Rational chebyshev spectral methods for unbounded solutions on an infinite interval using polynomial-growth special basis functions

AbstractIn the method of matched asymptotic expansions, one is often forced to compute solutions which grow as a polynomial in y as |y| → ∞. Similarly, the integral or repeated integral of a bounded function f(y) is generally unbounded also. The kth integral of a function f(y) solves . We describe a two-part algorithm for solving linear differential equations on y ϵ [−∞, ∞] where u(y) grows as a polynomial as |y| → ∞. First, perform an explicit, analytic transformation to a new unknown v so that v is bounded. Second, expand v as a rational Chebyshev series and apply a pseudospectral or Galerkin discretization. (For our examples, it is convenient to perform a preliminary step of splitting the problem into uncoupled equations for the parts of u which are symmetric and antisymmetric with respect to y = 0, but although this is very helpful when applicable, it is not necessary.) For the integral and interated integrals and for constant coefficient differential equations in general, the Galerkin matrices are banded with very low bandwidth. We derive an improvement on the “last coefficient error estimate” of the author's book which applies to series with a subgeometric rate of convergence, as is normally true of rational Chebyshev expansions

Asymptotic Fourier Coefficients for a C ∞ Bell (Smoothed-“Top-Hat”) & the Fourier Extension Problem

In constructing local Fourier bases and in solving differential equations with nonperiodic solutions through Fourier spectral algorithms, it is necessary to solve the Fourier Extension Problem. This is the task of extending a nonperiodic function, defined on an interval , to a function which is periodic on the larger interval . We derive the asymptotic Fourier coefficients for an infinitely differentiable function which is one on an interval , identically zero for , and varies smoothly in between. Such smoothed “top-hat” functions are “bells” in wavelet theory. Our bell is (for x ≥ 0) where where . By applying steepest descents to approximate the coefficient integrals in the limit of large degree j , we show that when the width L is fixed, the Fourier cosine coefficients a j of on are proportional to where Λ( j ) is an oscillatory factor of degree given in the text. We also show that to minimize error in a Fourier series truncated after the N th term, the width should be chosen to increase with N as . We derive similar asymptotics for the function f ( x )= x as extended by a more sophisticated scheme with overlapping bells; this gives an even faster rate of Fourier convergencePeer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43417/1/10915_2005_Article_9010.pd

The creation of large photon-number path entanglement conditioned on photodetection

Large photon-number path entanglement is an important resource for enhanced precision measurements and quantum imaging. We present a general constructive protocol to create any large photon number path-entangled state based on the conditional detection of single photons. The influence of imperfect detectors is considered and an asymptotic scaling law is derived.Comment: 6 pages, 4 figure

Renormalization Group Summation and the Free Energy of Hot QCD

Using an approach developed in the context of zero-temperature QCD to systematically sum higher order effects whose form is fixed by the renormalization group equation, we sum to all orders the leading log (LL) and next-to-leading log (NLL) contributions to the thermodynamic free energy in hot QCD. While the result varies considerably less with changes in the renormalization scale than does the purely perturbative result, a novel ambiguity arises which reflects the strong scheme dependence of thermal perturbation theory.Comment: 7 pages REVTEX4, 2 figures; v2: typos correcte

Symmetry of bound and antibound states in the semiclassical limit

We consider one dimensional scattering and show how the presence of a mild positive barrier separating the interaction region from infinity implies that the bound and antibound states are symmetric modulo exponentially small errors in 1/h. This simple result was inspired by a numerical experiment and we describe the numerical scheme for an efficient computation of resonances in one dimension

The pressure of QCD at finite temperatures and chemical potentials

The perturbative expansion of the pressure of hot QCD is computed here to order g^6ln(g) in the presence of finite quark chemical potentials. In this process all two- and three-loop one-particle irreducible vacuum diagrams of the theory are evaluated at arbitrary T and mu, and these results are then used to analytically verify the outcome of an old order g^4 calculation of Freedman and McLerran for the zero-temperature pressure. The results for the pressure and the different quark number susceptibilities at high T are compared with recent lattice simulations showing excellent agreement especially for the chemical potential dependent part of the pressure.Comment: 35 pages, 6 figures; text revised, one figure replace

Detection and quantification of oil under sea ice: the view from below

Traditional measures for detecting oil spills in the open-ocean are both difficult to apply and less effective in ice-covered seas. In view of the increasing levels of commercial activity in the Arctic, there is a growing gap between the potential need to respond to an oil spill in Arctic ice-covered waters and the capability to do so. In particular, there is no robust operational capability to remotely locate oil spilt under or encapsulated within sea ice. To date, most research approaches the problem from on or above the sea ice, and thus they suffer from the need to ‘see’ through the ice and overlying snow. Here we present results from a large-scale tank experiment which demonstrate the detection of oil beneath sea ice, and the quantification of the oil layer thickness is achievable through the combined use of an upward-looking camera and sonar deployed in the water column below a covering of sea ice. This approach using acoustic and visible measurements from below is simple and effective, and potentially transformative with respect to the operational response to oil spills in the Arctic marine environment. These results open up a new direction of research into oil detection in ice-covered seas, as well as describing a new and important role for underwater vehicles as platforms for oil-detecting sensors under Arctic sea ice

Two-loop HTL Thermodynamics with Quarks

We calculate the quark contribution to the free energy of a hot quark-gluon plasma to two-loop order using hard-thermal-loop (HTL) perturbation theory. All ultraviolet divergences can be absorbed into renormalizations of the vacuum energy and the HTL quark and gluon mass parameters. The quark and gluon HTL mass parameters are determined self-consistently by a variational prescription. Combining the quark contribution with the two-loop HTL perturbation theory free energy for pure-glue we obtain the total two-loop QCD free energy. Comparisons are made with lattice estimates of the free energy for N_f=2 and with exact numerical results obtained in the large-N_f limit.Comment: 33 pages, 6 figure

Noise reduction in 3D noncollinear parametric amplifier

We analytically find an approximate Bloch-Messiah reduction of a noncollinear parametric amplifier pumped with a focused monochromatic beam. We consider type I phase matching. The results are obtained using a perturbative expansion and scaled to a high gain regime. They allow a straightforward maximization of the signal gain and minimization of the parametric fluorescence noise. We find the fundamental mode of the amplifier, which is an elliptic Gaussian defining the optimal seed beam shape. We conclude that the output of the amplifier should be stripped of higher order modes, which are approximately Hermite-Gaussian beams. Alternatively, the pump waist can be adjusted such that the amount of noise produced in the higher order modes is minimized.Comment: 18 pages, 9 figures, accepted to Applied Physics

Chebyshev Solution of the Nearly-Singular One-Dimensional Helmholtz Equation and Related Singular Perturbation Equations: Multiple Scale Series and the Boundary Layer Rule-of-Thumb

The one-dimensional Helmholtz equation, ε 2 u xx − u = f ( x ), arises in many applications, often as a component of three-dimensional fluids codes. Unfortunately, it is difficult to solve for ε≪1 because the homogeneous solutions are exp (± x /ε), which have boundary layers of thickness O(1/ε). By analyzing the asymptotic Chebyshev coefficients of exponentials, we rederive the Orszag–Israeli rule [16] that Chebyshev polynomials are needed to obtain an accuracy of 1% or better for the homogeneous solutions. (Interestingly, this is identical with the boundary layer rule-of-thumb in [5], which was derived for singular functions like tanh([ x −1]/ε).) Two strategies for small ε are described. The first is the method of multiple scales, which is very general, and applies to variable coefficient differential equations, too. The second, when f ( x ) is a polynomial, is to compute an exact particular integral of the Helmholtz equation as a polynomial of the same degree in the form of a Chebyshev series by solving triangular pentadiagonal systems. This can be combined with the analytic homogeneous solutions to synthesize the general solution. However, the multiple scales method is more efficient than the Chebyshev algorithm when ε is very, very tiny.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45436/1/11075_2004_Article_2865.pd