Practical Distance Functions for Path-Planning in Planar Domains

Path planning is an important problem in robotics. One way to plan a path between two points $x,y$ within a (not necessarily simply-connected) planar domain $\Omega$, is to define a non-negative distance function $d(x,y)$ on $\Omega\times\Omega$ such that following the (descending) gradient of this distance function traces such a path. This presents two equally important challenges: A mathematical challenge -- to define $d$ such that $d(x,y)$ has a single minimum for any fixed $y$ (and this is when $x=y$), since a local minimum is in effect a "dead end", A computational challenge -- to define $d$ such that it may be computed efficiently. In this paper, given a description of $\Omega$, we show how to assign coordinates to each point of $\Omega$ and define a family of distance functions between points using these coordinates, such that both the mathematical and the computational challenges are met. This is done using the concepts of \emph{harmonic measure} and \emph{$f$-divergences}. In practice, path planning is done on a discrete network defined on a finite set of \emph{sites} sampled from $\Omega$, so any method that works well on the continuous domain must be adapted so that it still works well on the discrete domain. Given a set of sites sampled from $\Omega$, we show how to define a network connecting these sites such that a \emph{greedy routing} algorithm (which is the discrete equivalent of continuous gradient descent) based on the distance function mentioned above is guaranteed to generate a path in the network between any two such sites. In many cases, this network is close to a (desirable) planar graph, especially if the set of sites is dense

Quantum Mechanics and Discrete Time from "Timeless" Classical Dynamics

We study classical Hamiltonian systems in which the intrinsic proper time evolution parameter is related through a probability distribution to the physical time, which is assumed to be discrete. - This is motivated by the ``timeless'' reparametrization invariant model of a relativistic particle with two compactified extradimensions. In this example, discrete physical time is constructed based on quasi-local observables. - Generally, employing the path-integral formulation of classical mechanics developed by Gozzi et al., we show that these deterministic classical systems can be naturally described as unitary quantum mechanical models. The emergent quantum Hamiltonian is derived from the underlying classical one. It is closely related to the Liouville operator. We demonstrate in several examples the necessity of regularization, in order to arrive at quantum models with bounded spectrum and stable groundstate.Comment: 24 pages, 1 figure. Lecture given at DICE 2002. To be published in: Decoherence and Entropy in Complex Systems, Lecture Notes in Physics (Springer-Verlag, Berlin 2003). - Comprises quant-ph/0306096 and gr-qc/0301109, additional reference

Anomalous Thermostat and Intraband Discrete Breathers

We investigate the dynamics of a macroscopic system which consists of an anharmonic subsystem embedded in an arbitrary harmonic lattice, including quenched disorder. Elimination of the harmonic degrees of freedom leads to a nonlinear Langevin equation for the anharmonic coordinates. For zero temperature, we prove that the support of the Fourier transform of the memory kernel and of the time averaged velocity-velocity correlations functions of the anharmonic system can not overlap. As a consequence, the asymptotic solutions can be constant, periodic,quasiperiodic or almost periodic, and possibly weakly chaotic. For a sinusoidal trajectory with frequency $\Omega$ we find that the energy $E_T$ transferred to the harmonic system up to time $T$ is proportional to $T^{\alpha}$. If $\Omega$ equals one of the phonon frequencies $\omega_\nu$, it is $\alpha=2$. We prove that there is a full measure set such that for $\Omega$ in this set it is $\alpha=0$, i.e. there is no energy dissipation. Under certain conditions there exists a zero measure set such that for $\Omega \in this set the dissipation rate is nonzero and may be subdissipative$(0 \leq \alpha < 1)$or superdissipative$(1 <\alpha \leq 2)$. Consequently, the harmonic bath does act as an anomalous thermostat. Intraband discrete breathers are such solutions which do not relax. We prove for arbitrary anharmonicity and small but finite coupling that intraband discrete breathers with frequency$\Omega$exist for all$\Omega$in a Cantor set$\mathcal{C}(k)$of finite Lebesgue measure. This is achieved by estimating the contribution of small denominators appearing in the memory kernel. For$\Omega\in\mathcal{C}(k)$the small denominators do not lead to divergencies such that this kernel is a smooth and bounded function in$t$.Comment: Physica D in prin

A pseudospectral matrix method for time-dependent tensor fields on a spherical shell

We construct a pseudospectral method for the solution of time-dependent, non-linear partial differential equations on a three-dimensional spherical shell. The problem we address is the treatment of tensor fields on the sphere. As a test case we consider the evolution of a single black hole in numerical general relativity. A natural strategy would be the expansion in tensor spherical harmonics in spherical coordinates. Instead, we consider the simpler and potentially more efficient possibility of a double Fourier expansion on the sphere for tensors in Cartesian coordinates. As usual for the double Fourier method, we employ a filter to address time-step limitations and certain stability issues. We find that a tensor filter based on spin-weighted spherical harmonics is successful, while two simplified, non-spin-weighted filters do not lead to stable evolutions. The derivatives and the filter are implemented by matrix multiplication for efficiency. A key technical point is the construction of a matrix multiplication method for the spin-weighted spherical harmonic filter. As example for the efficient parallelization of the double Fourier, spin-weighted filter method we discuss an implementation on a GPU, which achieves a speed-up of up to a factor of 20 compared to a single core CPU implementation.Comment: 33 pages, 9 figure

Topological string in harmonic space and correlation functions in S3S^3 stringy cosmology

We develop the harmonic space method for conifold and use it to study local complex deformations of $T^{\ast}S^{3}$ preserving manifestly $SL(2,C)$ isometry. We derive the perturbative manifestly $SL(2,C)$ invariant partition function $\mathcal{Z}_{top}$ of topological string B model on locally deformed conifold. Generic $n$ momentum and winding modes of 2D $c=1$ non critical theory are described by highest $$ and lowest components $\upsilon_{(0,n)}$ of $SL(2,C)$ spin $s=\frac{n}{2}$ multiplets $_{(n-k,k)})$, $0\leq k\leq n$ and are shown to be naturally captured by harmonic monomials. Isodoublets ($n=1$) describe uncoupled units of momentum and winding modes and are exactly realized as the $SL(2,C)$ harmonic variables $U_{\alpha}^{+}$ and $V_{\alpha}^{-}$. We also derive a dictionary giving the passage from Laurent (Fourier) analysis on $T^{\ast}S^{1}$ ($S^{1}$) to the harmonic method on $T^{\ast}S^{3}$ ($S^{3}$). The manifestly $SU(2,C)$ covariant correlation functions of the $S^{3}$ quantum cosmology model of Gukov-Saraikin-Vafa are also studied.Comment: 91 page