optimal control of systems possessing symmetries it is shown that a symmetry in an optimization problem induces a decomposition of the optimal feedback control law into factors one factor can be calculated algebraically and depends only on the symmetry the other factor corresponds to a lower dimensional optimization problem this gives a priori information about the structure of the optimal feedback control law and indicates a possibly more efficient method for optimizing such systems

a first order system least squares method for the helmholtz equation we present a first order system least squares fosls method for the helmholtz equation at high wave number k which always leads to a hermitian positive definite algebraic system by utilizing a non trivial solution decomposition to the dual fosls problem which is quite different from that of the standard finite element methods we give an error analysis to the h p version of the fosls method where the dependence on the mesh size h the approximation order p and the wave number k is given explicitly in particular under some assumption of the boundary of the domain the l 2 norm error estimate of the scalar solution from the fosls method is shown to be quasi optimal under the condition that k h p is sufficiently small and the polynomial degree p is at least o log k numerical experiments are given to verify the theoretical results

stable and compatible polynomial extensions in three dimensions and applications to the p and h p finite element method polynomial extensions play a vital role in the analysis of the p and h p finite element method fem and the spectral element method we construct explicitly polynomial extensions on standard elements cubes and triangular prisms which together with the extension on tetrahedrons are used by the p and h p fem in three dimensions these extensions are proved to be stable and compatible with fem subspaces on tetrahedrons cubes and prisms and realize a continuous mapping h 1 2 00 t or h 1 2 00 s rightarrow h 1 omega st where omega st denotes one of these standard elements and t and s are their triangular and square faces applications of these polynomial extensions to the p and h p fem are illustrated

a new method for the calculation of the conductivity of inhomogeneous systems a new method for computing the conductivity of random irregular resistor networks is developed this method is a generalization of the transfer matrix technique proposed by derrida and vannimenus for regular 2d and 3d lattices at the same time for large systems the method presented in this paper is more efficient than the transfer matrix technique to demonstrate the method it is applied to a cubic lattice at the percolation threshold and away from it the conductivity has been found for lattices with size up to 324 3 the ratio between the conductivity exponent t and the correlation length exponent was estimated to be t 2 315 in good agreement with the literature data

monte carlo simulation of long time percolation diffusion on lattices above the threshold monte carlo simulations of percolation diffusion at and below the percolation threshold give results in accordance with theory above the percolation threshold however this is not the case it was thought that above the threshold a well defined crossover point separated the anomalous and classical diffusion regimes with the classical diffusion coefficient having the same critical exponent as the lattice conductivity but monte carlo simulations failed to confirm this expected behaviour analysis of new monte carlo results presented here for the square lattice shows that percolation diffusion is classical only strictly asymptotically instead of a crossover the approach to classical behaviour is better described as a very slow relaxation accounting for this relaxation enables the critical behaviour of percolation diffusion to be confirmed with a corresponding critical exponent

current distribution in the three dimensional random resistor network at the percolation threshold we study the multifractal properties of the current distribution of the three dimensional random resistor network at the percolation threshold for lattices ranging in size from 8 3 to 80 3 we measure the second fourth and sixth moments of the current distribution finding e g that t ensuremath nu 2 282 5 where t is the conductivity exponent and ensuremath nu is the correlation length exponent textcopyright 1996 the american physical society

conductivity exponent and backbone dimension in 2 d percolation abstract we present high statistics simulations for 2 d percolation clusters in the bus bar geometry at the critical point for site and for bond percolation we measured their backbone sizes and electrical conductivities for all sets of measurements we find large corrections to scaling most of which do not seem to be described by single powers using single power terms for the corrections to scaling of the backbone masses we would obtain fractal dimensions which are different for site and bond percolation while the correct result is d b 1 6432 0 0008 for both for the conductivity the corrections to scaling are strongly non monotonic for bond percolation the exponent t t is measured as 0 9826 0 0008 in disagreement with the alexander orbach and other conjectures

polynomial liftings on a tetrahedron and applications to the h p version of the finite element method in three dimensions we construct a stable polynomial lifting on a tetrahedron given polynomial traces defined on its faces satisfying continuity conditions along the edges we apply this result to establish the optimal convergence rate of the h p version of the finite element method in dimension three assuming the solution to be regular

precise calculation of the dynamical exponent of two dimensional percolation we report numerical data obtained on the special purpose computer percola for the exponent tilde t of the electrical conductivity of 2d percolation the extrapolation yields tilde t 0 9745 pm 0 0015 and a correction to the scaling exponent 1 2 0 2

low concentration series in general dimension we discuss recent work on the development and analysis of low concentration series for many models the recent breakthrough in the extremely efficient no free end method of series generation facilitates the derivation of 15th order series for multiple moments in general dimension the 15th order series have been obtained for lattice animals percolation and the edwards anderson ising spin glass in the latter cases multiple moments have been found from complete graph tables through to 13th order general dimension 13th order series have been derived for the resistive susceptibility the moments of the logarithms of the distribution of currents in resistor networks and the average transmission coefficient in the quantum percolation problem 11th order series have been found for several other systems including the crossover from animals to percolation the full resistance distribution nonlinear resistive susceptibility and current distribution in dilute resistor networks diffusion on percolation clusters the dilute ising model dilute antiferromagnet in a field and random field ising model and self avoiding walks on percolation clusters series for the dilute spin 1 2 quantum heisenberg ferromagnet are in the process of development analysis of these series gives estimates for critical thresholds amplitude ratios and critical exponents for all dimensions where comparisons are possible our series results are in good agreement with bothe expansion results near the upper critical dimension and with exact results when available in low dimensions and are competitive with other numerical approaches in intermediate realistic dimensions

polynomial extension operators part ii consider the tangential trace of a vector polynomial on the surface of a tetrahedron we construct an extension operator that extends such a trace function into a polynomial on the tetrahedron this operator can be continuously extended to the trace space of h mathbf curl furthermore it satisfies a commutativity property with an extension operator we constructed in part i of this series such extensions are a fundamental ingredient of high order finite element analysis

quasioptimality of some spectral mixed methods in this paper we construct a sequence of projectors into certain polynomial spaces satisfying a commuting diagram property with norm bounds independent of the polynomial degree using the projectors we obtain quasioptimality of some spectral mixed methods including the raviart thomas method and mixed formulations of maxwell equations we also prove some discrete friedrichs type inequalities involving curl

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

corrections to scaling for diffusion exponents on three dimensional percolation systems at criticality recent results of monte carlo simulations of the ant in the labyrinth method in three dimensional percolation lattices are reanalyzed in the light of more accurate corrections to scaling ansatz motivated by inconsistent results that have appeared in the literature the results are observed to be sensitive to the form of the scaling correction terms using a single correction term we estimate the valuek 0 197 0 004 for the anomalous diffusion exponent at criticality when two correction terms are included k 0 200 0 002 is obtained these new estimates are consistent with known theoretical bounds with recent series expansion results and with numerical calculations of the conductance of random resistor networks above criticality

shortest paths synthesis for a car like robot this paper deals with the complete characterization of the shortest paths for a car like robot previous works have shown that the search for a shortest path may be limited to a simple family of trajectories our work completes this study by providing a way to select inside this family an optimal path to link any two configurations we combine the necessary conditions given by pontryagin s maximum principle with a geometric reasoning this approach enables us to complete the local information with a global analysis of different wave fronts we construct a partition of the configuration space in regions where the same kind of path is optimal to reach the origin in other words we determine a shortest path synthesis by providing at each point an optimal control law to steer the robot to the origin

diffusion on random systems above below and at their percolation threshold in two and three dimensions a detailed monte carlo study is presented for classical diffusion random walks on random l l triangular and l l l simple cubic lattices with l up to 4096 and 256 respectively the speed of a cyber 205 vector computer is found to be about one order of magnitude larger than that of a usual cdc cyber 76 computer to reach the asymptotic scaling regime walks with up to 10 million steps were simulated with about 10 jl steps in total for l 256 at the percolation threshold we review and extend the dynamical scaling description for the distance traveled as function of time the diffusivity above the threshold and the cluster radius below earlier discrepancies between scaling theory and computer experiment are shown to be due to insufficient monte carlo data the conductivity exponent is found to be 2 0 0 2 in three and 1 28 0 02 in two dimensions our data in three dimensions follow well the finite size scaling theory below the threshold the approach of the distance traveled to its asymptotic value is consistent with theoretical speculations and an exponent 2 5 independent of dimensionality the correction to scaling exponent at pc seems to be larger in two than in three dimensions

monte carlo study of the site percolation model in two and three dimensions we investigate the site percolation problem on the square and simple cubic lattices by means of a monte carlo algorithm that in fact simulates systems with size l d 1 x infinity where l specifies the linear system size this algorithm can be regarded either as an extension of the hoshen kopelman method or as a special case of the transfer matrix monte carlo technique various quantities such as the magnetic correlation function are sampled in the finite directions of the above geometry simulations are arranged such that both bulk and surface quantities can be sampled on the square lattice we locate the percolation threshold at p c 0 592 746 5 4 and determine two universal quantities as q gbc 0 930 34 1 and q gsc 0 793 72 3 which are associated with bulk and surface correlations respectively these values agree well with the exact values 2 5 48 and 2 1 3 respectively which follow from conformal invariance on the simple cubic lattice we locate the percolation threshold at p c 0 311 607 7 4 we further determine the bulk thermal and magnetic exponents as y t 1 1437 6 and y h 2 5219 2 respectively and the surface magnetic exponent at the ordinary phase transition as y o hs 1 0248 3

molecular trajectory algorithm for random walks on percolation systems at criticality in two and three dimensions abstract random walk simulations based on a molecular trajectory algorithm are performed on critical percolation clusters the analysis of corrections to scaling is carried out it has been found that the fractal dimension of the random walk on the incipient infinite cluster is d w 2 873 0 008 in two dimensions and 3 78 0 02 in three dimensions if instead the diffusion is averaged over all clusters at the threshold not subject to the infinite restriction the corresponding critical exponent k is found to be k 0 3307 0 0014 for two dimensional space and 0 199 0 002 for three dimensional space moreover in our simulations the asymptotic behaviors of local critical exponents are reached much earlier than in other numerical methods

optimal planar turns under acceleration constraints this technical note considers the problem of finding optimal trajectories for a particle moving in a 2 d plane from a given initial position and velocity to a specified terminal heading under a magnitude constraint on the acceleration the cost functional to be minimized is the integral over time of a non negative power of the particle s speed special cases of such a cost functional include travel time and path length unlike previous work on related problems variations in the magnitude of the velocity vector are allowed pontryagin s maximum principle is used to show that the optimal paths possess a special property whereby the vector that divides the velocity and acceleration vectors in a specific ratio which depends on the cost functional is a constant this property is used to characterize the optimal acceleration vector and obtain the parametric equations of the corresponding optimal paths sufficiency conditions for optimality are used to show that trajectories satisfying the necessary conditions are actually optimal if the heading change required is sufficiently small solutions of the time optimal and the length optimal problems are obtained as special cases of the general problem

conductivity exponent in three dimensional percolation by diffusion based on molecular trajectory algorithm and blind ant rules abstract diffusion on random systems above and at their percolation threshold in three dimensions is carried out by a molecular trajectory method and a simple lattice random walk method respectively the classical regimes of diffusion on percolation near the threshold are observed in our simulations by both methods our monte carlo simulations by the simple lattice random walk method give the conductivity exponent 2 32 0 02 for diffusion on the incipient infinite clusters and 2 21 0 03 for diffusion on a percolating lattice above the threshold however while diffusion is performed by the molecular trajectory algorithm either on the incipient infinite clusters or on a percolating lattice above the threshold the result is found to be 2 26 0 02 in addition it takes less time step for diffusion based on the molecular trajectory algorithm to reach the asymptotic limit comparing with the simple lattice random walk