The Leading-Order Term in the Asymptotic Expansion of the Scattering Amplitude of a Collection of Finite Number of Dielectric Inhomogeneities of Small Diameter

2005 ◽  
Vol 3 (3) ◽  
pp. 285-296 ◽  
Author(s):  
Habib Ammari ◽  
Darko Volkov
Keyword(s):  
Asymptotic Expansion ◽  
Finite Number ◽  
Scattering Amplitude ◽  
Order Term ◽  
Small Diameter ◽  
Leading Order

scholarly journals Short-time heat diffusion in compact domains with discontinuous transmission boundary conditions

2015 ◽  
Vol 26 (01) ◽  
pp. 59-110 ◽  
Author(s):  
Claude Bardos ◽  
Denis Grebenkov ◽  
Anna Rozanova-Pierrat
Keyword(s):  
Asymptotic Expansion ◽  
Boundary Conditions ◽  
Order Term ◽  
Heat Content ◽  
Resistance Coefficient ◽  
Small Time ◽  
Heat Diffusion ◽  
Heat Problem ◽  
Short Time ◽  
Mathematical Justification

We consider a heat problem with discontinuous diffusion coefficients and discontinuous transmission boundary conditions with a resistance coefficient. For all bounded (ϵ, δ)-domains Ω ⊂ ℝn with a d-set boundary (for instance, a self-similar fractal), we find the first term of the small-time asymptotic expansion of the heat content in the complement of Ω, and also the second-order term in the case of a regular boundary. The asymptotic expansion is different for the cases of finite and infinite resistance of the boundary. The derived formulas relate the heat content to the volume of the interior Minkowski sausage and present a mathematical justification to the de Gennes' approach. The accuracy of the analytical results is illustrated by solving the heat problem on prefractal domains by a finite elements method.

Derivation of the maximum voltage drop in power grids of integrated circuits with an array bonding package

2012 ◽  
Vol 23 (6) ◽  
pp. 797-819
Author(s):  
M. AGUARELES ◽  
J. de HARO
Keyword(s):  
Integrated Circuits ◽  
Iterative Scheme ◽  
Voltage Drop ◽  
Order Term ◽  
Singular Limit ◽  
Power Grids ◽  
Leading Order ◽  
Conformal Maps ◽  
Maximum Value ◽  
Aligned Array

In this work we derive a formula for the maximum value of the voltage drop that takes place in power grids of integrated circuits with an array bonding package. We consider a simplified model for the power grid where the voltage is represented as the solution of the Poisson's equation in an infinite planar domain with a regularly aligned array of small square pads where the voltage is set to be zero. We study the singular limit where these pads' size tends to be zero and we derive an asymptotic formula in terms of a power series in ε, being 2ε the side of the square. We also discuss pads of more general shapes, for which we provide an expression for the leading order term. To deduce all these formulae we use the method of matched asymptotic expansions using an iterative scheme, along with conformal maps to compare this problem with the corresponding one when the pads are circular.

scholarly journals Material barriers to diffusive and stochastic transport

2018 ◽  
Vol 115 (37) ◽  
pp. 9074-9079 ◽  
Author(s):  
George Haller ◽  
Daniel Karrasch ◽  
Florian Kogelbauer
Keyword(s):  
Coherent Structures ◽  
Fluid Transport ◽  
Order Term ◽  
Velocity Fields ◽  
Leading Order ◽  
Transport Barrier ◽  
Material Surfaces ◽  
Stochastic Transport ◽  
Transport Barriers ◽  
Enhancer Detection

We seek transport barriers and transport enhancers as material surfaces across which the transport of diffusive tracers is minimal or maximal in a general, unsteady flow. We find that such surfaces are extremizers of a universal, nondimensional transport functional whose leading-order term in the diffusivity can be computed directly from the flow velocity. The most observable (uniform) transport extremizers are explicitly computable as null surfaces of an objective transport tensor. Even in the limit of vanishing diffusivity, these surfaces differ from all previously identified coherent structures for purely advective fluid transport. Our results extend directly to stochastic velocity fields and hence enable transport barrier and enhancer detection under uncertainties.

scholarly journals Extracting Hawking radiation near the horizon of AdS black holes

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Krishan Saraswat ◽  
Niayesh Afshordi
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Evaporation Rate ◽  
Order Term ◽  
Energy Condition ◽  
Null Energy Condition ◽  
Leading Order ◽  
Proper Distance ◽  
Ads Black Holes ◽  
A Cell

Abstract We study how the evaporation rate of spherically symmetric black holes is affected through the extraction of radiation close to the horizon. We adopt a model of extraction that involves a perfectly absorptive screen placed close to the horizon and show that the evaporation rate can be changed depending on how close to the horizon the screen is placed. We apply our results to show that the scrambling time defined by the Hayden-Preskill decoding criterion, which is derived in Pennington’s work (arXiv:1905.08255) through entanglement wedge reconstruction is modified. The modifications appear as logarithmic corrections to Pennington’s time scale which depend on where the absorptive screen is placed. By fixing the proper distance between the horizon and screen we show that for small AdS black holes the leading order term in the scrambling time is consistent with Pennington’s scrambling time. However, for large AdS black holes the leading order Log contains the Bekenstein-Hawking entropy of a cell of characteristic length equal to the AdS radius rather than the entropy of the full horizon. Furthermore, using the correspondence between the radial null energy condition (NEC) and the holographic c-theorem, we argue that the screen cannot be arbitrarily close to the horizon. This leads to a holographic argument that black hole mining using a screen cannot significantly alter the lifetime of a black hole.

A note on fractional Bessel equation and its asymptotics

2013 ◽  
Vol 16 (3) ◽  
Author(s):  
Wojciech Okrasiński ◽  
Łukasz Płociniczak
Keyword(s):  
Power Series ◽  
Asymptotic Formula ◽  
Asymptotic Analysis ◽  
Fractional Derivative ◽  
Order Term ◽  
High Accuracy ◽  
Fast Convergence ◽  
Leading Order ◽  
Bessel Equation ◽  
Integer Order

AbstractIn this note we propose a fractional generalization of the classical modified Bessel equation. Instead of the integer-order derivatives we use the Riemann-Liouville version. Next, we solve the fractional modified Bessel equation in terms of the power series and provide an asymptotic analysis of its solution for large arguments. We find a leading-order term of the asymptotic formula for the solution to the considered equation. This behavior is verified numerically and shows high accuracy and fast convergence. Our results reduce to the classical formulas when the order of the fractional derivative goes to integer values.

scholarly journals Kicked Burgers turbulence

2000 ◽  
Vol 416 ◽  
pp. 239-267 ◽  
Author(s):  
J. BEC ◽  
U. FRISCH ◽  
K. KHANIN
Keyword(s):  
Numerical Simulations ◽  
Power Law ◽  
Large Scale ◽  
Order Term ◽  
Inviscid Limit ◽  
Order Structure ◽  
Legendre Transform ◽  
Leading Order ◽  
Space And Time ◽  
Burgers Turbulence

Burgers turbulence subject to a force f(x, t) = [sum ]jfj(x)δ(t − tj), where tj are 'kicking times' and the 'impulses' fj(x) have arbitrary space dependence, combines features of the purely decaying and the continuously forced cases. With large-scale forcing this ‘kicked’ Burgers turbulence presents many of the regimes proposed by E et al. (1997) for the case of random white-noise-in-time forcing. It is also amenable to efficient numerical simulations in the inviscid limit, using a modification of the fast Legendre transform method developed for decaying Burgers turbulence by Noullez & Vergassola (1994). For the kicked case, concepts such as ‘minimizers’ and ‘main shock’, which play crucial roles in recent developments for forced Burgers turbulence, become elementary since everything can be constructed from simple two-dimensional area-preserving Euler–Lagrange maps.The main results are for the case of identical deterministic kicks which are periodic and analytic in space and are applied periodically in time. When the space integrals of the initial velocity and of the impulses vanish, it is proved and illustrated numerically that a space- and time-periodic solution is achieved exponentially fast. In this regime, probabilities can be defined by averaging over space and time periods. The probability densities of large negative velocity gradients and of (not-too-large) negative velocity increments follow the power law with −7/2 exponent proposed by E et al. (1997) in the inviscid limit, whose existence is still controversial in the case of white-in-time forcing. This power law, which is seen very clearly in the numerical simulations, is the signature of nascent shocks (preshocks) and holds only when at least one new shock is born between successive kicks.It is shown that the third-order structure function over a spatial separation Δx is analytic in Δx although the velocity field is generally only piecewise analytic (i.e. between shocks). Structure functions of order p ≠ 3 are non-analytic at Δx = 0. For even p there is a leading-order term proportional to [mid ]Δx[mid ] and for odd p > 3 the leading-order term ∝Δx has a non-analytic correction ∝Δx[mid ]Δx[mid ] stemming from shock mergers.

Asymptotic expansion for the scattering amplitude in scalar field theory

1989 ◽  
Vol 32 (5) ◽  
pp. 362-365
Author(s):  
S. A. Gadzhiev ◽  
R. K. Dzhafarov ◽  
A. I. Livashvili
Keyword(s):  
Field Theory ◽  
Asymptotic Expansion ◽  
Scalar Field ◽  
Scattering Amplitude ◽  
Scalar Field Theory

Nonlinear interactions in turbulence with strong irrotational straining

1997 ◽  
Vol 337 ◽  
pp. 333-364 ◽  
Author(s):  
N.K.-R. KEVLAHAN ◽  
J.C.R. HUNT
Keyword(s):  
Large Scale ◽  
Order Term ◽  
Initial Energy ◽  
Perturbation Series ◽  
Small Scale ◽  
Bluff Bodies ◽  
Leading Order ◽  
Viscous Effects ◽  
First Case ◽  
Maximum Amplification

The rate of growth of the nonlinear terms in the vorticity equation are analysed for a turbulent flow with r.m.s. velocity u0 and integral length scale L subjected to a strong uniform irrotational plane strain S, where (u0/L)/S=ε[Lt ]1. The rapid distortion theory (RDT) solution is the zeroth-order term of the perturbation series solution in terms of ε. We use the asymptotic form of the convolution integrals for the leading-order nonlinear terms when β= exp(−St)[Lt ]1 to determine at what time t and beyond what wavenumber k (normalized on L) the perturbation series in ε fails, and hence derive the following conditions for the validity of RDT in these flows. (a) The magnitude of the nonlinear terms of order ε depends sensitively on the amplitude of eddies with large length scales in the direction x2 of negative strain. (b) If the integral of the velocity component u2 is zero the leading-order nonlinear terms increase and decrease in the same way as the linear terms, even those that decrease exponentially. In this case RDT calculations of vorticity spectra become invalid at a time tNL∼L/u0k−3 independent of ε and the power law of the initial energy spectrum, but the calculation of the r.m.s. velocity components by RDT remains accurate until t= TNL∼L/u0, when the maximum amplification of r.m.s. vorticity is ω/S∼εexp(ε−1)[Gt ]1. (c) If this special condition does not apply, the leading-order nonlinear terms increase faster than the linear terms by a factor O(β−1). RDT calculations of the vorticity spectrum then fail at a shorter time tNL∼(1/S) ln(ε−1k−3); in this case TNL∼(1/S) ln(ε−1) and the maximum amplification of r.m.s. vorticity is ω/S∼1. (d) Viscous effects dominate when t[Gt ](1/S) ln(k−1(Re/ε)1/2). In the first case RDT fails immediately in this range, while in the second case RDT usually fails before viscosity becomes important. The general analytical result (a) is confirmed by numerical evaluation of the integrals for a particular form of eddy, while (a), (b), (c) are explained physically by considering the deformation of differently oriented vortex rings. The results are compared with small-scale turbulence approaching bluff bodies where ε[Lt ]1 and β[Lt ] 1.These results also explain dynamically why the intermediate eigenvector of the strain S aligns with the vorticity vector, why the greatest increase in enstrophy production occurs in regions where S has a positive intermediate eigenvalue; and why large-scale strain S of a small-scale vorticity can amplify the small-scale strain rates to a level greater than S – one of the essential characteristics of high-Reynolds-number turbulence.

scholarly journals A Sharp Version of Price’s Law for Wave Decay on Asymptotically Flat Spacetimes

2021 ◽  
Author(s):  
Peter Hintz
Keyword(s):  
Order Term ◽  
Angular Frequency ◽  
Leading Order ◽  
Full Generality ◽  
Asymptotically Flat ◽  
Low Energies ◽  
Flat Spacetimes ◽  
Explicit Constant ◽  
Sharp Version ◽  
Linear Scalar

AbstractWe prove Price’s law with an explicit leading order term for solutions $$\phi (t,x)$$ ϕ ( t , x ) of the scalar wave equation on a class of stationary asymptotically flat $$(3+1)$$ ( 3 + 1 ) -dimensional spacetimes including subextremal Kerr black holes. Our precise asymptotics in the full forward causal cone imply in particular that $$\phi (t,x)=c t^{-3}+{\mathcal {O}}(t^{-4+})$$ ϕ ( t , x ) = c t - 3 + O ( t - 4 + ) for bounded |x|, where $$c\in {\mathbb {C}}$$ c ∈ C is an explicit constant. This decay also holds along the event horizon on Kerr spacetimes and thus renders a result by Luk–Sbierski on the linear scalar instability of the Cauchy horizon unconditional. We moreover prove inverse quadratic decay of the radiation field, with explicit leading order term. We establish analogous results for scattering by stationary potentials with inverse cubic spatial decay. On the Schwarzschild spacetime, we prove pointwise $$t^{-2 l-3}$$ t - 2 l - 3 decay for waves with angular frequency at least l, and $$t^{-2 l-4}$$ t - 2 l - 4 decay for waves which are in addition initially static. This definitively settles Price’s law for linear scalar waves in full generality. The heart of the proof is the analysis of the resolvent at low energies. Rather than constructing its Schwartz kernel explicitly, we proceed more directly using the geometric microlocal approach to the limiting absorption principle pioneered by Melrose and recently extended to the zero energy limit by Vasy.

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