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Plane Poiseuille flow in viscous compressible fluid is known to be asymptotically stable if Reynolds number R and Mach number M are sufficiently small. On the other hand, for R and M being not necessarily small, an instability criterion for plane Poiseuille flow is known, and the criterion says that, when R increases, a pair of complex conjugate eigenvalues of the linearized operator cross the imaginary axis. In this paper it is proved that a spatially periodic traveling wave bifurcates from plane Poiseuille flow when the critical eigenvalues cross the imaginary axis.

In the first part of the paper, it is proved that for 1 < p < ∞ the couple (K , K ∞ ) of coinvariant subspaces of the shift operator on the unit circle is K-closed in the couple (L p (T),L∞ (T)). This property underlies basically all problems of real interpolation for the first couple. Also, a weighted analog of the above statement is established. In the second part it is shown that, given two closed ideals I and J in a uniform algebra such that the complex conjugate of I ∩ J is not included in some of them, the sum I + J is not closed. Though the methods of study in the two parts are quite different, the topics are related by the fact that the question treated in the second part emerged during the work on the first.

Let $\unicode[STIX]{x1D70C}\in (0,\infty ]$ be a real number. In this short note, we extend a recent result of Marques and Ramirez [‘On exceptional sets: the solution of a problem posed by K. Mahler’, Bull. Aust. Math. Soc.94 (2016), 15–19] by proving that any subset of $\overline{\mathbb{Q}}\cap B(0,\unicode[STIX]{x1D70C})$, which is closed under complex conjugation and contains $0$, is the exceptional set of uncountably many analytic transcendental functions with rational coefficients and radius of convergence $\unicode[STIX]{x1D70C}$. This solves the question posed by K. Mahler completely.

This paper aims at resolving the issues related to increased dimensionality of data in data mining. In this paper, Sparse Cholesky decomposition (SCD) is combined with Non-integer Matrix Factorization (NMF) to remove the problems arising due to increased data dimensionality. The increased data dimensionality in datasets is probably due to non-orthogonality of datasets. The complex conjugate values is used to remove the sparse matrix and a complex gradient algorithm reduces the sparse matrix by the extraction of conjugate values. The SCD-MNF extracts the feature vector and upper triangular matrix linearly maps the feature vector obtained from the SCD. Hence, NMF is employed with SCD for structuring the datasets and this helps to form a well-defined data geometry. The proposed system is evaluated against normalized mutual information and accuracy against different text datasets. The results prove that SCD-NMF attains better results than conventional methods in finding the instances related to the given query.

We resolve the complex conjugate ambiguity in spectral-domain optical coherence tomography (SD-OCT) by using achromatic two-harmonic method. Unlike previous researches, the optical phase of the fiber interferometer is modulated by an achromatic phase shifter based on an optical delay line. The achromatic phase modulation leads to a wavelength-independent scaling coefficient for the two harmonics. Dividing the mean absolute value of the first harmonic by that of the second harmonic in a B-scan interferogram directly gives the scaling coefficient. It greatly simplifies the determination of the magnitude ratio between the two harmonics without the need of third harmonic and cumbersome iterative calculations. The inverse fast Fourier transform of the complex-valued interferogram constructed with the scaling coefficient, first and second harmonics yields a full-range OCT image. Experimental results confirm the effectiveness of the proposed achromatic two-harmonic technique for suppressing the mirror artifacts in SD-OCT images.

In this paper, first we give a non-existence theorem for a real hypersurface M with anti-commuting shape operator S in the complex quadric Q m , that is, S ϕ + ϕ S = 0 , where ϕ denotes the structure tensor of M in Q m . By using such a non-existence theorem, we give a classification of Hopf hypersurfaces with recurrent normal Jacobi operator in the complex quadric Q m .

This work deals with the stabilization problem of a particular class of high-order unstable linear systems with time delay. In particular, systems with one unstable pole, q — 1 complex conjugate stable poles and one minimum phase zero. To solve this problem a modified version for the traditional PI controller, called in this work the Proporcional Integral Filtered PI f , is proposed. This new scheme includes a low-pass first order filter and allows improving the existing results on controlling high-order systems with time delay. Necessary and sufficient conditions for the existence of the PI f controller are expressed in terms of the parameters of the system and the maximum allowable time-delay magnitude. The proposed control scheme is illustrated by a numerical example applied to the temperature control of an unstable Continuously Stirred Tank Reactor (CSTR) linear model focused on the production of propylene glycol.

This work presents an alternating-direction implicit (ADI) finite-difference time-domain (FDTD) scheme for the study of structures that involve materials with arbitrary frequency dispersion. The material dispersion is fitted to the complex–conjugate pole-residue (CCPR) terms model, and a novel, to the best of our knowledge, numerical formulation is presented based on auxiliary differential equations and two-step ADI methodology. Additionally, the proposed technique is combined with the concept of the perfectly matched layer, and a new implicit scheme is introduced for the termination of media with CCPR dispersion in the ADI-FDTD framework. The ADI-FDTD formulation is compared with the explicit FDTD scheme for several benchmark two-dimensional problems in terms of accuracy and efficiency. The suggested algorithm is proven to be robust and capable of simulating applications in different frequency regions, spanning from microwaves to optical frequencies. It can provide a powerful tool for the analysis of nanostructures involving both strongly dispersive and nanosized materials, such as plasmonic metasurfaces, antennas, core–shell nanoparticle systems, light-trapping plasmonic solar cells, surface-enhanced Raman spectroscopy substrates, or nanodevices based on epsilon-near-zero materials.

In this paper, we propose a three-level linearized finite difference scheme for the high-dimensional nonlinear Ginzburg–Landau equation with fractional Laplacian. The Crank–Nicolson scheme is used for time discretization, and the fractional Laplacian is discretized by the fractional centered difference scheme. The proposed difference scheme (i.e. a linear system) can be solved efficiently by fast Fourier transform (FFT) and complex conjugate gradient method, since the coefficient matrix is a multi-level block Toeplitz matrix. Furthermore, we analyze the unique solvability and boundedness of solution of the difference scheme by the discrete energy method. It is also proved that the difference scheme is unconditionally stable and second-order accurate in time and space with respect to l ∞ -norm. Finally, several numerical examples are provided to validate the theoretical results.

In the present paper, first, resonant multi-soliton solutions of a combined pKP-BKP equation were obtained by carrying out the linear superposition principle and a group of 3D and contour images vividly represent the process of wave motion with physical interpretation. Next, M-breather solutions including 1-breather, 2-breather, 3-breather and hybrid solutions between breather and soliton have been constructed with their dynamic characters and physical structure via the complex conjugate method and symbolic computation. Finally, by combining the long wave limit method and the complex conjugate method in term of N-soliton solutions, M-lump solutions including 1-lump, 2-lump and 3-lump and two types of hybrid solutions between lumps and solitons were obtained. In addition, several groups of images exhibited their dynamic structure and physical properties with physical interpretation via symbolic computation. Furthermore, the obtained results have immensely augmented the exact solutions of a combined pKP-BKP equation on the available literature and enabled us to understand the nonlinear dynamic system deeply.

In order to solve the problem that the traditional DOA estimation algorithm has low angular resolution under low SNR and small snapshot conditions, a super-resolution DOA algorithm is proposed based on Pseudo data reconstruction. Firstly, using the odd function of the array steering vector function, a new matrix is constructed by using original array received covariance matrix and its complex conjugate. Then, the scanning source is introduced into the new matrix to construct a scanning covariance matrix. When the DOA of the scanning source is the same as the DOA of the target or the symmetric DOA of the target, the first noise eigenvalue of the scanning covariance matrix is twice as large as the first eigenvalue of the noise subspace corresponding to the original signal covariance matrix. Otherwise, there is no such relationship. In consequence, this property is used to construct the spatial spectrum for the DOA estimation. Then the interpolation method is used to overcome the shortcomings that the algorithm in this paper cannot be used for arbitrary arrays. Finally, simulation experiments verify the correctness and feasibility of the proposed algorithm.

The present paper describes a closed-form transfer function of low-delay band-pass maximally flat FIR digital differentiators. Band-pass maximally flat FIR digital differentiators provide extremely high-accuracy differentiation around the center frequency which is adjusted arbitrarily. At the same time, they can reduce noise in the frequency except around the center frequency. However, the conventional method for designing band-pass maximally flat FIR digital differentiators requires linear phase characteristics. In contrast, the proposed method can realize low-delay characteristics as well as linear phase characteristics and, therefore, is a general expression of band-pass maximally flat FIR digital differentiators. The proposed transfer function is achieved as the sum of two maximally flat complex FIR digital differentiators, the coefficients of which are complex conjugates of each other. The transfer functions of these complex differentiators are derived as closed-form solutions, so that the proposed transfer function is also described as a closed-form solution. Through design examples, the effectiveness of the proposed method is confirmed.

The problem of orbital stability of a periodic motion of an autonomous two-degreeof- freedom Hamiltonian system is studied. The linearized equations of perturbed motion always have two real multipliers equal to one, because of the autonomy and the Hamiltonian structure of the system. The other two multipliers are assumed to be complex conjugate numbers with absolute values equal to one, and the system has no resonances up to third order inclusive, but has a fourth-order resonance. It is believed that this case is the critical one for the resonance, when the solution of the stability problem requires considering terms higher than the fourth degree in the series expansion of the Hamiltonian of the perturbed motion. Using Lyapunov’s methods and KAM theory, sufficient conditions for stability and instability are obtained, which are represented in the form of inequalities depending on the coefficients of series expansion of the Hamiltonian up to the sixth degree inclusive.

The phenomenon of extraordinary acoustic transmission ( eat ) in a resonator, which has recently been investigated experimentally, is studied theoretically. It is shown that the combination of a single propagating mode and a symmetry orthogonal to the direction of propagation for a resonator leads to eat . This is accomplished by decomposing the problem using symmetry, the Blaschke product and the properties of functions of a single complex variable which have modulus one on the real axis. The conditions of a resonator requires that the solution has singularities in the analytic extension to complex frequencies (resonances) and it is precisely near these resonances that we observe eat . The condition of a Blaschke product requires that there is a zero at the complex conjugate of the singularity and eat occurs when the solution on the real axis passes between these complex conjugate pairs of poles and zeros. A detailed numerical study of the problem is conducted and we show that once the single mode of propagation or the symmetry is broken then eat (at least perfect transmission) no longer holds generally.

By using the hierarchical identification principle, the gradient-based iterative algorithm is suggested for solving a complex conjugate and transpose matrix equation. With the tools of the real representation of a complex matrix and the vec-operator, a new convergence proof is offered. The necessary and sufficient conditions for the convergence factor is determined to guarantee the convergence of the algorithm for any initial iterative matrix. Also a conjecture proposed by Wu et al. (2011) is solved. Two numerical examples are offered to illustrate the effectiveness of the suggested algorithm and conclusions.

We consider the Schwarz problem for J-analytic functions for the case in which the Jordan basis Q of the matrix J contains complex conjugate vectors. Conditions on the matrix Q are obtained under which there exists a unique solution of the Schwarz problem in Holder classes.

We introduce the notion of Reeb invariant Ricci tensor for real hypersurfaces in the complex quadric Q m = S O m + 2 ∕ S O m S O 2 . The Reeb invariant Ricci tensor implies that the unit normal vector field N becomes A -principal or A -isotropic. Then according to each case, we give a complete classification of real hypersurfaces in Q m = S O m + 2 ∕ S O m S O 2 with Reeb invariant Ricci tensor.

In this paper, we extend the discrete unified gas kinetic scheme (DUGKS) to simulate conjugate heat transfer flows. The conjugate interface that separates two media with different thermophysical properties is handled by a ghost-cell (GC) immersed boundary method. Specifically, fictitious cells are set in both domains to implement the continuity conditions of both temperature and heat flux at the conjugate interface. Information at the ghost cells is unavailable and should be reconstructed from the interface and neighboring field. With interface constraints involved in the reconstruction, the conjugate condition can be incorporated smoothly into the process of solving the full flow field. Those features make the present GC-DUGKS capture the conjugate interface, without introducing an additional term or modifying the equilibrium distribution function as in previous studies. Furthermore, the Strang-splitting scheme is applied for convenient handling of the thermal force term in the governing equation. Simulations of several well-established natural convection flows containing complex conjugate interfaces are performed to validate the GC-DUGKS. The results demonstrate the accuracy and feasibility of the present method for conjugate heat transfer problems.

Soliton molecules may exist experimentally and theoretically. In this work, we generate some soliton molecules for (2 + 1)–dimensional Sawada–Kotera equation by multiple soliton solutions and a subtle velocity resonance ansatz excited by an observation of bound state behavior of solitons. Asymmetric solitons can be derived by selecting specific parameters of soliton molecules. On the basis of multiple soliton solutions, some new interaction structures consisting of soliton molecules (SMs), breather molecules (BMs) and/or soliton-breather molecules (SBMs) are found by dealing with partial parameters under the velocity resonance and module resonance mechanisms along with complex conjugate condition assumptions. Abundant results show excellent performances and make the problem integrated solving approach a promising one.

We will consider the indefinite multidimensional truncated moment problem. Necessary and sufficient conditions for a given truncated multisequence to have a signed representing measure with minimal cardinality of its support are given by the existence of a rank preserving extension of a multivariate Hankel matrix (built from the given truncated multisequence) such that the corresponding associated polynomial ideal is real radical. This result is a special case of a more general characterisation of truncated multisequences with a minimal complex representing measure whose support is symmetric with respect to complex conjugation (which we will call quasi-complex). One motivation for our results is the fact that positive semidefinite truncated multisequences need not have a positive representing measure. Thus, our main result gives the potential for computing a signed representing measure μ with Jordan decomposition μ = μ + − μ − , where card μ − is small. We will illustrate this point on concrete examples.

The Arnoldi process can be applied to inexpensively approximate matrix functions of the form f ( A ) v and matrix functionals of the form v ∗ ( f ( A ) ) ∗ g ( A ) v , where A is a large square non-Hermitian matrix, v is a vector, and the superscript ∗ denotes transposition and complex conjugation. Here f and g are analytic functions that are defined in suitable regions in the complex plane. This paper reviews available approximation methods and describes new ones that provide higher accuracy for essentially the same computational effort by exploiting available, but generally not used, moment information. Numerical experiments show that in some cases the modifications of the Arnoldi decompositions proposed can improve the accuracy of v ∗ ( f ( A ) ) ∗ g ( A ) v about as much as performing an additional step of the Arnoldi process.

We report a time-reversal method based on the Richards-Wolf vectorial diffraction theory to generate a prescribed polarization topology on a defined trajectory within areas of relatively high intensity. An example is given to generate transversely oriented optical Möbius strips that wander around an axis perpendicular to the beam propagation direction. A number of sets of dipole antennae are purposefully positioned on a defined trajectory in the y = 0 plane and the radiation fields are collected by one high-NA objective lens. By sending the complex conjugate of the radiation fields in a time-reversed manner, the focal fields are calculated and the optical polarization topology on the trajectory can be tailored to form prescribed Möbius strips. The ability to control optical polarization topologies may find applications in nanofabrication, quantum communication, and light-matter interaction.

We show how to characterize the indistinguishability of up to four identical, bosonic or fermionic particles, which are rendered partially distinguishable through their internal degrees of freedom prepared in mixed states. This is accomplished via their counting statistics when subjected to a highly symmetric unitary acting upon their external (i.e. dynamical) degrees of freedom. For pure internal states, we further extract information on the particles’ collective phases, which ultimately allows for an experimental reconstruction of the full many-particle density operator up to complex conjugation.

By considering the nonlinear propagations of electron acoustic waves in quantum magnetoplasmas, we successfully derive a new ( )-dimensional nonlinear evolution equation (NLEE) via the standard reductive perturbation theory. Through introducing a suitable transformation, the Hirota's bilinear equation of the ( )-dimensional NLEE is first found analytically. Some interesting mathematical analytical results are further investigated, including some dynamics of solitary waves, breather waves, and hybrid waves. Based on the resulting bilinear form, an effective method is presented to find its N -solitary wave solutions. Moreover, by taking particular complex conjugate conditions, we further obtain its N -th–order breather wave solutions and general hybrid wave solutions which have not been reported yet. Finally, some significant characteristics of these nonlinear waves are illustrated to better understand their dynamical behavior. For these analytic solutions, the influences of each parameter on the dynamics of solitary waves, breather waves, and hybrid waves are discussed, and the method of how to control such nonlinear waves is also suggested.

The complex interaction between the volatility and the price trend is examined. We model stochastic asset prices using the asset flow model with randomness arising directly from supply and demand. We show that the volatility is smallest at the extrema of the expected logarithm of the price. Linearizing the stochastic differential equation (SDE) about equilibrium, we obtain an exact relation for the autocovariance function, and relate it to the (3 by 3) Jacobian of the linearized SDE. In particular, we find the conditions under which one has a pair of complex conjugate eigenvalues of the Jacobian resulting in oscillations. The frequency of the oscillations depends only on the imaginary part of the complex pair, while the decay rate depends only on the real eigenvalue and the real parts of the complex pair. For the deterministic system, oscillations typically decay rapidly. However, randomness induces oscillations to continue indefinitely with a frequency that depends on the parameters of the deterministic system. The computations and analytical results presented here demonstrate that volatility increases when traders place greater emphasis on trend, confirming a generally held belief among practitioners.

Our results indicate that continuous soliton families can exist and be stable in Kerr media with two-dimensional (2D) non-parity-time (PT)-symmetric complex potentials. There are several discrete eigenvalues in the linear spectra of these complex potentials. Fundamental solitons bifurcate from the largest discrete eigenvalue while the dipole solitons bifurcate from the second- or third- largest discrete eigenvalue. We further find that eigenvalues of the soliton linear-stability spectra emerge as complex conjugate pairs. The effect of different parameters of the complex potentials on soliton stability is discussed in detail. Moreover, we study the transverse energy flow vector of the solitons in these complex potentials.

A novel finite-difference time-domain formulation for the modeling of general anisotropic dispersive media is introduced in this work. The method accounts for fully anisotropic electric or magnetic materials with all elements of the permittivity and permeability tensors being non-zero. In addition, each element shows an arbitrary frequency dispersion described by the complex-conjugate pole–residue pairs model. The efficiency of the technique is demonstrated in benchmark numerical examples involving electromagnetic wave propagation through magnetized plasma, nematic liquid crystals and ferrites.

Problem of solving the strictly convex, quadratic programming problem is studied. The idea of conjugate directions is used. First we assume that we know the set of directions conjugate with respect to the hessian of the goal function. We apply n simultaneous directional minimizations along these conjugate directions starting from the same point followed by the addition of the directional corrections. Theorem justifying that the algorithm finds the global minimum of the quadratic goal function is proved. The way of effective construction of the required set of conjugate directions is presented. We start with a vector with zero value entries except the first one. At each step new vector conjugate to the previously generated is constructed whose number of nonzero entries is larger by one than in its predecessor. Conjugate directions obtained by means of the above construction procedure with appropriately selected parameters form an upper triangular matrix which in exact computations is the Cholesky factor of the inverse of the hessian matrix. Computational cost of calculating the inverse factorization is comparable with the cost of the Cholesky factorization of the original second derivative matrix. Calculation of those vectors involves exclusively matrix/vector multiplication and finding an inverse of a diagonal matrix. Some preliminary computational results on some test problems are reported. In the test problems all symmetric, positive definite matrices with dimensions from $$14\times 14$$ to $$2000\times 2000$$ from the repository of the Florida University were used as the hessians.

In this paper, we succeed to bilinearize the $\cal{PT}$-invariant nonlocal nonlinear Schr\"{o}dinger (NNLS) equation through a nonstandard procedure and present more general bright soliton solutions. We achieve this by bilinearizing both the NNLS equation and its associated parity transformed complex conjugate equation in a novel way. The obtained one and two soliton solutions are invariant under combined space and time reversal transformations and are more general than the known ones. Further, by considering the two-soliton solution we bring out certain novel interaction properties of the $\cal{PT}$-invariant multi-soliton solutions., Comment: 13 pages, 2 figures, Accepted for Publication in Physics Letters A

Recycling systems are a challenging problem from the control viewpoint due to their detrimental effects. This paper proposes a simple way to stabilize and control a class of high-order recycling systems with time delays, internal instability and possible complex conjugate poles. The strategy proposes P / PI / PD / PID -like delayed controllers. Conditions to guarantee the stability of the controlled closed-loop system are stated. The performance of the proposed strategy is illustrated by numerical simulation.

We consider the problem of testing whether a complex-valued vector random sequence is proper, i.e., if the sequence is uncorrelated with its complex conjugate. Previous nonparametric approaches to impropriety testing are limited to a sequence of independent random vectors, typically assumed to be Gaussian. In this paper, we extend the results to stationary vector sequences that can be non Gaussian. A binary hypothesis testing approach is formulated and a generalized likelihood ratio test (GLRT) is derived using the power spectral density estimator of an augmented sequence. An asymptotic analytical solution for calculating the test threshold is provided. The performance of the GLRT is analyzed by deriving an approximate asymptotic distribution of the GLRT statistic under the alternative hypothesis. The results are illustrated via simulations.

For a generalized system of the Cauchy–Riemann type with complex conjugation whose coefficients admit a strong singularity on a circle and a weak singularity at a point, we find an integral representation of the general solution.

The accidental crossing of energy levels is studied for a number of exactly solvable P T -symmetric potentials in one spatial dimension. This phenomenon occurs when the potential possesses two series of bound-state levels discriminated by the q = ± quasi-parity quantum number and a potential parameter is tuned to specific values. In contrast with the coalescing of two such real-energy levels with the same n quantum number and continuing as a complex conjugate pair, corresponding to the breakdown of P T symmetry, accidental crossing occurs for energy levels with different n and q . In this case the energy eigenvalues become degenerate, and the corresponding wave functions become linearly dependent. It is shown that besides the known examples, the P T -symmetric harmonic oscillator, Coulomb and Scarf II potentials, this phenomenon occurs for any member of the Natanzon potential class for which the q quantum number can be defined. Two such potentials are discussed as concrete examples: the P T -symmetric generalized Ginocchio potential and a four-parameter subset of the Natanzon potential class. These potentials have been described in detail previously, however, the accidental crossing of their energy eigenvalues has not been noticed then.

International audience; Partitioned approaches for the simulation of coupled conjugate heat transfer are gaining popularity in fields that require accurate thermal predictions. Considerable efforts have been put into determining stable coupling schemes, but performance enhancements have been neglected. This paper presents, for the first time, a detailed and comprehensive study of the numerical properties of Dirichlet-Robin coupling procedures, used in conjugate heat transfer simulation, with emphasis put on the optimal local coupling formulation that was recently derived from a stability analysis. This all-new optimal coupling approach provides local adaptability and has never been tested on a complex setup. This investigation looks to determine the relevance and the limitations of the theory when applied to complex conjugate heat transfer setups. The stability theory of the optimal Dirichlet-Robin coupling scheme is first recalled, then, a realistic 3D application, with complex geometry and flow structures, is used to evaluate the performance and sensitivity of Dirichlet-Robin couplings, with respect to various numerical parameters. This detailed study allows, for the first time, to evaluate the advantages and limitations of the recently proposed optimal procedure, when used on realistic 3D CHT problems. It turns out that the local optimal Dirichlet-Robin formulation outperforms all what is found in literature, and insures unconditional stability with monotone convergence for all considered setups of the 3D model.

First we introduce a new notion of pseudo-anti commuting Ricci tensor for real hypersurfaces in the complex quadric Q m = S O m + 2 / S O 2 S O m and give a complete classification of these hypersurfaces in the complex quadric Q m . Next as an application we give a complete classification of Ricci solitons on Hopf real hypersurfaces in the complex quadric Q m .

We prove that a totally complex algebraic number field K; having a conjugate which is not closed under complex conjugation, can be generated by a reciprocal integer, when the Galois group of its normal closure is contained in the hyperoctahedral group Bdeg(K)/2.Keywords: Reciprocal integers, unit primitive elements, reciprocal numbers

We investigate the concept of the pseudo-parity–time (pseudo- PT ) symmetry in periodic quantum systems. This pseudo parity–time symmetry manifests itself dynamically in the framework of the non-unitary evolution (Floquet) operator U ( τ ) = e − i L τ , over a period τ , which shows that the stability of the dynamics occurs when the PT -symmetry (or pseudo- PT ) of the time-independent non-Hermitian Hamiltonian L is unbroken i.e. its quasienergies E n are real. Nevertheless, when the PT -symmetry of the non-Hermitian Hamiltonian L is broken, which corresponds to the complex conjugate quasienergies E n , an instable dynamics arises. We investigate in greater detail a harmonic oscillator with imaginary time-dependent periodic driving term linear in x . The Floquet operator for the modulated system is pseudo- PT symmetric if the relative phase ϕ of the applied mode is not 0 or π .

We characterize continuous, nondegenerate multiplicative maps on the algebra of upper-triangular Toeplitz matrices. As an application, we show that the continuous multiplicative isometries of (for any norm) are necessarily similarities or similarities composed with complex conjugation; for unitarily invariant norms, such isometries are precisely the unitary similarities, or unitary similarities composed with complex conjugation. We partially extend these results to algebras generated by non-derogatory matrices.

In this paper we present an inter-temporal optimization problem of a representative R&D firm that simultaneously invests in horizontal and vertical innovations. We posit that learning-by-doing makes the process of quality improvements a positive function of the number of existing technologies with the function displaying a convex-concave form. We show that multiple steady-states can arise with two being saddle point stable and one unstable with complex conjugate eigenvalues. Thus, a threshold with respect to the variety of technologies exists that separates the two basins of attractions. From an economic point of view, this implies that a lock-in effect can occur such that it is optimal for the firm to produce only few technologies at a low quality when the initial number of technologies falls short of the threshold. Hence, history matters as concerns the state of development implying that past investments and innovations determine whether the firm produces a large or a small variety of high- or low-quality technologies, respectively.

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 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 behavior to the non-orthogonality of the eigenvectors and addresses the forced response of these system, highlighting how and in which cases, an unexpected amplification of the forced response may occur.

A linear steady third order system with one controlling (perturbing) action that is bounded in absolute magnitude is considered. The matrix specifying the system has one real and two complex conjugate eigenvalues. The behaviour of the boundary of the attainability region of this system as time increases is studied. It is shown that the structure of the boundary of the attainability region depends on the relation between the real eigenvalue and the real part of the complex conjugate eigenvalues.

In this article, we introduce the notion of pseudo-Einstein real hypersurfaces in the complex quadric Qm=SOm+2/SO2SOm and give a complete classification of such hypersurfaces.

Resonating systems show oscillatory characteristics. System identification of resonating systems and design of their model based control strategy always draw special attention. This work presents a Wiener type system identification technique for nonlinear resonating systems. Orthogonal basis function (henceforth termed as OBF) is employed to capture the linear dynamic part of the Wiener structure while the static nonlinear mapping is described by two means, viz., wavelet decomposition and least squares support vector machine. Use of OBF leads to a parsimonious nature in the resulting nonlinear model. Two types of OBFs have been used in this work viz. Laguerre filter and Kautz filter. The Kautz filter has capability of modelling systems with complex conjugate poles. A case study has been performed with continuous stirred tank reactor (henceforth referred as CSTR) which is a reasonably nonlinear resonating systems. Degree of nonlinearity as well as resonance increases with series-connected CSTR. Simulations are carried out using $$\hbox {MATLAB}^\circledR $$ software in order to evaluate the performances of various Wiener structures, and identify the OBF–Wiener model best suited for designing a model based controller.

We review some recent results on properties of tensor product and fusion coefficients under complex conjugation of one of the factors. Some of these results have been proven, some others are conjectures awaiting a proof, one of them involving hitherto unnoticed observations on ordinary representation theory of finite simple groups of Lie type, 8 pages, 2 figures

A feedback control is proposed in this paper for the control of wave energy converters. The proposed controller is a time-domain approximation for the complex conjugate control in the sense that it is an approach to an optimal solution through impedance matching. The theoretical underpinnings of complex conjugate control are fundamentally linear, and the proposed control exploits the linear control theory to provide a complex conjugate control in the time domain. The proposed control does not need wave prediction or wave measurements. The proposed control is novel in that it is a feedback strategy that has a multi-resonant generator. It targets both amplitude and phase through feedback that is constructed from individual frequency components that come from the spectral decomposition of the measurements signal. Each individual frequency uses a Proportional-Derivative control to provide both optimal resistive and reactive elements. By resonating each frequency component and summing them together the controller feedback effort that maximizes the amount of absorbed power is provided.

First, we introduce the notion of shape operator of Codazzi type for real hypersurfaces in the complex quadric \({Q^m}^* = SO^{o}_{m,2}/SO_mSO_2\). Next, we give a complete proof of non-existence of real hypersurfaces in \({Q^m}^* = SO^{o}_{m,2}/SO_mSO_2\) with shape operator of Codazzi type. Motivated by this result, we give a complete classification of real hypersurfaces in \({Q^m}^* = SO^{o}_{m,2}/SO_mSO_2\) with Reeb parallel shape operator.

This paper proposes a new motion parameter estimation algorithm, namely subband dual-frequency conjugate-Lv’s transform, for fast moving targets. This algorithm first constructs two subband signals with different central frequencies. Then, the two signals are shifted by different values and a new signal is constructed by multiplying one with the complex conjugate of the other. Finally, Keystone transform and LVT are performed to obtain the estimates. This method does not need any prior-knowledge of targets and can resolve the Doppler ambiguity problem when Keystone transform does not work, especially when the ambiguity number splits into two numbers. Both simulated and real data demonstrate the effectiveness of the proposed algorithm.