Holomorphic Lefschetz fixed point formula

A Superconducting QUantum Interference Device (SQUID) modulated by a fast oscillating magnetic flux can be used as a parametric amplifier, providing gain with very little added noise. Here, we develop linearized models to describe the parametrically flux-pumped SQUID in terms of an impedance. An unpumped SQUID acts as an inductance, the Josephson inductance, whereas a flux-pumped SQUID develops an additional, parallel element which we have coined the ``pumpistor.'' Parametric gain can be understood as a result of a negative resistance of the pumpistor. In the degenerate case, the gain is sensitive to the relative phase between the pump and signal. In the nondegenerate case, gain is independent of this phase. We develop our models first for degenerate parametric pumping in the three-wave and four-wave cases, where the pump frequency is either twice or equal to the signal frequency, respectively. We then derive expressions for the nondegenerate case where the pump frequency is not a multiple of the signal frequency, where it becomes necessary to consider idler tones which develop. For the nondegenerate three-wave case, we present an intuitive picture for a parametric amplifier containing a flux-pumped SQUID where current at the signal frequency depends upon the load impedance at an idler frequency. This understanding provides insight and readily testable predictions of circuits containing flux-pumped SQUIDs.

We obtain an analytical expression for the probability density of the Gaussian vector with a degenerate covariance matrix. The expression generalizes a similar expression for the nondegenerate case. A definition of the Gaussian vector on the basis of a probability density, which is common for the degenerate and nondegenerate cases, is given. Examples of using the obtained probability density for the synthesis of optimal signal-processing algorithms are presented.

In this paper we consider a geometric viewpoint to analyze the behaviour of the Conjugate Gradient (CG) method, for the solution of a symmetric linear system, when at current step a pivot breakdown possibly occurs (degenerate case). As well known this can occur when the system matrix is indefinite or singular. In the latter case the CG gets stuck, since the steplength along the current search direction cannot be computed. We show here that a simple geometric interpretation can be provided for the degenerate case, as long as some basics on projective geometry in the Euclidean space are considered.

This paper deals with multichannel time series, and extends the autocorrelation method for fitting the time series from the nondegenerate case to the general case. We shall show that the AR model obtained by solving the normal equations is always stable even in the degenerate case. The multichannel Levinson algorithm for solving the normal equations is extended from the nondegenerate case to the general case. In the nondegenerate case, the first n+1 values of the auto-correlation sequence of the obtained AR model of order n are identical with those of the original data. The next considered is the extrapolation problem of an autocorrelation sequence, that is, given data <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{R_{0}, R_{1},..., R_{n}}</tex> , it is to construct a model whose first n+1 values of the autocorrelation sequence are identical with these data. We shall show that, even in the degenerate case, when the given data <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{R_{0}, R_{1},..., R_{n}}</tex> are taken from the autocorrelation sequence of a purely nondeterministic time series, they are identical with the first n+1 values of the autocorrelation sequence of the AR model of order n.

Tomography vs quantum control for a three-level atom

We prove the existence of multidimensional traveling-wave solutions to the scalar equation for the transport of solutes (contaminants) with nonlinear adsorption and spatially periodic convection-diffusion-adsorption coefficients under the assumption that the nonlinear adsorption function satisfies the Lax and Oleinik entropy conditions. In the nondegenerate case, we also prove the uniqueness of the traveling waves. These traveling waves are analogues of viscous shock profiles. They propagate with effective speeds that depend on the periodic porous media only up to their mean states, and are given by an averaged Rankine-Hugoniot relation. This is a direct consequence of the fact that the transport equation is in conservation form. We use the sliding domain method, the continuation method, spectral theory, maximum principles, and a priori estimates. In the degenerate case, the traveling waves are weak solutions of a degenerate parabolic equation and are only Holder continuous. We obtain them by taking suitable limits on the non-degenerate traveling waves. The uniqueness of the degenerate traveling waves is open.

Asymptotic spreading of a time periodic diffusion equation with degenerate monostable nonlinearity

The compartmentalization of distinct templates in protocells and the exchange of templates between them (migration) are key elements of a modern scenario for prebiotic evolution. Here we use the diffusion approximation of population genetics to study analytically the steady-state properties of such a prebiotic scenario. The coexistence of distinct template types inside a protocell is achieved by a selective pressure at the protocell level (group selection) favoring protocells with a mixed template composition. In the degenerate case, where the templates have the same replication rate, we find that a vanishingly small migration rate suffices to eliminate the segregation effect of random drift and so to promote coexistence. In the nondegenerate case, a small migration rate greatly boosts coexistence as compared with the situation where there is no migration. However, increase of the migration rate beyond a critical value leads to the complete dominance of the more efficient template type (homogeneous regime). In this case, we find a continuous phase transition separating the homogeneous and the coexistence regimes, with the order parameter vanishing linearly with the distance to the transition point.

We investigate the dynamics of the wave packet formed by two codirected strongly interacting waves propagating in a medium with cubic nonlinearity. We obtain a soliton solution of the nonlinear Schrodinger equation in the degenerate case where the wave packet is described by a single partial momentum. In the nondegenerate case, we use the variational method to find the equation for the pulse duration, which turns out to be analogous to the equation for the coordinate in the Kepler problem. Solving it, we find the dependences of the pulse duration on the propagation distance in the cases of “finite” and “infinite” propagation regimes.

We propose a finite volume scheme for convection–diffusion equations with nonlinear diffusion. Such equations arise in numerous physical contexts. We will particularly focus on the drift-diffusion system for semiconductors and the porous media equation. In these two cases, it is shown that the transient solution converges to a steady-state solution as t tends to infinity. The introduced scheme is an extension of the Scharfetter–Gummel scheme for nonlinear diffusion. It remains valid in the degenerate case and preserves steady-states. We prove the convergence of the scheme in the nondegenerate case. Finally, we present some numerical simulations applied to the two physical models introduced and we underline the efficiency of the scheme to preserve long-time behavior of the solutions.

This paper exploits a general approach to geometric optics in inhomogeneous plasmas based on the properties of the local dielectric tensor ?. We express ? in terms of its eigenvalues ?j and eigenvectors ?. Then to zeroth order in the geometric optics approximation the determinant D = ?1?2?3 vanishes and the elements ?j vanish separately in pairs or simultaneously. It is shown that this branching in the dispersion relation changes the formulation of the geometric optics equations. The ray tracing and the transport of the amplitude of the wave in both degenerate and nondegenerate cases is described. The general procedure for transition through a boundary between degenerate and nondegenerate regions, where the rays split into two parts each following a different branch of the dispersion relation is also presented in this paper. We demonstrate our general method in a case, where radiation from a vacuum region enters an inhomogeneous magnetized plasma layer.

The spin-dependent Hall (SDH) effect in degenerate semiconductors is investigated theoretically. Starting from a two-component drift–diffusion equation, an expression for SDH voltage (VSDH) is derived, and drift and diffusive contributions to VSDH are studied. For the possible enhancement of the diffusive part, degenerate and nondegenerate cases are examined. We find that due to an increase in the diffusion coefficient VSDH increases in a degenerate semiconductor, consistent with the experimental observations. The expression for VSDH is reduced in three limiting cases, namely diffusive, drift–diffusion crossover and drift, and is analysed. The results agree with those obtained in recent theoretical investigations.

We investigate the time evolution of squeezing in the signal mode of the undamped parametric amplifier. Both the degenerate and nondegenerate cases are considered. These are the simplest models that give limits to squeezing caused by both the intrinsic pump noise and pump depletion. Conflicting analytic expressions have been reported. We resolve this by the use of exact numerical methods, employing both number state and stochastic techniques. Our central result is that in both cases we find that the minimum in the squeezed quadrature variance scales as N${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}^{\mathrm{\ensuremath{-}}1/2}$, where N\ifmmode\bar\else\textasciimacron\fi{} is the initial coherent pump photon number. This bound on the amount of squeezing obtainable, coupled with the loss in energy due to the conversion efficiency, serves to limit the phase information in the squeezed channel to no more than that of the coherent pump. We believe this to be a fundamental limit for using the squeezed radiation generated in coherently driven parametric amplifiers.

We present here a new approach to generalize supersymmetric quantum mechanics to treat multiparticle and multidimensional systems. We do this by introducing a vector superpotential in an orthogonal hyperspace. In the case of N distinguishable particles in three dimensions this results in a vector superpotential with 3N orthogonal components. The original scalar Schrödinger operator can be factored using a 3N-component gradient operator and introducing vector "charge" operators: Q(1) and Q(1)(dagger). Using these operators, we can write the original (scalar) Hamiltonian as H(1) = Q(1)(dagger) x Q(1) + E(0)((1)), where E(0)((1)) is the ground-state energy. The second sector Hamiltonian is a tensor given by H(2) = Q(1)Q(1)(dagger) + E(0)((1)) and is isospectral with H(1). The vector ground state of sector 2, psi(0)((2)), canbe used with the charge operator Q(1)(dagger) to obtain the excited-state wave function of the first sector. In addition, we show that H(2) can also be factored in terms of a sector 2 vector superpotential with components W(2j) = -(partial partial differential ln psi(0j)((2)))/partial partial differentialx(j). Here psi(0j)((2)) is the jth component of psi(0)((2)). Then one obtains charge operators Q(2) and Q(2)(dagger) so that the second sector Hamiltonian can be written as H(2) = Q(2)(dagger)Q(2) + E(0)((2)). This allows us to define a third sector Hamiltonian which is a scalar, H(3) = Q(2) x Q(2)(dagger) + E(0)((2)). This prescription continues with the sector Hamiltonians alternating between scalar and tensor forms, both of which can be treated by the variational method to obtain approximate solutions to both scalar and tensor sectors. We demonstrate the approach with examples of a pair of separable 1D harmonic oscillators and the example of a nonseparable 2D anharmonic oscillator (or equivalently a pair of coupled 1D oscillators). We consider both degenerate and nondegenerate cases. We also present a generalization to arbitrary curvilinear coordinate systems in the Appendix.

In this paper we study a fully discrete Semi-Lagrangian approximation of a second order Mean Field Game system, which can be degenerate. We prove that the resulting scheme is well posed and, if the state dimension is equals to one, we prove a convergence result. Some numerical simulations are provided, evidencing the convergence of the approximation and also the difference between the numerical results for the degenerate and non-degenerate cases.