Complex Plane Research Articles

Sharpening Littlewood subordination principle with univalent symbol

AbstractLet be a subharmonic function on the open unit disc , centered at the origin of the complex plane, and let be a holomorphic function such that . A classical result, known as Littlewood subordination principle, states , where and are integral means over the circle of radius centered at the origin, of the functions and , respectively. In this note, we obtain an unexpected improvement of Littlewood subordination principle in the case when the function is univalent, by proving that where . We also list some applications of this result including an improved variant of Rogosinski theorem with univalent symbol.

Asymptotic approximations of complex order tangent, Tangent-Bernoulli and Tangent-Genocchi polynomials

In this study, asymptotic formulas for complex order Tangent, Tangent-Bernoulli, and Tangent-Genocchi polynomials are obtained through the method of contour integration, strategically avoiding branch cuts in the process. By employing this technique, the paper elucidates the behavior of these polynomials in the complex plane. Additionally, the investigation expands upon the Taylor series expansion of these functions, revealing alternative asymptotic expansions. This dual approach not only enhances our understanding of the asymptotic properties of these polynomials but also offers alternative mathematical perspectives, enriching the existing body of knowledge in this field.

Multiple skin transitions in two-band non-Hermitian systems with long-range nonreciprocal hopping

Abstract Non-Hermitian skin effect is a prominent feature in non-Hermitian physics, leading to novel topological properties and expanding the traditional energy band theories. In this paper, we investigate a two-band non-Hermitian system in which multiple skin transitions are induced by long-range nonreciprocal hopping. The spectral winding number under periodic boundary conditions reveals the localization directions of skin states. Further, we present the analytical solution of transition points by tracing the self-intersecting points on the complex plane. Interestingly, the current system exhibits the abundant non-Hermitian skin effects, including the normal, W-shaped, and bipolar localization properties, which the eigenstate distributions and the generalized Brillouin zone can clearly illustrate. We also provide a phase diagram to represent the skin transition properties of the system comprehensively. Further, we demonstrate that the multimer non-Hermitian lattices also present the anomalous skin effect and multiple transitions, which occur in the region of the bulk band touching, the same as the two-band lattice. Moreover, a feasible scheme is proposed to realize the current non-Hermitian system with long-range nonreciprocal hopping by a topoelectrical circuit. This work further supplies the content of skin transitions and may help us explore more plentiful localization features in the two-band non-Hermitian systems.

NEW CONJECTURES FOR THE ENTIRE FUNCTIONS ASSOCIATED WITH FRACTIONAL CALCULUS

In this paper, we address the entire Fourier sine and cosine integrals related to the Mittag-Leffler function. We guess that the entire functions have the real zeros in the entire complex plane. They can be connected with the well-known conjectures in analytic number theory. They are considered as the special solutions for the time-fractional diffusion equation within the Caputo fractional derivative.

Nucleon resonance parameters from Roy–Steiner equations

A reliable determination of the pole parameters and residues of nucleon resonances is notoriously challenging, given the required analytic continuation into the complex plane. We provide a comprehensive analysis of such resonance parameters accessible with Roy–Steiner equations for pion–nucleon scattering—a set of partial-wave dispersion relations that combines the constraints from analyticity, unitarity, and crossing symmetry—most prominently of the Δ(1232) resonance. Further, we study the Roper, N(1440), resonance, which lies beyond the strict domain of validity, in comparison to Padé approximants, comment on the role of subthreshold singularities in the S-wave, and determine the residues of the f0(500), ρ(770), and f0(980) resonances in the t-channel process ππ→N¯N. The latter allows us to test—for the first time fully model independently in terms of the respective residues—universality of the ρ(770) couplings and the Goldberger–Treiman relation expected if the scalars behaved as dilatons, in both cases revealing large deviations from the narrow-resonance limit.

Underwater descattering with polarization line constraint

It is effective to use polarization information for underwater descattering. However, existing methods always consider only the degree of linear polarization (DoLP) and ignore the angle of linear polarization (AoLP). In this paper, we propose a novel polarization imaging model on the complex plane that considers both the DoLP and the AoLP. Our model unifies and generalizes the polarization imaging model. Based on the model, we design an effective descattering method that can restore the intensity and polarization information from degraded images. In our method, we propose a polarization quantization to reduce polarization states from 1.8×1016 to only thousands. We propose a polarization line constraint, by which each distinct polarization state produces a polarization line in a degraded image on the complex plane. By the constraint, the clear image and its polarization states can be restored from its corresponding degraded image. To validate our method, we give an image quality metric for evaluating the accuracy of the restored polarization information. Experimental results show that our method is competitive with other state-of-the-art methods and achieves the best performance in terms of the underwater metrics.

The Small-N Series in the Zero-Dimensional O(N) Model: Constructive Expansions and Transseries

AbstractWe consider the zero-dimensional quartic O(N) vector model and present a complete study of the partition function Z(g, N) and its logarithm, the free energy W(g, N), seen as functions of the coupling g on a Riemann surface. We are, in particular, interested in the study of the transseries expansions of these quantities. The point of this paper is to recover such results using constructive field theory techniques with the aim to use them in the future for a rigorous analysis of resurgence in genuine quantum field theoretical models in higher dimensions. Using constructive field theory techniques, we prove that both Z(g, N) and W(g, N) are Borel summable functions along all the rays in the cut complex plane $$\mathbb {C}_{\pi } =\mathbb {C}{\setminus } \mathbb {R}_-$$ C π = C \ R - . We recover the transseries expansion of Z(g, N) using the intermediate field representation. We furthermore study the small-N expansions of Z(g, N) and W(g, N). For any $$g=|g| e^{\imath \varphi }$$ g = | g | e ı φ on the sector of the Riemann surface with $$|\varphi |<3\pi /2$$ | φ | < 3 π / 2 , the small-N expansion of Z(g, N) has infinite radius of convergence in N, while the expansion of W(g, N) has a finite radius of convergence in N for g in a subdomain of the same sector. The Taylor coefficients of these expansions, $$Z_n(g)$$ Z n ( g ) and $$W_n(g)$$ W n ( g ) , exhibit analytic properties similar to Z(g, N) and W(g, N) and have transseries expansions. The transseries expansion of $$Z_n(g)$$ Z n ( g ) is readily accessible: much like Z(g, N), for any n, $$Z_n(g)$$ Z n ( g ) has a zero- and a one-instanton contribution. The transseries of $$W_n(g)$$ W n ( g ) is obtained using Möbius inversion, and summing these transseries yields the transseries expansion of W(g, N). The transseries of $$W_n(g)$$ W n ( g ) and W(g, N) are markedly different: while W(g, N) displays contributions from arbitrarily many multi-instantons, $$W_n(g)$$ W n ( g ) exhibits contributions of only up to n-instanton sectors.

On a multiplier operator induced by the Schwarzian derivative of univalent functions

AbstractIn this paper, we study a multiplier operator which is induced by the Schwarzian derivative of univalent functions with a quasiconformal extension to the extended complex plane. As applications, we show that the Brennan conjecture is satisfied for a large class of quasidisks. We also establish a new characterization of asymptotically conformal curves and of the Weil–Petersson curves in terms of the multiplier operator.

Period-doubling cascade route to chaos in an initially curved microbeam resonator exposed to fringing-field electrostatic actuation

AbstractThis paper explores the chaotic dynamics of a piezoelectrically laminated initially curved microbeam resonator subjected to fringing-field electrostatic actuation, for the first time. The resonator is fully clamped at both ends and is coated with two piezoelectric layers, encompassing both the top and bottom surfaces. The nonlinear motion equation which is obtained by considering the nonlinear fringing-field electrostatic force, includes geometric nonlinearities due to the mid-plane stretching and initial curvature. The motion equation is discretized using Galerkin method and the reduced order system is numerically integrated over the time for the time response. The variation of the first three natural frequencies with respect to the applied electrostatic voltage is determined and the frequency response curve is obtained using the combination of shooting and continuation methods. The bifurcation points have been examined and their types have been clarified based on the loci of the Floquet exponents on the complex plane. The period-doubled branches of the frequency response curves originating from the period doubling (PD) bifurcation points are stablished. It's demonstrated that the succession PD cascades leads to chaotic behavior. The chaotic behavior is identified qualitatively by constructing the corresponding Poincaré section and analyzing the response's associated frequency components. The bifurcation diagram is obtained for a wide range of excitation frequency and thus the exact range in which chaotic behavior occurs for the system is determined. The chaotic response of the system is regularized and controlled by applying an appropriate piezoelectric voltage which shifts the frequency response curve along the frequency axis.

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.

EXPONENTIAL ASYMPTOTICS USING NUMERICAL RATIONAL APPROXIMATION IN LINEAR DIFFERENTIAL EQUATIONS

Abstract Singularly perturbed ordinary differential equations often exhibit Stokes’ phenomenon, which describes the appearance and disappearance of oscillating exponentially small terms across curves in the complex plane known as Stokes lines. These curves originate at singular points in the leading-order solution to the differential equation. In many important problems, it is impossible to obtain a closed-form expression for these leading-order solutions, and it is therefore challenging to locate these singular points. We present evidence that the analytic leading-order solution of a linear differential equation can be replaced with a numerical rational approximation using the adaptive Antoulas–Anderson (AAA) method. Despite such an approximation having completely different singularity types and locations, we show that the subsequent exponential asymptotic analysis accurately predicts the exponentially small behaviour present in the solution. For sufficiently small values of the asymptotic parameter, this approach breaks down; however, the range of validity may be extended by increasing the number of poles in the rational approximation. We present a related nonlinear problem and discuss the challenges that arise due to nonlinear effects. Overall, our approach allows for the study of exponentially small asymptotic effects without requiring an exact analytic form for the leading-order solution; this permits exponential asymptotic methods to be used in a much wider range of applications.

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.