Positive Imaginary Part Research Articles

Intrinsic high-frequency characteristics of tunneling heterostructure devices.

A general numerical method has been developed to solve the periodic time-dependent Schr\odinger equation under a weak ac field where the quantum transmitting boundary method is employed to formulate boundary conditions of far-from-equilibrium open systems. Also derived are current components for ac small-signal analysis. We apply the method to a resonant tunneling diode (RTD) structure. Our calculations illustrate that the assumption of Lorentzian-like form of line shapes of the current functions is no longer valid at high frequencies. Thus a careful treatment to these integral functions is fundamental to obtain a physically reasonable result. Results of linear admittance, rectification coefficient, and second-harmonic generation coefficient are presented as a function of frequency and bias, at both positive differential resistance and negative-differential resistance (NDR) region. The calculations have shown that at high frequencies (several THz), the reactive feature of RTD, whether inductive or capacitive, depends on the bias and frequency. The capacitive feature, i.e., the positive imaginary part of the admittance, reaches maximum in the middle of the NDR region. This behavior can be ascribed to the confined electrons in the well. The characteristic of the transition from electron to optical behavior is observed when the frequency increases. The rectification coefficient and second-harmonic generation coefficient show a resonant enhancement at high frequencies. A comparison with the results obtained by the Wigner function is demonstrated. Different definitions of the ac reactive current component are discussed in order to clarify the confusion in the literature.

Frequency‐domain acoustic‐wave modeling and inversion of crosshole data: Part I—2.5-D modeling method

Crosshole experiments usually have sources and receivers confined to a plane, and it is assumed that there is negligible variation in the properties of the medium normal to this plane. Therefore, the problem appears two‐dimensional, except for the sources which are 3-D points rather than lines. This configuration is denoted as two‐and‐one‐half‐dimensional. We present a frequency‐domain approach to modeling acoustic wave propagation in such situations which allows correct treatment of point sources but takes advantage of the assumed 2-D nature of the medium to avoid full 3-D simulations. The approach uses a Fourier transform with respect to the out‐of‐plane coordinate to reduce the problem of modeling in 3-D to repeatedly solving a 2-D equation, which we accomplish using finite differences. The discrete inverse Fourier transform from the out‐of‐plane wavenumber implies the existence of an infinite number of spurious “ghost sources” spaced periodically in the out‐of‐plane direction. These sources generate significant artifacts on the time‐domain traces, even when their spatial period is much greater than the (in‐plane) dimensions of the survey area, because of time‐wrapping in the transform from the frequency domain. We describe two methods for reducing these artefacts, the more effective of which entails exponential damping by adding a positive imaginary part to the frequency, compensated by ramping of the wrapped time domain records. We check the modeling scheme by analysis of direct and scattered arrivals from simple models. The observed seismograms agree well with those calculated using Born theory, and so confirm the potential of this modeling method for use in inversion.

Evaluation of the elastic pp single-flip helicity amplitude at low momentum transfer and high energies.

We report on the evaluation of the single-flip helicity amplitude by analyzing proton-proton elastic polarization data in the four-momentum-transfer[minus][ital t] region from 1.5[times]10[sup [minus]3] to 0.6 (GeV/[ital c])[sup 2]. Recent results around 200 GeV/[ital c] in the low [vert bar][ital t][vert bar] range, where the interference of the electromagnetic and hadronic amplitudes is significant, allow a more reliable evaluation of the single-flip hadronic helicity amplitude [phi][sub 5][sup [ital h]]. Within the experimental errors, our analysis indicates a small and negative real part ([minus]1 to [minus]4 %) and a positive imaginary part (8 to 30 %) for the reduced hadronic single-flip helicity amplitude ([ital m]/ [radical][minus][ital t] )[phi][sub 5][sup [ital h]] with respect to the helicity-averaged hadronic nonflip amplitude. To study the stability of these results, we have also considered data at 45--300 GeV/[ital c] in the 0.057--0.6 (GeV/[ital c])[sup 2] momentum transfer range in conjunction with the aforementioned data.

"Blind" shape reconstruction from experimental data

A method for reconstructing the shape of a bounded impenetrable object from measured scattered field data is presented. The reconstruction algorithm is, in principle, the same as that used before for reconstructing the conductivity of a penetrable object and uses the fact that for high conductivity the skin depth of the scatterer is small, in which case the only meaningful information produced by the algorithm is the boundary of the scatterer. A striking increase in efficiency is achieved by incorporating into the algorithm the fact that for large conductivity, the contrast is dominated by a large positive imaginary part. This fact together with the knowledge that the scatterer is constrained in some test domain constitute the only a priori information about the scatterer that is used. There are no other implicit assumptions about the location, connectivity, convexity, or boundary conditions. The method is shown to successfully reconstruct the shape of an object from experimental scattered field data in a blind test.

Two‐dimensional location and shape reconstruction

A method for reconstructing the location and the shape of a bounded impenetrable object from measured scattered field data is presented. The algorithm is, in principle, the same as that used for reconstructing the conductivity of a penetrable object and uses the fact that for high conductivity the skin depth of the scatterer is small, in which case the only meaningful information produced by the algorithm is the boundary of the scatterer. A striking increase in efficiency is achieved by incorporating into the algorithm the fact that for large conductivity the contrast is dominated by a large positive imaginary part. This fact, together with the knowledge that the scatterer is constrained in some test domain, constitute the only a priori information about the scatterer that is used. There are no other implicit assumptions about the location, connectivity, convexity, or boundary conditions. Some refinements of the algorithm which reduce the number of points at which the unknown function is updated are incorporated to further increase efficiency. Results of a number of numerical examples are presented which demonstrate the effectiveness of the location and shape reconstruction algorithm.

Picosecond waveguide modulation in conjugated polymers

The technique of time-resolved waveguide modulation is applied to optical waveguides of the conjugated polymers, polydiacetylene-(CH 2) 4OCONHOCOC 4H 9 (poly-4BCMU) and poly(3-hexylthiophene (P3HT). This method yields values of the sign and magnitude of the complex intensity-dependent refractive index, n 2 , for guided be agt 630 nm ( poly-4 BCMU) and at 1.06 μm (P3HT), upon pumping near the principal absorption band. In poly-4BCMU, n 2 is real and negative, with a magnitude in the order of −10 −7 (MW/cm 2) −1. The decay of 2.5 ps closely matches the decay of the bleaching of the exiton absorption following resonant excitation. These results are consistent with phase-space filling by excitons as the principal nonlinear mechanism. In P3HT, fn 2 has a negative real part and a positive imaginary part, with |n 2|≌1×10 −4 ( MW/cm 2) −1 , for a pump photon energy of 2.06 eV. The positive imaginary part correllates with a photoinduced absorption at 1.06 μm (1.17 eV), indicating a shift of oscillator strength from the interband transition to localized absorptions in the infrared. The negative real part implies that the observed photoinduced absorption peaks at an energy below 1.17 eV.

Structure of the singularities of operator functions with a positive imaginary part

In the theory of linear operators in a Hilbert space an important role is played by the so-called operator-value~ R-functions. By the latter we mean [i] functions T(1), analytic in the upper half-plane iC+, whose values are bounded operators, acting in a Hilbert space H and having positive imaginary part (Im T(%) ~ 0, Im% > 0). We elucidate briefly the manner in which this object arises in the theory of perturbations of self-adjoint (and also nonself-adjoint) operators. Namely, assume that in a Hilbert space H there are given two self-adjoint operators A, A + V, where V ~ 0 is a bounded operator. Then from Hilbert's identity we obtain the following representation for the "bordered" resolvent of the "perturbed" operator A + V:

Singular solutions of a transmission problem in plane linear elasticity for wedge-shaped regions

In plane elasticity, when two different wedge-shaped elastic materials (isotropic, homogeneous) are bonded together along a common edge and subject to tractions on the boundary, the stress field ? will become infinite at the apex. In fact, asymptotically, the displacementu satisfies $$u \approx \sum\limits_{j = 1}^n {r^{ - i\lambda _j } } \sum\limits_{k = 0}^{\eta _j - 1} {\ln ^k } (r)\Phi _{jk} (\theta )asr \to 0,$$ where (r, ?) are the polar coordinates with origin at the wedge apex and the complex numbers? j are the roots with positive imaginary part and multiplicity? j of an explicitly given transcendental equation:G(?)=0. For applications it is desirable to understand how the number of terms that give rise to unbounded stresses in the above expansion depends on the elasticity constants and on the wedge angles. Setting $$N(q) = \sum\limits_{0< \operatorname{Im} (\lambda _j )< q} {\eta _j } $$ , the problem is then to studyN(1) as a function of the material constants and wedge angles. A rigorous analysis is presented here for the case in which the wedge angles are equal (symmetric domains). Using the fact thatG is an entire function of ? which depends continuously on the parameters of the problem, we develop a continuity method which enables us to evaluate the functions $$N\left( {\frac{\pi }{a}} \right)$$ and $$N\left( {\frac{\pi }{{2a}}} \right)$$ wherea is the measure of each wedge angle. This yields bounds onN(1) as well as some information on the zeros ofG that correspond to higher order singularities.

Picosecond nondegenerate waveguide modulation in a conjugated polymer

The technique of time-resolved waveguide modulation is applied to an optical waveguide of the conjugated polymer poly (3-hexylthiophene). This method yields values of the sign and magnitude of the complex intensity-dependent refractive index, n 2, for a guided beam at 1.06 μm (1.17 eV), upon pumping into the tail of the absorption. We find that the picosecond electronic contribution to n 2 has a negative real part, and a positive imaginary part, with ¦ n 2¦ ⋍ 1 × 10 -4 (MW/cm 2) -1, for a pump photon energy of 2.06 eV. The positive imaginary part correlates with the photoinduced absorption at 1.06 μm measured on the same sample, and consistent with earlier results on the same material, which indicate a shift of oscillator strength from the interband transition to localized absorptions in the infrared. The negative real part implies that the observed photoinduced absorption peaks at an energy below 1.17 eV. The time-resolved component of n 2 is accompanied by a slow component (> 12 ns) with positive real and imaginary parts and comparable magnitude, indicating a thermal increase in n (or equivalently the film thickness) and a long-lived photoinduced absorption.

The effect of anisotropic wall compliance on boundary-layer stability and transition

A fairly simple theoretical model of an anisotropic compliant wall has been developed. It has been used to undertake a comprehensive numerical study of boundary-layer stability over such walls. The study is based on linearized theory, makes the usual quasi-parallel-flow approximation, uses the Blasius profile as the basic undisturbed flow and assumes two-dimensional disturbances. An investigation is carried out of the effects of anisotropic wall compliance on the Tollmien–Schlichting waves and the two previously identified wall modes, namely travelling-wave flutter and divergence. In addition global convergence techniques are used to search for other possible instabilities.An asymptotic theory, valid for high Reynolds numbers, is also presented. This can provide accurate estimates of the eigenvalues. It is applicable to a much wider class of compliant walls than the relatively simple model used for the numerical study. An important use of the asymptotic theory is to help identify and elucidate the various energy-exchange mechanisms responsible for stabilization or destabilization of the instabilities. A reduction in the production of disturbance energy by the Reynolds shear stress is the main reason for the favourable effect of anisotropic wall compliance on instability growth. Other energy-exchange mechanisms, which have been found to make a significant contribution, include energy transfer from the disturbance to the mean flow due to the interaction of the fluctuating shear stress and the displaced mean flow, and the work done by the perturbations in wall pressure and shear stress.It is found that anisotropic wall compliance confers very considerable advantage with respect to reduction in instability growth rate and transition delay. Using a fairly conservative criterion an almost ten-fold rise in transitional Reynolds number is predicted for anisotropic walls having the appropriate properties. Anisotropic wall compliance makes travelling-wave flutter much more sensitive to viscous effects and has a considerable stabilizing influence. The application of global convergence methods has led to the discovery of an anomalous spatially growing eigenmode which, according to conventional interpretation, would represent an instability. Further study of an appropriate initial-value problem has revealed that the new eigenmode is probably not an instability and that, for compliant walls, complex wavenumbers with positive real and negative imaginary parts do not necessarily correspond to an instability.

Incomplete block-matrix factorization iterative methods for convection-diffusion problems

Standard Galerkin finite element methods or finite difference methods for singular perturbation problems lead to strongly unsymmetric matrices, which furthermore are in general notM-matrices. Accordingly, preconditioned iterative methods such as preconditioned (generalized) conjugate gradient methods, which have turned out to be very successful for symmetric and positive definite problems, can fail to converge or require an excessive number of iterations for singular perturbation problems. This is not so much due to the asymmetry, as it is to the fact that the spectrum can have both eigenvalues with positive and negative real parts, or eigenvalues with arbitrary small positive real parts and nonnegligible imaginary parts. This will be the case for a standard Galerkin method, unless the meshparameterh is chosen excessively small. There exist other discretization methods, however, for which the corresponding bilinear form is coercive, whence its finite element matrix has only eigenvalues with positive real parts; in fact, the real parts are positive uniformly in the singular perturbation parameter. In the present paper we examine the streamline diffusion finite element method in this respect. It is found that incomplete block-matrix factorization methods, both on classical form and on an inverse-free (vectorizable) form, coupled with a general least squares conjugate gradient method, can work exceptionally well on this type of problem. The number of iterations is sometimes significantly smaller than for the corresponding almost symmetric problem where the velocity field is close to zero or the singular perturbation parameter e=1.

Coupling between alfvén and slow magnetosonic waves in an inhomogeneous finite- β plasma—II. Eigenmode analysis of localized ballooning-interchange instability

Ballooning-interchange instability of guided hydromagnetic waves is examined by performing a numerical eigenmode analysis along a model field-line. This instability is driven by a combined effect of the plasma pressure gradient and the field-line curvature and can be considered as a candidate of the excitation mechanism of storm-time Pc5 pulsations in the outer magnetosphere. The system of eigenmode equations derived in an accompanying paper describes the coupling between Alfvén and slow magnetosonic modes due to inhomogeneity. Such a coupling results in oscillations with hybrid properties between the two, for example, the transverse magnetic perturbation in the radial direction and the out-of-phase relation in perturbed plasma and magnetic pressures ; they are important characteristics of the storm-time Pc5 pulsation. Eigenvalues of the fundamental mode oscillation have a positive imaginary part, that is, the fundamental eigenmode is unstable. The oscillation frequency of this unstable mode is proportional to the azimutha wave number, and is comparable with the ion drift frequency at the equator. These results suggest that the ion drift mode plays an important role in the unstable mode. Although the broad distribution of eigenfunctions along the field-line and a somewhat small value of the oscillation frequency are not consistent with observations, such inconsistencies are due to the oversimplification adopted in the model of plasma and magnetic fields. In practice, high energy particles can be expected to play a decisive role in determining the properties of the pulsations in the magnetosphere. Hence, it is concluded that the ballooning-interchange instability is a strong candidate for the excitation mechanism of the storm-time Pc5 pulsation.

Boundary values of analytic operator functions with a positive imaginary part

Let be the Schatten-von Neumann class, 0<ρ<∞ of operators in a Hilbert space. It is shown that for -valued R-functions the nontangential boundary values exist almost everywhere on ℝ and belong to the class if 0 1 the “boundary values” may be unbounded operators everywhere on ℝ. Finally, for ρ=1 the nontangential limits exist in the norm of the space , for any q>1. However, they belong to a class which, in general, cannot be smaller than the class, conjugate with the Matsaev ideal.

Uniqueness theorems for operator-valued functions with positive imaginary part, and the singular spectrum in the selfadjoint Friedrichs model

Analyse spectrale du modele de Friedrichs. On reduit l'etude du spectre a celles des racines d'une fonction operateur analytique avec la partie imaginaire positive. On donne des theoremes d'unicite pour des fonctions operateurs analytiques a partie imaginaire positive sur le cercle ou le demi-plan. On etudie en detail les racines la fonction operateur M(λ)

A decomposition theorem on differential polynomials of theta functions

Let τ = (τij) be a symmetric complex g = g matrix with the positive definite imaginary part. A theta function of level n means an entire function f(z) in g complex variables z = (zl …, zg) satisfying the difference relations:

3.4. Null-sets of operator functions with a positive imaginary part

3.4. Null-sets of operator functions with a positive imaginary part

An optical guided‐wave dye laser and directional coupling amplifier

AbstractThe properties of a new type active thin‐film optical waveguide with layers of Corning glass 7059, Pyrex, and Rh 6 G dye formed on a fused quartz substrate are discussed. It generates superradiant emission when excited by a pulsed N2UV laser. When the top layer is shaped as a square, resonant behavior is observed between the corner reflectors formed by the edges on the diagonal. When the optical path is displaced normal to the diagonal, a distinct resonance is observed. This structure was used as a directional coupler and the coupling efficiency was found to depend on the waveguide length. These properties can be treated theoretically relatively simply in terms of a complex propagation constant with a positive imaginary part.

An operator-theoretic approach to theorems of the Pick-Nevanlinna and Loewner types. I

In this paper we formulate a general operator-theoretic result from which we deduce some old and some new function-theoretic interpolation theorems. The theorems we are concerned with have as prototype the Pick-Nevanlinna theorem and the Loewner theorem on restrictions of functions with positive imaginary part in the upper half plane.

The modular symbol and continued fractions in higher dimensions

The modular symbol has proved a very useful tool for dealing with holomorphic automorphic forms on the upper half-plane. It serves, for instance, to compute the action of Hecke operators on automorphic forms, to give the value of twisted L-functions of certain elliptic curves in terms of a finite sum of modular symbols, and to prove that divisors of degree zero supported on the cusps of a modular curve have finite order in the Jacobian of that curve. For these matters, see [1], Chaps. IV and V, and the references cited there. Here is a definition of the modular symbol. Let H be the upper half-plane, that is, the set of all complex numbers with positive imaginary part. A cusp of H is defined to be any rational number (considered as a point of the real line) or oc. The cusps belong to the boundary of H in the extended complex plane. Let SL(2,1R) act on H in the usual way. For any subgroup F of finite index in SL(2, 9 ) we can form the quotient H/F. There is a closed, compact Riemann surface X(F) such that H/F is biholomorphic to the complement of some finite set of points x 1 . . . . ,x~ in X(F). These points x i are called the cusps of X(F). Let ~?(F) denote the holomorphic differential l-forms on X(F). Integration of forms over cycles yields a canonical N-linear map from H~(X(F);IR) to the complex dual space of ~(F). The theory of Riemann surfaces tells us that this map is an isomorphism. Now choose any two rational cusps a and b on the boundary of H. Let l be any reasonable arc beginning at a, ending at b, and staying in H in the interim for instance, l could be the geodesic in H from a to b. Then one easily shows that integration over the image of l in X(F) gives a linear map from ~?(F) to I12. This linear map gives an element of HI(X(F); IR) in the canonical way described above. We denote this homology class by [a, hi. We call [ ] the modular symbol. In summary, the "classical" modular symbol is a map from the set of pairs of cusps of H into H~(X(F);IR). This symbol enjoys properties which are the

Integral equations for the analytic extrapolation of scattering amplitudes with positivity constraints

We investigate the stable analytic extrapolation of the scattering amplitudes with positive imaginary parts from the whole boundary or from a part of it to interior points. By using the Lagrange technique in convex optimization, we derive a system of singular integral equations which completely solve the problem. In the case of the extrapolation from the whole boundary this system reduces to the Fredholm equation recently obtained by Aubersonet al., using different considerations, in connection with a related extremum problem. Our results give also the correct analytic-extrapolation formulae in the limiting case of scattering data which are exactly the restriction to the boundary of some analytic function.