Complex Conjugate Pairs Research Articles

Algebraic fundamental groups of fake projective planes

Fundamental groups of fake projective planes fall into fifty distinct isomorphism classes, one for each complex conjugate pair. We prove that this is not the case for their algebraic fundamental groups: there are only forty-six isomorphism classes. We show that there are four pairs of complex conjugate pairs of fake projective planes that are \operatorname{Aut}(\mathbb{C}) -equivalent and hence have mutually isomorphic algebraic fundamental groups. All other pairs of algebraic fundamental groups are shown to be distinct through explicit finite étale covers. As a by-product, this provides the first examples of commensurable but nonisomorphic lattices in a rank one semisimple Lie group that have isomorphic profinite completions.

ВЛИЯНИЕ ТОЧЕК ПОВОРОТА НА ЗАДЕРЖКУ РЕШЕНИЯ ВБЛИЗИ НЕУСТОЙЧИВОГО ПОЛОЖЕНИЯ РАВНОВЕСИЯ

This paper considers a system of equations consisting of three first-order equations of fast variables and one equation of a slow variable. The first approximation matrix of the system of fast variables has different eigenvalues, with one pair of complex conjugates. The eigenvalues have zeros and they are turning points. The system has an equilibrium position, and the stability of the equilibrium position is lost at a certain value of the slow variable. The problem is set to study the delay in solving a system of fast variables near an unstable equilibrium position and the influence of turning points on the delay. The delay of the solution near an unstable position is proven and the influence of turning points is clarified. The proof used geometric constructions using the level line of harmonic functions, methods of successive approximations, Laplace, stationary phase, integration by parts and comparison of infinitesimals.

Medium induced mixing, spatial modulations, and critical modes in QCD

The mixing between the chiral condensate and the density in hot and dense quantum chromodynamics (QCD) matter is familiar. We show that the mixing relevant for the ground state is considerably more extensive, and in particular also involves gluonic degrees of freedom. As a result, the Hessian of the QCD effective action is non-Hermitian, but retains a symmetry under combined charge and complex conjugation. This can lead to complex-conjugate pairs of eigenvalues of this Hessian, signaling regimes with spatially modulated correlations. Furthermore, based on the analytic structure of the quark determinant at a chiral critical point, we demonstrate that the corresponding massless critical mode is composed of the chiral condensate, the density and the Polyakov loops. Due to an avoided crossing, the critical mode turns out to be disconnected from the chiral condensate in vacuum. We present general arguments for all these features and illustrate them through explicit model calculations. Published by the American Physical Society 2024

Extreme Superposition: High-Order Fundamental Rogue Waves in the Far-Field Regime

This Memoir concerns fundamental rogue-wave solutions of the focusing nonlinear Schrödinger equation in the limit that the order of the rogue wave is large and the independent variables ( x , t ) (x,t) are proportional to the order (the far-field limit). We first formulate a Riemann-Hilbert representation of these solutions that allows the order to vary continuously rather than by integer increments. The intermediate solutions in this continuous family include also soliton solutions for zero boundary conditions spectrally encoded by a single complex-conjugate pair of poles of arbitrary order, as well as other solutions having nonzero boundary conditions matching those of the rogue waves albeit with far slower decay as x → ± ∞ x\to \pm \infty . The large-order far-field asymptotic behavior of the solution depends on which of three disjoint regions C \mathcal {C} (the “channels”), S \mathcal {S} (the “shelves”), and E \mathcal {E} (the “exterior domain”) contains the rescaled variables. On the region C \mathcal {C} , the amplitude is small and the solution is highly oscillatory, while on the region S \mathcal {S} , the solution is approximated by a modulated plane wave with a highly oscillatory correction term. The asymptotic behavior on these two domains is the same for all continuous orders. Assuming that the order belongs to the discrete sequence characteristic of rogue-wave solutions, the asymptotic behavior of the solution on the region E \mathcal {E} resembles that on S \mathcal {S} but without the oscillatory correction term. Solutions of other continuous orders behave quite differently on E \mathcal {E} .

Plug-and-play analytical paradigm for the scattering of plane waves by ‘layer-cake’ periodic systems

We investigate the scattering of scalar plane waves in two dimensions by a heterogeneous layer that is periodic in the direction parallel to its boundary. On describing the layer as a union of periodic laminae, we develop a solution of the scattering problem by merging the concept of propagator matrices and that of Bloch eigenstates featured by the unit cell of each lamina. The featured Bloch eigenstates are obtained by solving the quadratic eigenvalue problem (QEVP) that seeks a complex-valued wavenumber normal to the layer boundary given (i) the excitation frequency, and (ii) real-valued wavenumber parallel to the boundary—that is preserved throughout the system. Spectral analysis of the QEVP reveals sufficient conditions for discreteness of the eigenvalue spectrum and the fact that all eigenvalues come in complex-conjugate pairs. By deploying the factorization afforded by the propagator matrix approach, we demonstrate that the contribution of individual eigenvalues (and so eigenmodes) to the solution diminishes exponentially with absolute value of their imaginary part, which then forms a rational basis for truncation of the factorized Bloch (FB)-wave solution. The proposed methodology caters for the optimal design of rainbow traps, energy harvesters and metasurfaces, whose potency to manipulate waves is decided not only by the individual dispersion characteristics of the component laminae, but also by ordering and generally fitting of the latter into a composite layer. Using the FB approach, evaluation of trial configurations—as generated by the permutation and window translation/stretching of the component laminae—can be accelerated by decades.

Enhancing computation performance of rational approximation for frequency-dependent network equivalents with parallelism and complex vector fitting

This work examines two strategies for enhancing the rational approximation of Frequency-Dependent Network Equivalents (FDNE) using an 8-port FDNE featuring a frequency response marked by numerous peaks and valleys. Firstly, we employ Complex Vector Fitting (CVF), an alternative to the Vector Fitting (VF). CVF eliminates the constraint of complex conjugate pairs and was originally conceived for modeling baseband equivalents through scattering parameters. The implications of CVF for admittance (or impedance) matrix synthesis have not yet been previously reported in specialized literature. To enhance code performance and remove dependence on commercial software such as MATLAB®, VF and CVF were implemented in the C-language, utilizing a low-level linear algebra package and exploiting parallelism. We evaluated performance by varying the model order, number of ports, and frequency samples. The results confirm the feasibility of our approach, prompting a more in-depth exploration of the potential benefits regarding FDNE realization.

Spectral clustering of Markov chain transition matrices with complex eigenvalues

The Robust Perron Cluster Analysis (PCCA+) has become a popular spectral clustering algorithm for coarse-graining transition matrices of nearly decomposable Markov chains with transition states. Originally developed for reversible Markov chains, the algorithm only worked for transition matrices with real eigenvalues. In this paper, we therefore extend the theoretical framework of PCCA+ to Markov chains with a complex eigen-decomposition. We show that by replacing a complex conjugate pair of eigenvectors by their real and imaginary components, a real representation of the same subspace is obtained, which is suitable for the cluster analysis. We show that our approach leads to the same results as the generalized PCCA+ (GPCCA), which replaces the complex eigen-decomposition by a conceptually more difficult real Schur decomposition. We apply the method on non-reversible Markov chains, including circular chains, and demonstrate its efficiency compared to GPCCA. The experiments are performed in the Matlab programming language and codes are provided.

Experimental study of the nonlinear distortion of non-reciprocal transmission in nonlinear parity-time symmetric LC resonators

At the broken phase of a parity-time (PT)-symmetric dimer where the coupling is weak, the eigenfrequencies are complex conjugate pairs with non-vanishing real parts, leading to oscillation magnitudes with an exponentially growing mode and an exponentially decaying mode. If the large oscillation magnitudes are clamped due to the nonlinear gain of the PT-symmetric dimer, the exponentially growing mode eventually enters the stable oscillations. It was recently demonstrated that such a phenomenon can be utilized for non-reciprocal transmission. The distortion induced by nonlinearity is critical for the non-reciprocal transmission. Here, we experimentally explore the nonlinear distortion in PT-symmetric inductor–capacitor resonators by utilizing discrete components on a printed circuit board. It demonstrates that the IIP3 (the output-referred third-order intercept point) can achieve as high as 38.7 dBm at the frequency of 14.5 MHz corresponding to the maximum forward transmission. The noise figure of the system is measured to be about 11.25 dB.

Efficient Recursive Convolution Schemes for the Accurate FDTD Modeling of Graphene Interband Contribution via Complex Conjugate Pairs

The accurate time-domain numerical modeling of graphene surface conductivity due to the interband contribution is presented in this paper. Initially, the lower frequency limit for the interband conductivity inclusion is studied, highlighting that even in the far-infrared regime it should not be ignored. Then, a precise vector-fitting technique is utilized to decompose the analytical conductivity function into complex conjugate pairs. Every pair is connected to a complex surface current, while the frequency dispersion is evaluated via a proper recursive convolution scheme. Finally, a straightforward algebraic manipulation is conducted to eliminate the complex terms from the update equations of the electromagnetic components, concerning the finite-difference time-domain (FDTD) algorithm. The precision and efficiency of the proposed methodology are thoroughly validated by comparing the numerically extracted surface wave propagation characteristics to those obtained in terms of analytical expressions.

Extension of the Goldstone and the Englert-Brout-Higgs mechanisms to non-Hermitian theories

We discuss the extension of the Goldstone and Englert-Brout-Higgs mechanisms to non-Hermitian Hamiltonians that possess an antilinear PT symmetry. We study a model due to Alexandre, Ellis, Millington and Seynaeve and show that for the spontaneous breakdown of a continuous global symmetry we obtain a massless Goldstone boson in all three of the antilinear symmetry realizations: eigenvalues real, eigenvalues in complex conjugate pairs, and eigenvalues real but eigenvectors incomplete. In this last case we show that it is possible for the Goldstone boson mode to be a zero-norm state. For the breakdown of a continuous local symmetry the gauge boson acquires a non-zero mass by the Englert-Brout-Higgs mechanism in all realizations of the antilinear symmetry, except the one where the Goldstone boson itself has zero norm, in which case, and despite the fact that the continuous local symmetry has been spontaneously broken, the gauge boson remains massless.

Degeneracies and symmetry breaking in pseudo-Hermitian matrices

Real eigenvalues of pseudo-Hermitian matrices, such as real matrices and $\mathcal{PT}\ensuremath{-}\mathrm{symmetric}$ matrices, frequently split into complex conjugate pairs. This is accompanied by the breaking of certain symmetries of the eigenvectors and, typically, also a drastic change in the behavior of the system. In this paper, we classify the eigenspace of pseudo-Hermitian matrices and show that such symmetry breaking occurs if and only if eigenvalues of opposite kinds collide on the real axis of the complex eigenvalue plane. This enables a classification of the disconnected regions in parameter space where all eigenvalues are real---which correspond, physically, to the stable phases of the system. These disconnected regions are surrounded by exceptional surfaces, which comprise all the real-valued exceptional points of pseudo-Hermitian matrices. The exceptional surfaces, together with the diabolic points created by their intersections, comprise all points of pseudo-Hermiticity breaking. In particular, this clarifies that the degeneracy involved in symmetry breaking is not necessarily an exceptional point. We also discuss how our study relates to conserved quantities and derive the conditions for when degeneracies caused by external symmetries are susceptible to thresholdless pseudo-Hermiticity breaking. We illustrate our results with examples from photonics, condensed matter physics, and mechanics.

Ultralinear 140-GHz FMCW signal generation with optical parametric wideband frequency modulation enabling 1-mm range resolution.

We demonstrate ultralinear and ultrawideband frequency-modulated continuous-wave (FMCW) signal generation using an optical parametric wideband frequency modulation (OPWBFM) method. The OPWBFM method optically expands the bandwidths of FMCW signals beyond the electrical bandwidths of optical modulators via a cascaded four-wave mixing (FWM) process. Compared to the conventional direct modulation approach, the OPWBFM method simultaneously achieves high linearity and a short measurement time of the frequency sweep. On the other hand, it is also known that the OPWBFM method expands the phase noise of idlers as well as their bandwidths if an input conjugate pair has different phase noise. To avoid this phase noise expansion, it is crucial to synchronize the phase of an input complex conjugate pair of an FMCW signal using an optical frequency comb. For demonstration, we successfully generated an ultralinear 140-GHz FMCW signal by using the OPWBFM method. Moreover, we employ a frequency comb in the conjugate pair generation process, leading to the mitigation of phase noise expansion. By using a 140-GHz FMCW signal, we achieve a range resolution of ∼1 mm through fiber-based distance measurement. The results show the feasibility of an ultralinear and ultrawideband FMCW system with a sufficiently short measurement time.

Multitude of exceptional points in van der Waals magnets

Several works have recently addressed the emergence of exceptional points (EPs), i.e., degeneracies of non-Hermitian Hamiltonians, in the long-wavelength dynamics of coupled magnetic systems. Here, by focusing on the driven magnetization dynamics of a van der Waals ferromagnetic bilayer, we show that exceptional points can appear over extended portions of the first Brillouin zone as well. Furthermore, we demonstrate that the effective non-Hermitian magnon Hamiltonian, whose eigenvalues are purely real or come in complex conjugate pairs, respects an unusual wavevector-dependent pseudo-Hermiticity. Finally, for both armchair and zigzag nanoribbon geometries, we discuss both the complex and purely real spectra of the topological edge states and their experimental implications.

High Crosswind-Tolerance Airplane by Adjusting Dihedral Angle and Vertical Tail Volume

The present authors proposed the “quasi-neutral dihedral-effect and directional-stability” (QNDD) airplane, which is a modified neutral dihedral-effect and directional-stability (NDD) airplane for its realization. The QNDD airplane provides a superior tolerance to crosswinds compared to a conventional one by modifying geometric properties: dihedral angle of a main wing and vertical tail volume to achieve small values of and , which results in a small effect of crosswind gust on an airplane’s attitude. While the QNDD airplane yields small and , the NDD airplane yields and equal to zero. When the changes of and due to the wing deformations are considered, the NDD airplane is ideal, while the QNDD airplane is more practical. The QNDD airplane had four real eigenvalues rather than classically two reals and one pair of complex conjugates. The eigenvalues seen in a conventional airplane as a Dutch roll mode were seen to be reals in a QNDD airplane, and the mode was overdamped, which is different from a set of eigenvalues of a conventional one. From the Bode diagrams and unsteady simulations with a discrete gust applied as a crosswind, it was confirmed that the QNDD airplane had a higher tolerance to the crosswind.

Superharmonic instability of stokes waves

AbstractA stability of nearly limiting Stokes waves to superharmonic perturbations is considered numerically in approximation of an infinite depth. Investigation of the stability properties can give one an insight into the evolution of the Stokes wave. The new, previously inaccessible branches of superharmonic instability were investigated. Our numerical simulations suggest that eigenvalues of linearized dynamical equations, corresponding to the unstable modes, appear as a result of a collision of a pair of purely imaginary eigenvalues at the origin, and a subsequent appearance of a pair of purely real eigenvalues: a positive and a negative one that are symmetric with respect to zero. Complex conjugate pairs of purely imaginary eigenvalues correspond to stable modes, and as the steepness of the underlying Stokes wave grows, the pairs move toward the origin along the imaginary axis. Moreover, when studying the eigenvalues of linearized dynamical equations we find that as the steepness of the Stokes wave grows, the real eigenvalues follow a universal scaling law, that can be approximated by a power law. The asymptotic power law behavior of this dependence for instability of Stokes waves close to the limiting one is proposed. Surface elevation profiles for several unstable eigenmodes are made available through http://stokeswave.org website.

PT-symmetric potentials with imaginary asymptotic saturation

We point out that PT-symmetric potentials \(V_{\mathrm {PT}}(x)\) having imaginary asymptotic saturation, \(V_{\mathrm {PT}}(x=\pm \infty ) =\pm i V_1, V_1 \in {\mathbb {R}}\) are devoid of scattering states and spectral singularity. We show the existence of real (positive and negative) discrete spectrum both with and without complex conjugate pair(s) of eigenvalues (CCPEs). If the eigenstates are arranged in the ascending order of the real part of the discrete eigenvalues, the initial states have few nodes but latter ones oscillate fast. Both real and imaginary parts of \(\psi _n(x)\) vanish asymptotically, and \(|\psi _n(x)|\) are nodeless. For the CCPEs, these are asymmetric and peaking on the left (right) and for real energies these are symmetric and peaking at the origin. For CCPEs \(E_{\pm }\), the eigenstates \(\psi _{\pm }\) follow the interesting property, \(|\psi _+(x)|= N |\psi _-(-x)|, N \in {\mathbb {R}}^+\).

Full and Partial Eigenvalue Placement for Minimum Norm Static Output Feedback Control

The controller design for linear time-invariant state space systems seems to be straightforward and well established. This is not true for static output feedback control, which is still a challenging task. This paper deals with controller design based on eigenvalue assignment. We consider the placement of distinct as well as multiple real eigenvalues or complex conjugate pairs. The desired eigenvalue configurations are characterised in terms of algebraic divisibility of the characteristic polynomial of the closed-loop system. We also consider the problem of partial eigenvalue placement, where not all eigenvalues are fixed by feedback. Degrees of freedom in the controller design are used for the minimization of various matrix norms of the feedback gain matrix.

Pseudo-hermitian random matrix models: General formalism

Pseudo-hermitian matrices are matrices hermitian with respect to an indefinite metric. They can be thought of as the truncation of pseudo-hermitian operators, defined over some Krein space, together with the associated metric, to a finite dimensional subspace. As such, they can be used, in the usual spirit of random matrix theory, to model chaotic or disordered PT-symmetric quantum systems, or their gain-loss-balanced classical analogs, in the phase of broken PT-symmetry. The eigenvalues of pseudo-hermitian matrices are either real, or come in complex-conjugate pairs. In this paper we introduce a family of pseudo-hermitian random matrix models, depending parametrically on their metric. We apply the diagrammatic method to obtain its averaged resolvent and density of eigenvalues as explicit functions of the metric, in the limit of large matrix size N. As a concrete example, which is essentially an ensemble of elements of the non-compact unitary Lie algebra, we choose a particularly simple set of metrics, and compute the resulting resolvent and density of eigenvalues in closed form. The spectrum consists of a finite fraction of complex eigenvalues, which occupy uniformly two two-dimensional blobs, symmetric with respect to the real axis, as well as the complimentary fraction of real eigenvalues, condensed in a finite segment, with a known non-uniform density. The numbers of complex and real eigenvalues depend on the signature of the metric, that is, the numbers of its positive and negative eigenvalues. We have also carried thorough numerical analysis of the model for these particular metrics. Our numerical results converge rapidly towards the asymptotic analytical large-N expressions.

Real spectra of large real asymmetric random matrices.

When a randomness is introduced at the level of real matrix elements, depending on its particular realization, a pair of eigenvalues can appear as real or form a complex conjugate pair. We show that in the limit of large matrix size the density of such real eigenvalues is proportional to the square root of the asymptotic density of complex eigenvalues continuated to the real line. This relation allows one to calculate the real densities up to a normalization constant, which is then applied to various examples, including heavy-tailed ensembles and adjacency matrices of sparse random regular graphs.

Analysis of "big data" and mathematical modeling of the covid-19 epidemic in Europe

The regularities of the dynamics of the covid-19 pandemic in Ukraine and other European countries are studied by using the methods of analyzing "big data" in the form of time series and related information from open online sources. Statistical analysis of smoothed curves of new cases I (t), dead D (t), recovered R (t) and other time series has shown different types of dynamics: wave (i), quasi-wave with time shift (ii), stepwise (iii), with abnormally high or low amplitudes of local oscillations. The appropriate similarity trees have been constructed by using the nearest neighbor method. It is shown that the countries with different types of dynamics (i, ii, iii) are located in separate branches of the trees. The stability of zero and nonzero stationary points have been investigated on the basis of the popular mathematical model SIRS. The solutions of the linearized system have been obtained and the influence of the model parameters on the eigenvalues of the system matrix has been investigated. The presence of different types of dynamics is shown: with three negative real (a), one positive real (b), one real and a pair of complex conjugate(c) eigenvalues. The phase portraits have been constructed and the connection of the types of time series (i, ii, iii) and solutions (a, b, c) of the SIRS equations is shown. The obtained results allow us to estimate the dynamic behavior of the system, its stability or instability with the possibility of chaotic dynamics on the basis of the analysis of time series on any current day.