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In this article, the bilinear form, Backlund transformation, Lax pair and infinite conservation laws of the (3+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation with time-dependent coefficients are constructed based on the Bell polynomials approach. N-soliton solutions are studied by means of introducing the complex conjugate condition technique and selecting appropriate test functions and parameters, including the hybrid solution of the a-order kink waves, b-order periodic-kink waves and c-order periodic-breather waves. The homoclinic test method is applied to investigate their dynamical interaction properties between different forms of hybrid-type solutions. Besides, a number of examples are presented by choosing different types of interactions among the hybrid-type solutions. Finally, we analyze the wave propagation direction and velocity to reflect the novel evolutionary behaviors in the three-dimensional profile of the model. These results are helpful to the study of local wave interactions in nonlinear mathematical physics.

Under investigation in this letter is an (3+1)-dimensional Hirota–Satsuma–Ito-like equation, which provide strong support for studying the dynamic behavior of nonlinear waves. Based on a special Cole–Hopf transformation and Hirota bilinear method, the bilinear form of the equation is obtained and this form has never been given. High-order breather solutions, lump solutions and mixed solutions are obtained by using complex conjugate parameters and long-wave limit method. Then, the influence of the coefficient $$g_{t}(t)$$ of the bilinear equation on the interaction of these solutions is analyzed by means of images. It can be found that $$g_{t}(t)$$ changes the interaction of the solutions by influencing the positions and trajectories of higher-order breather solutions, lump solutions and mixed solutions. We find that different values of g(t) make the interaction of solutions different. Finally, the mixed solution of the equation including a breather wave and a line rogue wave is obtained by using the test function, and its dynamic properties are illustrated by means of images.

The present paper studied multi-breather, multi-lump and hybrid solutions of a novel KP-like equation by applying the Hirota bilinear method. The multi-breather solutions including one-, two- and three-breather and hybrid solutions between breathers and solitons were obtained through applying the complex conjugate method on the N-soliton solution. Via using the long wave limit method with respect to N-solution, the solution expression of multi-lump solutions was acquired; therefore, by choosing appropriate parameters on 2-, 3-, 4-, 5- and 6-soliton, one-, two-, three-lump and two types of hybrid solutions between lumps and solitons were derived. Additionally, the one-breather and one-lump solutions have been studied in-depth. In order to illustrate the wave trajectories, wave shapes and the fusion and fission process, several cases of plots with physical interpretation were given. Furthermore, the obtained solutions can be widely used to explain many interesting physical phenomena in the nature.

Let $G$ be a finite group and let $h$ be a positive integer. A $\text{BH}(G,h)$ matrix is a $G$-invariant $|G|\times |G|$ matrix $H$ whose entries are complex $h$th roots of unity such that $HH^*=|G|I_{|G|}$, where $H^*$ denotes the complex conjugate transpose of $H$, and $I_{|G|}$ is the identity matrix of order $|G|$. In this paper, we give three new constructions of $\text{BH}(G,h)$ matrices. The first construction is the first known family of $\text{BH}(G,h)$ matrices in which $G$ does not need to be abelian. The second and the third constructions are two families of $\text{BH}(G,h)$ matrices in which $G$ is a finite local ring.

In this paper, based on the bilinear form, we give multi-solitary wave solutions of the $$(4+1)$$ -dimensional Fokas equation. From the obtained multi-solitary wave solutions, with special parameters, we derive resonant solutions of N-solitary waves. For complex conjugate parameters, we analyze interactions of two periodic waves, interactions of a solitary wave and a periodic wave by making use of their phase shifts. Particularly, the intermediate processes of elastic interactions are analyzed in detail, and interesting fusional and fissionable phenomena are found. The asymptotic interactional behaviors for these solutions are analyzed theoretically and illustrated graphically.

We prove a generalization of the Shapiro–Shapiro conjecture on Wronskians of polynomials, allowing the Wronskian to have complex conjugate roots. We decompose the real Schubert cell according to the number of real roots of the Wronski map, and define an orientation of each connected component. For each part of this decomposition, we prove that the topological degree of the restricted Wronski map is given as an evaluation of a symmetric group character. In the case where all roots are real, this implies that the restricted Wronski map is a topologically trivial covering map; in particular, this gives a new proof of the Shapiro–Shapiro conjecture.

In this paper we prove that the time part of the germ of an analytic family of vector fields with a Hopf bifurcation is rigid in the parameter. Time parts are associated with the temporal invariant of the analytic classification. Because the eigenvalues at zero are complex conjugate, time parts usually unfold in the hyperbolic direction, where the singular points are linearizable. We first identify the time part of a generic conformal family and prove that any weak holomorphic conjugacy between two time parts yields a biholomorphism analytic in the parameter. The existence of Fatou coordinates in both the Siegel and in the Poincare domains plays a fundamental role in the proof of this result.

We prove two inequalities for the complex orientations of a separating non-singular real algebraic curve in $${\mathbb {RP}}^2$$ of any odd degree. We also construct a separating non-singular real (i.e., invariant under the complex conjugation) pseudoholomorphic curve in $${\mathbb {CP}}^2$$ of any degree congruent to 9 mod 12 which does not satisfy one of these inequalities. Therefore the oriented isotopy type of the real locus of each of these curves is algebraically unrealizable.

The South China block has been one of the most seismically quiescent regions in China, and the geometries and activities of the Quaternary faults have remained less studied due to the limited outcrops. Thus, source parameters of small-to-moderate earthquakes are important to help reveal the location, geometry distribution, and mechanical properties of the subsurface faults and thus improve the seismic risk assessment. On 12 October 2019, two earthquakes (the Ms 4.2 foreshock and the Ms 5.2 mainshock) occurred within 2 s and are located in southern South China block, near the junction region of the large-scale northeast-trending fault zones and the less continuous northwest-trending fault zones. We determined the point-source parameters of the two events via P-wave polarity analysis and regional waveform modeling, and the resolved focal mechanisms are significantly different with the minimum 3D rotation angle of 52°. We then resolved the rupture directivity of the two events by analyzing the azimuth variation of the source time duration and found the Ms 4.2 foreshock ruptured toward north-northwest for ∼1.0 km, and the Ms 5.2 mainshock ruptured toward east-southeast (ESE) for ∼1.5 km, implying conjugate strike-slip faulting. The conjugate causative faults have not been mapped on the regional geological map, and we infer that the two faults may be associated with the northwest-trending Bama-Bobai fault zone (the Shiwo section). These active faults are optimally oriented in the present-day stress field (northwest-southeast) and thus may now be potentially accumulating elastic strain to be released in a future large earthquake.

In the present paper, taking advantage of the bilinear method, we attained N-soliton solutions of the (3+1)-dimensional generalized Camassa–Holm Kadomtsev–Petviashvili (gCH-KP) equation. By combining the long wave limit method and the complex conjugate method to N-soliton solutions, M-lump wave solutions including one-lump wave, two-lump wave and three-lump wave and two types of hybrid solutions including hybrid solution between lump wave and solitons and between M-lump wave and soliton were obtained. In addition, several groups of images exhibited their dynamic structure and physical properties with physically interpreted via symbolic computation. Finally, resonant multi-soliton wave solutions were obtained by carrying out the linear superposition principle and the progresses of wave motions with physical interpretation represented graphically. Furthermore, the received results have immensely augmented the exact solutions of (3+1)-dimensional gCH-KP equation on the available literature and enabled us to understand the nonlinear dynamic system deeply. In addition, the proposed methods will be a strong boost to the calculation method of nonlinear equations.

Let $F$ be a CM field with totally real subfield $F^+$ and let $\pi$ be a $C$-algebraic cuspidal automorphic automorphic representation of $\mathrm{U}(a,b)(\mathbf{A}_{F^+})$ whose archimedean components lie in the (non-degenerate limit of) discrete series. We attach to $\pi$ a Galois representation $R_\pi:\mathrm{Gal}(\overline F/ F^+)\to{}^C\mathrm{U}(a,b)(\overline{\mathbf Q}_\ell)$ such that, for any complex conjugation element $c$, $R_\pi(c)$ is as predicted by the Buzzard--Gee conjecture. As a corollary, we deduce that the Galois representations attached to certain irregular, $C$-algebraic (essentially) conjugate self-dual cuspidal automorphic representations of $\mathrm{GL}_n(\mathbf A_F)$ are odd in the sense of Bella\"iche--Chenevier., Comment: 23 pages

The aim of this article is to apply a Floer theory to study symmetric periodic Reeb orbits. We define positive equivariant wrapped Floer homology using a (anti-)symplectic involution on a Liouville domain and investigate its algebraic properties. By a careful analysis of index iterations, we obtain a non-trivial lower bound on the minimal number of geometrically distinct symmetric periodic Reeb orbits on a certain class of real contact manifolds. This includes non-degenerate real dynamically convex starshaped hypersurfaces in $\mathbb{R}^{2n}$ which are invariant under complex conjugation. As a result, we give a partial answer to the Seifert conjecture on brake orbits in the contact setting., Comment: 42 pages, 4 figures, final version, to appear in Ergodic Theory and Dynamical Systems

In the classically unbroken phase, 3d $O(N)$ symmetric $\phi^4$ vector models admit two equivalent descriptions connected by a strong-weak duality closely related to the one found by Chang and Magruder long ago. We determine the exact analytic renormalization dependence of the critical couplings in the weak and strong branches as a function of the renormalization scheme (parametrized by $\kappa$) and for any $N$. It is shown that for $\kappa=\kappa_*$ the two fixed points merge and then, for $\kappa, Comment: 38 pages, 12 figures; v3: version to appear in JHEP

The phenomenon, which is classified as a complex resonance and takes place in electromagnetic structures described by nonself-adjoint boundary value problems, is considered. It is shown that the oscillation adjoint to a source is formed, when complex waves that are complex conjugate by wave numbers and amplitudes are excited in couples. This oscillation is described by the self-consistent boundary value problem for the adjoint Helmholtz equation, i.e., for the equation, whose right-hand side is the solution of the homogeneous boundary value problem. The distinctive feature of the complex resonance is in the fact that it exists in the whole range of complex waves, when the source is necessarily present. Theoretical and experimental results of the investigation of the considered phenomenon are presented.

Wu et al. (Applied Mathematics and Computation 217(2011)8343-8353) constructed a gradient based iterative (GI) algorithm to find the solution to the complex conjugate and transpose matrix equation $$\begin{aligned} A_{1}XB_{1}+A_{2}\overline{X}B_{2}+A_{3}X^{T}B_{3}+A_{4}X^{H}B_{4}=E \end{aligned}$$ and a sufficient condition for guaranteeing the convergence of GI algorithm was given for an arbitrary initial matrix. Zhang et al. (Journal of the Franklin Institute 354 (2017) 7585-7603) provided a new proof of GI method and the necessary and sufficient conditions was presented to guarantee that the proposed algorithm was convergent for an arbitrary initial matrix. In this paper, a relaxed gradient based iterative (RGI) algorithm is proposed to solve this complex conjugate and transpose matrix equation. The necessary and sufficient conditions for the convergence factor is determined to guarantee the convergence of the introduced algorithm for any initial iterative matrix. Numerical results are given to verify the efficiency of the new method. Finally, the application in time-varying linear system of the presented algorithm is provided.

Gradient-based iterative algorithm is suggested for solving a coupled complex conjugate and transpose matrix equations. Using the hierarchical identification principle and the real representation of a complex matrix, a convergence proof is offered. The necessary and sufficient conditions for the optimal convergence factor are determined. A numerical example is offered to validate the efficacy of the suggested algorithm.

In 1976, Mahler posed the question about the existence of a transcendental function f ∈ Z { z } with bounded coefficients and such that f ( Q ‾ ∩ B ( 0 , 1 ) ) ⊆ Q ‾ . In this paper, we prove, in particular, the existence of such a function but with the weaker requirement that the coefficients have only 2 and 3 as prime factors. More generally, we shall prove that any subset of Q ‾ ∩ B ( 0 , 1 ) , which is closed under complex conjugation and which contains the element 0, is the exceptional set of uncountably many transcendental functions in Z { z } with coefficients having only 2 and 3 as prime factors.

The complex-conjugate pole-residue (CCPR) model has been popularly adopted because CCPR-finite-difference time domain (FDTD) can reduce the memory requirement with the help of complex conjugate property of auxiliary variables. To fully utilize CCPR-FDTD, it is of great necessity to investigate its numerical stability since the FDTD method is conditionally stable. Nonetheless, the numerical stability conditions of CCPR-FDTD have not been studied because its derivation is not straightforward. In this communication, the numerical stability conditions of CCPR-FDTD are systematically derived by combining the von Neumann method with Routh–Hurwitz criterion. It is found that the numerical stability conditions of CCPR-FDTD are the same as those of the modified Lorentz-FDTD with bilinear transform. Moreover, the numerical accuracy of CCPR-FDTD is studied, and numerical examples are employed to validate this work.

A Bagnera-de Franchis variety $X = A/G$ is the quotient of an abelian variety $A$ by a free action of a finite cyclic group $G \subset Bihol(A)$, which does not contain only translations. Constructing explicit polarizations and using a method introduced by F. Catanese, we classify split Bagnera-de Franchis varieties up to complex conjugation in dimensions $\leq 4$.

Background: The knowledge of parameter estimation for interatomic potentials is useful in the computation of the vibrational structure of van der Waals molecules. Methods: On the estimation of the Generalized Morse and Classical Lennard-Jones potential energy functions, complex conjugates eigenvalues may be obtained. Different approaches can be used to solve this resulting problem. A method that uses the objective least squares function method to estimate parameters of the interatomic potentials is employed. Results: Numerical simulation of the systems using metal atoms yields complex conjugates eigenvalues at some initial point. Conclusion: Other approaches of solving the complex conjugates eigenvalues problem are discussed comprehensively.

Let $N$ and $p$ be prime numbers $\geq 5$ such that $p$ divides $N-1$. Let $I$ be Mazur’s Eisenstein ideal of level $N$ and $H_+$ be the plus part of $H_1(X_0(N), \mathbf Z_{p})$ for the complex conjugation. We give a conjectural explicit description of the group $I\cdot H_+/I^2\cdot H_+$ in terms of the 2nd $K$-group of the cyclotomic field $\mathbf Q(\zeta _N)$. We prove that this conjecture follows from a conjecture of Sharifi about some Eisenstein ideal of level $\Gamma _1(N)$. Following the work of Fukaya–Kato, we prove partial results on Sharifi’s conjecture. This allows us to prove partial results on our conjecture.

In this paper, a generalized $$(2+1)$$-dimensional nonlinear wave equation is obtained by extending the generalized $$(2+1)$$-dimensional Hirota bilinear equation into a more generalized form. The obtained new equation is useful in describing nonlinear wave phenomena in nonlinear optics, shallow water and oceanography. Based on the bilinear method, the N-soliton solutions of the generalized $$(2+1)$$-dimensional nonlinear wave equation are obtained. M-lump solutions are investigated by applying the long wave limit to the N-soliton solutions. The propagation orbits and velocities of the M-lump wave are analyzed. The high-order breather waves are obtained by establishing the complex conjugate relations in the parameters of the N-solitons. Furthermore, the interaction hybrid solutions are constructed, which contain hybrid solutions composed of breathers, solitons and lumps. The dynamical behaviors of the hybrid solutions are systematically analyzed via numerical simulations. The obtained results will enrich the study of theory of the nonlinear localized waves.

Wiener model belongs to a broad class of non-linear model structures called the block-oriented models. Various chemical, biological and electrical processes can be represented as a Wiener structure. There are vast numbers of approaches to identify the Wiener model; however, most of them are only theoretically sound. Therefore, there is a need for developing an industrial grade, simple identification scheme. The present research work attempts to explore the possibility of identifying Wiener models using relay feedback identification technique, which has been applied efficiently but limited mostly to linear structures. Here, the authors consider not all but wider probable cases of the linear subsystem, which include real roots, complex conjugate roots, first-order, integrating second-order and repeated roots. The proposed method is simple and requires less prior knowledge about the system. The technique being somehow general can be applied effectively to linear and unstable systems as well. Various simulation examples are used to demonstrate the efficacy of the proposed identification scheme.

Under investigation in this letter is an extended (3 + 1)-dimensional Jimbo–Miwa (eJM) equation, which can be used to describe many nonlinear phenomena in mathematical physics. With the aid of Hirota bilinear method and long-wave limit method, M-order lumps which describe multiple collisions of lumps are derived. The propagation orbit, velocity and extremum of the 1-order lump solutions on (x, y) plane are investigated in detail. Resorting to the extended homoclinic test technique, we obtain the breather–kink solutions, rational breather solutions and rogue wave solutions for the eJM equation. Meanwhile, through analysis and calculation, the amplitude and period of breather–kink solutions increase with p increasing and the extremum of rational breather solution and rogue waves are also derived. T-order breathers are obtained by means of choosing appropriate complex conjugate parameters on N-soliton solutions. Periods of the 1-order breather solutions on the (x, y) plane are determined by $$k_{12}$$ and $$k_{12}p_{11}+k_{11}p_{12}$$, while locations are determined by $$k_{11}$$ and $$k_{11}p_{11}-k_{12}p_{12}$$. Furthermore, hybrid solutions composed of the kink solitons, breathers and lumps for the eJM equation are worked out. Some figures are given to display the dynamical characteristics of these solutions.

This research work is motivated by the success of relay feedback in linear system domain thereby an attempt has been made at extending it to nonlinear Hammerstein models. The identification is carried out using a relay with hysteresis in feedback, the parameter estimation is done solely by solving state-space equations at various points on one cycle of the input and output signals. The work considers not all but wider probable cases of the linear subsystem-real roots, complex conjugate roots, first order plus dead time systems, integrating second order plus dead time and repeated roots. The proposed method is simple compared to other works in literature as well as applicable to unstable systems. Simulations of various examples are carried out to demonstrate the efficacy of the proposed identification scheme.

The deviation of continuous and discrete complex random variables from the traditional proper and symmetric assumption to a generalized improper and asymmetric characterization (accounting correlation between a random entity and its complex conjugate), respectively, introduces new design freedom and various potential merits. As such, the theory of impropriety has vast applications in medicine, geology, acoustics, optics, image and pattern recognition, computer vision, and other numerous research fields with our main focus on the communication systems. The journey begins from the design of improper Gaussian signaling in the interference-limited communications and leads to a more elaborate and practically feasible asymmetric discrete modulation design. Such asymmetric shaping bridges the gap between theoretically and practically achievable limits with sophisticated transceiver and detection schemes in both coded/uncoded wireless/optical communication systems. Interestingly, introducing asymmetry and adjusting the transmission parameters according to some design criterion render optimal performance without affecting the bandwidth or power requirements of the systems. This dual-flavored article initially presents the tutorial base content covering the interplay of reality/complexity, propriety/impropriety and circularity/noncircularity and then surveys majority of the contributions in this enormous journey., Comment: IEEE COMST (Early Access)

Multitask Gaussian processes (MTGPs) are a powerful approach for modeling dependencies between multiple related tasks or functions for joint regression. Current kernels for MTGPs cannot fully model nonlinear task correlations and other types of dependencies. In this article, we address this limitation. We focus on spectral mixture (SM) kernels and propose an enhancement of this type of kernels, called multitask generalized convolution SM (MT-GCSM) kernel. The MT-GCSM kernel can model nonlinear task correlations and dependence between components, including time and phase delay dependence. Each task in MT-GCSM has its GCSM kernel with its number of convolution structures, and dependencies between all components from different tasks are considered. Another constraint of current kernels for MTGPs is that components from different tasks are aligned. Here, we lift this constraint by using inner and outer full cross convolution between a base component and the reversed complex conjugate of another base component. Extensive experiments on two synthetic and three real-life data sets illustrate the difference between MT-GCSM and previous SM kernels as well as the practical effectiveness of MT-GCSM.

Multichannel noise reduction in the frequency domain is a fundamental problem in the areas of speech processing and speech recognition. In this paper, we address this problem and propose an alternative approach to retrieve a speech signal out of microphone array noisy observations. We focus on the spectral amplitude of the speech signal and assume that the spectral phase is less significant. The estimate of the spectral amplitude squared, that is the spectral power, is obtained by applying a complex linear filter to a modified version of the observations vector. This modified version is obtained as a Kronecker product of the complex conjugate of the observations vector and the original observations vector. The complex speech signal estimate is obtained by multiplying the spectral amplitude estimate with a complex exponential whose phase may be extracted from the minimum variance distortionless response beamformer. We present a modified optimization criterion according to which the proposed filters may be derived, and compare their performances to conventional multichannel noise reduction filters. We show that the new approach is preferable, in particular when the input signal-to-noise ratio (SNR) is low or the number of sensors is small.

So-called improper signals, i.e., signals which are correlated with their complex conjugates, can occur in many signal processing applications such as communication systems, medical imaging, audio and speech processing, analysis of oceanographic data, and many more. Being aware of potential impropriety can be crucial whenever we model signals as complex random quantities since an appropriate treatment of improper signals, e.g., by widely linear filtering, can significantly improve the system performance. After a brief introduction into the fundamentals of improper signals, this article focuses on the problem of quantifying the impropriety of complex random vectors and gives a survey of various impropriety measures in both the composite real representation and the augmented complex formulation. Unlike in previous publications, these two frameworks are presented side by side to reveal the differences and common points between them. Moreover, their applicability is compared in several practical examples. As additional aspects, we consider the problem of testing for impropriety based on measurement data, and the differential entropy of Gaussian vectors as an impropriety measure in information theoretic studies. The article includes a tutorial-style introduction, a collection of important formulae, a comparison of various mathematical approaches, as well as some new reformulations.

A new analytic method to calculate the coupling coefficient including self-inductances and mutual inductance between two arbitrary-shaped coils is proposed. The proposed method is applicable for all common coil geometries such as rectangular, square, polygonal, circular coils etc. In the formula, the coil function, which is a completely a new type of function, is defined. The differences in shape and position of the coils are only on the choice of the coil function and its complex conjugate. The layouts of the two coils including separation change and lateral misalignment are discussed. An experimental setup has been built to validate the analytic calculations. Theoretical and experimental results are then compared to be in good agreement, which verify the effectiveness of the proposed method.

Unitary symmetric eigenvalues (US-eigenvalues) of symmetric complex tensors and unitary eigenvalues (U-eigenvalues) for non-symmetric complex tensors are very important because of their background of quantum entanglement. US-eigenvalue is a generalization of Z-eigenvalue from the real case to the complex case, which is closely related to the best complex rank-one approximations to higher-order tensors. The problem of finding US-eigenpairs can be converted to an unconstrained nonlinear optimization problem with complex variables, their complex conjugate variables and real variables. However, optimization methods often need a first- or second-order derivative of the objective function, and cannot be applied to real valued functions of complex variables because they are not necessarily analytic in their argument. In this paper, we first establish the first-order complex Taylor series and Wirtinger calculus of complex gradient of real-valued functions with complex variables, their complex conjugate variables and real variables. Based on this theory, we propose a norm descent cautious BFGS method for computing US-eigenpairs of a symmetric complex tensor. Under appropriate conditions, global convergence and superlinear convergence of the proposed method are established. As an application, we give a method to compute U-eigenpairs for a non-symmetric complex tensor by finding the US-eigenpairs of its symmetric embedding. The numerical examples are presented to support the theoretical findings.

In the framework of the Stueckelberg-Wheeler-Feynman concept of a ``one-electron Universe'' we consider a worldline implicitly defined by a system of algebraic (precisely, polynomial) equations. Collection of pointlike ``particles'' of two kinds on the worldline (or its complex extension) is defined by the real (complex conjugate) roots of the polynomial system and detected then by an external inertial observer through the light cone connections. Then the observed collective dynamics of the particles' ensemble is, generally, subject to a number of Lorentz invariant conservation laws. Remarkably, this poperty follows from the Vieta's formulas for the roots of the generating polynomial system. At some discrete moments of the observer's proper time, mergings and subsequent transmutations of a pair of particles-roots take place simulating thus the processes of annihilation/creation of a particle/antiparticle pair, Comment: 7 pages, twoside, 1 figure

We extend the study of supersymmetric tridiagonal Hamiltonians to the case of non-Hermitian Hamiltonians with real or complex conjugate eigenvalues. We find the relation between matrix elements of the non-Hermitian Hamiltonian [Formula: see text] and its supersymmetric partner [Formula: see text] in a given basis. Moreover, the orthogonal polynomials in the eigenstate expansion problem attached to [Formula: see text] can be recovered from those polynomials arising from the same problem for [Formula: see text] with the help of kernel polynomials. Besides its generality, the developed formalism in this work is a natural home for using the numerically powerful Gauss quadrature techniques in probing the nature of some physical quantities such as the energy spectrum of [Formula: see text]-symmetric complex potentials. Finally, we solve the shifted [Formula: see text]-symmetric Morse oscillator exactly in the tridiagonal representation.

Accepting a time-symmetric quantum dynamical world with ontological wave functions or fields, we follow arguments that naturally lead to a two-boundary interpretation of quantum mechanics. The usual two boundary picture is a valid superdeterministic interpretation. It has, however, one unsatisfactory feature. The random selection of a chosen measurement path of the universe is far too complicated. To avoid it, we propose an alternate two-boundary concept called surjective mapping conjecture. It takes as fundamental a quantum-time running forward like the usual time on the wave-function side and backward on the complex conjugate side. Unrelated fixed arbitrary boundary conditions at the initial and the final quantum times then determine the measurement path of the expanding and contracting quantum-time universe in the required way.

In this paper, we introduce the notion of normal Jacobi operator of Codazzi type for real hypersurfaces in the complex hyperbolic quadric [Formula: see text]. The normal Jacobi operator of Codazzi type implies that the unit normal vector field [Formula: see text] becomes [Formula: see text]-principal or [Formula: see text]-isotropic. Then according to each case, we give a complete classification of Hopf real hypersurfaces in [Formula: see text] with normal Jacobi operator of Codazzi type. The result of the classification shows that no such hypersurfaces exist.

We propose and investigate several complex versions of extensions and restrictions of the Skyrme model with a well-defined Bogomolny-Prasad-Sommerfield (BPS) limit. The models studied possess complex kink, anti-kink, semi-kink, massless and purely imaginary compacton BPS solutions that all have real energies. The reality of the energies for a particular solution is guaranteed when a modified antilinear CPT-symmetry maps the Hamiltonian functional to its parity time-reversed complex conjugate and the solution field to itself or a new field with degenerate energy. In addition to the known BPS Skyrmion configurations we find new types that we refer to as step, cusp, shell, and purely imaginary compacton solutions., 21 pages, 7 figures