Complex Plane Research Articles

Realization of synchronized modulations by stator flux trajectory tracking for low‐carrier‐ratio induction motor drives

SummaryCompared with asynchronous voltage modulations, the synchronous space vector modulation (Syn‐SVPWM) can suppress current harmonics for high‐power induction motors (IM) by generating symmetrical voltage pulses. However, it is difficult to build a high‐performance current regulator based on Syn‐SVPWM because of the requirements of fixed voltage phases and the nonsymmetrical switching sequences in Syn‐SVPWM. Therefore, this manuscript proposes the equivalent realization of Syn‐SVPWM by the concept of stator flux trajectory tracking (SFTT). The new method controls the stator flux vector to move along evenly distributed flux samples in the complex plane. Then, the synchronization and symmetries of voltage pulses are maintained by organizing the switching sequence between adjacent flux samples. Meanwhile, the torque can be changed by online updating the control period based on the inverse torque model. Consequently, the proposed method can provide fast torque response with Syn‐SVPWM in a simple manner because it does not have to compensate the phase contradiction and to extract the fundamental of sampled current. Both simulations and experiments have verified the effectiveness of proposed method.

Numerical Investigation of Stratified Slope Failure Containing Rough Non-Persistent Joints Based on Distinct Element Method

To address the critical issue of slope stability in jointed rock masses with complex and rough structural planes, a rough joint network model using the Fourier transform was proposed and applied to the Shilu open pit mine. The on-site structural plane survey results were combined with MATLAB and PFC2D to establish numerical models for slope stability analysis considering both linear-jointed and rough-jointed rock slopes. By comparing the slip body and fracture distribution, it was found that the rough-jointed slope was stabler than the linear-jointed slope. In addition, the fracture patterns and slip displacement were significantly influenced by the inclination and undulation of the bedding planes. Slip failure patterns occurred when the angle of inclination was set at 60°. The joints played a crucial role in inducing the shear strength of slope rock masses, and the slide area was mainly observed in the shallow slope surface for inclination angles of 0° and 45°, and in the middle deep area for 60° and 90°. These results demonstrated the importance of considering rough non-persistent fractures when developing a new numerical model for slope failure modes.

Exploration of the double frequency complex plane used in Steinmetz’s method of calculation of complex electrical power in an alternating current circuit

Exploration of the double frequency complex plane used in Steinmetz’s method of calculation of complex electrical power in an alternating current circuit

Growth of meromorphic solution related to linear difference equations and value sharing

In this paper, we consider the growth of meromorphic solutions of complex linear difference equations of certain types and derive some results, which improve the results of Chiang and Feng [On the Nevanlinna characteristic of f ( z + η ) and difference equations in the complex plane. Ramanujan J. 2008;16(1):105–129; On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions. Trans Am Math Soc. 2009;361:3767–3791] and Chen [Growth and zeros of meromorphic solution of some linear difference equations. J Math Anal Appl. 2011;373:235–241]. Also in this paper, we consider the sharing value problems of entire functions with their difference operators and obtain a result which improves the result of Li, Yang and Yi [Entire functions sharing an entire function of smaller order with their shifts. Proc Jpn Acad. 89 (A) (2013), 34–39]. Moreover, we show by illustrating a number of examples that our results are best possible in certain senses.

Non-inertial interpretation of the Dirac oscillator

Non-inertial physics is seldom considered in quantum mechanics and this contrasts with the ubiquity of non-inertial reference frames. Here, we show an application to the Dirac oscillator which provides a fundamental model of relativistic quantum mechanics. The model emerges from a term linearly dependent on spatial coordinates added to the momentum of the free-particle Dirac Hamiltonian. The definition generates peculiar features (mutating vacuum energy, non-Hermitian momentum, accidental degeneracies of the spectrum, etc). We interpret these anomalies in terms of inertial effects. The demonstration is based on the decoupling of the Dirac equation from the stereographic projection that maps the 3D geometry of the dynamical problem to the complex plane. The decoupling shows that the fundamental mechanical model underpinning the Dirac oscillator reduces to the representation of the oscillator in the rotating reference frame attached to the orbital angular momentum. The resulting Coriolis-like contribution to the Hamiltonian accounts for the peculiarities of the model (mutating vacuum energy, form of the non-minimal correction to the momentum, classical intrinsic spin and gain of its quantum value, accidental degeneracies of the energy spectrum, supersymmetric potential). The suggested interpretation has an interdisciplinary character where stereographic geometry, classical physics of the Coriolis effect and quantum physics of Dirac particles contribute to the definition of one of the few exactly soluble models of relativistic quantum mechanics.

Double-Step Shape Invariance of Radial Jacobi-Reference Potential and Breakdown of Conventional Rules of Supersymmetric Quantum Mechanics

The paper reveals some remarkable form-invariance features of the ‘Jacobi-reference’ canonical Sturm–Liouville equation (CSLE) in the particular case of the density function with the simple pole at the origin. It is proven that the CSLE under consideration preserves its form under the two second-order Darboux–Crum transformations (DCTs) with the seed functions represented by specially chosen pairs of ‘basic’ quasi-rational solutions (q-RSs), i.e., such that their analytical continuations do not have zeros in the complex plane. It is proven that both transformations generally either increase or decrease by 2 the exponent difference (ExpDiff) for the mentioned pole while keeping two other parameters unchanged. The change is more complicated in the latter case if the ExpDiff for the pole of the original CSLE at the origin is smaller than 2. It was observed that the DCTs in question do not preserve bound energy levels according to the conventional supersymmetry (SUSY) rules. To understand this anomaly, we split the DCT in question into the two sequential Darboux deformations of the Liouville potentials associated with the CSLEs of our interest. We found that the first Darboux transformation turns the initial CSLE into the Heun equation written in the canonical form while the second transformation brings us back to the canonical form of the hypergeometric equation. It is shown that the first of these transformations necessarily places the mentioned ExpDiff into the limit-circle (LC) range and then the second transformation keeps the pole within the LC region, violating the conventional prescriptions of SUSY quantum mechanics.

L-functions and meromorphic functions satisfying the same Riemann-type functional equation and sharing sets

In 2023, Li–Du–Yi [Complex Var. Elliptic Equ., 68(10)2023, 1653–1677] proved that if two L-functions L 1 and L 2 in the extended Selberg class S # have positive degrees, satisfy the same functional equation with a ( 1 ) = 1 and E L 1 ( S ) = E L 2 ( S ) for a finite set S = { c 1 , c 2 , c 3 } , where c 1 , c 2 and c 3 are three distinct finite complex values, then L 1 = L 2 . We prove that if two L-functions L 1 and L 2 in the extended Selberg class S # have positive degrees, satisfy the same functional equation with a ( 1 ) = 1 and share a set { α 1 , α 2 , … , α t } CM, where t ≥ 1 is a positive integer, and α 1 , α 2 , ··· , α t are t distinct finite complex numbers, then L 1 = L 2 . The result also improves the corresponding results from Khoai–An [The Ramanujan J., 58(1)(2022),253-267]. We also prove that if a non-constant meromorphic function f with finitely many poles in the complex plane shares a set { α 1 , α 2 , … , α t } CM and some finite value c IM with a non-constant L-function L having a positive degree, then f = L. Some examples are provided to show that the main results obtained in this paper, in a sense, are best possible.

Effect of the boson peak and the ionic resonance in the dielectric properties of silicate materials at mm-wave and THz frequencies

In this paper, a broadband measurement of the dielectric properties of silicates has been done to cover the seldom characterized mm-wave spectrum and the low THz spectrum. The dielectric properties, i.e. loss and permittivity, show a frequency independent response up to approximately 40 GHz and the loss monotonically increases to the THz frequency range. Two resonance peaks appear at THz frequencies: the first is the boson peak, and the second one corresponds to the ionic resonance of the network former, which in this case is silicon. Even though the materials belong to different groups in the silicate family (crystalline and non-crystalline, with different levels of purity), they seem to have the same overall tendency in the dielectric properties. Argand plots are used for the first time at THz frequencies to provide physical insight into the boson peak and ionic resonance. The complex plane analysis reveals details on the polarization mechanisms underlying silicate network vibrations through direct visualization of their characteristic signatures, and simplifies the process of modeling the dielectric response, leading to a better fit.

Iterated Robin problem, the case of various parameters

ABSTRACT Prescribing Robin boundary conditions for the iterated Poisson equation ( ∂ z ∂ z ¯ ) n w = f leads to Robin-n problems. Extending previous results by allowing an independent choice for the parameters α k , β k for every iteration k , 1 ≤ k ≤ n , leads to explicit integral representations depending on the data of the Robin-n problem. Parting from these integral representation explicit solutions with their respective solvability conditions are derived. For the unit disc of the complex plane, the Robin functions for n = 2 and 3 are explicitly constructed.

Incorporating plan complexity into the statistical process control of volumetric modulated arc therapy pre-treatment verifications.

Statistical process control (SPC) is a powerful statistical tool for process monitoring that has been highly recommended in healthcare applications, including radiation therapy quality assurance (QA). The AAPM TG-218 report described the clinical implementation of SPC for Volumetric Modulated Arc Therapy (VMAT) pre-treatment verifications, pointing out the need to adjust tolerance limits based on plan complexity. However, the quantification of plan complexity and its integration into SPC remains an unresolvedchallenge. The primary aim of this study is to investigate the incorporation of plan complexity into the SPC framework for VMAT pre-treatment verifications. The study explores and evaluates various strategies for this incorporation, discussing their merits and limitations, and provides recommendations for clinicalapplication. A retrospective analysis was conducted on 309 VMAT plans from diverse anatomical sites using the PTW OCTAVIUS 4D device for QA measurements. Gamma Passing Rates (GPR) were obtained, and lower control limits were computed using both the conventional Shewhart method and three heuristic methods (scaled weighted variance, weighted standard deviations, and skewness correction) to accommodate non-normal data distributions. The 'Identify-Eliminate-Recalculate' method was employed for robust analysis. Eight complexity metrics were analyzed and two distinct strategies for incorporating plan complexity into SPC were assessed. The first strategy focused on establishing control limits for different treatment sites, while the second was based on the determination of control limits as a function of individual plan complexity. The study extensively examines the correlation between control limits and plan complexity and assesses the impact of complexity metrics on the controlprocess. The control limits established using SPC were strongly influenced by the complexity of treatment plans. In the first strategy, a clear correlation was found between control limits and average plan complexity for each site. The second approach derived control limits based on individual plan complexity metrics, enabling tailored tolerance limits. In both strategies, tolerance limits inversely correlated with plan complexity, resulting in all highly complex plans being classified as in control. In contrast, when plans were collectively analyzed without considering complexity, all the out-of-control plans were highlycomplex. Incorporating plan complexity into SPC for VMAT verifications requires meticulous and comprehensive analysis. To ensure overall process control, we advocate for stringent control and minimization of plan complexity during treatment planning, especially when control limits are adjusted based on plancomplexity.

A geometric property of quadrilaterals

Quadrilaterals in the complex plane play a significant part in the theory of planar quasiconformal mappings. Motivated by the geometric definition of quasiconformality, we prove that every quadrilateral with modulus in an interval \([1/K, K]\), where \(K>1\), contains a disk lying in its interior, of radius depending only on the internal distances between the pairs of opposite sides of the quadrilateral and on \(K\).

On the futility of the Fuoss-Kirkwood relation

The Fuoss-Kirkwood (FK) relation is used to derive analytical expressions for the distribution of uncorrelated relaxation times (DRT), particularly of probed biochemical and electrochemical systems, often without scrutiny. Its futility is proven simply by using the parity of impedance with complex frequencies. However, given the futile nature of the FK relation, these expressions are not suitable for validation of computed DRT spectra. Despite this, the need for such expressions persists. Addressing this need and the oversight of the dependency of the DRT value on the underlying data, the DRT is extended into the complex plane using the Hilbert transform (HT). It makes the DRT universal for any complex-valued quantity to not only quantify the extent of relaxations but also to assess their nature. keyword: distribution; integral transforms; relaxation time; superposition.

A new approach to the evaluation and solution of the relativistic kinetic dispersion relation and verification with continuum kinetic simulation

The present work describes a new approach to evaluation and root finding for the kinetic dispersion relation of Langmuir waves, which is central to the analytical understanding of collisionless damping in plasmas. The plasma dispersion function is solved to machine precision using direct integration in the complex plane in combination with an analytic evaluation of the residue to account for the deformation along the Landau contour. To efficiently attain machine precision, the contour is displaced in the complex plane prior to integration, and numerical subtleties related to the placement of the contour are discussed. The approach is generic in that it applies to arbitrary distribution functions, with the present manuscript focused on relativistic cases. Detailed verification of results via direct kinetic simulation in a variety of configuration space dimensions is also presented. Finally, the technique is applied to the challenging case of highly relativistic (i.e. extremely hot) plasmas. Here we show both qualitative agreement with prior work, as well as the disappearance of the Landau root which would have significant implication for real-life observation or experiment.

Comparison of K-spectral set bounds on norms of functions of a matrix or operator

We use results in [M. Crouzeix and A. Greenbaum, Spectral sets: numerical range and beyond, SIAM Jour. Matrix Anal. Appl., 40 (2019), pp. 1087-1101] to derive upper bounds on the norm of a function f of a matrix or other bounded linear operator A based on the infinity-norm of f on various regions in the complex plane. We compare these results to those that can be derived from a straightforward application of the Cauchy integral formula by replacing the norm of the integral by the integral of the resolvent norm. While, in some cases, the new upper bounds on ‖f(A)‖ are much tighter than those from the Cauchy integral formula, we show that in many cases of interest, the two bounds are of the same order of magnitude, with that from the Cauchy integral formula actually being slightly smaller. We give a partial explanation of this in terms of the numerical range of the resolvent at points near an ill-conditioned eigenvalue. At such points we show that the resolvent is close to a rank one matrix whose numerical range is a disk about a point near the origin, and we argue that in this case the two bounds are almost the same.

Some estimates for elliptic systems generalizing the Bitsadze system of equations

This article explores an elliptic system of n equations where the main part is the Bitsadze operator (the square of the Cauchy–Riemann operator) and the lower term is the product of a given matrix function by the conjugate of the desired vector function. The system was analyzed in the Banach space of vector functions that are bounded and uniformly H¨older continuous in the entire complex plane. It was revealed that the problem of solving the system in the specified space may not be Noetherian. An example of a homogeneous system with an infinite number of linearly independent solutions was given. As is known, for many classes of elliptic systems, the Noetherianity of boundary value problems in a compact domain is equivalent to the presence of a priori estimates in the corresponding spaces. In this regard, it is important to study the issues related to the establishment of a priori estimates for the system under consideration in the above space. In the case of coefficients weakly oscillating at infinity, necessary and sufficient conditions for the validity of the a priori estimate were found. These conditions were written out in the language of the spectrum of limit matrices formed by the partial limits of the coefficient matrix at infinity. Specific examples were provided to illustrate how the limit matrices are constructed and what the above conditions look like.

An efficient iterative method for matrix sign function with application in stability analysis of control systems using spectrum splitting

The goal of this study is to construct a novel iterative method to compute the matrix sign function using a different approach. It is discussed that the new method is globally convergent and asymptotically stable. It achieves the sixth order of convergence and only requires five matrix–matrix multiplications. The obtained results are extended to compute the number of eigenvalues of a matrix in a specified region of the complex plane. This is done by performing appropriate sequence of matrix sign computations. An application of this technique has been discussed in stability analysis of linear time‐invariant dynamic systems in control theory. Numerical results have been given to justify the effectual performance and superiority of the proposed method. Matrices of various sizes have been considered for this purpose.

Riemann zeros as quantized energies of scattering with impurities

We construct an integrable physical model of a single particle scattering with impurities spread on a circle. The S-matrices of the scattering with the impurities are such that the quantized energies of this system, coming from the Bethe Ansatz equations, correspond to the imaginary parts of the non-trivial zeros of the the Riemann ζ(s) function along the axis \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak{R}\\left(s\\right)=\\frac{1}{2}$$\\end{document} of the complex s-plane. A simple and natural generalization of the original scattering problem leads instead to Bethe Ansatz equations whose solutions are the non-trivial zeros of the Dirichlet L-functions again along the axis \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak{R}\\left(s\\right)=\\frac{1}{2}$$\\end{document}. The conjecture that all the non-trivial zeros of these functions are aligned along this axis of the complex s-plane is known as the Generalised Riemann Hypothesis (GRH). In the language of the scattering problem analysed in this paper the validity of the GRH is equivalent to the completeness of the Bethe Ansatz equations. Moreover the idea that the validity of the GRH requires both the duality equation (i.e. the mapping s → 1 – s) and the Euler product representation of the Dirichlet L-functions finds additional and novel support from the physical scattering model analysed in this paper. This is further illustrated by an explicit counterexample provided by the solutions of the Bethe Ansatz equations which employ the Davenport-Heilbronn function \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{D}\\left(s\\right)$$\\end{document}, i.e. a function whose completion satisfies the duality equation χ(s) = χ(1 – s) but that does not have an Euler product representation. In this case, even though there are infinitely many solutions of the Bethe Ansatz equations along the axis \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak{R}\\left(s\\right)=\\frac{1}{2}$$\\end{document}, there are also infinitely many pairs of solutions away from this axis and symmetrically placed with respect to it.

Anti-Instability of Complex Ghost

Abstract We argue that Lee–Wick’s complex ghost appearing in any higher derivative theory is stable and its asymptotic field exists. It may be more appropriate to call it “anti-unstable”, in the sense that the more the ghost "decays" into lighter ordinary particles, the larger the probability that the ghost remains as itself becomes. This is explicitly shown by analyzing the two-point functions of the ghost Heisenberg field which is obtained as an exact result in the N → ∞ limit in a massive scalar ghost theory with light O(N)-vector scalar matter. The anti-instability is a consequence of the fact that the poles of the complex ghost propagator are located on the physical sheet in the complex plane of four-momentum squared. This should be contrasted with the case of the ordinary unstable particle, whose propagator has no pole on the physical sheet.

$K^{th}$-order Differential Subordination Results of Analytic Functions in the Complex Plane

$K^{th}$-order Differential Subordination Results of Analytic Functions in the Complex Plane

On the convolution of convex 2-gons

We study the convolution of functions of the formfα(z):=(1+z1−z)α−12α, which map the open unit disk of the complex plane onto polygons of 2 edges when α∈(0,1). Inspired by a work of Cima, we study the limits of convolutions of finitely many fα and the convolution of arbitrary unbounded convex mappings. The analysis for the latter is based on the notion of angle at infinity, which provides an estimate for the growth at infinity and determines whether the convolution is bounded or not. A generalization to an arbitrary number of factors shows that the convolution of n randomly chosen unbounded convex mappings has a probability of 1/n! of remaining unbounded. We provide the precise asymptotic behavior of the coefficients of the functions fα.