Complex Plane Research Articles

Impact of random geometric errors in an elliptical superconducting magnet with application to a compact heavy-ion synchrotron

A compact heavy-ion synchrotron is under development for next-generation cancer therapy. A superconducting magnet is designed with a main dipole field of 3.5 T and a field error less than 5 × 10−4 to maintain compactness. The coil is wound directly with monolithic Nb-Ti wire on a curved elliptical mandrel, comprising 22 laminated layers to achieve the desired magnetic field. Given the critical need for field quality, it is imperative to determine the tolerance of coil fabrication and to devise a method to eliminate field error during the design stage. This paper presents a numerical investigation of possible random geometric errors stemming from fabrication and assembly tolerances in a curved elliptical superconducting magnet. We first provide an analytical formulation derived in a complex plane to estimate the field error of a straight elliptical coil. We then conduct a Monte Carlo simulation to assess cases involving misalignment of individual wires and coil block sectors. Additionally, we compare simulation results with field measurements obtained from a short model magnet tested previously to predict potential random errors in manufacturing. Finally, we calculate the beam dynamic aperture to assess the effects of random geometric errors on beam loss using Monte Carlo simulation results. This method enables the prediction of tolerances for fabricating high-quality high-field magnets and aids in making design decisions concerning the utilization of active shim coils.

Multiplication operators on the weighted Sobolev disk algebra

Let ⅅ be the unit disk in the complex plane ℂ. For α > −1, the weighted Sobolev disk algebra SA(ⅅ, dAα ) consists of all analytic functions in the weighted Sobolev space W 2,2(ⅅ, dAα ). In this paper, we prove that the multiplication operator is similar to on SA(ⅅ, dAα ) if and only if n = m, where n, m are positive integers. Then we characterize when a bounded operator P on SA(ⅅ, dAα ) belongs to the commutant of by using the matrix representation of P. In addition, we compute the exact norm of Mz on SA(ⅅ, dAα ). Finally, we prove that on the unweighted Sobolev disk algebra SA(ⅅ) the restrictions of to different invariant subspaces zkSA(ⅅ) (k ≥ 1) are not unitarily equivalent, and the restrictions of (n ≥ 2) to different invariant subspaces Sj (0 ≤ j < n) are also not unitarily equivalent.

Subgroups and Homomorphism Structures of Complex Pythagorean Fuzzy Sets

This research introduces the notion of complex Pythagorean fuzzy subgroup (CPFSG). Both complex fuzzy subgroup (CFSG) and complex intuitionistic fuzzy subgroup (CIFSG) have significance in assigning membership grades in the unit disk in the complex plane. CFSG has a limitation solved by CIFSG, while CIFSG deals with a limited range of values. The important novelty of the CPFSG lies in its ability to solve the above limitations simultaneously and gets a wider range of values to be engaged in CPFSG. This work has introduced and investigated CPFSG as a new algebraic structure via the conditions that the sum of the square membership and non-membership lies on the unit interval for both the amplitude term and phase term. The result as any CIFSG is CPFSG but the convers is not true has been proved. Complex Pythagorean fuzzy coset has been defined and complex Pythagorean fuzzy normal subgroup (CPFNSG) and their algebraic characteristic has been demonstrated. Homomorphism on the CPFSG is shown. Some results as the inverse image of CPFSG and CPFNSG under isomorphism function are also a CPFSG and CPFNSG, respectively.

Finite Sum of Integral Operators from the Fractional Cauchy Spaces to Bloch-Type and Zygmund-Type Spaces

The family of fractional Cauchy transforms, defined on the open unit disc in the complex plane, is of classical and modern interest. Membership of an analytic function in the family is determined by the requirement that the function can be expressed as an integral of a certain kernel against a complex Borel measure on the disc. Such an integral representation imposes a growth condition on the function and its derivatives. This exposes a connection between the families of Cauchy transforms and familiar spaces of analytic functions, such as the Bloch spaces and the Zygmund space. The notion of a composition operator has been a fruitful area of study. More generally, many authors have studied weighted composition operators, the differentiation operator, integral-type operators, and various products of such operators, acting from one normed linear space of analytic functions to another such space. A common theme of such works is to characterize the operator-theoretic notions of boundedness and compactness in terms of the inducing symbols of the operator. We extend these studies to a specific linear transformation which will be defined as the sum of finitely many integral operators. Our conclusions include a complete characterization of boundedness and compactness of the integral sum, acting from the fractional Cauchy spaces to the Bloch-type and Zygmund-type spaces.

Functions of a Complex Variable with a Large Parameter and Construction of Regions

The study of the asymptotic behavior of solutions of singularly perturbed equations in complex domains comes down to the study of integrals of exponential functions containing a large parameter. Such integrals differ significantly from the integrals to which the saddle point method is applicable. The saddle point method is not applicable to study such integrals. Thus, the problem arises of constructing domains and choosing integration paths for studying such integrals. In this work, specific examples of integrals show the construction of regions in the complex plane and the choice of integration paths. The chosen integration paths ensure that the integrals are bounded over a large parameter as this parameter tends to infinity. When constructing the domain and choosing integration paths, level lines of some harmonic functions that have zeros and singular points were used. The principle of symmetry is also used. In early works, cases were considered when the eigenvalues ​​of the first approximation matrix of a singularly perturbed equation had only zeros or only poles. Cases where the eigenvalues ​​have zeros and poles were not considered.

Possible molecules of triple-heavy pentaquarks within the extended local hidden gauge formalism

In this study, we explore the interactions between mesons and baryons in the open heavy sectors to identify potential triple-heavy molecular pentaquarks. We derive the meson-baryon interaction potentials using the vector meson exchange mechanism within the extended local hidden gauge formalism. The scattering amplitudes are computed by solving the coupled-channel Bethe-Salpeter equation, revealing several bound systems. By analyzing the poles of these amplitudes in the complex plane, we determine the masses and widths of these bound states. Additionally, we evaluate the couplings and compositeness of different channels within each bound system to assess their molecular characteristics. Our predictions include 4 Ωccc-like states, 4 Ωbbb-like states, 14 Ωbcc-like states, and 10 Ωbbc-like states, which could be targets for future experimental investigations. Published by the American Physical Society 2024

Out-of-Plane Equilibrium Points in the Photogravitational Hill Three-Body Problem

This paper investigates the movement of a negligible mass body (third body) in the vicinity of the out-of-plane equilibrium points of the Hill three-body problem under the effect of radiation pressure of the primaries. We study the effect of the radiation parameters through the factors qi,i=1,2 on the existence, position, zero-velocity curves and stability of the out-of-plane equilibrium points. These equilibrium positions are derived analytically under the action of radiation pressure exerted by the radiating primary bodies. We determined that these points emerge in symmetrical pairs, and based on the values of the radiation parameters, there may be two along the Oz axis and either none or two on the Oxz plane (outside the axes). A thorough numerical investigation found that both radiation factors have a strong influence on the position of the out-of-plane equilibrium points. Our results also reveal that the parameters have impact on the geometry of the zero-velocity curves. Furthermore, the stability of these points is examined in the linear sense. To do so, the spatial distribution of the eigenvalues on the complex plane of the linearized system is visualized for a wide range of radiation parameter combinations. By a numerical investigation, it is found that all equilibrium points are unstable in general.

Constraints on co- and cross-polarization reflection and transmission of Babinet-complementary metasurfaces

A study of constraints on reflection and transmission coefficients for Babinet-complementary metasurfaces is presented in this work. These coefficients have been found to describe circular paths in the complex plane, whose centers and radii are affected by losses. Additionally, it has been shown that it is possible to fully control the cross-polarization coefficients by rotating the metasurface with respect to the polarization direction. Our findings have been verified through numerical simulations and experiments. The properties discussed in this paper could be useful in limiting the types of possible responses of metasurfaces.

Floor plan generation: The interplay among data, machine, and designer

Recent advancements in machine learning (ML) in architectural design led to new developments in automated generation of floor plans. However, critical evaluation of ML-based generated floor plans has not progressed proportionally due to the subjectivity and complexity of the assessment, particularly for large and more complex floor plans. Accordingly, a hybrid (quantitative and qualitative) floor plan evaluation scheme is introduced in this study, focusing on multiple architectural aspects. To verify the effectiveness of the proposed framework, the evaluation scheme is applied on the generated floor plans resulting from two baseline computer vision models. The models have been trained on a newly introduced large-scale floor plan dataset called Modified Swiss Dwellings (MSD). The results showed that despite the progression of computer vision models for floor plan generation, they still have difficulty capturing the more complex architectural qualities. In addition, the prospect of floor plan generation and evaluation and possible future developments are discussed.

Robust Controller Design Ensuring the Desired Aperiodic Stability Degree of a Control System with Affine Uncertainty

This paper considers a system whose characteristic polynomial coefficients are linear combinations of the interval parameters of a plant forming a parametric polytope. A linear robust controller is parametrically designed to place a dominant pole of the system within the desired interval of the negative real semi-axis and ensure an aperiodic transient in the system. The parametric design procedure involves a low-order controller with dependent and free parameters: the former serve to place the dominant pole within the desired interval on the complex plane whereas the latter to shift the other poles to some localization regions beyond a given bound (to the left of the dominant pole to satisfy the pole dominance principle). To evaluate the dependent parameters of the controller, the originals of the interval bounds of the dominant pole are determined for the plant’s parametric polytope based on a corresponding theorem (see below). The free parameters of the controller are chosen using the robust vertex or edge D-partition method, depending on the boundary edge branches of the localization regions of the free poles. A numerical example of the parametric design procedure is provided: a PID controller is built to ensure an acceptable aperiodic transient time in a load-lifting mechanism with interval values of cable length and load weight.

Powers of Karpelevič arcs and their sparsest realising matrices

The region in the complex plane containing the eigenvalues of all n×n stochastic matrices was described by Karpelevič in 1951, and it is since then known as the Karpelevič region. The boundary of the Karpelevič region is the union of arcs called the Karpelevič arcs. We provide a complete characterization of the Karpelevič arcs that are powers of some other Karpelevič arc. Furthermore, we find the necessary and sufficient conditions for a sparsest stochastic matrix associated with the Karpelevič arc of order n to be a power of another stochastic matrix.

7437 Harnessing Diabetes Technology for Management of Immune Checkpoint Inhibitor Associated Diabetes

Abstract Disclosure: A. Knape: None. A. Kotwal: None. D. Rohlfsen: None. A. Patel: None. W. Goldner: None. Background: Immune checkpoint inhibitors (ICIs) are a key treatment modality for many cancers. One percent of patients who receive ICIs can develop insulin deficient ICI associated diabetes (ICI-DM). These patients often have complex oncologic therapy plans making treatment of newly diagnosed insulin deficient diabetes challenging which often results in additional healthcare-related burden for the patient. Methods: This is a retrospective case series of five adults who developed ICI-DM after receiving either PD-1, CTLA-4 or PDL-1 inhibitors or a combination. Data on demographics, medical history and labs were collected in addition to mode of glucose monitoring and treatment after diagnosis of ICI-DM. Results: Patients were age 44-72 at diagnosis of ICI-DM, three were female. Two were admitted to the hospital in DKA, one was diagnosed in the ER with severe hyperglycemia without DKA, two had outpatient hyperglycemia without DKA. Glucose at diagnosis ranged from 215-510 mg/dL. C-peptide was low for all patients. Fingerstick blood sugar (FSBS) monitoring and basal/bolus insulin was immediately initiated for all patients. All were transitioned from FSBS to continuous glucose monitor (CGM) and 2/5 ultimately transitioned to sensor-augmented insulin pump therapy (AID). Time in range on CGM improved in four patients with percent time in range improving between 10-48%. In two patients, hypoglycemia frequency declined by 6% and 3%. Hyperglycemia frequency declined by 13% and 48% in patients on AID therapy and by 14% and 25% in two of those on basal/bolus insulin therapy. All patients preferred CGM over FSBS and remained on CGM therapy. Those who chose AID preferred it over basal/bolus insulin therapy stating it was less complicated than basal/bolus insulin with CGM. Conclusions: We report successful treatment of five cases of ICI-DM with CGM + basal/bolus insulin or AID. 80% demonstrated improved time in range and decreased overall frequency of hyperglycemia. 40% had decreased frequency of hypoglycemia. CGM allows for dynamic adjustment of insulin therapy in a population with complex medical regimens resulting in fluctuating glucose levels. The integrated pump and sensor technology in AID was less burdensome than basal/bolus in those who chose it. Managing ICI-DM is complicated. The burden of this entity can be lessened by harnessing the power of diabetes technology. Presentation: 6/2/2024

Solitons moving on background waves of the focusing nonlinear Schrödinger equation with step-like initial condition

This work concerns the long-time asymptotic behaviors of the focusing nonlinear Schrödinger equation with step-like initial condition in present of discrete spectrum. The exact step initial-value problem with non-vanishing boundary on one side has been solved, while the step-like initial-value problem with solitons emerging remains open. We study this problem and explore the solitons moving on background waves of the focusing nonlinear Schrödinger equation by classifying all possible locations of discrete spectrum associated with the spectral functions. It is shown that there are five kinds of zones for the discrete spectrum in complex plane, which are called dumbing zone, trapping zone, trapping/waking zone, transmitting/waking zone and transmitting zone, respectively. By means of Deift–Zhou nonlinear steepest-descent method for Riemann–Hilbert problems, the long-time asymptotics of the solution along with the locations of the solitons for each case are formulated. Numerical simulations match very well with the theoretical analysis.

Forecast, analytical and trend analysis of the development of the agro-industrial complex of Russia in the main agri-food markets as an area of state strategic guidelines

Abstract The modern structure of the Russian agro-industrial complex has the properties of consolidation, expansion and prevalence of agricultural organizations and agricultural holdings, as evidenced by the factor of their provision for about two-thirds of the total agricultural production. This circumstance states the growing dominance of large agricultural business and the progressive evolution of the agricultural sector, as well as the reduction in the importance of individual farms of the population. In the course of the study, the current state of development of the agro-industrial complex of Russia is considered, a forecast, analytical and trend analysis of the development of the domestic agro-industrial complex is carried out through the prism of the fundamental agri-food markets and the contours of the subsequent progress in the development of the agricultural sector in the context of the evolving restructuring of the social, economic and political situation are determined. The novelty of the research lies in the presentation of forecast estimates and trends in the development of the main agri-food markets of the Russian agro-industrial complex, which determine the presumptive transformation of the agrarian mechanism of Russia in the medium and long term: destabilization of international political and trade-economic cooperation, intensification of economic restrictive measures, as well as fundamentally new state priorities. The agricultural sector of the economy tends to react extremely sharply to the current situation in recent years, manifested in unstable international economic relations and foreign policy crises, which directly negatively affects the conditions for the functioning of the agricultural business. The presented forecast scenario for the development of agri-food markets will contribute to the formation of directions for state strategic management, planning and regulation of the agro-industrial complex for the medium and long term.

Neurologists' Attitudes and Perceptions on Palliative Care: A Qualitative Study.

Despite significant advances in the treatment of neurologic disorders, many conditions require complex care planning and advanced care planning. Neurologists are in a unique position because they are integral in providing patient centered care, understanding neurologic disease and illness trajectory, and how disease can affect patients' sense of self and values. Currently, little is known about neurologists' perceptions and challenges in care planning and palliative care for their patients. Neurologists from one Canadian academic institution participated in a 30-minute semistructured interview from November 2022 to April 2023. Interviews were conducted until saturation was reached and confirmed. Interviews occurred online through a secure platform or in-person and were recorded. Data were analyzed using a constant comparative method using constructivist grounded theory. Member checking was conducted post interview. Ten neurologists participated across a broad spectrum of neurology experience and subspecialties. We developed a detailed theory of understanding neurologists' attitudes and perceptions of palliative care. When neurologists delay or fail to initiate care planning discussions or palliative care, it results from a complex interplay between patient, physician, and resource accessibility factors. Certain contextual factors, such as a first visit or follow-up, inpatient vs outpatient setting, clinic culture, and the type of clinic practice, are factors that can influence these conversations. As a result, physicians may fail to use available resources, or they may involve other care providers or refer to subspecialty neurologic clinics. However, this delay can still lead to patient and provider harm. Opportunities to improve care exist with continuing education opportunities for trainees and staff, collaboration with palliative care specialists, and health systems support, such as increasing public awareness to address misconceptions about palliative care and resource availability. Our findings identify that failure or delay to initiate care planning and palliative care by neurologists results from a complex interplay between local culture, experience, context, practice type, and patient factors. Opportunities to improve care include increasing educational opportunities, building integrated and collaborative practices, and dedicated health systems support.

Observation of chaotic fluctuations in turbulent plasma

Turbulence is a prevalent phenomenon in space and astrophysical plasmas, often characterized by stochastic fluctuations. While laboratory experiments and numerical simulations have revealed chaotic behavior, in situ observations of turbulent plasmas in natural environments have predominantly shown highly stochastic signatures. Here, we present unprecedented in situ evidence of chaotic fluctuations in the turbulent solar wind plasma downstream of the Earth's bow shock. By analyzing the relative location of magnetic-field fluctuations on the permutation entropy–complexity plane, we demonstrate that turbulence in the magnetosheath plasma exhibits characteristics of chaotic fluctuations rather than stochastic behavior, diverging from the expected traits of well-developed turbulence. This finding challenges established notions of plasma turbulence and reveals the need for caution when using the magnetosheath as a laboratory for studying plasma turbulence.

Suppression of synchronous current of active magnetic bearings considering radial displacement inconsistency

Suppressing rotor synchronous current is vital for achieving precise control in active magnetic bearing (AMB) rotor systems. The displacement amplitude inconsistency in orthogonal channels presents new challenges for synchronous current suppression. This paper proposes an improved complex least mean square (CLMS) algorithm with error compensation to effectively mitigate rotor unbalanced vibration. Firstly, a dynamic model of the AMB-rotor system with an unbalanced mass is established, and the displacement amplitude inconsistency within the orthogonal channels of the radial magnetic bearings is analyzed. Subsequently, the rotor displacement signal from the orthogonal channels is converted into a complex plane signal. By calculating the negative sequence weight component, the displacement signal is reconstructed, and an error compensation channel based on the CLMS is introduced to enable real-time tracking and compensation of displacement amplitude inconsistencies in the orthogonal channels. Moreover, system stability across the full speed range is ensured through the application of the phase compensation angle method. Finally, experimental validation on a magnetic levitation molecular pump prototype confirms the effectiveness of the proposed algorithm.

Sinc method in spectrum completion and inverse Sturm–Liouville problems

Cardinal series representations for solutions of the Sturm–Liouville equation with a complex‐valued potential are obtained, by using the corresponding transmutation operator. Consequently, partial sums of the series approximate the solutions uniformly with respect to in any strip of the complex plane. This property of the obtained series representations leads to their applications in a variety of spectral problems. In particular, we show their applicability to the spectrum completion problem, consisting in computing large sets of the eigenvalues from a reduced finite set of known eigenvalues, without any information on the potential as well as on the constants from boundary conditions. Among other applications this leads to an efficient numerical method for computing a Weyl function from two finite sets of the eigenvalues. This possibility is explored in the present work and illustrated by numerical tests. Finally, based on the cardinal series representations obtained, we develop a method for the numerical solution of the inverse two‐spectra Sturm–Liouville problem and show its numerical efficiency.

Algorithm 1046: An Improved Recurrence Method for the Scaled Complex Error Function

Calculation of the scaled complex error function \(w(z)\) by recurrence is discussed, and a new method for determining the number of steps required to achieve a given accuracy is introduced. This method is found to work throughout the complex plane, except for a short section of the real line, centred at the origin. An algorithm based on this analysis is implemented; Taylor series with stored coefficients are used to compute \(w(z)\) in a small region where recurrence is not efficient. The new algorithm is tested extensively and found to outperform earlier recurrence-based codes. It also performs favourably against recent codes based on other methods.

The Riemann–Hilbert Approach to the Higher-Order Gerdjikov–Ivanov Equation on the Half Line

The Fokas method exhibits remarkable versatility in solving boundary value problems associated with both linear and nonlinear partial differential equations, particularly when conventional approaches encounter challenges in handling intricate boundary conditions. The existing literature often lacks the incorporation of unconventional boundary conditions, and this study addresses this issue by extending the application of the Fokas method to the higher-order Gerdjikov-Ivanov equation on the half line (−∞,0]. We have demonstrated the exclusive representation of the potential function u(z,t) in the higher-order Gerdjikov–Ivanov equation through the solution of a Riemann–Hilbert problem. The characteristic function is partitioned on the complex plane, and we obtain the jump matrix between each partition based on the positive and negative values of the partition as well as the spectral matrix determined by the initial data and boundary value data. The findings suggest that the spectral functions are not mutually independent; instead, they conform to a global relationship. The novel aspect of this study is the application of the Fokas method to a previously unexplored case, contributing to the theoretical and practical understanding of complex partial differential equations and filling a gap in the treatment of boundary conditions.