Nonzero Imaginary Part Research Articles

On preconditioned generalized shift-splitting iteration methods for saddle point problems

We study a preconditioned generalized shift-splitting iteration method for solving saddle point problems. The unconditional convergence theory of the preconditioned generalized shift-splitting iteration method is established. When the splitting matrix is used as a preconditioner, we analyze eigenvalue distribution of the preconditioned saddle point matrix. It is proved that complex eigenvalues having nonzero imaginary parts of the preconditioned matrix are located in an intersection of two circles and the real parts of all eigenvalues of the preconditioned matrix are located in a positive interval. Numerical experiments are used to verify our theoretical results and illustrate effectiveness of the proposed iteration method and the corresponding splitting preconditioner.

Eigenvalue bounds of the shift-splitting preconditioned singular nonsymmetric saddle-point matrices

For singular nonsymmetric saddle-point problems, a shift-splitting preconditioner was studied in (Appl. Math. Comput. 269:947-955, 2015). To further show the efficiency of the shift-splitting preconditioner, we provide eigenvalue bounds for the nonzero eigenvalues of the shift-splitting preconditioned singular nonsymmetric saddle-point matrices. For real parts of the eigenvalues, the bound is provided by valid inequalities. For eigenvalues having nonzero imaginary parts, the bound is a combination of two inequalities proving their clustering in a confined region of the complex plane. Finally, two numerical examples are presented to verify the theoretical results.

Spectral analysis of the generalized shift-splitting preconditioned saddle point problem

A shift-splitting preconditioner was recently proposed for saddle point problems, which is based on a generalized shift-splitting of the saddle point matrix. We provide a new analysis to prove that the corresponding shift-splitting iteration method is unconditional convergent. To further show the efficiency of the shift-splitting preconditioner, the eigenvalue distribution of the shift-splitting preconditioned saddle point matrix is investigated. We show that all eigenvalues having nonzero imaginary parts are located in an intersection of two circles and all real eigenvalues are located in a positive interval. Numerical examples are given to confirm our theoretical results.

Sparse-Iteration 4D CORDIC Algorithms for Multiplying Quaternions

Novel 4D CORDIC algorithms and hardware architecture for multiplying quaternions are presented, aimed at constant-coefficient multipliers. In our solution, microrotations are multiplications by hypercomplex numbers with only one non-zero imaginary part. Such transformations can be described using sparse matrices and implemented using less hardware resources than the iteration of the known quaternion CORDIC algorithm. Chip area can be saved because sparse microrotations can be computed using a permutation network and two-operand adders, without four-operand additions, which are necessary in the known solution. The obtainable area savings depend on the implementation technology and the architectures of adders and bit shifters but can be as large as 45 and 25 percent for ASIC and FPGA circuits, respectively. The circuitry simplifications come at the cost of slowing down computations: our approach improves the area-delay product, and a single sparse iteration can be executed up to 15 percent faster than the conventional one, but more microrotations are usually necessary to achieve the required accuracy. On the other hand, a sparse 4D iteration can be identified with microrotations of two 2D CORDICs which work in parallel. This makes convergence analysis easier than in the previous 4D solution, as it is sufficient to consider only two dimensions.

Silicon electro-optic modulator based on an ITO-integrated tunable directional coupler

Indium-tin-oxide (ITO) has attracted great attention because of its electrically-induced epsilon-near-zero (ENZ) characteristics, which has allowed us to develop electro-absorption optical modulators. As an extended application of ITO in a silicon optical modulator, we propose a silicon electro-optic modulator based on an ITO-integrated tunable directional coupler. An ITO block placed at the center of the two-core silicon directional coupler plays a key role in attenuating the optical power of the guided modes. Strong confinement of the optical field at the ENZ layer of ITO with the nonzero imaginary part of ITO’s permittivity let the guiding light experience high absorption as it propagates along the directional coupler. Numerical simulations reveal that the highest modulation depth of 5 dB is achievable at the ENZ region of ITO at a 1.55 μm wavelength. We can modulate the optical signals on an entire C-band ranging from a 1.530 to 1.565 μm wavelength with an on/off extinction ratio of larger than 4.6 dB.

Dip or nothingness of a Higgs resonance from the interference with a complex phase

We show that new resonance shapes---a pure dip, nothingness, and an enhanced pure peak---can be produced from the interference between the resonance and continuum with a relative phase. Production conditions of those new shapes are derived based on a general parametrization of the interference. The narrow width approximation is modified to work with the nonzero imaginary part of interference, and the correction factor can characterize the resonance shape. We demonstrate that the new resonance shapes of heavy Higgs bosons, ${H}^{0}$ and ${A}^{0}$ in the type-II aligned two Higgs doublet model, generally show up in $gg\ensuremath{\rightarrow}{H}^{0}/{A}^{0}\ensuremath{\rightarrow}t\overline{t}$ as well as $b\overline{b}$ and $\ensuremath{\gamma}\ensuremath{\gamma}$ channels. The pure ${A}^{0}$ resonance dip in the $t\overline{t}$ channel is a particularly interesting signal as it can be probed by the current search techniques that do not even take into account interferences; the high-luminosity LHC 14 TeV can perhaps probe a large part of its parameter space.

Quasiparticle scattering rate in a strongly polarized Fermi mixture

We analyse the scattering rate of an impurity atom in a Fermi sea as a function of momentum and temperature in the BCS-BEC crossover. The cross section is calculated using a microscopic multichannel theory for the Feshbach resonance scattering, including finite range and medium effects. We show that pair correlations significantly increase the cross section for strong interactions close to the unitarity regime. These pair correlations give rise to a molecule pole of the cross section at negative energy on the BEC side of the resonance, which smoothly evolves into a resonance at positive scattering energy with a non-zero imaginary part on the BCS side. The resonance is the analogue of superfluid pairing for the corresponding population balanced system. Using Fermi liquid theory, we then show that the low temperature scattering rate of the impurity atom is significantly increased due to these pair correlations for low momenta. We demonstrate that finite range and mass imbalance effects are significant for the experimentally relevant $^6$Li-$^{40}$K mixture, and we finally discuss how the scattering rate can be measured using radio-frequency spectroscopy and Bose-Fermi mixtures.

Nonlinear multicore waveguiding structures with balanced gain and loss

We study existence, stability, and dynamics of linear and nonlinear stationary modes propagating in radially symmetric multicore waveguides with balanced gain and loss. We demonstrate that, in general, the system can be reduced to an effective $\mathcal{PT}\ensuremath{-}\text{symmetric}$ dimer with asymmetric coupling. In the linear case, we find that there exist two modes with real propagation constants before an onset of the $\mathcal{PT}\ensuremath{-}\text{symmetry}$ breaking while other modes have always the propagation constants with nonzero imaginary parts. This leads to a stable (unstable) propagation of the modes when gain is localized in the core (ring) of the waveguiding structure. In the case of nonlinear response, we show that an interplay between nonlinearity, gain, and loss induces a high degree of instability, with only small windows in the parameter space where quasistable propagation is observed. We propose a novel stabilization mechanism based on a periodic modulation of both gain and loss along the propagation direction that allows bounded light propagation in the multicore waveguiding structures.

Hierarchical spin glasses in a magnetic field: A renormalization-group study

By using renormalization-group (RG) methods, we study a non-mean-field model of a spin glass built on a hierarchical lattice, the hierarchical Edwards-Anderson model in a magnetic field. We investigate the spin-glass transition in a field by studying the existence of a stable critical RG fixed point (FP) with perturbation theory. In the parameter region where the model has a mean-field behavior - corresponding to $d \geq 4$ for a $d$-dimensional Ising model - we find a stable FP that corresponds to a spin-glass transition in a field. In the non-mean-field parameter region the FP above is unstable, and we determined exactly all other FPs: to our knowledge, this is the first time that all perturbative FPs for the full set of RG equations of a spin glass in a field have been characterized in the non-mean-field region. We find that all potentially stable FPs in the non-mean-field region have a nonzero imaginary part: this constitutes, to the best of our knowledge, the first demonstration for a spin glass in a field that there is no perturbative FP corresponding to a spin-glass transition in the non-mean-field region. Finally, we discuss the possible interpretations of this result, such as the absence of a phase transition in a field, or the existence of a transition associated with a non-perturbative FP.

Electromagnetic effects in the pion dispersion relation at finite temperature

We investigate the charged-neutral pion self-energy difference at finite temperature. Within Chiral Perturbation Theory we extend previous analysis in the chiral and soft pion limits. Our analysis with physical masses leads to additional contributions for temperatures typical of a meson gas, including a momentum-dependent function for the self energy. In addition, a nonzero imaginary part arises, which we define consistently in the Coulomb gauge and comes from an infrared enhanced contribution due to thermal bath photons. For distributions typical of a heavy-ion meson gas, the charged and neutral pion masses and their difference depend on temperature through slowly increasing functions. Chiral symmetry restoration is ultimately responsible for keeping the corrections smooth and compatible with observed pion spectra. We study phenomenological effects related to the electromagnetic damping leading to corrections for transport coefficients and neutral-charged mean free times differences. An important aspect is the connection with chiral symmetry restoration through the relation of the pion mass difference with the vector-axial spectral function difference, which holds at $T=0$ due to a sum rule in the chiral and soft pion limits. We analyze the modifications of that sum rule including nonzero pion masses and temperature, up to order $T^2$ and $M_\pi^2$. Both effects produce terms making the pion mass difference grow against chiral-restoring decreasing contributions. Finally, we analyze the corrections to the previous ChPT and sum rule results within the resonance saturation framework at finite temperature, including explicitly $\rho$ and $a_1$ exchanges. Our results show that the ChPT result is robust at low and intermediate temperatures, the leading corrections being of order $T^2 M_\pi^2/M_R^2$ with $M_R$ the involved resonance masses.

Tests for real and complex unit roots in vector autoregressive models

The article proposes new tests for the number of real and complex unit roots in vector autoregressive models. The tests are based on the eigenvalues of the sample companion matrix. The limiting distributions of the eigenvalues converging to the unit eigenvalues turn out to be of a non-standard form and expressible in terms of Brownian motions. The tests are defined such that the null distributions related to eigenvalues ± 1 are the same. The tests for the unit eigenvalues with nonzero imaginary part are defined independently of the angular frequency. When the tests are adjusted for deterministic terms, the null distributions usually change. Critical values are tabulated via simulations. Also some simulation based finite sample properties are presented together with comparisons with corresponding likelihood ratio tests. The relation of the unit roots to cointegration is discussed. An empirical example is provided to show how to use the test with real data.

Towards many-dimensional real-time quantum theory for heavy-particle dynamics. II. Beyond semiclassics by quantum smoothing of the singularity in quantum-classical correspondence

A theory of many-dimensional real-time quantum dynamics is studied in terms of action decomposed function (ADF), a class of quantum wave function. In the preceding companion paper [S. Takahashi and K. Takatsuka, Phys. Rev. A 89, 012108 (2014)], we showed that semiclassical dynamics for ADF in the Lagrange picture of phase flow can be described in terms of what we call deviation determinant and associated quantum phases without use of the stability matrix. Consequently, the Hessian of the involved potential functions is not required in this formalism. This paper is devoted to an analysis of the mechanism of quantum diffusion (quantum smoothing) that removes the singularity inherent in the semiclassical ADF: We derive a Lorentzian form for the amplitude factor of ADF. The real part of its denominator comes from the deviation determinant, while the imaginary part reflects quantum diffusion and is proportional to the Planck constant. The presence of the nonzero imaginary part smooths out the singularity and removes the divergence. Besides, this imaginary part can be obtained through a Wronskian relation with the deviation vectors, which can be solved rather easily at each space-time point on a classical trajectory. A number of theoretical advantages of the Lorentzian form and the Wronskian relation are illustrated theoretically and numerically. It turns out that there is no essential difficulty in applications to many-dimensional heavy-particle systems such as molecules. The theory is examined with stringent numerical tests.

Physical curl forces: dipole dynamics near optical vortices

The force on a particle with complex electric polarizability is known to be not derivable from a potential, so its curl is non-zero. This ‘curl force’ is studied in detail for motion near an anisotropic optical vortex of arbitrary strength. Fundamental questions are raised by the fact that although the curl force requires the polarizability to have a non-zero imaginary part, reflecting absorption or scattering (‘dissipation’) in the internal dipole dynamics, the particle motion that it generates is non-dissipative (volume-preserving in the position-velocity state space).

Optical determination of thick graphene layer number based on surface plasmon resonance

An optical method of measuring the number of layers in a graphene sample is formulated and compared with the conventional surface plasmon resonance (SPR) detection scheme, the latter being appropriate only for a very few graphene layers. Numerical results based on transfer-matrix method support that an alternative method, wherein the SPR substrate includes a dielectric overlayer, is feasible over a wide range of graphene layer numbers. While the multilayer graphene may lead to a broad and shallow SPR curve owing to the nonzero imaginary part in its relative permittivity, the dielectric overlayer makes the resonant surface plasmons less affected by graphene, resulting in a strong and deep absorption band at resonance. Linear regression analysis shows that the measurable graphene layer number can be as high as 50.

Functional and effective connectivity in subthalamic local field potential recordings of patients with Parkinson’s disease

In Parkinson’s disease (PD) levodopa-associated changes in the power and long-range temporal correlations of beta oscillations have been demonstrated, yet the presence and modulation of genuine connectivity in local field potentials (LFP) recorded from the subthalamic nucleus (STN) remains an open question. The present study investigated LFP recorded bilaterally from the STN at wakeful rest in ten patients with PD after overnight withdrawal of levodopa (OFF) and after a single dose levodopa administration (ON). We utilized connectivity measures being insensitive to volume conduction (functional connectivity: non-zero imaginary part of coherency; effective connectivity: phase–slope index). We demonstrated the presence of neuronal interactions in the frequency range of 10–30Hz in STN–LFP without a preferential directionality of interactions between different contacts along the electrode tracks. While the direction of neuronal interactions per se was preserved after levodopa administration, functional connectivity and the ventral–dorsal information flow were modulated by medication. The OFF–ON differences in functional connectivity were correlated with the levodopa-induced improvement in clinical Unified Parkinson’s Disease Rating Scale scores. We hypothesize that regional neuronal interactions, as reflected in STN–LFP connectivity, might represent a basis for the intra-nuclear spatial specificity of deep brain stimulation. Moreover, our results suggest the potential use of volume conduction-insensitive measures of connectivity in STN–LFP as a marker of clinical motor symptoms in PD.

Spectral approximation of quadratic operator polynomials arising in photonic band structure calculations

Galerkin spectral approximation theory for non-self-adjoint quadratic operator polynomials with periodic coefficients is considered. The main applications are complex band structure calculations in metallic photonic crystals, periodic waveguides, and metamaterials. We show that the spectrum of the considered operator polynomials consists of isolated eigenvalues of finite multiplicity with a nonzero imaginary part. The spectral problem is equivalent to a non-compact block operator matrix and norm convergence is shown for a block operator matrix having the same generalized eigenvectors as the original operator. Convergence rates of finite element discretizations are considered and numerical experiments with the $$p$$ p -version and the $$h$$ h -version of the finite element method confirm the theoretical convergence rates.

Consensus and performance optimisation of multi‐agent systems with position‐only information via impulsive control

In this study, the consensus problem of second-order multi-agent systems (MAS) with position-only information is studied. Allowable sampling period for which second-order consensus can be achieved is obtained with two impulsive consensus algorithms. It is shown that if there is at least one eigenvalue of the Laplcian matrix with a non-zero imaginary part, consensus cannot be achieved for sufficiently small or large impulsive periods for both algorithms. Furthermore, the convergence performance of the MAS is optimised. Convergence speed, asymptotical decay factor and per-step decay factor of the error energy are utilised to investigate the convergence performance, and the relationship among impulsive period, topology structure and convergence performance is derived. Finally, numerical examples are given to validate our theoretical results.

Band structure of photonic crystal with dispersive and lossy materials using Dirichlet-to-Neumann wave vector eigen equation method

We present the Dirichlet-to-Neumann wave vector eigen equation (DtN-WVEE) method to calculate the band structure of photonic crystal (PhC) with dispersive and lossy materials, and then use finite-difference time-domain method to verify the validity of the results of the DtN-WVEE method. Since the material is lossy with a non-zero imaginary part of epsilon, the wave vector of PhC band structure also has a non-zero imaginary part which results in the reduction of the intensity of the mode. We further present the band structure which gives the relationship among the normalized frequency, the real part of the wave vector, and the imaginary part of the wave vector. To our knowledge, the band structure of lossy PhC has not been well studied yet, and our results can be useful in designing the devices using surface plasmon polariton effect.

Analysis of the scattering by an unbounded rough surface

This paper is concerned with the mathematical analysis of the solution for the wave propagation from the scattering by an unbounded penetrable rough surface. Throughout, the wavenumber is assumed to have a nonzero imaginary part that accounts for the energy absorption. The scattering problem is modeled as a boundary value problem governed by the Helmholtz equation with transparent boundary conditions proposed on plane surfaces confining the scattering surface. The existence and uniqueness of the weak solution for the model problem are established by using a variational approach. Furthermore, the scattering problem is investigated for the case when the scattering profile is a sufficiently small and smooth deformation of a plane surface. Under this assumption, the problem is equivalently formulated into a set of two‐point boundary value problems in the frequency domain, and the analytical solution, in the form of an infinite series, is deduced by using a boundary perturbation technique combined with the transformed field expansion approach. Copyright © 2012 John Wiley & Sons, Ltd.

Guiding structures with multiply connected cross sections: Evolution of propagation in external fields at complex Robin parameters

Properties of the two-dimensional ring and three-dimensional infinitely long straight hollow waveguide with unit width and inner radius ρ0 in the superposition of the longitudinal uniform magnetic field B and the Aharonov–Bohm flux are analyzed within the framework of the scalar Helmholtz equation under the assumption that the Robin boundary conditions at the inner and outer confining walls contain extrapolation lengths Λin and Λout, respectively, with nonzero imaginary parts. As a result of this complexity, the self-adjointness of the Hamiltonian is lost, its eigenvalues E, in general, become complex too and the discrete bound states of the annulus, which are characteristic for the real Λ, turn into the corresponding quasibound states with their lifetime defined by the imaginary parts Ei of the eigenenergies while the current along the wire exponentially amplifies/attenuates with the distance depending on the sign of Ei. It is shown that, compared to the disk geometry, the annulus opens up additional possibilities of varying magnetization and currents by tuning imaginary components of the Robin parameters on each confining circumference; in particular, the possibility of restoring a lossless longitudinal flux by zeroing Ei is discussed from mathematical and physical points of view. For each ρ0, the energy E turns real under a special correlation between the imaginary parts of Λin and Λout with the opposite signs being what physically corresponds to the equal transverse fluxes through the inner and outer interfaces of the annulus. In the asymptotic case of the very large radius, ρ0→∞, simple expressions are derived and applied to the analysis of the dependence of the real energy E on Λin and Λout. New features also emerge in the magnetic field influence; for example, if, for the quantum disk, the imaginary energy Ei is quenched by the strong intensities B, then for the annulus this takes place only when the inner Robin distance Λin is real; otherwise, it almost quadratically depends on B with the corresponding enhancement of the reactive scattering. The closely related problem of the hole in the otherwise uniform medium is also addressed for the real and complex extrapolation lengths with the emphasis on the comparative analysis with its dot counterpart.