Complex Coefficients Research Articles

Approximation of complex coefficient filters

Approximation of complex coefficient filters

AI driven fault diagnosis approach for stator turn to turn faults in induction motors

Induction motors (IMs) are vital in industrial applications. Although all motor faults can disrupt its operation significantly, stator turn to turn faults (ITFs) are the most challenging one due to their detection difficulties. This paper introduces an AI-based approach to detect ITFs and assess their severity. A simulation based on an accurate mathematical model of the IM under ITFs is employed to generate the training data. Recognizing that ITFs directly affect the motor’s current balance, complex current unbalance coefficient is identified and used as the key feature for detecting ITFs. Since unbalanced supply voltage (USV) can also disrupt current balance, the AI models are trained to account for USV by incorporating complex voltage unbalance coefficient that helps to distinguish between ITF-induced and voltage-induced imbalances. After feature extraction, the AI models are trained and validated with simulation data. The approach’s effectiveness is further tested using an experimental setup, where measurements from motors under various fault conditions, including USV scenarios, are considered. The results indicate that the gradient boosting model outperforms other ML models in detecting ITFs in IMs and assessing their severity. In the pursuit of achieving highest possible performance, DNN is tested and compared with ML models. The study reveals that DNN demonstrates superior performance in all tested scenarios including USV making DNN the top performer that to be used in the proposed approach. The proposed AI-based approach based on DNN offers high accuracy in fault detection and can effectively distinguish between ITFs and USV-induced anomalies, maintaining low estimation errors and robust performance across different operational conditions.

Precise Measurement of Complex Coefficient Filters by Using Square Waves†

Usually, the frequency characteristics of the complex filter can be examined by applying a pair of quadrature sinusoidal waves ( and ) to a FUT (filter under test). While this method is recognized for its accuracy, it requires two sinusoidal test signals with tightly matched amplitudes and a strict phase shift over a broad frequency range. This becomes particularly challenging at higher frequencies. To address this issue, this paper proposes a new measurement method for the complex filter. Instead of sinusoidal waves, the proposed method adopts square waves which can be easily generated by using a digital divider as the test signals to the FUT. The output signals are observed using a spectrum analyzer. Since the Fourier‐series expansion of a square wave is decomposed into its fundamental and harmonic components, the frequency response is measured from the fundamental component using a spectrum analyzer equipped with a high‐resolution bandpass filter. This paper describes the proposed square‐wave method from theoretical aspects. Through experiment, the validity of the proposed method is confirmed. © 2025 The Author(s). IEEJ Transactions on Electrical and Electronic Engineering published by Institute of Electrical Engineers of Japan and Wiley Periodicals LLC.

Five Open Problems on the Dirichlet-to-Neumann Operator with Variable Coefficients

We consider the Dirichlet-to-Neumann operator N associated with an elliptic operator Au=-∑k,l=1d∂k(ckl∂lu)with possibly complex coefficients. We present five interesting open problems related to N.

Non-symmetrical vortex beam shaping in VECSEL laser arrays

We propose and numerically test, to our knowledge, a novel concept for asymmetric vortex beam generation in a degenerate vertical external cavity surface emitting laser (DVECSEL). The method is based on a phase-locking ring array of lasers created inside a degenerate cavity with a binary amplitude mask containing circular holes. The diffraction engineering of the mask profile allows for controlling the complex coupling between the lasers. The asymmetry between different lasers is introduced by varying the hole diameters corresponding to different lasers. Several examples of masks with non-uniform or uniform circular holes are investigated numerically and analytically to assess the impact of non-uniform complex coupling coefficients on the degeneracy between the vortex and anti-vortex steady states of the ring laser arrays. It is found that the in-phase solution always dominates irrespective of non-uniform masks. The only solution to make one particular vortex solution dominant over other possible steady-state solutions consists of imprinting the necessary phase shift among neighboring lasers in the argument of their coupling coefficients. We also investigate the role of the Henry factor inherent to the use of a semiconductor active medium in the probabilities to generate vortex solutions. Analytical calculations are performed to generalize a formula previously reported [ Opt. Express 30, 15648 (2022)OPEXFF1094-408710.1364/OE.456946], for the limiting Henry factor to cover the case of complex couplings.

On well-posedness for parabolic Cauchy problems of Lions type with rough initial data

Abstract We establish a complete picture for well-posedness of parabolic Cauchy problems with time-independent, uniformly elliptic, bounded measurable complex coefficients. We exhibit a range of p for which tempered distributions in homogeneous Hardy–Sobolev spaces $$\dot{H}^{s,p}$$ H ˙ s , p with regularity index $$s \in (-1,1)$$ s ∈ ( - 1 , 1 ) are initial data. Source terms of Lions’ type lie in weighted tent spaces, and weak solutions are built with their gradients in weighted tent spaces as well. A similar result can be achieved for initial data in homogeneous Besov spaces $$\dot{B}^{s}_{p,p}$$ B ˙ p , p s .

A Type of Eneström-Kakeya Theorem for Quaternionic Polynomials Involving Monotonicity with a Reversal

The Eneström-Kakeya theorem states that if $P(z)=\sum_{\ell =0}^n a_\ell z^\ell$ is a polynomial of degree $n$ with real coefficients satisfying $0\leq a_0\leq a_1\leq \cdots\leq a_n$, then all zeros of $P$ lie in $|z|\leq 1$ in the complex plane. Motivated by recent results concerning an Eneström-Kakeya "type" condition on the real and imaginary parts of complex coefficients, we give similar results with hypotheses concerning the real and imaginary parts of the coefficients of a quaternionic polynomial. We give bounds on the moduli of quaternionic zeros of such polynomials.

Double star arrangement and the pointed multinet

Abstract Let be a hyperplane arrangement in a complex projective space. It is an open question if the degree one cohomology jump loci (with complex coefficients) are determined by the combinatorics of . By the work of Falk and Yuzvinsky [Compositio Math. 143 (2007), no. 4, 1069–1088] and Marco‐Buzunáriz [Graphs Combin. 25 (2009), no. 4, 469–488], all the irreducible components passing through the origin are determined by the multinet structure, which is combinatorially determined. Denham and Suciu introduced the pointed multinet structure, which is combinatorially determined, to obtain examples of arrangements with translated positive‐dimensional components in the degree one cohomology jump loci [Proc. Lond. Math. Soc. (3) 108 (2014), no. 6, 1435–1470]. Suciu asked the question if all translated positive‐dimensional components appear in this manner [Ann. Fac. Sci. Toulouse Math. (6) 23 (2014), no. 2, 417–481]. In this paper, we show that the double star arrangement introduced by Ishibashi, Sugawara, and Yoshinaga [Adv. in Appl. Math. 162 (2025), Paper No. 102790, Example 3.2] gives a negative answer to this question.

Adaptive hybrid PSO-GD optimized phase-locked loop for robust grid synchronization in renewable energy systems

PurposeThis paper aims to address the challenges of maintaining accurate grid synchronization in renewable energy systems under dynamic grid disturbances. Grid-connected renewable energy systems require precise synchronization with the utility grid to ensure stable power transfer and maintain power quality. Phase-locked loops (PLLs) serve as critical components by accurately tracking the grid’s phase angle, frequency and voltage parameters in real time. However, the increasing penetration of renewable sources introduces significant grid disturbances including voltage sags, phase jumps, frequency variations and harmonic distortions that challenge conventional PLL performance. Under these dynamic conditions, maintaining accurate grid synchronization becomes essential for preventing system instability, ensuring power quality and meeting grid code requirements.Design/methodology/approachA novel hybrid PLL method is proposed, combining particle swarm optimization (PSO) and gradient descent (GD) algorithms. The adaptive model continuously tunes PLL parameters, enabling real-time responsiveness to grid fluctuations, including voltage sags, phase shifts and harmonic distortions. The mathematical framework incorporates state-space modeling and Lyapunov stability analysis. Advanced loop filter architectures combining multiple delayed signal cancellation (MDSC) with complex coefficient filtering (CCF) provide enhanced harmonic rejection. Real-time parameter adaptation ensures optimal performance under varying grid conditions.FindingsThe hybrid PSO-GD PLL demonstrates superior performance over traditional synchronization techniques (SRF-PLL, DDSRF-PLL, MSOGI-PLL) across multiple metrics. In steady-state operation, it achieves frequency error of ±0.1 Hz and phase error of ±0.5° compared to ±0.5 Hz and ±2.0° for SRF-PLL. Dynamic response shows significant improvements with 25 ms settling time and ±0.3 Hz frequency overshoot, versus 40 ms and ±2.0 Hz in SRF-PLL. The system exhibits enhanced stability with 65° phase margin, 120 Hz bandwidth and excellent performance under grid fault conditions through auto-tuning capability.Originality/valueThis work introduces an adaptive, hybrid optimization framework for PLL, specifically enhancing grid-tied renewable systems’ synchronization accuracy and resilience. Integrating PSO’s global search capabilities with GD’s local refinement creates a unique dual-stage optimization process that surpasses traditional single-algorithm approaches. The mathematical framework’s incorporation of advanced filtering techniques (MDSC and CCF) with Lyapunov-based stability analysis represents a novel approach to PLL design. Moreover, the auto-tuning capability through real-time parameter adaptation addresses the critical challenge of maintaining synchronization in increasingly complex grid environments with high renewable penetration.

Solving Quadratic Equations with Complex Coefficients & Determining the Square Root for any Complex Number

This paper presents a definitive solution for expressing the second root (z1/2) of any complex number z as a function of z and Δ1/2 as a function of Δ in the context of quadratic equations with complex coefficients. This research has resulted in two direct algorithms and associated software. These algorithms are not only accurate but also exceptionally fast, providing a powerful tool for efficiently determining the second root of any complex number and solving quadratic equations with complex coefficients. This innovative approach addresses a long-standing challenge in the field, significantly contributing to the accuracy and speed of calculations in complex coefficient scenarios either manually or using a programmable computing machine such as smart phones or computers.

Analysis of Power Quality Based on Machine Learning Methods for Low-Voltage Electrical Distribution Lines

The main objective of this paper is to propose two innovative monitoring methods for electrical disturbances in low-voltage networks. The two approaches present a focus on the classification of voltage signals in the frequency domain using machine learning techniques. The first technique proposed here uses the Fourier transform (FT) of the voltage waveform and classifies the corresponding complex coefficients through a multilayered neural network with multivalve neurons (MLMVN). In this case, the classifier structure has three layers and a small number of neurons in the hidden layer. This allows complex-valued inputs to be processed without the need for pre-coding, thus reducing computational cost and keeping training time short. The second technique involves these of the short-time Fourier transform (STFT) and a convolution neural network (CNN) with 2Dconvolutions in each layer for feature extraction and dimensionality reduction. The voltage waveform perturbations taken into consideration are: voltage sag, voltage swell, harmonic pollution, voltage notch, and interruption. The comparison between the two proposed techniques is developed in two phases: initially, the simulated data used during the training phase are considered and, subsequently, various experimental measurements are processed, obtained both through an artificial disturbance generator and through a variable load. The two techniques represent an innovative approach to this problem and guarantee excellent classification results

Null controllability of some stochastic complex parabolic equations with singular potentials

This paper is devoted to proving the null controllability of some forward stochastic complex parabolic equations perturbed by an inverse-square singular potential. By the standard duality technique, it suffices to establish suitable observability inequalities for the corresponding backward stochastic system. For this purpose, we derive a new global Carleman estimate of backward singular stochastic complex parabolic equations by a new auxiliary function. In order to overcome the difficulties caused by randomness and singularity, some conditions are proposed for complex coefficients. Moreover, when the singular stochastic complex parabolic equations degenerate to stochastic heat equations without singularity, the corresponding null controllability still holds.

Iterative Solutions for Certain Complex Coefficient Linear Systems: Jacobi and Gauss-Seidel Methods

In this study, the performance of the Jacobi and Gauss-Seidel iteration methods for solving systems of linear equations with complex coefficients is analyzed. The coefficient matrix of the system is transformed into a real coefficient system by separating the real and imaginary parts. The study aims to compare the accuracy and computational efficiency of these methods within the context of selected examples, while also evaluating their convergence behavior. The findings demonstrate that, for the examples considered, the Gauss-Seidel method converges faster and with lower initial errors compared to the Jacobi method.

Consensus control of homogeneous networks of general second order discrete-time systems

We consider the problem of obtaining linear state feedback controllers to achieve consensus in a homogeneous network of general second-order discrete-time systems whose graph is directed. Consensus is achieved if certain polynomials associated with the closed-loop network are Schur stable. The coefficients of these polynomials depend on the eigenvalues of the network Laplacian matrix and can therefore be complex. This paper provides a new and simple proof and condition for a second-order polynomial with complex coefficients to be Schur. This condition was applied to obtain controllers which achieve consensus with a guaranteed rate of convergence. We show that consensus can always be achieved for marginally stable systems and discretized systems. Simple conditions for consensus achieving controllers are obtained when the Laplacian eigenvalues are all real. We also demonstrate how these results can be applied to some robust control problems.

L2-type invariants and cohomology jump loci for complex smooth quasi-projective varieties

Let [Formula: see text] be a complex smooth quasi-projective variety with a fixed epimorphism [Formula: see text]. We consider the asymptotic behavior of invariants such as Betti numbers with any field coefficients and the order of the torsion part of singular integral homology associated to [Formula: see text], known as the [Formula: see text]-type invariants. At homological degree one, we give concrete formulas for these limits by the geometric information of [Formula: see text] when [Formula: see text] is orbifold effective. The proof relies on a study about cohomological degree one jump loci of [Formula: see text]. We extend part of Arapura’s result for cohomological degree one jump loci of [Formula: see text] with complex field coefficients to the one with positive characteristic field coefficients. As an application, when [Formula: see text] is a hyperplane arrangement complement, a combinatoric upper bound is given for the number of parallel positive-dimensional components in cohomological degree one jump loci with complex coefficients. Another application is that we give a positive answer to a question posed by Denham and Suciu for hyperplane arrangement.

A New General Fast Neurodynamics (GFN) for Solving Complex Generalized Sylvester Equation With Power Systems Application

ABSTRACTThe study on topics related to solving matrix equations with complex coefficients initiates an interesting topic for research in the previous decades. In this research document, a novel and general‐fast recurrent neural network (RNN) model, based on an extended activation function (AF), will be designed for solving the generalized Sylvester equation with complex coefficients. The key innovation of the new extended activation function, compared to existing functions, is based on a set of constants in the new AF which reduces the fixed convergence time of the corresponding general‐fast RNN formula. Convergence analysis and numerical experiments in Simulink will show the efficiency and the accelerated convergence ability of the proposed dynamical system. Furthermore, the novel neurodynamics is applied for the control of a single‐machine bus model and for computing the different currents in electrical circuits.

Natural vibrations of a viscoelastic three-layer cylindrical body

The natural vibrations of a viscoelastic coaxial cylindrical body are considered; the space between the shells is filled with a viscoelastic material. The relationship between stress and strain satisfies the Boltzmann–Voltaire hereditary integral. As an example of a viscoelastic material, a three-parametric relaxation kernel with a weak Rzhanitsyn–Koltunov singularity is used. The problems of small vibrations of the mechanical system under consideration are solved. Equations of small vibrations of the aggregate in displacements are obtained on the basis of Lame’s differential equations of the theory of viscoelasticity with complex coefficients. The equations of vibration of the outer and inner shells, which are made of viscoelastic material, satisfy the equations of motion of the shell, subject to the Kirchhoff–Love hypotheses. The problem is solved using the Green–Lamb transformation and the complex amplitude method. The stresses and displacements of each shell and filler are expressed through special functions of the Bessel and Neumann complex argument of an arbitrary order. A frequency equation with a complex parameter is obtained, which is solved numerically using the Muller method. For structurally inhomogeneous mechanical systems, the dependences of several modes of the complex natural frequency (real and imaginary parts) on various parameters of three-layer bodies are comparatively assessed. The application of asymptotic and numerical methods for solving frequency equations with a complex-output parameter is also comparatively assessed.

On stability of linear autonomous difference equations with complex coefficients

We study the stability of linear autonomous scalar difference equations with complex coefficients. For an equation with an arbitrary number of delays, we propose a simple proof of the linear connectivity of the stability region in the space of coefficients. This result allows us to assert that the stability region of the equation in the space of coefficients is the region of the D-decomposition of this space containing the origin of coordinates. Further, we consider some equations with two delays and complex coefficients, for which we give detailed analytic and geometric descriptions of the regions of uniform and exponential stability.

Sparse systems with high local multiplicity

Abstract Consider a sparse system of Laurent polynomials in variables with complex coefficients and support in a finite lattice set . The maximal number of isolated roots of the system in the torus is known to be the normalized volume of the convex hull of (the BKK bound). We explore the following question: if the cardinality of equals , what is the maximum local intersection multiplicity at one point in the torus in terms of and ? This study was initiated by Gabrielov [13] in the multivariate case. We give an upper bound that is always sharp when and, under a technical hypothesis, it is considerably smaller than the previous upper bound for any dimension and codimension . We also present, for any value of and , a particular sparse system with high local multiplicity with exponents in the vertices of a cyclic polytope and we explain the rationale of our choice. Our work raises several interesting questions.

Mechanism and effect of seal vibration of a radial gate at small opening heights: Fluid–structure interaction simulation

Self-excited vibrations (SEVs) of water seals of hydraulic gates may induce dangerous flow-induced vibration (FIV) of the gate structure at small gate openings, thus revealing the mechanism and impact of seal SEV is crucial. In this study, we simulated the FIV characteristics of a large radial gate by using the three-dimensional fluid–structure interaction (FSI) model, analyzed the SEV mechanism of two seal types by using a one-dimensional theoretical model, and recommended the seal type that is safer by comparing the effects of the common J-seal and Semi-seal types. The simulated FSI features and the theoretical formulation show that the water hammer effect can be considered by a complex coefficient in the vibration equation, which can describe the SEV phenomenon of seals. The increasing seal deformation and hydraulic oscillation are due to the absolute value of the water hammer reflection coefficient at seals exceeding 1.0. The amplified deformation and oscillation, which aggravate overall gate FIV, can be stable when the water hammer reflection coefficient gradually reduces to 1.0. Since the Semi-seal exhibits higher inherent frequencies and limited deformation, its SEV occurs in a smaller range and causes a lower effect on the gate than the J-seal. The deformation amplitude of the gate FIV induced by Semi-seal SEV is reduced by 50% compared to J-seal. Therefore, it is recommended to use the Semi-seal type to reduce the resonance risk of the gate at small openings.