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We consider the light cone (`retardation') equation (LCE) of an inertially moving observer and a single worldline parameterized by arbitrary rational functions. Then a set of apparent copies, R- or C-particles, defined by the (real or complex conjugate) roots of the LCE will be detected by the observer. For any rational worldline the collective R-C dynamics is manifestly Lorentz-invariant and conservative; the latter property follows directly from the structure of Vieta formulas for the LCE roots. In particular, two Lorentz invariants, the square of total 4-momentum and total rest mass, are distinct and both integer-valued. Asymptotically, at large values of the observer's proper time, one distinguishes three types of the LCE roots and associated R-C particles, with specific locations and evolutions; each of three kinds of particles can assemble into compact large groups - clusters. Throughout the paper, we make no use of differential equations of motion, field equations, etc.: the collective R-C dynamics is purely algebraic, 5 pages, twocolumn

The condition under that the circular waveguide, loaded with a co-axial dielectric cylinder and a latching ferrite toroid, magnetized azimuthally, exhibits properties of a nonreciprocal digital phase shifter for the normal $\boldsymbol{TE}_{01}$ mode, is derived. A numerical approach is developed and applied to study the dependence of the width of the area in which the structure regarded produces differential phase shift on its material and geometry parameters. It makes use of a specific set of roots of the characteristic equation of configuration, deduced earlier by complex Kummer and Tricomi or by Tricomi and its complex conjugate confluent hypergeometric functions, and by real zeroth and first order Bessel ones, and by the real positive $\overline{\boldsymbol{L}}_{4\pm}$ numbers, linked with another set of roots of the equation said. Results are given for a chosen value of the off-diagonal ferrite permeability tensor element and a changing dielectric cylinder to transmission line radius ratio. It is accepted that the relative permittivities of the two media are the same.

We study unequal rapidity correlators in the stochastic Langevin picture of Jalilian-Marian-Iancu-McLerran-Weigert-Leonidov-Kovner (JIMWLK) evolution in the Color Glass Condensate effective field theory. We discuss a diagrammatic interpretation of the long-range correlators. By separately evolving the Wilson lines in the direct and complex conjugate amplitudes, we use the formalism to study two-particle production at large rapidity separations. We show that the evolution between the rapidities of the two produced particles can be expressed as a linear equation, even in the full nonlinear limit. We also show how the Langevin formalism for two-particle correlations reduces to a BFKL picture in the dilute limit and in momentum space, providing an interpretation of BFKL evolution as a stochastic process for color charges., Comment: 18 pages, 1 figure

This paper addresses the problem of designing asymptotically optimal improper constellations with a given circularity coefficient (correlation coefficient between the constellation and its complex conjugate). The designed constellations are optimal in the sense that, at high signal-to-noise-ratio (SNR) and for a large number of symbols, yield the lowest probability of error under an average power constraint for additive white Gaussian noise channels. As the number of symbols grows, the optimal constellation is the intersection of the hexagonal lattice with an ellipse whose eccentricity determines the circularity coefficient. Based on this asymptotic result, we propose an algorithm to design finite improper constellations. The proposed constellations provide significant SNR gains with respect to previous improper designs, which were generated through a widely linear transformation of a standard M-ary quadrature amplitude modulation constellation. As an application example, we study the use of these improper constellations by a secondary user in an underlay cognitive radio network. The work of Jesús A. López-Fernández and R. G. Ayestarán was partly supported by the Ministerio de Ciencia, Innovación y Universidades under project TEC2017-86619-R (ARTEINE), and by the Gobierno del Principado de Asturias under project GRUPIN-IDI2018-000191. The work of I. Santamaria was partly supported by the Ministerio de Economía y Competitividad (MINECO) of Spain, and AEI/FEDER funds of the E.U., under grant TEC2016-75067-C4-4-R (CARMEN) and TEC2015-69648-REDC (Red COMONSENS). The work of C. Lameiro was supported by the German Research Foundation (DFG) under grant LA 4107/1-1.

In this paper we propose a wider class of symmetries including the Galilean shift symmetry as a subclass. We will show how to construct ghost-free nonlocal actions, consisting of infinite derivative operators, which are invariant under such symmetries, but whose functional form is not simply given by exponentials of entire functions. Motivated by this, we will consider the case of a scalar field and discuss the pole structure of the propagator which has infinitely many complex conjugate poles, but satisfies the tree-level unitarity. We will also consider the possibility to construct UV complete Galilean theories by showing how the ultraviolet behavior of loop integrals can be ameliorated. Moreover, we will consider kinetic operators respecting the same symmetries in the context of linearized gravity. In such a scenario, the graviton propagator turns out to be ghost-free and the spacetime metric generated by a point-like source is nonsingular. These new nonlocal models can be seen as an infinite derivative generalization of Lee-Wick theories and open a new branch of nonlocal theories., 19 pages, 2 figures. Version accepted for publication in PRD

There are two definitions of the inner product of modal spatial functions used in the literature. Both definitions integrate the product of the modal spatial functions over a line, area, or volume. The only difference is that one of the definitions takes the complex conjugate of one of the modal spatial functions before multiplying the modes together. The definitions are the same if the modal spatial functions are real. If the modal spatial functions are complex, only the definition which takes the complex conjugate is an inner product. If the specific acoustic impedance of the boundaries has a real part, then the modes are only orthogonal with the definition which does not take the complex conjugate, although this definition is not strictly an inner product because the modal spatial functions are complex in this situation. However, this definition of "inner product" can be used to calculate the coefficients in the modal expansion of the system response. On the other hand, when it comes to calculating the mean pressure squared and the mean sound intensity, the modal spatial functions cross-products cannot be ignored because the modes are not orthogonal for the definition which takes the complex conjugate.

In this work, I study the deformation of the topological string by Ω ¯ , the complex conjugate of the Ω-deformation. Namely, I identify Ω ¯ in terms of a physical state in the string spectrum and verify that the deformed Yang-Mills and ADHM actions are reproduced. This completes the study initiated in [1] where we show that Ω ¯ decouples from the one-loop topological amplitudes in heterotic string theory. Similarly to the N = 2 ⋆ deformation, I show that the quadratic terms in the effective action play a crucial role in obtaining the correct realisation of the full Ω-deformation. Finally, I comment on the differences between the graviphoton and the Ω-deformation in general and discuss possible Ω ¯ remnants at the boundary of the string moduli space.

This work proposes a dead-time compensator (DTC) with dead-beat disturbance rejection response. The proposed strategy is based on the simplified dead-time compensator (SDTC) because of its simplicity and good performance. A new robustness filter to obtain a dead-beat disturbance rejection response is proposed. The dead-beat robustness filter is then implemented using complex conjugate poles and an additional zero. Simulation results comparing the proposed methodology with the SDTC show that the new obtained controller presents faster disturbance rejection with finite time response.

This paper links the celebrated Cauchy&acute, s interlacing theorem of eigenvalues for partitioned updated sequences of Hermitian matrices with stability and convergence problems and results of related sequences of matrices. The results are also applied to sequences of factorizations of semidefinite matrices with their complex conjugates ones to obtain sufficiency-type stability results for the factors in those factorizations. Some extensions are given for parallel characterizations of convergent sequences of matrices. In both cases, the updated information has a Hermitian structure, in particular, a symmetric structure occurs if the involved vector and matrices are complex. These results rely on the relation of stable matrices and convergent matrices (those ones being intuitively stable in a discrete context). An epidemic model involving a clustering structure is discussed in light of the given results. Finally, an application is given for a discrete-time aggregation dynamic system where an aggregated subsystem is incorporated into the whole system at each iteration step. The whole aggregation system and the sequence of aggregated subsystems are assumed to be controlled via linear-output feedback. The characterization of the aggregation dynamic system linked to the updating dynamics through the iteration procedure implies that such a system is, generally, time-varying.

We derive general relationships between the number of complex poles of a propagator and the sign of the spectral function originating from the branch cut in the Minkowski region under some assumptions on the asymptotic behaviors of the propagator. We apply this relation to the mass-deformed Yang-Mills model with one-loop quantum corrections, which is identified with a low-energy effective theory of the Yang-Mills theory, to show that the gluon propagator in this model has a pair of complex conjugate poles or "tachyonic" poles of multiplicity two, in accordance with the fact that the gluon field has a negative spectral function, while the ghost propagator has at most one "unphysical" pole. Finally, we discuss implications of these results for gluon confinement and other non-perturbative aspects of the Yang-Mills theory., 18 pages, 20 figures, final version published in PRD

This paper discusses and analyzes the quasistatic optimal plasmonic dipole resonance of a small dielectric particle embedded in a lossy surrounding medium. The optimal resonance at any given frequency is defined by the complex valued dielectric constant that maximizes the absorption of the particle under the quasistatic approximation and a passivity constraint. In particular, for an ellipsoid aligned along the exciting field, the optimal material property is given by the complex conjugate of the pole position associated with the polarizability of the particle. In this paper, we employ the classical Mie theory to analyze this approximation for spherical particles in a lossy surrounding medium. It turns out that the quasistatic optimal plasmonic resonance is valid, provided that the electrical size of the particle is sufficiently small at the same time as the external losses are sufficiently large. Hence, it is important to note that this approximation cannot be used for a lossless medium, and which is also obvious, since the quasistatic optimal dipole absorption becomes unbounded for this case. Moreover, it turns out that the optimal normalized absorption cross sectional area of the small dielectric sphere has a very subtle limiting behavior and is, in fact, unbounded even in full dynamics when both the electrical size and the exterior losses tend to zero at the same time. A detailed analysis is carried out to assess the validity of the quasistatic estimation of the optimal resonance, and numerical examples are included to illustrate the asymptotic results.This paper discusses and analyzes the quasistatic optimal plasmonic dipole resonance of a small dielectric particle embedded in a lossy surrounding medium. The optimal resonance at any given frequency is defined by the complex valued dielectric constant that maximizes the absorption of the particle under the quasistatic approximation and a passivity constraint. In particular, for an ellipsoid aligned along the exciting field, the optimal material property is given by the complex conjugate of the pole position associated with the polarizability of the particle. In this paper, we employ the classical Mie theory to analyze this approximation for spherical particles in a lossy surrounding medium. It turns out that the quasistatic optimal plasmonic resonance is valid, provided that the electrical size of the particle is sufficiently small at the same time as the external losses are sufficiently large. Hence, it is important to note that this approximation cannot be used for a lossless medium, and which is also...

Phase-amplitude reduction is a widely applied technique in the study of limit cycle oscillators with the ability to represent a complicated and high-dimensional dynamical system in a more analytically tractable set of coordinates. Recent work has focused on the use of isostable coordinates, which characterize the transient decay of solutions toward a periodic orbit, and can ultimately be used to increase the accuracy of these reduced models. The breadth of systems to which this phase-amplitude reduction strategy can be applied, however, is still rather limited. In this work, the theory of phase-amplitude reduction using isostable coordinates is further developed to accommodate a broader set of dynamical systems. In the first part, limit cycles of piecewise smooth dynamical systems are considered and strategies are developed to compute the associated reduced equations. In the second part, the notion of isostable coordinates for complex-valued Floquet multipliers is introduced, resulting in one phaselike coordinate and one amplitudelike coordinate for each pair of complex conjugate Floquet multipliers. Examples are given with relevance to piecewise smooth representations of excitable cardiomyocytes and the relationship between the reduced coordinate system and the emergence of cardiac alternans is discussed. Also, phase-amplitude reduction is implemented for a chaotic, externally forced pendulum with complex Floquet multipliers and a resulting control strategy for the stabilization of its periodic solution is investigated.

In recent work Alexandre, Ellis, Millington and Seynaeve have extended the Goldstone theorem to non-Hermitian Hamiltonians that possess a discrete antilinear symmetry such as $PT$. They restricted their discussion to those realizations of antilinear symmetry in which all the energy eigenvalues of the Hamiltonian are real. Here we extend the discussion to the two other realizations possible with antilinear symmetry, namely energies in complex conjugate pairs or Jordan-block Hamiltonians that are not diagonalizable at all. In particular, we show that under certain circumstances it is possible for the Goldstone boson mode itself to be one of the zero-norm states that are characteristic of Jordan-block Hamiltonians. While we discuss the same model as Alexandre et al. our treatment is quite different, though their main conclusion that one can have Goldstone bosons in the non-Hermitian case remains intact. In their paper Alexandre et al. presented a variational procedure for the action in which the surface term played an explicit role, to thus suggest that one has to use such a procedure in order to establish the Goldstone theorem in the non-Hermitian case. However, by taking certain fields that they took to be Hermitian to actually either be anti-Hermitian or be made so by a similarity transformation, we show that we are then able to obtain a Goldstone boson using a completely standard variational procedure. Since we use a standard variational procedure we can readily extend our analysis to a continuous local symmetry by introducing a gauge boson. We show that when we do this the gauge boson acquires a non-zero mass by the 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., 21 pages, revtex4. Final version, Phys. Rev D 99, 045006 (2019). Title of paper changed in journal

The elements of the coherency matrix give the strength of the components of a Mueller matrix in the coherency basis. The Z-matrix (called the polarization-coupling matrix or state-generating matrix) represents a partial sum of the coherency expansion. For transmission through a deterministic medium, the coherency elements can be used directly as generators to calculate the development of polarization upon propagation. The commutation properties of the coherency elements are investigated. New matrices that we call the W-matrix and the X-matrix, both different representations of the Z-matrix in a Jones basis, are introduced. The W-matrix controls the transformation of the Jones vector and also the covariance matrix. The product of the X-matrix with its complex conjugate gives the matrix representation of the Mueller matrix in the Jones basis. The development of Mueller matrix and coherency matrix elements upon propagation through some examples of a uniform medium is investigated. It is shown that the coherency matrix is more easily interpreted than the Mueller matrix. Analytic expressions are presented to calculate the elementary polarization properties from coherency matrix elements or Mueller matrix parameters.

International audience; In conjugate heat transfer (CHT) analysis, the fluid-solid heat transfer calculation is realized by expanding the fluid capability to include heat conduction in solid regions neighboring the fluid. Stability conditions of CHT analysis can be provided by a simplified model problem and from this model, two fundamental parameters are introduced in the coupled approach: a "numerical" Biot number, and an optimal coefficient. The first parameter defines the nature of the fluid-solid interaction. The optimal coefficient ensures unconditional stability. Results presented in this paper consider a wide range of steady thermal phenomena from weak to very strong fluid-solid interaction (insulating materials). At the end of the paper, an example of transient CHT is also shown. The methods proposed in this paper illustrate that a model problem can provide effective and practical solutions directly applicable to complex conjugate heat transfer problems.

Currently, with the advancement of sophisticated SAR imaging modes, such as multitemporal interferometry and polarimetry, the return backscatter results are presented in two or more channels [1, 2]. The SAR interferogram, achieved by multiplying the first image by the complex conjugate of the second one [3, 4], has become an important tool for multiple-channel SAR image interpretation.

We study unequal rapidity correlators in the stochastic Langevin picture of Jalilian-Marian-Iancu-McLerran-Weigert-Leonidov-Kovner (JIMWLK) evolution in the color glass condensate effective field theory. We discuss a diagrammatic interpretation of the long-range con elators. By separately evolving the Wilson lines in the direct and complex conjugate amplitudes, we use the formalism to study two-particle production at large rapidity separations. We show that the evolution between the rapidities of the two produced particles can be expressed as a linear equation, even in the full nonlinear limit. We also show how the Langevin formalism for two-particle correlations reduces to a Balitsky-Fadin-Kuraev-Lipatov (BFKL) picture in the dilute limit and in momentum space, providing an interpretation of BFKL evolution as a stochastic process for color charges.

Let K be a finite abelian group and let exp ⁡ ( K ) denote the least common multiple of the orders of the elements of K. A BH ( K , h ) matrix is a K-invariant | K | × | K | matrix H whose entries are complex hth roots of unity such that H H ⁎ = | K | I , where H ⁎ denotes the complex conjugate transpose of H, and I is the identity matrix of order | K | . Let ν p ( x ) denote the p-adic valuation of the integer x. Using bilinear forms on K, we show that a BH ( K , h ) exists whenever (i) ν p ( h ) ≥ ⌈ ν p ( exp ⁡ ( K ) ) / 2 ⌉ for every prime divisor p of | K | and (ii) ν 2 ( h ) ≥ 2 if ν 2 ( | K | ) is odd and K has a direct factor Z 2 . Employing the field descent method, we prove that these conditions are necessary for the existence of a BH ( K , h ) matrix in the case where K is cyclic of prime power order.

We discuss the analytic continuation of the gluon propagator from the Euclidean region to the complex squared-momentum plane towards the Minkowski region from a viewpoint of gluon confinement. For this purpose, we investigate the massive Yang-Mills model with one-loop quantum corrections, which is to be identified with a low-energy effective theory of the Yang-Mills theory in the sense that the confining decoupling solution for the Euclidean gluon and ghost propagators of the Yang-Mills theory in the Landau gauge obtained by numerical simulations on the lattice are reproduced with good accuracy from the massive Yang-Mills model by taking into account one-loop quantum corrections. We show that the gluon propagator in the massive Yang-Mills model has a pair of complex conjugate poles or "tachyonic" poles of multiplicity two, in accordance with the fact that the gluon field has a negative spectral function, while the ghost propagator has at most one "unphysical" pole. These results are consistent with general relationships between the number of complex poles of a propagator and the sign of the spectral function originating from the branch cut in the Minkowski region under some assumptions on the asymptotic behaviors of the propagator. Consequently, we give an analytical proof for violation of the reflection positivity as a necessary condition for gluon confinement for any choice of the parameters in the massive Yang-Mills model, including the physical point. Moreover, the complex structure of the propagator enables us to explain why the gluon propagator in the Euclidean region is well described by the Gribov-Stingl form., Comment: 1+6 pages, plenary talk presented at Light Cone 2019 - QCD on the light cone: from hadrons to heavy ions, 16-20 September 2019 held at Ecole Polytechnique, Palaiseau, France

We investigate the statistical properties of eigenvalues of pseudo-Hermitian random matrices whose eigenvalues are real or complex conjugate. It is shown that when the spectrum splits into separated sets of real and complex conjugate eigenvalues, the real ones show characteristics of an intermediate incomplete spectrum, that is, of a so-called thinned ensemble. On the other hand, the complex ones show repulsion compatible with cubic-order repulsion of non normal matrices for the real matrices, but higher order repulsion for the complex and quaternion matrices.

We determine the gluon and ghost spectral functions along with the analytic structure of the associated propagators from numerical data describing gauge correlators at space-like momenta obtained by either solving the Dyson-Schwinger equations or through lattice simulations. Our novel reconstruction technique shows the expected branch cut for the gluon and the ghost propagator, which, in the gluon case, is supplemented with a pair of complex conjugate poles. Possible implications of the existence of these poles are briefly addressed., Comment: 7 pages, 5 figures. Extended version which includes the analysis of lattice unquenched 2+1+1 Landau gauge data and quenched Rxi gauge data. Matches the published version in PLB

To overcome the traditional disadvantages of single source points detection methods in underdetermined blind source separation problem, this paper proposes a novel algorithm to detect single source points for the linear instantaneous mixed model. First, the algorithm utilizes a certain relationship between the time-frequency coefficients and the complex conjugate factors of the observation signal to realize single source points detection. Then, the algorithm finds more time-frequency points that meets the requirements automatically and cluster them by utilizing a clustering algorithm based on the improved potential function. Finally, the estimation of the mixed matrix is achieved by clustering the re-selected single source points. Simulation experiments on linear mixture model demonstrates the efficiency and feasibility for estimating the mixing matrix.

Although parity-time (PT)-symmetric systems can exhibit real spectra in the exact PT-symmetry regime, PT-symmetry is actually not a necessary condition for the real spectra. Here, we show that non-PT-symmetric photonic crystals (PCs) carrying Dirac-like cone dispersions can always exhibit real spectra as long as the average non-Hermiticity strength within the unit cell for the eigenstates is zero. By building a non-Hermitian Hamiltonian model, we find that the real spectra of the non-PT-symmetric system can be explained using the concept of pseudo-Hermiticity. We demonstrate using effective medium theories that, in the long-wavelength limit, such non-PT-symmetric PCs behave like the so-called complex conjugate medium (CCM) whose refractive index is real but whose permittivity and permeability are complex numbers. The real refractive index for this effective CCM is guaranteed by the real spectrum of the PCs, and the complex permittivity and permeability come from non-PT-symmetric loss-gain distributions. We show some interesting phenomena associated with CCM, such as the lasing effect.

We consider the standard hypergeometric differential operator $D$ regarded as an operator on the complex plane $C$ and the complex conjugate operator $\overline D$. These operators formally commute and are formally adjoint one to another with respect to an appropriate weight. We find conditions when they commute in the Nelson sense and write explicitly their joint spectral decomposition. It is determined by a two-dimensional counterpart of the Jacobi transform (synonyms: generalized Mehler--Fock transform, Olevskii transform). We also show that the inverse transform is an operator of spectral decomposition for a pair of commuting difference operators defined in terms of shifts in imaginary direction., 55p, typos were corrected

Most discussions of propagators in Lee-Wick theories focus on the presence of two massive complex conjugate poles in the propagator. We show that there is in fact only one pole near the physical region, or in another representation three pole-like structures with compensating extra poles. The latter modified Lehmann representation is useful caculationally and conceptually only if one includes the resonance structure in the spectral integral., 10 pages, 8 figures. Version 2 has an improved drawing of Fig. 3

The complex eigenvalues of some non-Hermitian Hamiltonians, e.g., parity-time-symmetric Hamiltonians, come in complex-conjugate pairs. We show that for non-Hermitian scattering Hamiltonians (of a structureless particle in one dimension) possessing one of four certain symmetries, the poles of the S-matrix eigenvalues in the complex momentum plane are symmetric about the imaginary axis, i.e., they are complex-conjugate pairs on the complex-energy plane. This applies even to states which are not bounded eigenstates of the system, i.e., antibound or virtual states, resonances, and antiresonances. The four Hamiltonian symmetries are formulated as the commutation of the Hamiltonian with specific antilinear operators. Example potentials with such symmetries are constructed and their pole structures and scattering properties are calculated., Basque Country Government; MINECO/FEDER, UE; Basque Government predoctoral program; Turkish Academy of Sciences

If $\Gamma\lt \operatorname{PSL}(2,\mathbb{C})$ is a lattice, we define an invariant of a representation $\Gamma\rightarrow\operatorname{PSL}(n,\allowbreak\mathbb{C})$ using the Borel class $\beta(n)\in\mathrm{H}_{\mathrm{c}}^{3}(\operatorname{PSL}(n,\mathbb{C}),\mathbb{R})$ . We show that this invariant satisfies a Milnor–Wood type inequality and its maximal value is attained precisely by the representations conjugate to the restriction to $\Gamma$ of the irreducible complex $n$ -dimensional representation of $\operatorname{PSL}(2,\mathbb{C})$ or its complex conjugate. Major ingredients of independent interest are the study of our extension to degenerate configurations of flags of a cocycle defined by Goncharov, as well as the identification of $\mathrm{H}_{\mathrm{b}}^{3}(\operatorname{SL}(n,\mathbb{C}),\mathbb{R})$ as a normed space.

Fundamental upper bounds are given for the plasmonic multipole absorption and scattering of a rotationally invariant dielectric sphere embedded in a lossy surrounding medium. A specialized Mie theory is developed for this purpose and when combined with the corresponding generalized optical theorem, an optimization problem is obtained which is explicitly solved by straightforward analysis. In particular, the absorption cross section is a concave quadratic form in the related Mie (scattering) parameters and the convex scattering cross section can be maximized by using a Lagrange multiplier constraining the absorption to be non-negative. For the homogeneous sphere, the Weierstrass preparation theorem is used to establish the existence and the uniqueness of the plasmonic singularities and explicit asymptotic expressions are given for the dipole and the quadrupole. It is shown that the optimal passive material for multipole absorption and scattering of a small homogeneous dielectric sphere embedded in a dispersive medium is given approximately as the complex conjugate and the real part of the corresponding pole positions, respectively. Numerical examples are given to illustrate the theory, including a comparison with the plasmonic dipole and quadrupole resonances obtained in gold, silver and aluminum nanospheres based on some specific Brendel-Bormann (BB) dielectric models for these metals. % of gold, silver and aluminum. Based on these BB-models, it is interesting to note that the metal spheres can be tuned to optimal absorption at a particular size at a particular frequency., arXiv admin note: text overlap with arXiv:1806.08960

This paper introduces biquaternion eigen-decomposition theory (via Peirce decomposition) with respect to a selected quaternion with a non-zero vector part. The eigen-decomposition allows evaluation of polynomials and power series with real coefficients as functions of quaternions. This extension of analytic functions to functions of quaternions requires only standard complex function evaluation. The theory also applies to quaternion rotations. The theory uses biquaternion calculations indicated by matrix methods via the algebraic isomorphism between Hamilton’s biquaternions and appropriate $$4\times 4$$ complex matrices. The isomorphism preserves algebraic structure. In particular, the left and right biquaternion multiplication by the selected quaternion maps to left and right matrix multiplication, respectively. This unifies the representation of the left and right quaternion multiplication as a linear map into a single matrix form. This matrix, as a linear operator, acts on matrices, so that the eigenvectors have matrix form that maps into the biquaternions. Use of an alternate quaternion basis results in a similarity transform of the representation matrix, preserving eigenvalues across change of basis. The similarity transform allows simple eigenvector calculation. The matrix for the selected quaternion has two identical, complex conjugate pairs of eigenvalues. Each pair corresponds to two complex conjugate pairs of eigenvector biquaternions, an idempotent pair and a nilpotent pair. Idempotent and nilpotent eigenvectors correspond to the commuting and non-commuting parts, respectively, of quaternion multiplication.

We reelaborate on the basic properties of PT symmetry from a geometrical Perspective. Thetransfer matrix associated with these systems induces a Möbius transformation in the complex plane.The trace of this matrix classifies the actions into three types that represent rotations, translations, andparallel displacements. We find that a PT invariant system can be pictured as a complex conjugationfollowed by an inversion in a circle. We elucidate the physical meaning of these geometrical operationsand link them with measurable properties of the system.

This paper presents approximate and exact stability analysis to determine the critical time delays of one and two degree of freedom (DOF) single-actuator real-time hybrid substructuring (RTHS). The approximate stability analysis is based on applying the Routh-Hurwitz stability criterion using a first-order Taylor expansion of the actuator dynamics represented by a constant gain and pure time delay. The exact stability analysis involves solving the critical frequencies first by canceling the exponential delay term of the closed-loop characteristic equation through multiplication by its complex conjugate. The exact critical time delays are then found using the phase of the closed-loop characteristic equation evaluated at the critical frequencies. These stability analysis techniques are applied to several 1-DOF and 2-DOF mass-spring configurations of single-actuator RTHS. Results show that the critical time delays are highly dependent on the mass, stiffness, and damping partitioning of the physical and numerical substructures as well as the actuator gain. This information provides useful insight into the stability behavior of a particular substructure partitioning configuration to evaluate the feasibility of conducting RTHS tests of these systems.

This study examines the Hopf (double Hopf) bifurcations and transitions of two dimensional quasi-geostrophic (QG) flows that model various large-scale oceanic and atmospheric circulations. Using the Kolmogorov function to represent an external forcing in the tropical region, it is shown that the equilibrium of the QG model loses its stability if the combination of the Rossby number, the Ekman number, and the eddy viscosity satisfies a specific condition. Further use of the center manifold technique reveals two different types of the dynamical transition from either a pair of simple complex eigenvalues or a double pair of complex conjugate eigenvalues. These dynamical transitions are confirmed in the numerical analyses of the QG dynamics at the equilibrium, which capture Hopf (double Hopf) bifurcations due to the existence of a nonzero imaginary part of the first eigenvalue. The transition from a pair of simple complex eigenvalues is of particular interest, because it indicates the existence of a stable periodic pattern that is similar to atmospheric easterly waves and related tropical cyclone formation in the tropical atmosphere.

Let [Formula: see text] be an algebraic integer of degree [Formula: see text]. Let [Formula: see text] be the [Formula: see text] complex conjugate of [Formula: see text]. Assume that the Galois group [Formula: see text] is isomorphic to the symmetric group [Formula: see text]. We give a [Formula: see text]-basis and the discriminant of the order [Formula: see text]. We end up with an open question showing that this problem seems much harder when we assume that [Formula: see text] is already Galois or even cyclic.

The features of the resonant soliton of the 3 × 3 operator is studied and detailed. The scattering data of this operator contains four transmission coefficients, two in each half complex ζ-plane, where ζ is the spectral parameter. When the potential matrix has anti-hermitian symmetry ( Q α , β = − Q β , α * ), each of the two transmission coefficients in the lower half plane become equal to the complex conjugates of one of the two in the upper half plane, leaving only two independent transmission coefficients, which is taken to be those in the upper half plane. The bound state scattering data for this operator consists of the zeros of these two transmission coefficients (bound state eigenvalues) and a normalization coefficient associated with each eigenvalue. With two transmission coefficients, there is a wider variety of possible soliton solutions than in the Zakharov–Shabat (ZS) 2 × 2 case. First, the eigenvalues could belong to only one transmission coefficient, with the other transmission coefficient having none. In this case the soliton solution will only exist in one of the two conjugate pairs of components of Q lying next to the diagonal, with all other components of Q vanishing. These soliton solutions will be equivalent to those in the ZS case, up to scaling factors. If one would distribute the eigenvalues among both transmission coefficients so that each transmission coefficient would have at least one eigenvalue, the soliton structure becomes more complex. In this case, any one eigenvalue will be found to generally make contributions to all non-diagonal components of Q. Of particular interest, inside this class is a special class of solitons which exist only when these two transmission coefficients have exactly equal eigenvalues. Equality of eigenvalues from different transmission coefficients give rise to “resonant solitons”. This latter class is the case which is treated here. It is found that there are two different states for the resonant soliton solution, both of which arise from a bifurcation. The bifurcation is shown to arise from the algebraic structure of the 3 × 3 scattering matrix. The linear dispersion relations (the solution of the inverse scattering problem) are also extended to the case when resonant solitons are present. It is further shown that the same result could be obtained by taking the limit of the eigenvalues in each transmission coefficient approaching the other. The general soliton solution under anti-hermitian symmetry is studied when there is only one eigenvalue in each transmission coefficient, with each unequal to the other. The asymptotics of this solution is detailed and then the same is done when the eigenvalues are exactly equal, showing that the latter contains the well known parametric interactions of “up-conversion” and “down-conversion”. Lastly, it is pointed out how this equality of eigenvalues in different transmission coefficients can be interpreted as a “nonlinear resonance condition”.

It is a problem when modeling the thin layer of periodic structures with numerical method by directly discretizing the inner part, because the fine mesh applied to simulate the thin layer would result in great computational data. The internal network boundary condition (INBC) incorporated in the transmission line matrix (TLM) scheme is proposed in this paper, to avoid the directly discretization of the thin layer and achieve a high accuracy. The thin layer of periodic structure is regarded as an easily analyzed two-port network. Vector fitting approach is used to approximate the network parameters into a series of rational expressions with either real term or complex conjugate pole–residue pairs. The INBC equation with discrete-time form is derived and then incorporated in TLM scheme by introducing the approximate pole–residue pairs to the TLM update equation. Compared with the extremely fine-discretized TLM unit cell used to deal with the thin layer by the conventional TLM scheme, a much larger coarse TLM unit cell is accurate enough to be used to discretize the structure by the proposed method. Compared with the conventional TLM scheme, substantial savings in computational storage are achieved. A common frequency selective surface (FSS) structure is first analyzed for the validation of the method. Good agreement in scattering characteristics is observed in frequency domain between the proposed method and simulation by HFSS software. To further demonstrate the validity of the proposed method, an aperture-coupled resonator-based FSS is designed and fabricated. Good agreement among the S-parameters from the proposed method, simulation, and measurement is observed, verifying the accuracy of the proposed method.

We introduce the notion of harmonic curvature for real hypersurfaces in the complex quadric Q m = S O m + 2 / S O m S O 2 . We give a complete classification, in terms of their A -principal or their A -isotropic unit normal vector fields, of real hypersurfaces in Q m = S O m + 2 / S O m S O 2 having harmonic curvature tensor.

We propose a class of multidimensional higher derivative theories of gravity without extra real degrees of freedom besides the graviton field. The propagator shows up the usual real graviton pole and extra complex conjugates poles that do not contribute to the absorptive part of the physical scattering amplitudes. Indeed, they may consistently be excluded from the asymptotic observable states of the theory making use of the Lee-Wick and Cutkoski, Landshoff, Olive and Polkinghorne prescription for the construction of a unitary S-matrix. Therefore, the spectrum consists on the graviton and short lived elementary unstable particles that we named "anti-gravitons" because of their repulsive contribution to the gravitational potential at short distance. However, another interpretation of the complex conjugate pairs is proposed based on the Calmet's suggestion, i.e. they could be understood as black hole precursors long established in the classical theory. Since the theory is CPT invariant, the complex conjugate of the micro black hole precursor has received as a white hole precursor consistently with the t'Hooft complementary principle. It is proved that the quantum theory is super-renormalizable in even dimension, i.e. only a finite number of divergent diagrams survive, and finite in odd dimension. Furthermore, turning on a local potential of the Riemann tensor we can make the theory finite in any dimension. The singularity-free Newtonian gravitational potential is explicitly computed for a range of higher derivative theories. Finally, we propose a new super-reneromalizable or finite Lee-Wick standard model of particle physics., Comment: LaTeX, 12 pages

Spherical arrays have many advantages for direction-of-arrival (DOA) estimation in 3D space. In this paper, a new real-valued method is proposed to estimate DOAs for spherical arrays. It exploits the property of the complex conjugate of spherical harmonics to find a unitary matrix which can transform the complex array steering matrix into a real matrix. Based on the unitary transformation, a new real-valued array model is constructed to keep the same dimension as the complex-valued model. Then variational sparse Bayesian learning (VSBL) is used to model the joint sparsity between the real part and imaginary part of original data. We get the approximate posterior of the sparse components. The real-valued DOA estimation method acquires good estimation performance and simultaneously decreases the computational cost considerably. Simulation results demonstrate the performance of the proposed method. A new real-valued model is constructed by unitary transform for spherical arrays.Variational sparse Bayesian method is used to estimate DOAs.The real-valued DOA estimation decreases the computational load considerably.

A wide class of autonomous real quadratic dynamic system without real (but with two complex conjugate or even imaginary) equilibrium points is considered. For any system of this class, a new idea of the uniquely definite degenerate autonomous real quadratic dynamic system having exactly one real double equilibrium point (there are no complex equilibrium points) is introduced. It is shown that if the degenerate system demonstrates the chaotic behavior, then for the original (not degenerate) system, a similar chaotic behavior also takes place. The idea of the degenerate system for researches of the real quadratic systems, for which number of complex conjugate equilibrium points more than two, is also used. The same idea can be adapted to research of any autonomous real quadratic system having at least one pair complex conjugate equilibrium points. An attempt to apply some derived results to a search problem of hidden chaotic attractors was undertaken. Examples are given.

This brief demonstrates a method to realize complex conjugate poles using passive-switched-capacitor networks. In addition, a simplified continuous-time model will be introduced so that accurate transfer functions and output noise can be obtained without the need for complicated charge-balance equations and thereby allow a better intuitive understanding of these structures. The developed theory is demonstrated through the design of a second-order low-pass Butterworth biquad with applications intended in the design of wireless RF receivers.

Gyroscopic conservative dynamical systems may exhibit flutter instability that leads to a pair of complex conjugate eigenvalues, one of which has a positive real part and thus leads to a divergent free response of the system. When dealing with non-conservative systems, the pitch fork bifurcation shifts toward the negative real part of the root locus, presenting a pair of eigenvalues with equal imaginary parts, while the real parts may or may not be negative. Several works study the stability of these systems for relevant engineering applications such as the flutter in airplane wings or suspended bridges, brake squeal, etc. and a common approach to detect the stability is the complex eigenvalue analysis that considers systems with all negative real part eigenvalues as stable systems. This paper studies analytically and numerically the cases where the free response of these systems exhibits a transient divergent time history even if all the eigenvalues have negative real part thus usually considered as stable, and relates such a behaviour to the non orthogonality of the eigenvectors. Finally, a numerical method to evaluate the presence of such instability is proposed.

Similarities and differences between the force fields of a classical real dipole and a complex dipole are analyzed. The complex dipole is a pair of points equipped with complex conjugate masses and situated in a complex domain. The results of this analysis are used in the problem of motion of a material point in the field of attraction of a triangle uniformly rotating in its plane about its center of mass. It is assumed that a complex dipole is assigned to each vertex of the triangle. The existence and stability of libration points are studied. In particular, it is shown that there exist libration points outside the plane of the triangle.

First we introduce the notion of parallel normal Jacobi operator for real hypersurfaces in the complex quadric . Next we give a complete classification of real hypersurfaces in the complex quadric with parallel normal Jacobi operator.

In this paper, we investigate the free damped vibration of a nanobeam resting on viscoelastic foundation. Nanobeam and viscoelastic foundation are modeled using nonlocal elasticity and fractional order viscoelasticity theories. Motion equation is derived using D’Alambert’s principle and involves two retardation times and fractional order derivative parameters regarding to a nanobeam and viscoelastic foundation. The analytical solution is obtained using the Laplace transform method and it is given as a sum of two terms. First term denoting the drift of the system’s equilibrium position is given as an improper integral taken along two sides of the cut of complex plane. Two complex conjugate roots located in the left half-plane of the complex plane determine the second term describing the damped vibration around equilibrium position. Results for complex roots of characteristic equation obtained for a single nanobeam without viscoelastic foundation, where imaginary parts represent damped frequencies, are validated with the results found in the literature for natural frequencies of a single-walled carbon nanotube obtained from molecular dynamics simulations. In order to examine the effects of nonlocal parameter, fractional order parameters and retardation times on the behavior of characteristic equation roots in the complex plane and the time-response of nanobeam, several numerical examples are given.

In this paper we prove that a dynamically convex starshaped hypersurface in $\mathbb{C}^2$ which is invariant under complex conjugation admits a global surface of section which is invariant under conjugation as well. We obtain this invariant global surface by embedding $\mathbb{C}^2$ into $\mathbb{CP}^2$ and applying a stretching argument to real holomorphic curves in $\mathbb{CP}^2$. The motivation for this result arises from recent progress in applying holomorphic curve techniques to gain a deeper understanding on the dynamics of the restricted three body problem., 36 pages

In this study, surface impedance boundary condition (SIBC) incorporated in conformal finite-difference time-domain (CFDTD) to efficiently simulate the thin conductive layer of periodic structure is proposed. The method is based on the vector fitting (VF) technique to approximate the frequency-dependent surface impedance of the thin conductive layer by a series of partial fractions in terms of real or complex conjugate pole-residue pairs. A discrete-time equation of SIBC is then derived and incorporated in CFDTD update equations by Laplace transform and time central difference methods. The proposed method takes the following advantages, (i) the time step is no longer bounded by the fine resolution of the thin layer; (ii) The complicated recursive convolution approach is avoided, and the approximation of the frequency-dependent surface impedance of the thin conductive layer can be easy to handle with the VF method. This method is numerically verified for the time- and frequency-dependent scattering characteristics of several periodic structures behaved under normal and oblique incident waves.

An improved continuous cycling method is proposed for PID controllers for unstable systems with two unstable poles and time delay. The method involves the determination of the controller settings by solving the magnitude and the phase angle criteria for the system with a proportional controller. Subsequently, incorporating the Proportional-Derivative-Integral (PID) controller transfer function, with unity proportional gain and pre-determined values of reset time and derivative time in the first step, with the system model and again solving the amplitude and the phase angle criteria, we get the updated gain of the controller. The method is applied by simulation on (i) a second order system with time delay and two unstable poles and (ii) a non-linear model of a CSTR with complex conjugate unstable poles and time delay. The controller settings significantly enhance the performances of both the servo and the regulatory problems and give robust performances.

This paper considers mixing matrix estimation for underdetermined blind source separation. First, we propose an effective detection algorithm to identify single source points where only one source occurs. The detection algorithm finds single source points by utilizing the time---frequency coefficients of mixed signals and the complex conjugates of the coefficients. Then, a method based on probability density is proposed in order to find more reliable single source points and cluster them. Finally, the mixing matrix is obtained through re-selecting and clustering single source points. The experimental results indicate that the algorithm can accurately estimate the mixing matrix when there are fewer sensors than sources.