Large Imaginary Part Research Articles

On the roots of the equation $$\zeta (s)=a$$ ζ ( s ) = a

Given any complex number \(a\), we prove that there are infinitely many simple roots of the equation \(\zeta (s)=a\) with arbitrarily large imaginary part. Besides, we give a heuristic interpretation of a certain regularity of the graph of the curve \(t\mapsto \zeta ({1\over 2}+it)\). Moreover, we show that the curve \(\mathbb {R}\ni t\mapsto (\zeta ({1\over 2}+it),\zeta '({1\over 2}+it))\) is not dense in \(\mathbb {C}^2\).

D¯andBmesons in a nuclear medium

We discuss the mass modifications of $\bar{\mathrm{D}}$ and $\bar{\mathrm{D}}^{\ast}$ ($\mathrm{B}$ and $\mathrm{B}^{\ast}$) mesons in nuclear medium. The heavy quark symmetry for $\bar{\mathrm{D}}$ and $\bar{\mathrm{D}}^{\ast}$ ($\mathrm{B}$ and $\mathrm{B}^{\ast}$) mesons is adopted, and the interaction between a $\bar{\mathrm{D}}$ or $\bar{\mathrm{D}}^{\ast}$ ($\mathrm{B}$ or $\mathrm{B}^{\ast}$) meson and a nucleon is supplied from the pion exchange. We find the negative mass shifts for $\bar{\mathrm{D}}$ meson and $\mathrm{B}$ meson, and hence that the $\bar{\mathrm{D}}$ and $\mathrm{B}$ mesons are bound in the nuclear medium. As applications, we consider the atomic nuclei with a $\bar{\mathrm{D}}$ meson, $^{40}_{\bar{\mathrm{D}}}\mathrm{Ca}$ and $^{208}_{\bar{\mathrm{D}}}\mathrm{Pb}$, and investigate the energy levels of the $\bar{\mathrm{D}}$ meson in each nucleus. We also discuss the mass shifts in the isospin asymmetric nuclear medium, and present the possible phenomenon about distribution of isospin density around a $\bar{\mathrm{D}}$ or $\mathrm{B}$ meson in nuclear medium. We find that the mass shifts of $\bar{\mathrm{D}}^{\ast}$ and $\mathrm{B}^{\ast}$ mesons have large imaginary parts, which would prevent precise study of the energy levels of $\bar{\mathrm{D}}^{\ast}$ and $\mathrm{B}^{\ast}$ mesons in nuclei.

Quiescent phases with distributed exit times

Diffusive coupling of a dynamical system to a quiescent (zero) phase, with the same rates for all variables, stabilizes against oscillations. When the coupling rates are increased then, at a stationary point, the eigenvalues of the Jacobian matrix with positive real parts and large imaginary parts may move towards the imaginary axis of the complex plane and eventually enter the left half-plane. Diffusive coupling means that holding times in the active and in the quiescent phase are exponentially distributed. Here, we ask whether similar phenomena occur if the exponential distributions are replaced by other distributions. A general stability result can be shown for arbitrary distributions, and several more specific results for Gamma distributions and delta peaks (leading to delay equations). Some of the results apply to traveling fronts in reaction diffusion equations with quiescent phase.

On Born approximation in black hole scattering

A massless field propagating on spherically symmetric black hole metrics such as the Schwarzschild, Reissner-Nordstr\"{o}m and Reissner-Nordstr\"{o}m-de Sitter backgrounds is considered. In particular, explicit formulae in terms of transcendental functions for the scattering of massless scalar particles off black holes are derived within a Born approximation. It is shown that the conditions on the existence of the Born integral forbid a straightforward extraction of the quasi normal modes using the Born approximation for the scattering amplitude. Such a method has been used in literature. We suggest a novel, well defined method, to extract the large imaginary part of quasinormal modes via the Coulomb-like phase shift. Furthermore, we compare the numerically evaluated exact scattering amplitude with the Born one to find that the approximation is not very useful for the scattering of massless scalar, electromagnetic as well as gravitational waves from black holes.

On Stable Self-Similar Blow up for Equivariant Wave Maps: The Linearized Problem

We consider co-rotational wave maps from (3 + 1) Minkowski space into the three-sphere. This is an energy supercritical model which is known to exhibit finite time blow up via self-similar solutions. The ground state self-similar solution f 0 is known in closed form and based on numerics, it is supposed to describe the generic blow up behavior of the system. In this paper we develop a rigorous linear perturbation theory around f 0. This is an indispensable prerequisite for the study of nonlinear stability of the self-similar blow up which is conducted in the companion paper (Donninger in Commun. Pure Appl. Math., 64(8), 2011). In particular, we prove that f 0 is linearly stable if it is mode stable. Furthermore, concerning the mode stability problem, we prove new results that exclude the existence of unstable eigenvalues with large imaginary parts and also, with real parts larger than $${\frac{1}{2}}$$ . The remaining compact region is well-studied numerically and all available results strongly suggest the nonexistence of unstable modes.

Commentary: Arbitrarily polarized long-range surface-plasmon-polariton waves

designs, a NAD can plot the field ratio that is determined by the interference effect in the multilayer. 5 A design method that is based on NAD is utilized to design a multilayered system for the propagation of an LRSPP wave with a predetermined angle of incidence θi (tangential componentofwavevector)inaprism-couplingconfiguration.WhentheLRSPPwaveisexcited, the locus of the metal film in NAD is characterized as a projected locus that traverses a huge distance to the terminal point and intersects the real axis far from the origin, as shown in Fig. 1. The projected locus is caused by a multilayer under the metal film that can provide a large imaginary part of normalized admittance η LRSPP(s) as the start of the locus. The analysis of LRSPP in a NAD supports the design of a new configuration for exciting either s -o rppolarized LRSPP waves. Furthermore, a configuration for exciting s- and p-polarized LRSPP waves simultaneously can be realized using multilayers on both sides of the metal film. 6 In the traditional Kretschmann prism/metal film/air configuration, the p-polarized light can be applied to excite the SPP wave with lower initial admittance η SPP ip compared with η LRSPP ip and the associated locus of the metal film is plotted in Fig. 1. However, the s-polarized light cannot excite the SPP wave because the s-polarized admittance of air η SPP is always negative imaginary when total reflection occurs. The locus of a metal film would never let the locus point go toward the characteristic admittance of a prism to reach a coupling effect. However, the restriction on an s-polarized LRSPP wave can be overcome using a symmetrical cell that is equivalent to a single layer with designed characteristic admittance and phase thickness. The initial admittance can be provided using a periodic multilayer composed of several symmetric cells of the ABA or ABCBA type with designable equivalent characteristic admittance. 5 The initial admittances for s- and p-polarization are functions of the angle of incidence θi, the refractive index, and the thickness of each layer in the unit cell. To excite an LRSPP wave on a metal film at a designated angle of incidence θi, both positive imaginary admittances with values larger than the absolute imaginary part of characteristic admittance of metal can be designed by tuning parameters in

Semi-analytic results for quasi-normal frequencies

The last decade has seen considerable interest in the quasi-normal frequencies [QNFs] of black holes (and even wormholes), both asymptotically flat and with cosmological horizons. There is wide agreement that the QNFs are often of the form ω n = (offset) + in (gap), though some authors have encountered situations where this behaviour seems to fail. To get a better understanding of the general situation we consider a semi-analytic model based on a piecewise Eckart (Pöschl-Teller) potential, allowing for different heights and different rates of exponential fall off in the two asymptotic directions. This model is sufficiently general to capture and display key features of the black hole QNFs while simultaneously being analytically tractable, at least for asymptotically large imaginary parts of the QNFs. We shall derive an appropriate “quantization condition” for the asymptotic QNFs, and extract as much analytic information as possible. In particular, we shall explicitly verify that the (offset) + in (gap) behaviour is common but not universal, with this behaviour failing unless the ratio of rates of exponential falloff on the two sides of the potential is a rational number. (This is “common but not universal” in the sense that the rational numbers are dense in the reals.) We argue that this behaviour is likely to persist for black holes with cosmological horizons.

Spectra that are Newton after extension or translation

The appending of real numbers, and also conjugate pairs, to Newton spectra is studied to understand circumstances in which the Newton inequalities are preserved. Appending to a non-Newton spectrum to achieve the Newton inequalities is also studied. Finally the translations of Newton spectra that are Newton are also studied. A sample result is that any number of positive real numbers may be appended to a Newton spectrum, to retain the Newton property, when the Newton coefficients are positive, while any Newton spectrum may be made non-Newton by appending a conjugate pair with positive real part and sufficiently large imaginary part.

Analysis of data on proton-proton scattering in the energy range 100–1300 MeV

The results of a partial-wave analysis of data on proton-proton interaction in the energy range 100–1300 MeV are presented. The real parts of phase shifts were found for states of orbital angular momentum up to L = 9, while their imaginary parts were determined for states of orbital angular momentum up to L = 5. The sixth parameter of the S matrix was introduced in order to describe states of total angular momentum J = 2 and 4. The inelasticity thresholds were chosen individually for each state and were found to be substantially different from one another. The resulting solution was characterized by χ2 = 1.155 per point in the case where the number of experimental data was 12 841 and by a large imaginary part of the phase shift in the 3P2-wave state at the edge of the energy range. Special features of the interaction in orbital states are discussed along with the energy dependence of integrated amplitudes and amplitudes of the scattering matrix at zero angle.

A Multilevel Jacobi–Davidson Method for Polynomial PDE Eigenvalue Problems Arising in Plasma Physics

The simulation of drift instabilities in the plasma edge leads to cubic polynomial PDE eigenvalue problems with parameter dependent coefficients. The aim is to determine the wave number which leads to the maximum growth rate of the amplitude of the wave. This requires the solution of a large number of PDE eigenvalue problems. Since we are only interested in a smooth eigenfunction corresponding to the eigenvalue with largest imaginary part, the Jacobi–Davidson method can be applied. Unfortunately, a naive implementation of this method is much too expensive for the large number of problems that have to be solved. In this paper we will present a multilevel approach for the construction of an appropriate initial search space. We will also discuss the efficient solution of the correction equation, and we will show how optimal scaling helps to accelerate the convergence.

Optical polariton modes in a nanoscale semiconductor

Because of the existence of an exciton-biexciton transition inside an optically excited semiconductor, the dielectric constant is modified to be a wave-vector-dependent function. The spatial dispersion relation leads to three propagating modes of polariton, for which two additional boundary conditions will be required. In the vicinity of a resonance, the mathematical study shows that two modes among the three dominate, and the third wave with a large imaginary part can be neglected without affecting the essential physics. Based on the study, a criterion is developed for selecting the appropriate modes. Numerical results are presented for a thin semiconducting film.

A Numerically Stable Analysis Method for Complex Multilayer Waveguides Based on Modified Transfer-Matrix Equations

This paper discusses a method of analyzing complex one-dimensional multilayer waveguides in a numerically stable way. When a multilayer waveguide contains a quite thick layer or a layer whose refractive index has a very large real or imaginary part, conventional analysis methods tend to fail to find out its modes. In order to solve such a problem, the discussed analysis method is based on modified transfer matrix equations. The method consists of three steps. At the first step, two types of modified transfer matrix equations are used to derive two types of characteristic equations. At the next step, the iteration method comprised of initial root-finding and root-tracking is used to find the roots of the characteristic equations. Since the two characteristic equations have different features, the initial root-finding is carried out with one of them, and the root-tracking with the other. Finally, the field profiles of waveguide modes are calculated by using the found roots and employing a stable routine based on the modified transfer matrix equations. A few presented examples show that the discussed method works well. In addition to the method, it is shown that one of the characteristic equations can be expressed as an explicit form rather than a conventional recursive matrix relation because of the simple form of one of the transfer matrix equations.

Computational method for radar absorbing composite lattice grids

Composite lattice grids reinforced by glass fibers (GFRC) and carbon fibers (CFRC) filled with spongy materials can be designed as lightweight radar absorbing structures (RAS). In the present paper, a computational approach based on periodic moment method (PMM) has been developed to calculate reflection coefficients of radar absorbing composite lattice grids. Total reflection backing (TRB) is considered directly in our PMM program by treating it as a dielectric material with large imaginary part of permittivity. Two different mechanisms of reflection reduction for radar absorbing lattice grids are revealed. At low frequency, reflection coefficients increase with the volume fraction of the grid cell wall. At high frequency, several grating lobes propagate away from the doubly periodic plane, and reflection coefficients depend on both the cell wall volume fraction and interelement distance.

THREE-DIMENSIONAL METAMATERIAL MICROWAVE ABSORBERS COMPOSED OF COPLANAR MAGNETIC AND ELECTRIC RESONATORS

In this paper, a 3-dimensional metamaterial absorber operating at 11.8GHz was presented. The metamaterial absorber is composed of coplanar magnetic and electric resonators, with the latter in the center part of the former. By carefully adjusting structural dimensions of magnetic and electric resonators, absorbance per unit cell can reach up to 96% at 11.8GHz with a 6% FWHM (Full Width at Half Maximum). The full-wave simulations conflrmed nearly equal permeability and permittivity and large imaginary part of the refractive index at 11.8GHz and thus proved the efiectiveness of the proposed 3-dimensional metamaterial absorber for microwave applications.

Long-range surface polaritons in ultra-thin films of silicon

We present an experimental and theoretical study of the optical excitation of long-range surface polaritons supported by thin layers of amorphous silicon (a-Si). The large imaginary part of the dielectric constant of a-Si at visible and ultraviolet (UV) frequencies allows the excitation of surface polariton modes similar to long-range surface plasmon polaritons on metals. Propagation of these modes along considerable distances is possible because the electric field is largely excluded from the absorbing thin film. We show that by decreasing the thickness of the Si layer these excitations can be extended up to UV frequencies, opening the possibility to surface polariton UV optics compatible with standard Si technology.

Loss-induced transition of the Goos-Hänchen effect for metals and dielectrics

We report a unifying approach to the Goos-Hänchen (GH) shifts on external optical reflection for metals and dielectrics in particular for the case of high losses, that is for a large imaginary part of the dielectric constant. In this regime metals and dielectrics have a similar GH shift which is in contrast to the low-loss regime where the metallic and dielectric forms of the GH shift are very different. When going from the low-loss to the high-loss regime we find that metals show a much more prominent transition; we present a condition on the dielectric constant which characterizes this transition. We illustrate our theoretical analysis with a realistic example of seven lossy materials.

Comparing different approaches for solving optimizing models with significant nonlinearities

The property of saddle-path instability often arises in economic models derived from optimizing behavior by individual agents. In the case when underlying functional forms are nonlinear, it is likely that the stable and unstable arms defining the saddle-path dynamics will also have nonlinear properties. While closed-form analytic solutions can always be derived for linearized deterministic versions of these models, it will be necessary to use numerical techniques to derive the dynamic properties of calibrated versions of the associated nonlinear models. There are a range of different approaches by which it is possible to solve the dynamics of nonlinear models with the saddle-path property. Alternative solution methods using numerical techniques are discussed. In all cases, the success of the solution can be assessed by evaluating whether or not the chosen solution gives a time-path for each variable that goes from the chosen initial condition to (a small neighborhood of) the steady-state. Unfortunately, however, there is generally no way of evaluating whether intermediate points on the path between the initial condition and the steady-state are close approximations to the “true” solution. In order to evaluate the “goodness-of-fit” along the entire dynamic path, it is necessary to have a closed-form solution of the entire solution path. This can only be achieved in special cases. For one special case, we demonstrate how it is possible to construct a model with complex-valued eigenvalues with large imaginary parts and hence significant cycles. Such a model can be employed as a benchmark to compare the properties of solutions derived using a range of solution algorithms.

Enhanced crystal-field splitting and orbital-selective coherence induced by strong correlations inV2O3

We present a study of the paramagnetic metallic and insulating phases of vanadium sesquioxide by means of the $N$th order muffin-tin orbital implementation of density functional theory combined with dynamical mean-field theory. The transition is shown to be driven by a correlation-induced enhancement of the crystal field splitting within the $t_{2g}$ manifold, which results in a suppression of the hybridization between the $a_{1g}$ and $e_g^{\pi}$ bands. We discuss the changes in the effective quasi-particle band structure caused by the correlations and the corresponding self-energies. At temperatures of about 400 K we find the $a_{1g}$ orbitals to display coherent quasi-particle behavior, while a large imaginary part of the self-energy and broad features in the spectral function indicate that the $e_g^{\pi}$ orbitals are still far above their coherence temperature. The local spectral functions are in excellent agreement with recent bulk sensitive photoemission data. Finally, we also make a prediction for angle-resolved photoemission experiments by calculating momentum-resolved spectral functions.

Scaling of the microwave magneto-impedance in Tl 2Ba 2CaCu 2O 8+ δ thin films

We present measurements of the magnetic field-induced microwave complex resistivity changes at 47 GHz in Tl 2Ba 2CaCu 2O 8+ δ (TBCCO) thin films, in the ranges 58 K < T < T c and 0 < μ 0 H < 0.8 T. The large imaginary part Δ ρ 2( H) points to strong elastic response, but the data are not easily reconciled with a rigid vortex model. We find that, over a wide range of temperatures, all the pairs of curves Δ ρ 1( H) and Δ ρ 2( H) can be collapsed on a pair of scaling curves Δ ρ 1[ H/ H ∗( T)], Δ ρ 2[ H/ H ∗( T)], with the same scaling field H ∗( T). We argue that H ∗( T) is related to the loss of vortex rigidity due to a vortex transformation.

Multiloop PI/PID control system improvement via adjusting the dominant pole or the peak amplitude ratio

Multiloop control systems designed by simple methods such as the biggest log-modulus tuning may be too sluggish or too oscillatory. Sluggish responses are due to dominant poles near the origin, and oscillatory responses are due to dominant poles with large imaginary parts or large peak amplitude ratios in the closed-loop systems. A method that adjusts such dominant poles and amplitude ratios through the continuation method is proposed. Several examples show that the improved multiloop control system can be obtained without resorting to complicated design methods.