Positive Imaginary Part Research Articles

Modal expansion of optical far-field quantities using quasinormal modes

We discuss an approach for modal expansion of optical far-field quantities based on quasinormal modes (QNMs). The issue of the exponential divergence of QNMs is circumvented by contour integration of the far-field quantities involving resonance poles with negative and positive imaginary parts. A numerical realization of the approach is demonstrated by convergence studies for a nanophotonic system.

Comparative study of the flexoelectricity effect with a highly/weakly interface in distinct piezoelectric materials (PZT-2, PZT-4, PZT-5H, LiNbO3, BaTiO3)

A comparative study of Love-type wave vibrations in a distinct piezoelectric (PE) material thin film with a highly and weakly dielectric interface with an elastic pre-stressed elastic plate is investigated analytically. Using the PZT-2, PZT-4, PZT-5H, LiNbO3 and BaTiO3 numerical data for the PE material, we achieved significant outcomes of PE and flexoelectric effect in the study. Dispersion equations are achieved analytically for both electrically open and short cases together with highly and weekly conducting interfaces in the transcendental complex form. Outcomes specify that microstructures having flexoelectricity has a significant influence on dispersion features. The existence of flexoelectricity shows that complex phase velocity with a positive imaginary part increases the phase velocity of considered waves with time. Numerical results are highlighted graphically and validated with existing results. The present study is a prior attempt to show the flexoelectricity influence with weekly and highly conducting interfaces and also to show that the outcomes are significant to obtain better surface acoustic wave device’s performance.

Anatomy of flexoelectricity in micro plates with dielectrically highly/weakly and mechanically complaint interface

The present study analytically investigates the Love-type wave vibrations in a piezoelectric thin film with highly and weakly dielectric conducting interface with pre-stressed elastic plate. Dispersion equations are obtained analytically for both electrically open and short cases together with highly and weekly interfaces in the transcendental complex form. Outcomes specify that microstructures consisting flexoelectricity has an essential influence on dispersion features of the Love-type wave and also affected by the conducting interface. As a remarkable outcome, complex phase velocity with positive imaginary part increases with time. Moreover, the dispersion relation depends substantially on the flexoelectric coefficients and width of the guiding plate. The present study is the prior attempt to show the flexoelectricity influence with weekly and highly conducting interfaces.

Electromagnetic modes and resonances of two-dimensional bodies

The electromagnetic modes and the resonances of homogeneous, finite size, two-dimensional bodies are examined in the frequency domain by a rigorous full wave approach based on an integro-differential formulation of the electromagnetic scattering problem. Using a modal expansion for the current density that disentangles the geometric and material properties of the body the integro-differential equation for the induced surface (free or polarization) current density field is solved. The current modes and the corresponding resonant values of the surface conductivity (eigen-conductivities) are evaluated by solving a linear eigenvalue problem with a non-Hermitian operator. They are inherent properties of the body geometry and do not depend on the body material. The material only determines the coefficients of the modal expansion and hence the frequencies at which their amplitudes are maximum (resonance frequencies). The eigen-conductivities and the current modes are studied in detail as the frequency, the shape and the size of the body vary. Open and closed surfaces are considered. The presence of vortex current modes, in addition to the source-sink current modes (no whirling modes), which characterize plasmonic oscillations, is shown. Important topological features of the current modes, such as the number of sources and sinks, the number of vortexes, the direction of the vortexes are preserved as the size of the body and the frequency vary. Unlike the source-sink current modes, in open surfaces the vortex current modes can be resonantly excited only in materials with positive imaginary part of the surface conductivity. Eventually, as examples, the scattering by two-dimensional bodies with either positive or negative imaginary part of the surface conductivity is analyzed and the contributions of the different modes are examined.

Complex Photoconductivity Reveals How the Nonstoichiometric Sr/Ti Affects the Charge Dynamics of a SrTiO3 Photocatalyst.

Strontium titanate (SrTiO3) is a perovskite that is important in water-splitting photocatalytic chemistry. Although excess Sr is known to improve the photocatalytic activity, its effect on charge dynamics remain largely unaddressed. Herein, we present a detailed analyses of gigahertz complex transient photoconductivity (Δσ) measured using time-resolved microwave conductivity (TRMC). We show that charge carrier trapping associated with the emergence of an anomalous positive imaginary part and the first-order rate constant of the normal positive real part of Δσ dramatically decreased with increasing Sr/Ti ratio. The second-order rate constant attributed to charge recombination simultaneously decreased, and these rate constants were well correlated with the improved hydrogen evolution rate of aqueous SrTiO3 suspensions with a Pt co-catalyst. These findings provide a fresh perspective on the stoichiometry-carrier dynamics relationship paramount for the optimization of composition-engineered photocatalysts and reveal the broad implications for mechanistic studies based on TRMC evaluation.

Influence of viscosity temperature dependence on the spectral characteristics of the thermoviscous liquids flow stability equation

The flow of a viscous model fluid in a flat channel with a non-uniform temperature field is considered. The problem of the stability of a thermoviscous fluid is solved on the basis of the derived generalized Orr-Sommerfeld equation by the spectral decomposition method in Chebyshev polynomials. The effect of taking into account the linear and exponential dependences of the fluid viscosity on temperature on the spectral characteristics of the hydrodynamic stability equation for an incompressible fluid in a flat channel with given different wall temperatures is investigated. Analytically obtained profiles of the flow rate of a thermovisible fluid. The spectral pictures of the eigenvalues of the generalized Orr-Sommerfeld equation are constructed. It is shown that the structure of the spectra largely depends on the properties of the liquid, which are determined by the viscosity functional dependence index. It has been established that for small values of the thermoviscosity parameter the spectrum compares the spectrum for isothermal fluid flow, however, as it increases, the number of eigenvalues and their density increase, that is, there are more points at which the problem has a nontrivial solution. The stability of the flow of a thermoviscous fluid depends on the presence of an eigenvalue with a positive imaginary part among the entire set of eigenvalues found with fixed Reynolds number and wavenumber parameters. It is shown that with a fixed Reynolds number and a wave number with an increase in the thermoviscosity parameter, the flow becomes unstable. The spectral characteristics determine the structure of the eigenfunctions and the critical parameters of the flow of a thermally viscous fluid. The eigenfunctions constructed in the subsequent works show the behavior of transverse-velocity perturbations, their possible growth or decay over time.

Negative capacitance or inductive loop? – A general assessment of a common low frequency impedance feature

Impedance analysis is not an easy task. This becomes even more obvious if unusual shapes appear in the impedance diagram. Here, a low frequency hook with positive imaginary part is discussed that often leaves researchers clueless of how to handle it, especially if there is no common model for the device under test that includes the hook, which is actually true for most electrochemical systems. The low frequency hook is explained from a systems theory perspective and a very basic explanation is given of what has to happen phenomenologically for this feature to appear. The most common equivalent circuit model (ECM) for such a low frequency hook is reviewed and the relation to a general empirical low-pass filter type model is discussed. A common electrochemical example and an abstract minimal example are introduced to facilitate the interpretation of the low frequency hook for an arbitrary (photo-) electrochemical system.

Action of complex symplectic matrices on the Siegel upper half space

The Siegel upper half space, Sn, the space of complex symmetric matrices, Z with positive definite imaginary part, is the generalization of the complex upper half plane in higher dimensions. In this paper, we study a generalization of linear fractional transformations, ΦS, where S is a complex symplectic matrix, on the Siegel upper half space. We partially classify the complex symplectic matrices for which ΦS(Z) is well defined. We also consider Sn and S‾n as metric spaces and discuss distance properties of the map ΦS from Sn to Sn and S‾n respectively.

Prime lattice points in ovals

We study the distribution of lattice points with prime coordinates lying in the dilate of a convex planar domain having smooth boundary, with nowhere vanishing curvature. Counting lattice points weighted by a von Mangoldt function gives an asymptotic formula, with the main term being the area of the dilated domain, and our goal is to study the remainder term. Assuming the Riemann Hypothesis, we give a sharp upper bound, and further assuming that the positive imaginary parts of the zeros of the Riemann zeta functions are linearly independent over the rationals allows us to give a formula for the value distribution function of the properly normalized remainder term.

Diamagnetism and the dispersion of the magnetic permeability

It is well known that the usual Kramers--Kronig relations for the relative permeability function $\mu(\omega)$ are not compatible with diamagnetism ($\mu(0)<1$) and a positive imaginary part ($\text{Im}\,\mu(\omega)>0$ for $\omega>0$). We demonstrate that a certain physical meaning can be attributed to $\mu$ for all frequencies, and that in the presence of spatial dispersion, $\mu$ does not necessarily tend to 1 for high frequencies $\omega$ and fixed wavenumber $\mathbf k$. Taking the asymptotic behavior into account, diamagnetism can be compatible with Kramers--Kronig relations even if the imaginary part of the permeability is positive. We provide several examples of diamagnetic media and metamaterials for which $\mu(\omega,\mathbf k)\not\to 1$ as $\omega\to\infty$.

Substrate Integrated Waveguide Filter With Flat Passband Based on Complex Couplings

A lossy filter with flat passband and asymmetric response is proposed by using complex couplings. The complex coupling coefficients with positive and negative imaginary parts are realized with transmission lines loaded with series and shunt resistors, respectively. Design formulas are derived for the critical parameters. A filter prototype is developed at 4.97 GHz and substrate integrated waveguide resonators with unloaded $Q$ -factor of 450 are used. One transmission zero is located in its upper stopband. The 0.2-dB bandwidth of 279 MHz is achieved and the equivalent $Q$ -factor is about 1900. Good agreement is obtained among the synthesized, simulated, and measured results.

Transmission singularities in resonant electron tunneling through double complex potential barrier

Tunneling of electrons through a barrier with complex potential is investigated. We focus on two cases, symmetric double rectangular barrier and double delta potential barrier, and give expressions for resonant transmission probability for both cases. Expressions for reflection amplitude and absorption are also obtained in the case of delta potential. It will be shown that for given incident particle energy, parameters of the potential structure can be obtained so that the resonant transmission probability approaches infinity. The phenomenon of infinite transmission occurs only for potentials which have positive imaginary part.

On a Liouville-type theorem for the Ginzburg–Landau system

We prove that entire, complex valued solutions to the Ginzburg–Landau system with positive real and imaginary parts are constant in any spatial dimension. This property was shown very recently only in the planar case.

English

We consider solving complex symmetric linear systems with multiple right-hand sides. We assume that the coefficient matrix has indefinite real part and positive definite imaginary part. We propose a new block conjugate gradient type method based on the Schur complement of a certain 2-by-2 real block form. The algorithm of the proposed method consists of building blocks that involve only real arithmetic with real symmetric matrices of the original size. We also present the convergence property of the proposed method and an efficient algorithmic implementation. In numerical experiments, we compare our method to a complex-valued direct solver, and a preconditioned and nonpreconditioned block Krylov method that uses complex arithmetic.

The search for a Hamiltonian whose energy spectrum coincides with the Riemann zeta zeroes

Inspired by the Hilbert–Polya proposal to prove the Riemann Hypothesis we have studied the Schroedinger QM equation involving a highly nontrivial potential, and whose self-adjoint Hamiltonian operator has for its energy spectrum one which approaches the imaginary parts of the zeta zeroes only in the asymptotic (very large [Formula: see text]) region. The ordinates [Formula: see text] are the positive imaginary parts of the nontrivial zeta zeroes in the critical line :[Formula: see text]. The latter results are consistent with the validity of the Bohr–Sommerfeld semi-classical quantization condition. It is shown how one may modify the parameters which define the potential, and fine tune its values, such that the energy spectrum of the (modified) Hamiltonian matches not only the first two zeroes but the other consecutive zeroes. The highly nontrivial functional form of the potential is found via the Bohr–Sommerfeld quantization formula using the full-fledged Riemann–von Mangoldt counting formula (without any truncations) for the number [Formula: see text] of zeroes in the critical strip with imaginary part greater than [Formula: see text] and less than or equal to [Formula: see text].

Free Pick functions: Representations, asymptotic behavior and matrix monotonicity in several noncommuting variables

R. Nevanlinna showed that elements of the Pick class, the set of complex analytic functions taking the upper half plane into itself, have certain integral representations which reflect their asymptotic behavior at infinity. Later, Löwner connected the Pick class to matrix monotone functions. We generalize the Nevanlinna representation theorems and Löwner's theorem on matrix monotone functions to the free Pick class, the collection of functions that map tuples of matrices with positive imaginary part into the matrices with positive imaginary part which obey the free functional calculus.

Integral Equation Formulations for Characteristic Modes of Dielectric and Magnetic Bodies

Five types of surface integral equation (IE) formulations are presented to determine the characteristic modes of dielectric and magnetic bodies. It should be noted that these five formulations are derived in a unified manner. Instead of directly using the IE coefficient matrices of scattering problems, this paper presents the formulations in a more natural and rigorous way. The IE formulations result in the generalized real symmetric eigenvalue equations for the characteristic modes. It is found that the eigenvalues of the first four formulations indicate the negative or positive imaginary part of the complex power of the characteristic modes, which are zero at resonance. The first two IE formulations result in two new generalized eigenvalue equations, while the last three IE formulations lead to the same generalized eigenvalue equations as in the references. Moreover, the first two formulations can be immune from spurious modes from which the third and fourth formulations suffer.

Causal Stroh formalism for uniformly-moving dislocations in anisotropic media: Somigliana dislocations and Mach cones

In this work, Stroh’s formalism is endowed with causal properties on the basis of an analysis of the radiation condition in the Green tensor of the elastodynamic wave equation. The modified formalism is applied to dislocations moving uniformly in an anisotropic medium. In practice, accounting for causality amounts to a simple analytic continuation procedure whereby to the dislocation velocity is added an infinitesimal positive imaginary part. This device allows for a straightforward computation of velocity-dependent field expressions that are valid whatever the dislocation velocity–including supersonic regimes–without needing to consider subsonic and supersonic cases separately. As an illustration, the distortion field of a Somigliana dislocation of the Peierls–Nabarro–Eshelby-type with finite-width core is computed analytically, starting from the Green’s tensor of elastodynamics. To obtain the result in the form of a single compact expression, use of the modified Stroh formalism requires splitting the Green’s function into its reactive and radiative parts. In supersonic regimes, the solution obtained displays Mach cones, which are supported by Dirac measures in the Volterra limit. From these results, an explanation of Payton’s ‘backward’ Mach cones (Payton, 1995) is given in terms of slowness surfaces, and a simple criterion for their existence is derived. The findings are illustrated by full-field calculations from analytical formulas for a dislocation of finite width in iron, and by Huygens-type geometric constructions of Mach cones from ray surfaces.

On Properties of Certain Special Zeros of Functions in the Selberg Class

In this paper, assuming generalized Riemann hypothesis, we give an upper bound for the multiplicity of eventual zero at central point 1 / 2 and location of the first zero with positive imaginary part of function in a certain subclass of the extended Selberg class. We apply our results to automorphic L-functions attached to irreducible unitary automorphic representations of $$GL_N(\mathbb {Q})$$ .

Resolution of a conjecture on variance functions for one-parameter natural exponential families

We prove a conjecture of Bar-Lev, Bshouty and Enis stating that a polynomial with a simple root at 0 and a complex root with positive imaginary part is the variance function of a one-parameter natural exponential family with mean domain (0,∞) if and only if the real part of the complex root is not positive.