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We assess the calculation of hyperfine coupling (HFC) tensors by different variants of Projected Hartree-Fock (PHF) theory. For a set of small main-group S = ½ radicals (BO, CO+, CN, AlO, vinyl, me...

We perform complex analyses of the gluon propagator at nonzero quark chemical potential in the long-wavelength limit, using an effective model with a gluon mass term of the Landau-gauge Yang-Mills theory, which is a Landau-gauge limit of the Curci-Ferrari model with quantum corrections being included within the one-loop level. We mainly investigate complex poles of the gluon propagator, which could be relevant to confinement. Around typical values of the model parameters, we show that the gluon propagator has one or two pairs of complex conjugate poles depending on the value of the chemical potential. In addition to a pair similar to that in the case of zero chemical potential, a new pair appears near the real axis when the chemical potential is roughly between the effective quark mass and the effective gluon mass of the model. We discuss possible interpretations of these poles. Additionally, we prove the uniqueness of analytic continuation of the Matsubara propagator to a class of functions that vanish at infinity and are holomorphic except for a finite number of complex poles and singularities on the real axis., 18 pages, 14 figures, final version published in PRD

A new polynomial \(Q_G(x)\) associated to graphs G is defined and studied. The main theorems represent \(Q_G(x)\) as a quasi-specialization of the rank-generating polynomial \(S_G(x,y)\) of Oxley and Whittle, J Comb Theory Ser B 59:210–244, 1993, [10] and show that \(Q_G\) is likewise a generalized Tutte–Grothendieck invariant. The value \(Q_G(1)\) gives the amplitude of acceptance for a class of quantum circuits with associated graphs G. This class, called stabilizer circuits or Clifford circuits, has long been known to have deterministic polynomial time simulation, so \(Q_G(1)\) is polynomial time computable, given G as input. The specialization has \(y = -\sqrt{2}i\), which (along with its complex conjugate) is the only choice that invalidates formulas in a theorem by Noble, Comb. Probab. Comput 15:449–461, 2006, [9] classifying hard and easy real points of \(S_G\), so the complexity of other points \(Q_G(x)\) is open. We reduce the base cases for \(S_G\) by adjoining multiple kinds of isolated nodes and draw possible further implications of the connections between matroid theory and quantum computing developed by this work.

For decades, researchers have sought to understand how the irreversibility of the surrounding world emerges from the seemingly time symmetric, fundamental laws of physics. Quantum mechanics conjectured a clue that final irreversibility is set by the measurement procedure and that the time reversal requires complex conjugation of the wave function, which is overly complex to spontaneously appear in nature. Building on this Landau-Wigner conjecture, it became possible to demonstrate that time reversal is exponentially improbable in a virgin nature and to design an algorithm artificially reversing a time arrow for a given quantum state on the IBM quantum computer. However, the implemented arrow-of-time reversal embraced only the known states initially disentangled from the thermodynamic reservoir. Here we develop a procedure for reversing the temporal evolution of an arbitrary unknown quantum state. This opens the route for general universal algorithms sending temporal evolution of an arbitrary system backwards in time., Comment: 5 pages

In Fourier domain optical coherence tomography (FDOCT), the depth profile is mirrored about the zero delay between the sample and reference optical paths, limiting the imaging depth to half of the entire ranging space and undermining the optimal sensitivity window. We present a new method, to the best of our knowledge, to remove the complex conjugate artifact by using circularly polarized light as reference. Quadrature detection of the complex fringe is achieved by utilizing the intrinsic λ / 4 delay between two polarization channels. We use passive broadband polarization optics to control the polarization state of the light in the reference and sample arms and a balanced polarization diversity detection unit to simultaneously detect phase-shifted fringes. We demonstrate a 40 dB artifact suppression ratio with a swept-source optical coherence tomography system. Our proposed method is immune to sample motion and laser phase noise, and imposes no restrictions to the source bandwidth, imaging speed, or computational power. In vivo images of the human finger, as well as the cornea and retina of a non-human primate, were demonstrated.

International audience; Considerable effort has been expended on the design of control systems for wave energy converters (WECs) over the past two decades. Working from the fundamental requirement of impedance matching, a variety of conceptual and practical algorithms have emerged, which bring various levels of realism to the original complex conjugate ideal, facilitating maximum power transfer. Simplifications can be introduced, such as passive control and causal control, while some enhanced algorithms allow physical constraints to be observed, nonlinear model dynamics to be articulated, or nonideal power takeoff systems to be recognized. However, in general, model-based WEC control systems are evaluated in tandem with identical simulation models, while the sensitivity of controller performance to modeling errors is ignored. In addition, the WEC model utilized by the controller rarely, if ever, fully represents the nonlinear nature of the true WEC dynamics. This paper articulates this model sensitivity issue for different WEC control system architectures and shows that it is of potentially greater impact than for traditional regulation/servomechanism control problems. Recommendations are given on the best WEC control architecture to adopt from a sensitivity/robustness perspective.

In this article, we study two extended higher-order KdV-type models, namely, the extended Sawada-Kotera (eSK) and the extended Lax (eLax) equations. These models successfully describe propagation of dimly nonlinear long waves in fluids, ion-acoustic waves in harmonic sparklers. We firstly derive multi-soliton solutions of the models. We then construct interection solutions in-terms of hyperbolic and sinusoidal functions using the multi-soliton solutions with appropriate complex conjugate parameters. Such parameters influence and control the phase shifts, propagation direction and energies of the waves. In particularly, we present their collision solutions in the identical plane with different parametric constraints, which degenerate to the line rogue waves, x-shaped rogue waves, cnoidal periodic waves, interactions of rogue and bell waves, line breather and double breather waves. The dynamical characteristics of the wave solutions has been plotted by choosing particular values of the parameters in graphically.

Impulsive differential equations demonstrate rather more complex behavior of solutions than ordinary differential equations. This complexity is due to discontinuities of the integral curves at the moments of impulse actions. We consider a periodic impulsive system in the critical case, when the monodromy matrix of the linear approximation of the system at an equilibrium point has a pair of complex conjugate multipliers on the unit circle. An algorithm for computing of the first Lyapunov value is proposed.

In this paper, we consider a new generalized KP equation which is obtained by adding the extra term $$u_{tz}$$ in the previous equation. Based on these detailed discussions in the previous reference documentations, we know that solitons, breathers, lump waves, and rogue waves are four typical local waves. Therefore, we mainly focus on investigating the multi-soliton solutions, high-order breather solutions, and high-order rational solutions. The high-order breather solutions can be derived by taking complex conjugate parameters in the multi-soliton solutions. Applying the long wave limit method to the multi-soliton solutions, we conclude Theorem 3.1 which can be used directly to obtain high-order rational solutions. Meanwhile, for the case of three-soliton and five-soliton, the elastic interaction solutions among two parallel breathers and one soliton as well as between one breather and one soliton also can be derived, respectively. For all these types of exact solutions, we provide corresponding graphics to illustrate their dynamical characteristics in the end.

In analog receivers, the use of matched filtering with linear frequency modulated signals constitutes an effective means for detecting the presence of an incoming frame as well as for estimating the symbol timing offset (STO). However, in digital receivers, the discrete nature of the corresponding polyphase codes leads, under certain conditions of carrier frequency offsets, to a significant degradation at the output of the matched filter, which, consequently, can result in erroneous STO estimations. Exploiting the symmetry of a reference sequence consisting of a polyphase code and its complex conjugate, this work proposes a new algorithm based on reversed autocorrelation (RC) to improve the accuracy of the STO estimation obtained by matched filtering. On the one hand, the theoretical analysis demonstrates the superiority of matched filtering over RC in terms of detection performance in the low signal-to-noise ratio regime. On the other hand, it is shown by means of numerical evaluation that the RC can efficiently resolve the STO estimation ambiguity.

In the field of meteor study using radio techniques, the transmitting antenna of the radio beacons is most of the time a classical turnstile antenna design. As an alternative, it is proposed a pyramidal shaped omnidirectional antenna with complex conjugate impedance, which exhibits similar performance. By design, this type of antenna does not require additional electrical components for obtaining the circular polarization and the impedance matching. Furthermore, the pyramidal shape allows the practical implementation of the antenna using a simple and light mechanical structure.

In this paper, we introduce a quadratic approach for single-channel noise reduction. The desired signal magnitude is estimated by applying a linear filter to a modified version of the observations’ vector. The modified version is constructed from a Kronecker product of the observations’ vector with its complex conjugate. The estimated signal magnitude is multiplied by a complex exponential whose phase is obtained using a conventional linear filtering approach. We focus on the linear and quadratic maximum signal-to-noise ratio (SNR) filters and demonstrate that the quadratic filter is superior in terms of subband SNR gains. In addition, in the context of speech enhancement, we show that the quadratic filter is ideally preferable in terms of perceptual evaluation of speech quality (PESQ) and short-time objective intelligibility (STOI) scores. The advantages, compared to the conventional linear filtering approach, are particularly significant for low input SNRs, at the expanse of a higher computational complexity. The results are verified in practical scenarios with nonstationary noise and in comparison to well-known speech enhancement methods. We demonstrate that the quadratic maximum SNR filter may be superior, depending on the nonstationary noise type.

Fake projective planes are smooth complex surfaces of general type with Betti numbers equal to those of the usual projective plane. They come in complex conjugate pairs and have been classified as quotients of the two-dimensional ball by explicitly written arithmetic subgroups. In this paper we find equations of a projective model of a conjugate pair of fake projective planes by studying the geometry of the quotient of such surface by an order seven automorphism., Comment: This is a full version of "Research announcement: equations of a fake projective plane", arXiv:1710.04501. Key tables and some M2 and Magma code from the paper are included in separate files for convenience

The study of the optical properties of plasmonic nanostructures in the stationary regime has greatly benefited from the development of numerical methods, among which Finite Difference Time Domain (FDTD) is popular. In contrast, the use of these numerical tools for assessing the transient plasmonic optical response triggered by ultrashort laser pulses is hampered by the difficulty to address small variations of the material optical properties with reasonable computational time. Yet, many of the developments based on this ultrashort response rely on the dynamics of the near-field topography around the nanostructures. In this article, we present a way to bridge this gap with the complex-conjugate pole-residue pair (CCPRP) approach. A CCPRP-based FDTD simulator has been developed. First, a simple methodology to check the end-to-end accuracy of the FDTD simulation is provided. Then, in conjunction with a three-temperature model, the approach enables us to calculate the ultrafast transient near field inside and around a gold nanoparticle (AuNP) upon absorption of a subpicosecond laser pulse. The transient variation of the field intensity inside and around the AuNP is compared with the one determined by the Mie theory. The dependence of the transient field intensity on the distance away from the nanoparticle surface and on the delay time after laser pulse absorption is finally analyzed.

In Fourier domain OCT, the depth profile is mirrored about the zero delay, limiting the imaging depth to half of the entire ranging space. We present a novel configuration for OCT to robustly remove the complex conjugate artifact. Our method utilizes the intrinsic delay of circularly polarized light in two polarization channels, using only passive broadband polarization optics and conventional polarization diversity detection unit. Our method is immune to sample motion and adds no restrictions to source bandwidth, imaging speed or computational load. 45 dB suppression of the mirror artifact is demonstrated by an SSOCT with some in-vivo images.

Let $(X,J) $ be an almost complex manifold with a (smooth) involution $\sigma:X\to X$ such that $Fix(\sigma)\neq \emptyset$. Assume that $\sigma$ is a complex conjugation, i.e, the differential of $\sigma$ anti-commutes with $J$. The space $P(m,X):=\mathbb{S}^m\times X/\!\sim$ where $(v,x)\sim (-v,\sigma(x))$ is known as a generalized Dold manifold. Suppose that a group $G\cong \mathbb Z_2^s$ acts smoothly on $X$ such that $g\circ \sigma =\sigma\circ g$ for all $g\in G$. Using the action of the diagonal subgroup $D=O(1)^{m+1}\subset O(m+1)$ on the sphere $\mathbb S^{m}$ for which there are only finitely many pairs of antipodal points that are stablized by $D$, we obtain an action of $\mathcal G=D\times G$ on $\mathbb S^m\times X$, which descends to a (smooth) action of $\mathcal G$ on $P(m,X)$. When the stationary point set $X^G$ for the $G$ action on $X$ is finite, the same also holds for the $\mathcal G$ action on $P(m,X)$. The main result of this note is that the equivariant cobordism class $[P(m,X),\mathcal G]$ vanishes if and only if $[X,G]$ vanishes. We illustrate this result in the case when $X$ is the complex flag manifold, $\sigma$ is the natural complex conjugation and $G\cong (\mathbb Z_2)^n$ is contained in the diagonal subgroup of $U(n)$., Comment: 10 pages

The acoustical radiosity model (ARM) is a typical algorithm in geometrical acoustics to simulate the sound field in a room of ideally diffusely reflecting boundary. Even in such a room, the sound field decay is a complex process, as the relaxation of the sound field observed in simulations has shown. Based on the Laplace transform of the acoustical radiosity equation, this paper gives a set of properties of the ARM. It shows the system has a series of real or complex conjugate L-eigenvalues and corresponding L-eigenfunctions. Under the relaxation condition, the sound energy decay on the room boundary, generated by finite initial excitation, can be expanded as a summation of decay components, which are composed of real and/or complex conjugate decay modes. Each decay mode is a decaying and oscillating L-eigenfunction corresponding to an L-eigenvalue. The real part of the L-eigenvalue is the exponential decay rate, and the image part is the angular frequency of the oscillation. The reverberant sound field inside the room space has a similar decay structure to the boundary. As an example, the decay structure in a sphere is analyzed. The relaxation of the sound field is explained by the geometrical significance of the sound field decay.

This paper extends the concept of Complex Conjugate Control (CCC) of linear wave energy converters (WECs) to nonlinear WECs by designing optimal limit cycles with Hamiltonian Surface Shaping and Power Flow Control (HSSPFC). It will be shown that CCC for a regular wave is equivalent to a power factor of one in electrical power networks, equivalent to mechanical resonance in a mass-spring-damper (MSD) system, and equivalent to a linear limit cycle constrained to a Hamiltonian surface defined in HSSPFC. Specifically, the optimal linear limit cycle is defined as a second-order center in the phase plane projection of the constant energy orbit across the Hamiltonian surface. This concept of CCC described by a linear limit cycle constrained to a Hamiltonian surface will be extended to nonlinear limit cycles constrained to a Hamiltonian surface for maximum energy harvesting by the nonlinear WEC. The case studies presented confirm increased energy harvesting which utilizes nonlinear geometry realization for reactive power generation.

In view of the expectation that the existence of complex poles is a signal of confinement, we investigate the analytic structure of the gluon, quark, and ghost propagators in the Landau gauge QCD and QCD-like theories by employing an effective model with a gluon mass term of the Yang-Mills theory, which we call the massive Yang-Mills model. In this model, we particularly investigate the number of complex poles in the parameter space of the model consisting of gauge coupling constant, gluon mass, and quark mass for the gauge group $SU(3)$ and various numbers of quark flavors $N_F$ within the asymptotic free region. Both the gluon and quark propagators at the best-fit parameters for $N_F=2$ QCD have one pair of complex conjugate poles, while the number of complex poles in the gluon propagator varies between zero and four depending on the number of quark flavors and quark mass. Moreover, as a general feature, we argue that the gluon spectral function of this model with nonzero quark mass is negative in the infrared limit. In sharp contrast to gluons, the quark and ghost propagators are insensitive to the number of quark flavors within the current approximations adopted in this paper. These results suggest that details of the confinement mechanism may depend on the number of quark flavors and quark mass., 22 pages, 15 figures, revised version, ref. 54 corrected

We show that for each [math] there is a monic integer polynomial [math] of degree [math] which is irreducible over [math] and has two complex conjugate roots as close to the unit circle as is allowed by the Liouville-type inequality.

We consider a setting in which it is desired to find an optimal complex vector $${\mathbf {x}}\in {\mathbb {C}}^N$$ x ∈ C N that satisfies $${\mathcal {A}}({\mathbf {x}}) \approx {\mathbf {b}}$$ A ( x ) ≈ b in a least-squares sense, where $${\mathbf {b}} \in {\mathbb {C}}^M$$ b ∈ C M is a data vector (possibly noise-corrupted), and $${\mathcal {A}}(\cdot ): {\mathbb {C}}^N \rightarrow {\mathbb {C}}^M$$ A ( · ) : C N → C M is a measurement operator. If $${\mathcal {A}}(\cdot )$$ A ( · ) were linear, this reduces to the classical linear least-squares problem, which has a well-known analytic solution as well as powerful iterative solution algorithms. However, instead of linear least-squares, this work considers the more complicated scenario where $${\mathcal {A}}(\cdot )$$ A ( · ) is nonlinear, but can be represented as the summation and/or composition of some operators that are linear and some operators that are antilinear. Some common nonlinear operations that have this structure include complex conjugation or taking the real-part or imaginary-part of a complex vector. Previous literature has shown that this kind of mixed linear/antilinear least-squares problem can be mapped into a linear least-squares problem by considering $${\mathbf {x}}$$ x as a vector in $${\mathbb {R}}^{2N}$$ R 2 N instead of $${\mathbb {C}}^N$$ C N . While this approach is valid, the replacement of the original complex-valued optimization problem with a real-valued optimization problem can be complicated to implement, and can also be associated with increased computational complexity. In this work, we describe theory and computational methods that enable mixed linear/antilinear least-squares problems to be solved iteratively using standard linear least-squares tools, while retaining all of the complex-valued structure of the original inverse problem. An illustration is provided to demonstrate that this approach can simplify the implementation and reduce the computational complexity of iterative solution algorithms.

The goal of this article is to give invariant necessary and sufficient conditions for a quadratic system, presented in whatever normal form, to have anyone of 17 out of the 20 phase portraits of the family of quadratic systems with two complex conjugate invariant lines intersecting at a finite real point. The systems in this family have a maximum of one limit cycle. Among the 17 phase portraits we have two with limit cycles. We also give invariant necessary and sufficient conditions for a system to have one of the three remaining phase portraits, out of which one has a limit cycle and another one a homoclinic loop. In the region $${\mathcal {R}}$$ determined by these last conditions, due to the presence of systems with a homoclinic loop, an analytic condition, the three phase portraits cannot be separated by algebraic conditions in terms of invariant polynomials. We also give the bifurcation diagram of this family, outside the region $${\mathcal {R}}$$ , in the twelve parameter space of coefficients of the systems.

Nebe, Rains and Sloane studied the polynomial invariants for real and complex Clifford groups and they relate the invariants to the space of complete weight enumerators of certain self-dual codes. The purpose of this paper is to show that very similar results can be obtained for the invariants of the complex Clifford group $\mathcal{X}_m$ acting on the space of conjugate polynomials in $2^m$ variables of degree $N_1$ in $x_f$ and of degree $N_2$ in their complex conjugates $\overline{x_f}$. In particular, we show that the dimension of this space is $2$, for $(N_1,N_2)=(5,5)$. This solves the Conjecture 2 given in Zhu, Kueng, Grassl and Gross affirmatively. In other words if an orbit of the complex Clifford group is a projective $4$-design, then it is automatically a projective $5$-design., Comment: 12 pages

Photonic pseudospin-1/2 systems, which exhibit Dirac cone dispersion at Brillouin zone corners in analogy to graphene, have been extensively studied in recent years. However, it is known that a linear band crossing of two bands cannot emerge at the center of Brillouin zone in a two-dimensional photonic system respecting time reversal symmetry. Using a square lattice of elliptical magneto-optical cylinders, we constructed an unpaired Dirac point at the Brillouin zone center as the intersection of the second and third bands corresponding to the monopole and dipole excitations. Effective medium theory can be applied to the two linearly crossed bands with the effective constitutive parameters numerically calculated using the boundary effective medium approach. It is shown that only the effective permittivity approaches zero while the determinant of the nonzero effective permeability vanishes at the Dirac point frequency, showing a different behavior from the double-zero index metamaterials obtained from the pseudospin-1 triply degenerate points for time reversal symmetric systems. Exotic phenomena, such as the Klein tunneling and Zitterbewegung, in the pseudospin-1/2 system can be well understood from the effective medium description. When the Dirac point is lifted, the edge state dispersion near the $\Gamma$ point can be accurately predicted by the effective constitutive parameters. We also further realized magneto-optical complex conjugate metamaterials for a wide frequency range by introducing a particular type of non-Hermittian perturbations which make the two linear bands coalescence to form exceptional points at the real frequency., Comment: 48 pages, 14 figures

A conformal map is one that preserves angles. In the case of mappings from one connected domain in \(\mathbb {C}\) to another, such a map is holomorphic, or else its complex conjugate is holomorphic.

In this paper, first, we investigate the commuting property between the normal Jacobi operator~${\bar R}_N$ and the structure Jacobi operator~$R_{\xi}$ for Hopf real hypersurfaces in the complex quadric~$Q^m = SO_{m+2}/SO_mSO_2$, $m \geq 3$, which is defined by ${\bar R}_N R_{\xi} = R_{\xi}{\bar R}_N$. Moreover, a new characterization of Hopf real hypersurfaces with $\mathfrak A$-principal singular normal vector field in the complex quadric~$Q^{m}$ is obtained. By virtue of this result, we can give a remarkable classification of Hopf real hypersurfaces in the complex quadric~$Q^{m}$ with commuting Jacobi operators., Comment: 20 pages. arXiv admin note: text overlap with arXiv:1907.04661, arXiv:1605.05316

Complex surface methods are widely used in various industrial tools such as cutting tools, shell of television, turbine blades etc. Researchers tend to concentrate on conjugate surfaces, which can be produced by digital method to optimize any process. Conjugate surface deals with relative motion. Most of the cases pertain to power transmitting devices such as gears. Providing conjugate surface is very difficult, and to overcome this problem, digital conjugate method has been introduced. It produces event complex conjugate surfaces. In such cases, researchers have focused on Digital Gear Tooth Surface (DGTS). It causes conjugate motion between the gear teeth, which is represented through discrete points, and with these discrete points, the digital surface of conjugate moments is resolved. Computer simulated examples generate and machine the non-standard and complex shapes of digital conjugate tooth surfaces. This technique is not only useful for machining discrete digital gear tooth surfaces and gear tooth surface with complex design, but also other 3D digital surfaces with optimized process.

Continuum models for time-reversal (TR) invariant topological insulators (TIs) in $d \geq 3$ dimensions are provided by harmonic oscillators coupled to certain $SO(d)$ gauge fields. These models are equivalent to the presence of spin-orbit (SO) interaction in the oscillator Hamiltonians at a critical coupling strength (equivalent to the harmonic oscillator frequency) and leads to flat Landau Level (LL) spectra and therefore to infinite degeneracy of either the positive or the negative helicity states depending on the sign of the SO coupling. Generalizing the results of Haaker et al. to $d \geq 4$, we construct vector operators commuting with these Hamiltonians and show that $SO(d,2)$ emerges as the non-compact extended dynamical symmetry. Focusing on the model in four dimensions, we demonstrate that the infinite degeneracy of the flat spectra can be fully explained in terms of the discrete unitary representations of $SO(4,2)$, i.e. the {\it doubletons}. The degeneracy in the opposite helicity branch is finite, but can still be explained exploiting the complex conjugate {\it doubleton} representations. Subsequently, the analysis is generalized to $d$ dimensions, distinguishing the cases of odd and even $d$. We also determine the spectrum generating algebra in these models and briefly comment on the algebraic organization of the LL states w.r.t to an underlying "deformed" AdS geometry as well as on the organization of the surface states under open boundary conditions in view of our results., Comment: 22 pages

We report the hosting of two complex conjugate exceptional points (EPs) in two complementary optical waveguides and explore the nonadiabatic chiral and reverse-chiral modal dynamics in two waveguides following dynamical EP encirclement schemes.

Let $X$ be a complex projective variety defined over $\mathbb R$. Recently, Bernardi and the first author introduced the notion of admissible rank with respect to $X$. This rank takes into account only decompositions that are stable under complex conjugation. Such a decomposition carries a label, i.e., a pair of integers recording the cardinality of its totally real part. We study basic properties of admissible ranks for varieties, along with special examples of curves; for instance, for rational normal curves admissible and complex ranks coincide. Along the way, we introduce the scheme theoretic version of admissible rank. Finally, analogously to the situation of real ranks, we analyze typical labels, i.e., those arising as labels of a full-dimensional Euclidean open set. We highlight similarities and differences with typical ranks., Comment: 18 pp

This paper provides necessary and sufficient conditions for the existence of a pair of complex conjugate roots, each of multiplicity two, in the spectrum of a linear time-invariant single-delay equation of retarded type. This pair of roots is also shown to be always strictly dominant, determining thus the asymptotic behavior of the system. The proof of this result is based on the corresponding result for real roots of multiplicity four, continuous dependence of roots with respect to parameters, and the study of crossing imaginary roots. We also present how this design can be applied to vibration suppression and flexible mode compensation., Comment: arXiv admin note: substantial text overlap with arXiv:2002.06146

This chapter provides a self-contained introduction to quantum groups or Hopf algebras and their associated braided categories of (co)modules in the (co)quasitriangular case and crossed modules in the case of invertible antipode. We then cover the construction of differential structures on them and on associated algebras, including the theory of the quantum Lie algebra of a bicovariant calculus and of braided-Lie algebras in the coquasitriangular case. Basic examples include q-SU2 and the associated q-sphere. The chapter ends with the notion of a bar category needed to formulate complex conjugation and *-operations in a more categorical way.

This paper is describing solutions for singularly perturbed linear systems which are considered in a particularly critical case. The matrix of a linear system has complex conjugate eigenvalues. The eigenvalues ​​of the matrix system under consideration do not have zeros on the boundary of the region under consideration and outside of this region. Imaginary parts of the eigenvalues ​​of the matrix are positive with the exception of boundary points in the considered domain. For evaluation of functions, a proved lemma was used. A uniform approximation was constructed for the solution of the initial Cauchy problem in particularly critical case with a certain degree of accuracy.

A state-specific multireference coupled-cluster (MRCC) method based on Arponen's bivariational principle is presented, the bivar-MRCC method. The method is based on single-reference theory and therefore has a relatively straightforward formulation and modest computational complexity. The main difference from established methods is the bivariational formulation, in which independent parameterizations of the wave function (ket) and its complex conjugate (bra) are made. Importantly, this allows manifest multiplicative separability of the state (exact in the extended bivar-MRECC version of the method and approximate otherwise), and additive separability of the energy, while preserving polynomial scaling of the working equations. A feature of the bivariational principle is that the formal bra and ket references can be included as bivariational parameters, which eliminates much of the bias toward the formal reference. A pilot implementation is described, and extensive benchmark calculations on several standard problems are performed. The results from the bivar-MRCC method are comparable to established state-specific multireference methods. Considering the relative affordability of the bivar-MRCC method, it may become a practical tool for non-experts.

Open systems with gain, loss, or both, described by non-Hermitian Hamiltonians, have been a research frontier for the past decade. In particular, such Hamiltonians which possess parity-time ($\mathcal{PT}$) symmetry feature dynamically stable regimes of unbroken symmetry with completely real eigenspectra that are rendered into complex conjugate pairs as the strength of the non-Hermiticity increases. By subjecting a $\mathcal{PT}$-symmetric system to a periodic (Floquet) driving, the regime of dynamical stability can be dramatically affected, leading to a frequency-dependent threshold for the $\mathcal{PT}$-symmetry breaking transition. We present a simple model of a time-dependent $\mathcal{PT}$-symmetric Hamiltonian which smoothly connects the static case, a $\mathcal{PT}$-symmetric Floquet case, and a neutral-$\mathcal{PT}$-symmetric case. We analytically and numerically analyze the $\mathcal{PT}$ phase diagrams in each case, and show that slivers of $\mathcal{PT}$-broken ($\mathcal{PT}$-symmetric) phase extend deep into the nominally low (high) non-Hermiticity region., Comment: 9 pages, 3 figures

The S-transform (ST), which is an important tool in signal processing, is a conceptual principle of the Fourier transform (FT) with a Gaussian window function. It has been observed from literature study that only linearity, scaling, time-shifting and convolution theorem of ST is documented. This led to the finding of remaining properties of ST in order to establish it as complete transform technique. Along with this a new better definition of convolution theorem for ST is also presented. Therefore this paper is focused on formulation of properties such as time-reversal, time-derivatives, complex conjugate, convolution theorem, correlation theorem and Parseval’s theorem. A comparative analysis of existing convolution theorem with proposed convolution theorem is also given in this manuscript.

The experimental realization of balanced gain and loss in a quantum system has been a long standing goal in quantum mechanics since the introduction of the concept of $\mathcal{PT}$ symmetry and has only recently been achieved. In this paper we analyze balanced gain and loss in Gaussian multi-well potentials with either only gain or loss in each well. By means of symmetrization via matrix models we can construct asymmetric extended potentials with partially real or complex conjugate spectra. This will be demonstrated explicitly for double-well and triple-well systems. Such systems can be realized with Bose-Einstein condensates in optical trapping potentials in the presence of localized particle gain and loss. The usage of asymmetric potentials in the process is more versatile and is considered beneficial in real experimental implementations., 10 pages, 6 figures, accepted for publication in Phys. Rev. A

This paper addresses the problem of designing universal quantum circuits to transform $k$ uses of a $d$-dimensional unitary input-operation into a unitary output-operation in a probabilistic heralded manner. Three classes of protocols are considered, parallel circuits, where the input-operations can be simultaneously, adaptive circuits, where sequential uses of the input-operations are allowed, and general protocols, where the use of the input-operations may be performed without a definite causal order. For these three classes, we develop a systematic semidefinite programming approach that finds a circuit which obtains the desired transformation with the maximal success probability. We then analyse in detail three particular transformations; unitary transposition, unitary complex conjugation, and unitary inversion. For unitary transposition and unitary inverse, we prove that for any fixed dimension $d$, adaptive circuits have an exponential improvement in terms of uses $k$ when compared to parallel ones. For unitary complex conjugation and unitary inversion we prove that if the number of uses $k$ is strictly smaller than $d-1$, the probability of success is necessarily zero. We also discuss the advantage of indefinite causal order protocols over causal ones and introduce the concept of delayed input-state quantum circuits., Closer to the published version. Typos were corrected and more details on SDP methods are now provided. This paper includes results that were removed from the first version our previous work (arXiv:1810.06944). Matlab code to accompany this article can be found at https://github.com/mtcq/unitary_inverse

To solve the high complexity of the subspace-based direction-of-arrival (DOA) estimation algorithm, a super-resolution DOA algorithm is built in this paper. However, in this method, matrix decomposition is required for each search angle. Therefore, in this paper, real-valued processing is used to reduce the scanning range by half, which is less effective in algorithm complexity. The super-resolution algorithm mainly uses the conservation of energy. By exploring the relationship between the covariance matrix and its complex conjugate, we constructed the real-valued matrix and introduced a real-valued searching source to make the operation of the matrix real-valued. Finally, the simulation experiments show that the proposed algorithm not only reduces the spectral search range by half but also has a higher angular resolution than the traditional algorithm.

It was understood that Chern insulators cannot be realized in the presence of PT symmetry. In this paper, we reveal a new class of PT-symmetric Chern insulators, which has internal degrees of freedom forming real representations of a symmetry group with a complex endomorphism field. As a generalization to the conventional 2n-dimensional Chern insulators with integer n ≥ 1, these PT-symmetric Chern insulators have the n-th complex Chern number as their topological invariant, and have a ℤ classification given by the equivariant orthogonal K theory. Thus, in a fairly different sense, there exist ubiquitously Chern insulators with PT symmetry. By generalizing the Thouless charge pump argument, we find that, for a PT-symmetric Chern insulator with Chern number ν, there are equally many ν flavors of coexisting left- and right-handed chiral modes. Chiral modes with opposite chirality are complex conjugates to each other as complex representations of the internal symmetry group, but are not isomorphic. For the physical dimensionality d = 2, the PT-symmetric Chern insulators may be realized in artificial systems including photonic crystals and periodic mechanical systems.

International audience; We analyse the properties of the inter-event times for planar linear time-invariant systems controlled by an event-triggered state-feedback law. The triggering rule is given by the relative threshold strategy and we assume that the tunable triggering parameter is small. Several cases are distinguished depending on the nature of the eigenvalues of the (continuous-time) closed-loop system matrix in absence of sampling. When these eigenvalues are real, it is shown that the inter-event times lie in a neighborhood of a given constant for all positive times or converge to the neighborhood of a given constant as time grows. When the eigenvalues are complex conjugates, the inter-event times oscillate with a varying period for which we give an estimate. Moreover, the values taken by the inter-event times over this varying period are approximately the same for all initial conditions. As a consequence, one can run a single simulation over a given interval of time to infer properties of the inter-event times for all initial conditions and all positive times. Numerical simulations are provided to support the presented theoretical guarantees. These results help to understand the behaviour of the inter-event times, instead of solely relying on numerical simulations, and can be exploited to evaluate the performance of the considered triggering condition in terms of average inter-transmission times.

Complex-valued neural networks (CVNNs) have become a hot research topic in neural network community for their powerful ability in dealing with complex-valued signal problems. However, as one of the most popular training methods for CVNNs, complex gradient method still suffers from slow convergence and easily traps into the local minima. To this end, in this paper we combine the conjugate gradient method and the Barzilai-Borwein method, and propose a complex conjugate gradient training method with Barzilai-Borwein stepsize for complex-valued neural networks. The superiority of the proposed method is substantiated with two numerical examples.

New approach is developed in this paper for mode matching of LTI models at different grid points such that a Linear Parameter Varying model can be obtained through interpolation. The approach is based on matching eigenvector associated with each eigenvalue such that state consistency can be guaranteed for local LTI models. The novelty of the developed approach is the ability to handle mode veering phenomena, where two distinct real poles converge into two repeated ones then separate into complex conjugate poles and vise versa. Linear interpolation procedure based on Least-Squares errors is also presented to interpolate local LTI models. The proposed approach is demonstrated through numerical example.

A three dimensional Lorentzian hypersurface x : M 1 3 → R 1 4 is called conformally flat if its induced metric is conformal to the flat Lorentzian metric, this property is preserved under the conformal transformation of R 1 4 . Using the projective light-cone model, for those ones whose shape operators have a pair of complex conjugate eigenvalues, we study the integrability condition by constructing a scalar conformal invariant and a canonical moving frame in this paper. It follows that these hypersurfaces can be determined by the solutions to a system of three-order partial differential equations.

Conjugation spaces are topological spaces equipped with an involution such that their fixed points have the same mod $2$ cohomology (as a graded vector space, a ring, and even an unstable algebra) but with all degrees divided by two, generalizing the classical examples of complex projective spaces under complex conjugation. Spaces which are constructed from unit balls in complex Euclidean spaces are called spherical and are very well understood. Our aim is twofold. We construct "exotic" conjugation spaces and study the realization question: which spaces can be realized as real loci, i.e., fixed points of conjugation spaces. We identify obstructions and provide examples of spaces and manifolds which cannot be realized as such., Comment: 16 pages

The recently realized photonic crystal Fano laser constitutes the first demonstration of passive pulse generation in nanolasers [Nat. Photonics $\boldsymbol{11}$, 81-84 (2017)]. We show that the laser operation is confined to only two degrees-of-freedom after the initial transition stage. We show that the original 5D dynamic model can be reduced to a 1D model in a narrow region of the parameter space and it evolves into a 2D model after the exceptional point, where the eigenvalues transition from being purely to a complex conjugate pair. The 2D reduced model allows us to establish an effective band structure for the eigenvalue problem of the stability matrix to explain the laser dynamics. The reduced model is used to associate a previously unknown origin of instability with a new unstable periodic orbit separating the stable steady-state from the stable periodic orbit., 12 pages, 7 figures, journal, Phys. Rev. A, before editorial correction

The traditional Cohen's class time-frequency representation is extended to the linear canonical domain by using a well-established closed-form instantaneous cross-correlation function (CICF) type of linear canonical transform (LCT) free parameters embedded approach. The derived CICF type of Cohen's class (CICFCC) unifies some well-known Cohen's classes in linear canonical domains including the affine characteristic, basis function, convolution expression and instantaneous crosscorrelation function types of Cohen's classes, and can be considered as the Cohen's class's closed-form representation in linear canonical domains. A fundamental theory about the CICFCC's essential properties, such as marginal distribution, energy conservation, unique reconstruction, Moyal formula, complex conjugate symmetry, time reversal symmetry, scaling property, time shift property, frequency shift property, and LCT invariance, is then established. Possible applications are also carried out to illustrate that the CICFCC outperforms the traditional one in nonstationary signal separation and detection.

Service load replication performed on multiaxial hydraulic test rigs has been widely applied in automotive engineering for durability testing in laboratory. The frequency-domain off-line iterative learning control is used to generate the desired drive file, i.e. the input signals which drive the actuators of the test rig. During the iterations an experimentally identified linear frequency-domain system model is used. As the durability test rig and the specimen under test have a strong nonlinear behavior, a large number of iterations are needed to generate the drive file. This process will cause premature deterioration to the specimen unavoidably. In order to accelerate drive file construction, a method embedding complex conjugate gradient algorithm into the conventional off-line iterative learning control is proposed to reproduce the loading conditions. The basic principle and monotone convergence of the method is presented. The drive signal is updated according to the complex conjugate gradient and the optimal learning gain. An optimal learning gain can be obtained by an estimate loop. Finally, simulations are carried out based on the identified parameter model of a real spindle-coupled multiaxial test rig. With real-life spindle forces from the wheel force transducer in the proving ground test to be replicated, the simulation results indicate that the proposed conventional off-line iterative learning control with complex conjugate gradient algorithm allows generation of drive file more rapidly and precisely compared with the state-of-the-art off-line iterative learning control. Few have been done about the proposed method before. The new method is not limited to the durability testing and can be extended to other systems where repetitive tracking task is required.

We discuss the use of pairs of optical masks for setting tunable optical focalizers and for implementing controllable absorption masks. For phase-only masks, one element of a given pair has a complex amplitude transmittance that is equal to the complex conjugate of the other element. For the absorption masks, we use a suitable attenuation offset. Then, from the attenuation offset value, one element has the opposite absorption profile that the other element. These methods are useful for generating varifocal lenses, governable prisms, tunable axicons, controllable axilenses, for tuning field depth, and for controlling Super Gaussian beams.

We develop a complete theory of symmetry and topology in non-Hermitian physics. We demonstrate that non-Hermiticity ramifies the celebrated Altland-Zirnbauer symmetry classification for insulators and superconductors. In particular, charge conjugation is defined in terms of transposition rather than complex conjugation due to the lack of Hermiticity, and hence chiral symmetry becomes distinct from sublattice symmetry. It is also shown that non-Hermiticity enables a Hermitian-conjugate counterpart of the Altland-Zirnbauer symmetry. Taking into account sublattice symmetry or pseudo-Hermiticity as an additional symmetry, the total number of symmetry classes is 38 instead of 10, which describe intrinsic non-Hermitian topological phases as well as non-Hermitian random matrices. Furthermore, due to the complex nature of energy spectra, non-Hermitian systems feature two different types of complex-energy gaps, point-like and line-like vacant regions. On the basis of these concepts and K-theory, we complete classification of non-Hermitian topological phases in arbitrary dimensions and symmetry classes. Remarkably, non-Hermitian topology depends on the type of complex-energy gaps and multiple topological structures appear for each symmetry class and each spatial dimension, which are also illustrated in detail with concrete examples. Moreover, the bulk-boundary correspondence in non-Hermitian systems is elucidated within our framework, and symmetries preventing the non-Hermitian skin effect are identified. Our classification not only categorizes recently observed lasing and transport topological phenomena, but also predicts a new type of symmetry-protected topological lasers with lasing helical edge states and dissipative topological superconductors with nonorthogonal Majorana edge states. Furthermore, our theory provides topological classification of Hermitian and non-Hermitian free bosons., Comment: 52 pages, 7 figures, 16 tables