners coincides with a quantumr matrix up to a diagonal matrix and give the commutation relations of the intertwiners we also introduce an analogue of the universalr matrix for the kashiwara algebra

linear transmitter with correlated noises abstract a linear transmitter with correlated gaussian white additive and multiplicative noises and a periodic signal coupled either additively or multiplicatively is considered the correlations have a weak destructive effect in case of the additive signal and a strong constructive effect in case of the multiplicative one analytical results for the linear transmitter with the additive signal agree with those obtained previously by a different approach we also show how to analytically calculate certain expectation values involving exponentials of gaussian processes

boundary regularity and the dirichlet problem for harmonic maps in a previous paper 10 we developed an interior regularity theory for energy minimizing harmonic maps into riemannian manifolds in the first two sections of this paper we prove boundary regularity for energy minimizing maps with prescribed dirichlet boundary condition we show that such maps are regular in a full neighborhood of the boundary assuming appropriate regularity on the manifolds the boundary and the data the reader may refer to theorem 2 7 for a statement of the precise result it is not surprising that the boundary regularity is actually stronger than the partial regularity we obtained for the interior this is due to the fact that there are no nontrivial smooth harmonic maps from hemispheres s which map the boundary s 9s to a point i j n 2 and is analogous to the fact that in certain cases we were able to obtain complete regularity in the interior many authors have worked on boundary regularity for this general type of problem we mention hildebrandt and widman 5 and hamilton 4 as having obtained important results specifically for harmonic maps morrey had obtained the boundary regularity for domain dimension n 2 in conjunction with his investigation of the plateau problem in riemannian manifolds 8 in 3 of this paper we observe that the direct method gives solvability of the dirichlet problem under reasonable hypotheses on the manifolds we give as an application an amusing proof of a theorem of sacks and uhlenbeck 9 on the existence of minimal 2 spheres representing the second homotopy group of a manifold the same method gives smooth harmonic representations for k n for a certain class of manifolds n these are characterized by the nonexistence of lower dimensional harmonic spheres whose homogeneous extensions are minimal see proposition 3 4 in the last section of the paper we discuss approximation of l maps by smooth maps we give a simple example of an l map from the three dimensional ball to the two sphere which is not an l limit of continuous maps we