bluuebunny/crossref_metadata_2025_split huggingface dataset:
On the Piano Movers' problem: IV. Various decomposable two‐dimensional motion‐planning problems
Authors: Sharir Micha , Ariel‐Sheffi Elka 
Published: April 2007

AbstractVarious special motion‐planning problems involving arbitrarily many degrees of freedom are shown to admit relatively simple solutions by techniques based on the connectivity graph approach described by Schwartz and Sharir. The solutions exploit the particularly simple configuration space structure of the robot systems considered. A typical problem is that of planning motions for a 2‐dimensional robot system consisting of several arms all jointed at one common endpoint and free to rotate past each other. The algorithm given for solving this problem runs in time O(nk+4), where k is the number of arms.
The inversion problem and applications of the generalized radon transform
Authors: Beylkin Gregory 
Published: April 2007

AbstractWe prove that under certain conditions the inversion problem for the generalized Radon transform reduces to solving a Fredholm integral equation and we obtain the asymptotic expansion of the symbol of the integral operator in this equation.We consider applications of the generalized Radon transform to partial differential equations with variable coefficients and provide a solution to the inversion problem for the attenuated and exponential Radon transforms.
Translation representations for automorphic solutions of the wave equation in non‐euclidean spaces. II
Authors: Lax Peter D. , Phillips Ralph S. 
Published: April 2007

AbstractIn this part we prove that the translation representation defined in Part I for the non‐Euclidean wave equation is complete. We show how one can derive from this translation representation a spectral representation for the Laplace‐Beltrami operator over geometrically finite fundamental polyhedra with infinite volume, without the use of Eisenstein series.
On the piano movers' problem: V. The case of a rod moving in three‐dimensional space amidst polyhedral obstacles
Authors: Schwartz Jacob T. , Sharir Micha 
Published: April 2007

AbstractThis paper, a fifth in a series, solves some additional 3‐D special cases of the „piano movers”︁ problem, which arises in robotics. The main problem solved in this paper is that of planning the motion of a rod moving amidst polyhedral obstacles. We present polynomial‐time motion‐planning algorithms for this case, using the connectivity‐graph technique described in the preceding papers. We also study certain more general polyhedral problems, which arise in the motion planning problem considered here but have application to other similar problems. Application of these techniques to the problem of planning the motion of a general polyhedral body moving in 3‐space amidst polyhedral obstacles is also described.
On the essential spectrum of ideal magnetohydrodynamics. Dedicated to harold grad on the occasion of his sixtieth birthday
Authors: Hameiri Eliezer 
Published: April 2007

AbstractThe essential spectrum of magnetohydrodynamics (MHD) is shown to arise from waves propagating one‐dimensionally along magnetic field lines. Different polarizations of these waves give rise to the “Alfven” and “ballooning” spectra. The essential spectrum of an axisymmetric equilibrium when a single azimuthal mode number is considered consists of the Alfven spectrum only, while the ballooning modes appear as intervals of accumulation of discrete eigenvalues with different mode numbers. We derive some necessary and one sufficient conditions for stability and show some examples where the criteria coincide to yield a simple condition for stability.
Translation representations for automorphic solutions of the wave equation in non‐euclidean spaces. III
Authors: Lax Peter D. , Phillips Ralph S. 
Published: April 2007

AbstractIn Part III we show that the translation representations defined in Part I, and shown to be complete in Part II, are incoming and outgoing in the sense of propagation of signals along rays. We give a new proof of the absolute continuity of the spectrum below ‐¼n2, and point out its implications for local energy decay. We give a new proof of completeness in dimensions 2 and 3 when energy is positive. Finally, we define and prove completeness of the translation representations when the metric is perturbed on a compact set.
Existence of multiple periodic orbits on star‐shaped hamiltonian surfaces
Authors: Berestycki Henri , Lasry Jean‐Michel , Mancini Giovanni , Ruf Bernhard 
Published: April 2007

AbstractConsider the Hamiltonian system (HS) i = 1, …, N. Here, H ϵ C2(ℝ2N, ℝ). In this paper, we investigate the existence of periodic orbits of (HS) on a given energy surface Σ = {z ϵ ℝ2N; H(z) = c} (c > o is a constant). The surface Σ is required to verify certain geometric assumptions: Σ bounds a star‐shaped compact region ℛ and αℰ ⊂ ℛ ⊂ βℰ for some ellipsoid ℰ ⊂ ℝ2N, o < α < β. We exhibit a constant δ > O (depending in an explicit fashion on the lengths of the main axes of ℰ and one other geometrical parameter of Σ) such that if furthermore β2/α2 < 1 + δ, then (HS) has at least N distinct geometric orbits on Σ. This result is shown to extend and unify several earlier works on this subject (among them works by Weinstein, Rabinowitz, Ekeland‐Lasry and Ekeland). In proving this result we construct index theories for an S1 ‐action, from which we derive abstract critical point theorems for S1 ‐invariant functionals. We also derive an estimate for the minimal period of solutions to differential equatious.
A new approach to queer differential equations
Authors: Laurence Peter , Stredulinsky E. W. 
Published: April 2007

AbstractWe study the variational problem magnified image Where Ψ* is the increasing rearrangement of Ψ. An approximate problem is introduced which involves a variational problem with n free boundaries (n → ∞). Various estimates are established. In particular, when Ω is convex we show that the solution to the approximate problem is superharmonic and has bounded gradient.
Interfaces with a corner point in one‐dimensional porous medium flow
Authors: Aronson D. G. , Caffarelli L. A. , Vazquez J. L. 
Published: April 2007

AbstractWe prove that for a large class of initial distributions the solutions of the initial value problem for the one‐dimensional porous medium equation have interfaces which start to move abruptly after a positive waiting time. This happens, for example, if the initial pressure is o(|x|2) as x → 0. We also give sufficient conditions for the smoothness of the interface which improve previous results.
Brownian motion in a wedge with oblique reflection
Authors: Varadhan S. R. S. , Williams R. J. 
Published: April 2007

AbstractThis work is concerned with the existence and uniqueness of a strong Markov process that has continuous sample paths and the following additional properties:
The state space is an infinite two‐dimensional wedge, and the process behaves in the interior of the wedge like an ordinary Brownian motion.
The process reflects instantaneously at the boundary of the wedge, the angle of reflection being constant along each side.
The amount of time that the process spends at the comer of the wedge is zero (i.e., the set of times for which the process is at the comer has Lebesgue measure zero).
Hereafter, let ξ be the angle of the wedge (0 < ξ < 2π), let θ1 and θ2 be the angles of reflection on the two sides of the wedge, measured from the inward normals, the positive angles being toward the corner (‐½π < θ1, θ2 ½π), and set α = (θ1 + θ2)/ξ.The question of existence and uniqueness is recast as a submartingale problem in the style used by Stroock and Varadhan (Diffusion processes with boundary conditions, Comm. Pure Appl. Math. 24, 1971, pp. 147‐225), for diffusions on smooth domains with smooth boundary conditions. It is shown that no solution exists if α ≧ 2. In this case, there is a unique continuous strong Markov process satisfying (i)‐(ii) above; it reaches the corner of the wedge almost surely and it remains there. If α < 2, however, then there is a unique continuous strong Markov process statisfying (i)‐(iii). It is shown that starting away from the corner this process does not reach the corner of the wedge if α ≦ 0, and does reach the corner if 0 < α < 2.The general theory of multi‐dimensional diffusions does not apply to the above problem because in general the boundary of the state space is not smooth and there is a discontinuity in the direction of reflection at the corner. For some values of α, the process arises from diffusion approximations to storage systems and queueing networks.
(i) The state space is an infinite two‐dimensional wedge, and the process behaves in the interior of the wedge like an ordinary Brownian motion. (ii) The process reflects instantaneously at the boundary of the wedge, and the angle of reflection being constant along each side. (iii) The amount of time that the process spends at the corner of the wedge is zero (i.e., the set of times for which the process is at the corner has Lebesgue measure zero).
Probabilistic approach to the neumann problem
Authors: Hsu Pei 
Published: April 2007

AbstractThe basic problem considered in this paper is to solve the following Neumann boundary value problem probabilistically:magnified image where we assume that q is in a certain functional class to be specified below, and φ is a bounded measurable function on the boundary. We give a martingale formulation of the Neumann problem and show that this formulation is essentially equivalent to the classical formulation. The paper culminates in an explicit formula for the solution of this problem in terms of reflecting Brownian motion and its boundary local time.
A factored implicit scheme for numerical weather prediction
Authors: Augenbaum J. M. , Cohn S. E. , Isaacson E. , Dee D. P. , Marchesin D. 
Published: April 2007

AbstractThe nonlinear partial differential equations of atmospheric dynamics govern motion on two time scales, a fast one and a slow one. Only the slow‐scale motions are relevant in predicting the evolution of large weather patterns. Implicit numerical methods are therefore attractive for weather prediction, since they permit a large time step chosen to resolve only the slow motions.To develop an implicit method which is efficient for problems in more than one spatial dimension, one must approximate the problem by smaller, usually one‐dimensional problems. A popular way to do so is to approximately factor the multidimensional implicit operator into one‐dimensional operators. The factorization error incurred in such methods, however, is often unacceptably large for problems with multiple time scales.We propose a new factorization method for numerical weather prediction which is based on factoring separately the fast and slow parts of the implicit operator. We show analytically that the new method has small factorization error, which is comparable to other discretization errors of the overall scheme. The analysis is based on properties of the shallow water equations, a simple two‐dimensional version of the fully three‐dimensional equations of atmospheric dynamics.
A simple one‐dimensional model for the three‐dimensional vorticity equation
Authors: Constantin P. , Lax P. D. , Majda A. 
Published: April 2007

AbstractA simple qualitative one‐dimensional model for the 3‐D vorticity equation of incompressible fluid flow is developed. This simple model is solved exactly; despite its simplicity, this equation retains several of the most important structural features in the vorticity equations and its solutions exhibit some of the phenomena observed in numerical computations for breakdown for the 3‐D Euler equations.
Finite sections of toeplitz‐like matrices
Authors: Percus J. K. 
Published: April 2007

AbstractDeterminants of large finite sections of Toeplitz and Toeplitz‐like matrices are evaluated by an expansion in which the deviation from the identity is parametrically increased. Classical results are reproduced and the inverse matrix expanded as well. The expansion is asymptotically valid only for low‐order terms, and so a reordered expansion is introduced. It has the desired asymptotic character and suggests that sharp bounds are also available.
A random‐walk interpretation of the incompressible navier‐stokes equations
Authors: Peskin Charles S. 
Published: April 2007

AbstractWe introduce a stochastic process that roughly corresponds to a random walk of a single molecule in an incompressible fluid, and we consider the force (per unit volume) exerted by such a “molecule” on the rest of the fluid. Averaging this force over an ensemble of such molecules and taking an appropriate limit, we obtain the Navier‐Stokes equations. The origin of time‐irreversibility in these equations is discussed in terms of the model.
A uniqueness theorem for an inverse boundary value problem in electrical prospection
Authors: Sylvester John , Uhlmann Gunther 
Published: April 2007

AbstractWe show that a near constant conductivity of a two‐dimensional body can be uniquely determined by steady state direct current measurements at the boundary. Mathematically, we show that the coefficient γ in the operator ∇.γ∇ is uniquely determined by its Dirichlet integrals.
The milne and kramers problems for the boltzmann equation of a hard sphere gas
Authors: Bardos Claude , Caflisch Russel E. , Nicolaenko Basil 
Published: April 2007

AbstractExistence, uniqueness and properties of asymptotic behavior are proved for solutions of the Milne and Kramers problems for the linearized Boltzmann equation for a gas of hard spheres. The proof uses energy‐like estimates and follows the analysis of Bardos, Santos and Sentis for the equations of neutron transport.
Generalized voronoi diagrams for moving a ladder. I: Topological analysis
Authors: Ó'Dunlaing Colm , Sharir Micha , Yap Chee K. 
Published: April 2007

AbstractGiven a bounded open subset Ω of the plane whose boundary is the union of finitely many polygons, and a real number d > 0, a manifold FP (the [free placements]) may be defined as the set of placements of a closed oriented line‐segment B (a [ladder]) of length d inside Ω. FP is a three‐dimensional manifold. A [Voronoi complex] in this manifold, a two‐dimensional cell complex, is defined by analogy with the classical geometric construction in the plane; within this complex a one‐dimensional subcomplex N, called the skeleton, is defined. It is shown that every component of FP contains a unique component of N, and canonical motions are given to move the ladder to placements within N. In this way, general motion planning is reduced to searching in a suitable representation of N as a (combinatorial) graph. Efficient construction of N is described in a companion paper.
Monte carlo calculation of transport for three‐dimensional magnetohydrodynamic equilibria
Authors: Bauer F. , Betancourt O. , Garabedian P. 
Published: April 2007

AbstractA Monte Carlo algorithm has been implemented which couples the results of a three‐dimensional magnetohydrodynamic equilibrium code with particle transport calculations similar to those of Boozer and Kuo‐Petravic. The equilibrium results are in a form which lends itself to easy and accurate computation of the coefficients required to follow particle orbits in flux variables at finite pressure. In particular, the parallel current and the Clebsch potential for the current are obtained from Fourier series solutions of first‐order partial differential equations with constant coefficients.Confinement times are estimated from the exponential decay of expected values of appropriately chosen functionals of the particle distribution. The observation that these functionals satisfy boundary conditions helps us to compute confinement times over a wide range of collision frequencies, including cases where losses due to particle trapping are very high. Initial conditions are chosen to optimize the Monte Carlo calculation by using known information about the expected particle distribution.Results for actual and proposed stellarator experiment are given. Electron and ion confinement times are compared, since this is relevant to the issue of the effect of ambipolar electric fields on stellarator confinement. A spectral method is given to determine the electric field from charge neutrality, and numerical evidence is presented to suggest that anomalous electron transport may be due to small resonant terms in the electric potential. Also, comparisons are made with a simplified theory of particle transport already incorporated in the equilibrium code.
Long time existence for a slightly perturbed vortex sheet
Authors: Caflisch Russel E. , Orellana Oscar F. 
Published: April 2007

AbstractConsider a flat two‐dimensional vortex sheet perturbed initially by a small analytic disturbance. By a formal perturbation analysis, Moore derived an approximate differential equation for the evolution of the vortex sheet. We present a simplified derivation of Moore's approximate equation and analyze errors in the approximation. The result is used to prove existence of smooth solutions for long time. If the initial perturbation is of size ϵ and is analytic in a strip |𝒥m γ| < ρ, existence of a smooth solution of Birkhoff's equation is shown for time t < k2p, if ϵ is sufficiently small, with κ → 1 as ϵ → 0. For the particular case of sinusoidal data of wave length π and amplitude e, Moore's analysis and independent numerical results show singularity development at time tc = |log ϵ| + O(log|log ϵ|. Our results prove existence for t < κ|log ϵ|, if ϵ is sufficiently small, with k κ → 1 as ϵ → 0. Thus our existence results are nearly optimal.
Geometric bases and topological equivalence
Authors: Szczepanski Susan 
Published: April 2007

AbstractThere are many algebraic and topological invariants associated to a singular point of a complex analytic function. The intent here is to discuss some of these invariants and the topological classification of singularities. Specifically, we establish that the topological type is determined by the Lefschetz vanishing cycles obtained by unfolding the singularity and certain local monodromy operators defined by Gabrielov. In Brieskorn's terminology singularities with the same geometric bases are topologically indistinguishable. Thus the higher invariants in the hierarchy of Brieskorn are necessary to understand the geometry of higher singularities. As a corollary to our main theorem, we obtain the result of Lê‐Ramanujam which states that the topological type is constant in a oneparameter family of singularities with constant Milnor number.
Nonlinear waves in friedman‐robertson‐walker space‐times
Authors: Kovalyov Mikhail 
Published: April 2007

AbstractIn this paper we consider the initial value problem for the nonlinear wave equation □u = F(u, u′) in Friedman‐Robertson‐Walker space‐time, □ being the D'Alambertian in local coordinates of space‐time. We obtain decay estimates and show that the equation has global solutions for small initial data. We do it by reducing the problem to an initial value problem for the wave equation over hyperbolic space. As byproduct we derive decay and global existence for solutions of the wave equation over the hyperbolic space with small initial data. The same technique with some auxiliary lemmas similar to the ones proved in [6], [7] can be used to generalize the result to the case when F depends also on second derivatives of u in a certain way.
Existence, uniqueness, and stability of stationary barotropic flow with forcing and dissipation
Authors: Wolansky G. 
Published: April 2007

AbstractWe study a simplified model of barotropic flow. The governing equation is the two‐dimensional Euler equation for incompressible flow, subject to Ekman dissipation and arbitrary forcing, in a bounded domain. This equation is converted to an equivalent functional differential equation via a semi‐lagrangian formalism. We use Hölder Cα estimates and compactness arguments for the latter equation to deduce existence, uniqueness and asymptotic stability of stationary solutions under the assumption of small forcing with respect to the dissipation constant.In case the forcing is of one sign only throughout the domain, we are able to deduce existence without any further assumptions. The same result holds in the limit of the dissipation constant approaching zero while the ratio of this constant to the forcing is fixed. Thus we get the existence of special types of stationary solutions to the inviscid Euler equation consistent with the given forcing.
Multiple impulse solutions to McKean's caricature of the nerve equation. I—existence
Authors: Wang Wei‐Ping 
Published: April 2007

AbstractMcKean's caricature of the nerve equation:

equation image

is considered. The H in (1) is the Heaviside function. We prove the existence of multiple impulse solutions consisting of any finite number of pulses. These solutions are referred to as n‐ple impulse solutions, where n is an arbitrary positive integer.
The modulator of the local time
Authors: Berman Simeon M. 
Published: April 2007

AbstractLet x(t), 0 ≦ t ≦ 1, be a real measurable function having a local time α(x, t) which is a continuous function of t for almost all x. It is also assumed that, for some m ≧ 2 and some real interval B, αm(x, 1) is integrable over B. The modulator is a function Mm(t, B), t > 0, denned in terms of α. It is shown that the modulator serves as a measure of the smoothness of the Lm(B)‐valued function α(., t) with respect to t. Then it is shown that the modulator plays a central role in precisely describing certain irregularity properties of x(t). The results are applied to the case where x(t) is the sample function of a real stochastic process. In this way new results are obtained for large classes of Gaussian and Markov processes.
Dirichlet and neumann spectrum
Authors: Winkler Andrew M. 
Published: April 2007

AbstractIt is well known that the critical points of the Rayleigh quotient for the Dirichlet form relative to the square integral norm are eigenfunctions of the Laplace operator with Neumann boundary conditions and eigenvalue equal to the critical value.Clearly then, if we fix a critical point and consider the corresponding value and restrict the Rayleigh quotient to the space of eigenfunctions of the Laplace operator with that given eigenvalue, then the critical point remains a critical point of the restriction.In this paper we answer the converse question. We show that if a given number occurs as a critical value on the subspace of eigenfunctions with that same number as eigenvalue, then either it is a critical value for the quotient on the entire space or it is a critical value on the subspace of all functions satisfying Dirichlet boundary conditions. Moreover, a precise relation holds between the corresponding critical points.A similar relation holds for any elliptic operator in divergence form. The relation stems from a lattice identity induced by a certain indefinite quadratic form, involving certain subspaces.This theory has been successfully applied to calculating the cuspidal spectrum in a certain range for the Laplace‐Beltrami operator for certain finite volume hyperbolic surfaces, addressing a conjecture of Roelcke and Selberg, and recent conjectures of Sarnak; see [1].The numerical method used can be applied quite generally, typically to equations related to underlying symmetries where decompositions of the group give rise to lots of formal solutions.
A relative morse index for the symplectic action
Authors: Floer Andreas 
Published: April 2007

AbstractThe notion of a Morse index of a function on a finite‐dimensional manifold cannot be generalized directly to the symplectic action function a on the loop space of a manifold. In this paper, we define for any pair of critical points of a a relative Morse index, which corresponds to the difference of the two Morse indices in finite dimensions. It is based on the spectral flow of the Hessian of a and can be identified with a topological invariant recently defined by Viterbo, and with the dimension of the space of trajectories between the two critical points.