Zero-order Term Research Articles

Convergence of vibrational angular momentum terms within the Watson Hamiltonian

Vibrational angular momentum terms within the Watson Hamiltonian are often considered negligible or are approximated by the zeroth order term of an expansion of the inverse of the effective moment of inertia tensor. A multimode expansion of this tensor up to second order has been used to study the impact of first and second order terms on the vibrational transitions of N(2)H(2) and HBeH(2)BeH. Comparison with experimental data is provided. The expansion of the tensor can be exploited to introduce efficient prescreening techniques.

Exponential Decay Rate of the Perturbed Energy of the Wave Equation with Zero Order Term

In this paper, we consider the wave equation with zero order term. We use the compactness uniqueness argument and some result of I. Lasiecka and D. Tataru in [4] to prove, directly, the exponential decay rate of the perturbed energy.

An Eulerian Surface Hopping Method for the Schrödinger Equation with Conical Crossings

In a nucleonic propagation through conical crossings of electronic energy levels, the codimension two conical crossings are the simplest energy level crossings, which affect the Born–Oppenheimer approximation in the zeroth order term. The purpose of this paper is to develop the surface hopping method for the Schrodinger equation with conical crossings in the Eulerian formulation. The approach is based on the semiclassical approximation governed by the Liouville equations, which are valid away from the conical crossing manifold. At the crossing manifold, electrons hop to another energy level with the probability determined by the Landau–Zener formula. This hopping mechanics is formulated as an interface condition, which is then built into the numerical flux for solving the underlying Liouville equation for each energy level. While a Lagrangian particle method requires the increase in time of the particle numbers, or a large number of statistical samples in a Monte Carlo setting, the advantage of an Euleria...

Multiple solutions of semilinear degenerate elliptic boundary value problems

The purpose of this paper is to study a class of semilinear elliptic boundary value problems with degenerate boundary conditions which include as particular cases the Dirichlet problem and the Robin problem. The approach here is based on the super-sub-solution method in the degenerate case, and is distinguished by the extensive use of an Lp Schauder theory elaborated for second-order, elliptic differential operators with discontinuous zero-th order term. By using Schauder's fixed point theorem, we prove that the existence of an ordered pair of sub- and supersolutions of our problem implies the existence of a solution of the problem. The results extend an earlier theorem due to Kazdan and Warner to the degenerate case. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Equal-order finite elements with local projection stabilization for the Darcy–Brinkman equations

For the Darcy–Brinkman equations, which model porous media flow, we present an equal-order H 1-conforming finite element method for approximating velocity and pressure based on a local projection stabilization technique. The method is stable and accurate uniformly with respect to the coefficients of the viscosity and the zeroth order term in the momentum equation. We prove a priori error estimates in a mesh-dependent norm as well as in the L 2-norm for velocity and pressure. In particular, we obtain optimal order of convergence in L 2 for the pressure in the Darcy case with vanishing viscosity and for the velocity in the general case with a positive viscosity coefficient. Numerical results for different values of the coefficients in the Darcy–Brinkman model are presented which confirm the theoretical results and indicate nearly optimal order also in cases which are not covered by the theory.

Iterative method for zero-order suppression in off-axis digital holography

We propose a method to suppress the so-called zero-order term in a hologram, based on an iterative principle. During the hologram acquisition process, the encoded information includes the intensities of the two beams creating the interference pattern, which do not contain information about the recorded complex wavefront, and that can disrupt the reconstructed signal. The proposed method selectively suppresses the zero-order term by employing the information obtained during wavefront reconstruction in an iterative procedure, thus enabling its suppression without any a priori knowledge about the object. The method is analyzed analytically and its convergence is studied. Then, its performance is shown experimentally. Its robustness is assessed by applying the procedure on various types of holograms, such as topographic images of microscopic specimens or speckle holograms.

Comparison and regularity results for degenerate elliptic equations related to Gauss measure

This article deals with a Dirichlet problem relative to a linear elliptic equation with zero-order terms, whose ellipticity condition is given in terms of the function ν(x)ϕ(x), where is the density of Gauss measure, ν(x) is a nonnegative integrable function. Using the notion of rearrangement with respect to Gauss measure, we prove a comparison result with a problem of the same type defined in a half space, with data depending only on the first variable. Also, some regularity results for the solutions in Lorentz–Zygmund spaces are obtained by us.

Comparison results for solutions of nonlinear parabolic equations

We prove a comparison result for the solutions of Cauchy–Dirichlet problems for a nonlinear parabolic equation. Using Schwarz spherical symmetrization, we compare the concentration of solutions to such problems with the concentration of solutions to conveniently symmetrized problems. The result takes into account, in a sharp form, the influence of the zero-order term.

Homogenization of nonlinear reaction-diffusion equation with a large reaction term

This paper deals with the homogenization of a second order parabolic operator with a large nonlinear potential and periodically oscillating coefficients of both spatial and temporal variables. Under a centering condition for the nonlinear zero-order term, we obtain the effective problem and prove a convergence result. The main feature of the homogenized equation is the appearance of a non-linear convection term.

Convergence of a Time Discretisation for Doubly Nonlinear Evolution Equations of Second Order

The convergence of a time discretisation with variable time steps is shown for a class of doubly nonlinear evolution equations of second order. This also proves existence of a weak solution. The operator acting on the zero-order term is assumed to be the sum of a linear, bounded, symmetric, strongly positive operator and a nonlinear operator that fulfils a certain growth and a Holder-type continuity condition. The operator acting on the first-order time derivative is a nonlinear hemicontinuous operator that fulfils a certain growth condition and is (up to some shift) monotone and coercive.

Simultaneous B0‐ and B1+‐Map acquisition for fast localized shim, frequency, and RF power determination in the heart at 3 T

High-field (>or=3 T) cardiac MRI is challenged by inhomogeneities of both the static magnetic field (B(0)) and the transmit radiofrequency field (B(1)+). The inhomogeneous B fields not only demand improved shimming methods but also impede the correct determination of the zero-order terms, i.e., the local resonance frequency f(0) and the radiofrequency power to generate the intended local B(1)+ field. In this work, dual echo time B(0)-map and dual flip angle B(1)+-map acquisition methods are combined to acquire multislice B(0)- and B(1)+-maps simultaneously covering the entire heart in a single breath hold of 18 heartbeats. A previously proposed excitation pulse shape dependent slice profile correction is tested and applied to reduce systematic errors of the multislice B(1)+-map. Localized higher-order shim correction values including the zero-order terms for frequency f(0) and radiofrequency power can be determined based on the acquired B(0)- and B(1)+-maps. This method has been tested in 7 healthy adult human subjects at 3 T and improved the B(0) field homogeneity (standard deviation) from 60 Hz to 35 Hz and the average B(1)+ field from 77% to 100% of the desired B(1)+ field when compared to more commonly used preparation methods.

Suppression of the zero-order term in off-axis digital holography through nonlinear filtering

We present experimental validation of a new reconstruction method for off-axis digital holographic microscopy (DHM). This method effectively suppresses the object autocorrelation, namely, the zero-order term, from holographic data, thereby improving the reconstruction bandwidth of complex wavefronts. The algorithm is based on nonlinear filtering and can be applied to standard DHM setups with realistic recording conditions. We study the robustness of the technique under different experimental configurations, and quantitatively demonstrate its enhancement capabilities on phase signals.

Generalized Liouville time-dependent perturbation theory

A generalized time-dependent perturbation theory is derived for superoperators. Instead of using the ``standard'' breakup of the Hamiltonian into a known zeroth order term and a correction, we use the approximate superpropagator to define the correction superoperator which is then used to obtain a series representation of the exact Liouville operator. The theory reduces to known limits and may be used for a perturbation expansion of classical Wigner and Husimi dynamics as well as for recent phase-space-based semiclassical approximations. The theory is demonstrated for a model quartic potential.

Scaling and correlation of vorticity fluctuations in turbulent channels

Data from direct numerical simulations have been correlated to examine the effects of Reynolds number on vorticity fluctuation profiles. Three different scalings are found: two in the inner region and the third in the outer region. In the inner region a two-term asymptotic expansion is required to represent the mean square vorticity profiles; ⟨ωω⟩#=⟨ωω⟩0#+⟨ωω⟩1+u∗/U0. The first scaling ⟨ωω⟩0#=⟨ωω⟩0/(u∗3U0/ν2) is for inactive motions that do not generate Reynolds shear stress. It applies to the streamwise and spanwise components. However, the zero-order term is zero for the normal vorticity component. The scrubbing of the inactive motions over the wall generates vorticity, which is a maximum at the wall, and diffuses out to about y+=50 before it decays. The fluctuating viscous wall shear stress is due entirely to this motion, and as a result the stress ratio (rms/mean) τ′/τ0=CU0/u∗ depends on the Reynolds number. The second scaling ⟨ωω⟩1+=⟨ωω⟩1/(u∗4/ν2), the same scaling as the Reynolds shear stress, is active motions. All three components participate in these motions, which are zero at the wall, rise to a peak at about y+=13–20, and fall to zero at about y+=400. This is consistent with the fact that the Reynolds shear stress is identically zero at the wall. The third scaling correlates data in the outer region using the Kolmogorov time scale ⟨ωω⟩/(u∗3/hν). Matching between the inner and outer regions has a common part (the equivalent of the log law) of ∼C/y+ or ∼C/Y for all three components. All components decrease further and become isotropic as the centerline is approached. Scaling with the Kolmogorov scale implies that dissipation is the important activity. The correlated data are fitted to simplified equations that can be formed into composite expansions to predict profiles at any Reynolds number.

Frozen Gaussian series representation of the imaginary time propagator theory and numerical tests

Thawed Gaussian wavepackets have been used in recent years to compute approximations to the thermal density matrix. From a numerical point of view, it is cheaper to employ frozen Gaussian wavepackets. In this paper, we provide the formalism for the computation of thermal densities using frozen Gaussian wavepackets. We show that the exact density may be given in terms of a series, in which the zeroth order term is the frozen Gaussian. A numerical test of the methodology is presented for deep tunneling in the quartic double well potential. In all cases, the series is observed to converge. The convergence of the diagonal density matrix element is much faster than that of the antidiagonal one, suggesting that the methodology should be especially useful for the computation of partition functions. As a by product of this study, we find that the density matrix in configuration space can have more than two saddle points at low temperatures. This has implications for the use of the quantum instanton theory.

A comparative study of Laplacians and Schrödinger– Lichnerowicz–Weitzenböck identities in Riemannian and antisymplectic geometry

We introduce an antisymplectic Dirac operator and antisymplectic gamma matrices. We explore similarities between, on one hand, the Schrödinger–Lichnerowicz formula for spinor bundles in Riemannian spin geometry, which contains a zeroth–order term proportional to the Levi–Civita scalar curvature, and, on the other hand, the nilpotent, Grassmann-odd, second-order Δ operator in antisymplectic geometry, which, in general, has a zeroth-order term proportional to the odd scalar curvature of an arbitrary antisymplectic and torsion-free connection that is compatible with the measure density. Finally, we discuss the close relationship with the two-loop scalar curvature term in the quantum Hamiltonian for a particle in a curved Riemannian space.

Two-component description of dynamical systems that can be approximated by solitons: The case of the ion acoustic wave equations of plasma physics

A new approach to the perturbative analysis of dynamical systems, which can be described approximately by soliton solutions of integrable non-linear wave equations, is employed in the case of small-amplitude solutions of the ion acoustic wave equations of plasma physics. Instead of pursuing the traditional derivation of a perturbed KdV equation, the ion velocity is written as a sum of two components: elastic and inelastic. In the single-soliton case, the elastic component is the full solution. In the multiple-soliton case, it is complemented by the inelastic component. The original system is transformed into two evolution equations: An asymptotically integrable Normal Form for ordinary KdV solitons, and an equation for the inelastic component. The zero-order term of the elastic component is a single-soliton or multiple-soliton solution of the Normal Form. The inelastic component asymptotes into a linear combination of single-soliton solutions of the Normal Form, with amplitudes determined by soliton interactions, plus a second-order decaying dispersive wave. Satisfaction of a conservation law by the inelastic component and of mass conservation by the disturbance to the ion density is determined solely by the initial data and/or boundary conditions imposed on the inelastic component. The electrostatic potential is a first-order quantity. It is affected by the inelastic component only in second order. The charge density displays a triple-layer structure. The analysis is carried out through the third order.

Initial value technique for singularly perturbed two point boundary value problems using an exponentially fitted finite difference scheme

In this paper, we present an approximate method (Initial value technique) for the numerical solution of quasilinear singularly perturbed two point boundary value problems in ordinary differential equations having a boundary layer at one end (left or right) point. It is motivated by the asymptotic behavior of singular perturbation problems. The original problem is reduced to an asymptotically equivalent first order initial value problem by approximating the zeroth order term by outer solution obtained by asymptotic expansion, and then this initial value problem is solved by an exponentially fitted finite difference scheme. Some numerical examples are given to illustrate the given method. It is observed that the presented method approximates the exact solution very well for crude mesh size h .

Practical Error Estimates for Reynolds' Lubrication Approximation and its Higher Order Corrections

Reynolds' lubrication approximation is used extensively to study flows between moving machine parts, in narrow channels, and in thin films. The solution of Reynolds' equation may be thought of as the zeroth order term in an expansion of the solution of the Stokes equations in powers of the aspect ratio $\epsilon$ of the domain. In this paper, we show how to compute the terms in this expansion to arbitrary order on a two-dimensional, $x$-periodic domain and derive rigorous, a-priori error bounds for the difference between the exact solution and the truncated expansion solution. Unlike previous studies of this sort, the constants in our error bounds are either independent of the function $h(x)$ describing the geometry, or depend on $h$ and its derivatives in an explicit, intuitive way. Specifically, if the expansion is truncated at order $2k$, the error is $O(\epsilon^{2k+2})$ and $h$ enters into the error bound only through its first and third inverse moments $\int_0^1 h(x)^{-m} dx$, $m=1,3$ and via the max norms $\big\|\frac{1}{\ell!} h^{\ell-1} \partial_x^\ell h\big\|_\infty$, $1\le\ell\le2k+2$. We validate our estimates by comparing with finite element solutions and present numerical evidence that suggests that even when $h$ is real analytic and periodic, the expansion solution forms an asymptotic series rather than a convergent series.

Approximate Analysis of Power Offset over Spatially Correlated MIMO Channels

Power offset is zero-order term in the capacity versus signal-to-noise ratio curve. In this paper, approximate analysis of power offset is presented to describe MIMO system with uniform linear antenna arrays of fixed length. It is assumed that the number of receive antenna is larger than that of transmit antenna. Spatially Correlated MIMO Channel is approximated by tri-diagonal toeplitz matrix. The determinant of tri-diagonal toeplitz matrix, which is fitted by elementary curve, is one of the key factors related to power offset. Based on the curve fitting, the determinant of tri-diagonal toeplitz matrix is mathematically tractable. Consequently, the expression of local extreme points can be derived to optimize power offset. The simulation results show that approximation above is accurate in local extreme points of power offset. The proposed expression of local extreme points is helpful to approach optimal power offset.