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In this article, by the parqueting-reflection technique, we obtain a covering of the complex plane from reflecting the Eclipse domain. Then, by the Cauchy–Pompeiu formula, we derive the generalized Cauchy–Pompeiu representation formula in the Eclipse domain. In addition, we consider the Schwarz boundary value problem (BVP) for the nonhomogeneous Cauchy–Riemann equation and the explicit expressions of solution and the condition of solvability. Besides, by using the Schwarz BVP, we investigate the inhomogeneous Dirichlet and homogeneous Neumann BVPs. [ABSTRACT FROM AUTHOR]

This work aims to explore the solution of a nonlinear fractional integro-differential equation in the complex domain through the utilization of both analytical and numerical approaches. The demonstration of the existence and uniqueness of a solution is established under certain appropriate conditions with the use of Banach fixed point theorems. To date, no research effort has been undertaken to look into the solution of this integro equation, particularly due to its fractional order specification within the complex plane. The validation of the proposed methodology was performed by utilizing a novel strategy that involves implementing the Rationalized Haar wavelet numerical method with the application of the Bernoulli polynomial technique. The primary reason for choosing the proposed technique lies in its ability to transform the solution of the given nonlinear fractional integro-differential equation into a representation that corresponds to a linear system of algebraic equations. Furthermore, we conduct a comparative analysis between the outcomes obtained from the suggested method and those derived from the rationalized Haar wavelet method without employing any shared mathematical methodologies. In order to evaluate the precision and effectiveness of the proposed method, a series of numerical examples have been developed. [ABSTRACT FROM AUTHOR]

In this extensive note, various differential-type operators in certain domains of the complex plane will be first introduced, a number of their comprehensive characteristic properties will be next pointed out and an extensive theorem dealing with some argument properties for several multivalent(ly) analytic functions will be also presented. In addition, numerous implications and suggestions, which can be obtained with the help of general result, will be determined. [ABSTRACT FROM AUTHOR]

This paper examines the mysterious non-trivial zeros of the Riemann zeta function ... and explains their role, e.g., in the computation of the error term in Riemann's J function for estimating the quantity of primes less than a given number. The paper also explains the close connection between the Riemann zeta function ... and the prime numbers. [ABSTRACT FROM AUTHOR]

In this article, by the parqueting-reflection technique, we obtain a covering of the complex plane from reflecting the Eclipse domain. Then, by the Cauchy–Pompeiu formula, we derive the generalized Cauchy–Pompeiu representation formula in the Eclipse domain. In addition, we consider the Schwarz boundary value problem (BVP) for the nonhomogeneous Cauchy–Riemann equation and the explicit expressions of solution and the condition of solvability. Besides, by using the Schwarz BVP, we investigate the inhomogeneous Dirichlet and homogeneous Neumann BVPs. [ABSTRACT FROM AUTHOR]

The conformal mapping technique has long been used to obtain exact solutions to Laplace's equation in two-dimensional domains with awkward geometries. However, a major limitation of the technique is that it is only directly compatible with Dirichlet and zero-flux Neumann boundary conditions. It would be useful to have a means of adapting the technique to handle more general boundary conditions, for example, Robin or nonlinear flux conditions. Boundary tracing is an unconventional method for tackling boundary value problems with generic flux boundary conditions, where one takes a known solution to the field equation and seeks new boundaries satisfying the prescribed boundary condition. In this paper, we adapt boundary tracing for compatibility with conformal mapping to produce a new prescription for studying Laplace's equation coupled with general flux boundary conditions. We illustrate the procedure via two simple examples involving heat transfer. In both cases, we demonstrate how to construct infinite families of nontrivial domains in which the solution to the chosen flux boundary value problem is exactly equal to a selected harmonic function. [ABSTRACT FROM AUTHOR]

This paper touches on the part played by the non-trivial zeros of the Riemann zeta function ζ, providing many important information and insights in the process, including some approaches to the Riemann hypothesis. [ABSTRACT FROM AUTHOR]

We consider the equation Ψ (z + h) = M (z) Ψ (z) , where z ∈ C , h>0 is a parameter, and M : C ↦ S L (2 , C) is a given analytic function. We get asymptotics of its analytic solutions as h → 0. The asymptotic formulas contain an analog of the geometric (Berry) phase well known in the quasiclassical analysis of differential equations. [ABSTRACT FROM AUTHOR]

This article proposes a novel algorithm employing space vector ellipse (SVE) in a complex plane to classify and visualize power quality disturbance-events (PQDEs). In the proposed method, at first, the time-domain signal and a reference signal, which are separated by 90°, are mapped in a complex 2D coordinates. Thus, the tip of resultant rotating vector traces an ellipse, from which three parameters, namely, semi-major axis, semi-minor axis and inclination angle, are obtained. Then, the ellipse parameters are exploited to classify and visualize nine types of PQDEs, namely, voltage sag, swell, interruption, harmonic, sag-harmonic, swell-harmonic, notch, flicker and transient. To validate the practicability of the proposed approach, an extensive real-time simulation study is carried out on RTDS platform using a test microgrid network to generate a large number of PQDEs. The test events were successfully classified and visualized in complex plane. Moreover, the noisy and practical signals, recorded by IEEE 1159.2 Working Group, were successfully classified to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

By this research notes, the well-known Tremblay operator and certain core knowledge therewith are firstly introduced and an extensive result containing numerous (analytic and) geometric properties (of possible applications of the related operator) along with a number of special implications are then constituted. As method for proving, the well-known assertion proposed by [8] is also considered there. [ABSTRACT FROM AUTHOR]

This scientific investigation deals with introducing certain basic information relating to the error functions in z--plane, establishing extensive relations between various series expansions of the complex error functions and presenting a number of their implications. [ABSTRACT FROM AUTHOR]

We approximate the solution of the nonlinear Fredholm integro-differential equation (NFID) in the complex plane by periodic quasi-wavelets (PQWs). This kind of wavelets possesses orthonormality properties, the numbers of terms in the decomposition and reconstruction formulas are strictly limited, and the localization is not emphasized. To the best of our knowledge, there are no numerical methods to obtain the solution of the NFID by PQWs. Here, we attempt to obtain the numerical solution of the NFID based on B-spline functions. Finally, the simulation results are shown for three examples. [ABSTRACT FROM AUTHOR]

As a generalization of the well-known concept of Pascal’s snail for two circles a geometric notion of Pascal’s snail of order n is introduced in the paper for n circles. It is solved the problem of its algebraic description in the form of a polynomial with complex coefficients under the condition that the module of the complex variable is equal to 1. The role and importance of the information component is demonstrated by an example of animation drawings in GeoGebra environment, which have been used during the experiments. Some of the accompanying proofs turn out to be obvious in verbal sense and the realibility of the assertions is confirmed by animation drawing models. [ABSTRACT FROM AUTHOR]

In this paper, we present some new methods for the estimates below the modulus of some integrals of the unit circle in the complex plane. All these methods are the best connection between the real and complex case. The results of this paper could be applied in many other areas of mathematics and natural sciences. [ABSTRACT FROM AUTHOR]

In this scientific note, an operator, which is the well-known Tremblay operator in the literature, is first introduced and some of its applications to certain analytic complex functions, which are normalized and analytic in the open unit disk, are then determined. In addition, certain special results of the related applications are also emphasized. [ABSTRACT FROM AUTHOR]

This paper explicates the Riemann hypothesis and proves its validity; it explains why the non-trivial zeros of the Riemann zeta function £ will always be on the critical line Re(s) = 1 2 and not anywhere else on the critical strip bounded by Re(s) = 0 and Re(s) = 1. Much exact calculations are presented, instead of approximations, for the sake of accuracy or precision, clarity and rigor. [ABSTRACT FROM AUTHOR]

This work is based on using wavelet for calculating one‐dimensional nonlinear Volterra‐Hammerstein and mixed Volterra‐Fredholm‐Hammerstein integral equation of the second kind in a complex plane. So far, as we know, no study has yet been attempted for solving this integral equation in the complex plane. The main specificity of this method is to avoid solving any linear or algebra system for approximated of integral equations. In Section 2, we introduce the integral operator for RH wavelet and use it in our numerical methods. In Section 3, we show that our problems have a unique solution. Furthermore, we give an upper bound for the error analysis. Finally, we make some example in Section 4 and solve them. [ABSTRACT FROM AUTHOR]

This work is based on using wavelet for calculating one‐dimensional nonlinear Volterra‐Hammerstein and mixed Volterra‐Fredholm‐Hammerstein integral equation of the second kind in a complex plane. So far, as we know, no study has yet been attempted for solving this integral equation in the complex plane. The main specificity of this method is to avoid solving any linear or algebra system for approximated of integral equations. In Section 2, we introduce the integral operator for RH wavelet and use it in our numerical methods. In Section 3, we show that our problems have a unique solution. Furthermore, we give an upper bound for the error analysis. Finally, we make some example in Section 4 and solve them. [ABSTRACT FROM AUTHOR]

This study examines a modal condition to cause inter‐area, low‐frequency, electromechanical power oscillations in an interconnected power system with two subsystems. The modal condition being examined is the closeness of two electromechanical oscillation modes from each subsystem on the complex plane, which is referred to as open‐loop modal resonance. The analysis in this study indicates that under the condition of open‐loop modal resonance, the electromechanical oscillation modes of the interconnected power system are located at approximately opposite positions on the complex plane with respect to the resonant electromechanical oscillation modes of the subsystems. Hence, when open‐loop modal resonance occurs, one electromechanical oscillation mode of the interconnected power system must have reduced damping. In this study, the damping reduction in the inter‐area, low‐frequency power oscillations caused by open‐loop modal resonance is demonstrated using example power systems. An incident of inter‐area, low‐frequency electromechanical power oscillations that occurred in an actual large‐scale power system in China is examined. The examination results indicate that the poorly damped, inter‐area, low‐frequency power oscillation incident was caused by open‐loop modal resonance. [ABSTRACT FROM AUTHOR]

Thermodynamics of the quark-gluon plasma at finite density is studied in the framework of the Field Correlator Method, where thermodynamical effects of Polyakov loops and color magnetic confinement are taken into account. Having found good agreement with numerical lattice data for zero density, we calculate pressure , for MeV and MeV. For the first time, the explicit integral form is found in this region, demonstrating analytic structure in the complex plane. The resulting multiple complex branch points are found at the Roberge-Weiss values of , with defined by the values of Polyakov lines and color magnetic confinement. [ABSTRACT FROM AUTHOR]

The aim of this research is first to reveal some novel and nonlinear applications relating to some equations and inequalities in the complex plane and then to present a number of consequences thereof. AMS subject classifications: 30D05, 26A33, 34A08, 35G10, 35F05, 30C55, 30C45, 30A10 [ABSTRACT FROM AUTHOR]

This study presents a design method of power system stabilisers (PSSs) to damp system oscillations. The design is based on fuzzy logic and linear matrix inequality (LMI) techniques. The design guarantees a robust pole placement in a desired region in the complex plane for different loading conditions. Due to system non‐linearity and load dependency, the whole wide range of operating condition is divided into three subregions. For each subregion, a norm‐bounded uncertainty linear model is developed and a robust LMI‐based pole placer is designed considering controller constraint. Takagi–Sugeno fuzzy model is used to guarantee smoothly switching between the LMI‐based controllers of the three subregions. The proposed technique avoids solving many LMIs as in the polytypic approach and avoids the infeasibility problem that might arise from solving the LMIs for the complete region of operation. In addition, the proposed fuzzy logic switching controller has reduced drastically the fuzzy rules due to the use of two‐dimensional (2D) membership functions. Particle swarm optimisation is used to tune the proposed 2D membership functions to guarantee smooth switching of the controllers while maintaining the desired constraints. The simulation results of both single‐machine and multi‐machine power systems confirm the effectiveness of the proposed PSS design. [ABSTRACT FROM AUTHOR]

Extended linear matrix inequality (LMI) conditions, ensuring the stability of commensurate fractional‐order linear systems by static‐output feedbacks, are given. It is assumed that the system uncertainties are constant and possibly present in all the system matrices. The stabilising static‐output feedback is conceived to overcome the system uncertainty and place the poles of the closed‐loop system in a well‐defined domain that is formed by the intersection of many regions in the complex plane. The control design is formulated as the solution of a set of linear matrix inequality conditions. The validity of the obtained results is testified through an example of a fractional‐order system with polytopic uncertainties. [ABSTRACT FROM AUTHOR]

Inequalities involving Mittag-Leffler functions play important roles in themselves and in their diverse applications. Many inequalities involving Mittag-Leffler functions have been established by a number of authors in many different ways. In this paper by making use of a novel idea and technique, a comprehensive result which contains several differential inequalities (in the complex plane C) relating to the Mittag-Leffler (type) functions: Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. (α, γ,β, z ? C; Re(α) > 0; Re(γ) > 0; Re(β) > 0) (in its kernel (λ)υ being the familier Pochhammer symbol) is first established. A number of corollaries and consequences of the main result are then presented. [ABSTRACT FROM AUTHOR]

A closed algebraic embedding of C ⁎ = C 1 ∖ { 0 } into C 2 is sporadic if for every curve A ⊆ C 2 isomorphic to an affine line the intersection with C ⁎ is at least 2. Non-sporadic embeddings have been classified. There are very few known sporadic embeddings. We establish geometric and algebraic tools to classify them based on the analysis of the minimal log resolution ( X , D ) → ( P 2 , U ) , where U is the closure of C ⁎ on P 2 . We show in particular that one can choose coordinates on C 2 in which the type at infinity of the C ⁎ and the self-intersection of its proper transform on X are sharply limited. [ABSTRACT FROM AUTHOR]

The problem of measuring the depth of nonmagnetic conducting coatings on nonmagnetic substrates with simultaneous detuning of the gap and substrate conductivity is considered. It is shown that the gap can be detuned by introducing a compensating voltage with a fixed phase and an amplitude that is adjusted proportionally to the high-frequency channel signal. [ABSTRACT FROM AUTHOR]

This article takes a fresh look at reswitching. When two production techniques are compared, reswitching occurs when one technique is more viable than the other at a high interest rate, switches to being less viable at a lower rate, and reswitches to being more viable again at even lower rates. For some, reswitching undermines the foundations of neoclassical economics because it belies the idea of a monotonic relationship between relative capital values and factor price. The reswitching equation is an nth degree polynomial having n roots, implying the existence of n interest rates. Conventional analysis uses one interest rate but ignores the others. We argue that the others should not be ignored because all rates are determined simultaneously, and when one rate shifts, all rates shift. We demonstrate that the Samuelson reswitching model possesses a ‘dual’ expression containing every interest rate, the rates being compressed into a composite, interest-rate variable, thereby establishing a role for interest rates previously thought lacking in use and meaning. The relationship between this composite interest rate and capital value does not exhibit reswitching. The notion of a composite interest rate has implications for economics beyond reswitching. [ABSTRACT FROM PUBLISHER]

This study presents an off‐the‐shelf error feedback controller that can completely rejects the sinusoidal disturbances whose magnitudes, phases, bias, frequencies and even the number of frequencies are all unknown. A closed‐form solution is given under the assumptions that the plant has no uncertainty and has hyperbolic zero dynamics (i.e. there is no zero on the imaginary axis of the complex plane), and that the number of unknown frequencies has upper bound which is known. A non‐minimum phase plant can be dealt with. The proposed controller is tested by a simulation, which verifies its effectiveness. [ABSTRACT FROM AUTHOR]

If F is a family of functions, we define a sequence by alternative recurrence with respect to the family F a sequence (an)n that verifies the relation an+1 ∊ {f(an-1) ∣ f ∊ F} for any n ⩾ 1, where the first two terms 0 1 a , a are given. We prove that for particular cases of family we can obtain the regular networks in the complex plane. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we use a numerical procedure for solving Fredholm integral equations of the second kind in complex plane. The periodic quasi-wavelets (PQWs) constructed on are utilized as a basis in collocation method to reduce the solution of linear integral equations to a system of algebraic equations. Convergence analysis is derived and we used some numerical examples to illustrate the accuracy and the implementation of the method. [Copyright &y& Elsevier]

The electrical and magnetic properties of polycrystalline as-sintered and annealed Co0.3Mn0.2Zn0.5Fe2O4 prepared by using conventional electroceramic processing steps in the laboratory are presented. The microstructures are examined via scanning electron micrographs and energy dispersive X-ray (EDX) analysis. The ac electrical behavior represented via complex plane plots in the frequency range 160-211 MHz reveals inductive contribution via resonance originated within the material system. The ac permeability is presented in the frequency range 50-300 MHz affirmed resonance behavior. The inductive response in the complex planes (impedance Z* and modulus M*) reflecting resonance behavior due to magnetic softness is obvious around 140 MHz whereas in the permeability dispersion negative domain is observed at about 150 MHz attributing to the spin rotation. [ABSTRACT FROM AUTHOR]

An in-depth investigation of oscillation modes in ring oscillators is presented. The small-signal stability is initially considered, demonstrating that the poles associated with the dc regime are uniformly distributed on a circle on the complex plane, with increasing density for a higher number of stages. The existence of multiple pairs of poles on the right-hand side of the complex plane gives rise to different oscillation modes, related here to the eigenvectors of the circulant active-device immitance matrix. This paper shows that the stability properties of detected modes depend on both the order of appearance from dc regime, in a sequence of Hopf bifurcations, and the bifurcations undergone by each steady-state mode until reaching the final operation point, when changing a bias voltage, for instance. Thus, the stability analysis must combine a bifurcation analysis from dc regime and a bifurcation analysis of each individual oscillation mode in large-signal regime. The large-signal stability analysis presented shows the possible stabilization mechanisms, which lead to the common physical observation of some of these modes. The stabilization of the desired mode, using concepts from bifurcation theory, is also presented. All techniques have been successfully applied to a ring oscillator at 12.6 GHz. [ABSTRACT FROM AUTHOR]

Abstract: If X is an arc in the complex plane with 0 as an endpoint, then the preimage of X under is also an arc, and the endpoints of are the points in the preimage of the nonzero endpoint of X. In this paper, the author explores necessary and sufficient conditions under which a chainable continuum in has chainable preimage under f. The paper contains an example of a chainable continuum X (the simple three-fold Knaster continuum) embedded in the complex plane in such a way that 0 is an endpoint of X and the preimage of X under the square map is not chainable. [ABSTRACT FROM AUTHOR]

Abstract: The problems of computing single-valued, analytic branches of the logarithm and square root functions on a bounded, simply connected domain S are studied. If the boundary of S is a polynomial-time computable Jordan curve, the complexity of these problems can be characterized by counting classes , MP (or MidBitP), and : The logarithm problem is polynomial-time solvable if and only if . For the square root problem, it has been shown to have the upper bound and lower bound . That is, if then the square root problem is polynomial-time solvable, and if then the square root problem is not polynomial-time solvable. [Copyright &y& Elsevier]

For the many aspects of psychological time, there are many approaches to understanding, including cultural theories, psychological studies, philosophical analyses and mathematical models. A useful approach is to assume that there are many kinds of psychological time in coexistence, and to distinguish their different cultural and meaning-based contexts, as do Yamada and Kato (2006). From a different perspective, Rudolph (2006) presents rigorous mathematical models of psychological time, in the hope that psychologists will apply them as ways to gain insights into experience and change. This discussion enjoins some qualities of time, attention and ambivalence (Rudolph's themes) to properties of dialectical structures (Hood, 1995), with fluctuation as a source of individuality within the spiraling cycles of reversible time proposed by Yamada and Kato. Fluctuations at a border, especially a fractal border, may show the way in which a present experience contains traces of the past and constructions of the future. Applications of mathematical models in cultural contexts serve here to more fully elaborate the qualities of psychological time, rather than to reduce them as statistical products. This approach suggests a reconsideration of the distinctive roles of different kinds of time in the study of life cycles. [ABSTRACT FROM AUTHOR]

The problem of the diffraction of an electromagnetic plane wave by a convex cone of arbitrary smooth cross‐section with impedance (Leontovich) boundary conditions is studied. The vector problem is reduced to that for the Debye potentials. By means of Kontorovich–Lebedev integrals, two spectral functions are introduced and the corresponding boundary value problem is formulated. The spectral functions for the potentials are found to satisfy the Helmholtz equations on the unit sphere and to be coupled through non‐traditional boundary conditions of the impedance type with shifts on the spectral variable. The use of the Green theorem permits us to establish an integral formulation of the boundary value problem for the spectral functions. The formal asymptotic solution of the problem is then given for the case of a narrow cone. For this, two different methods are given: a method of perturbation applied to the spectral integral equations and an adaptation of the method of matching the asymptotic series in spectral domain. Both methods lead to the same closed‐form result for the leading term of the scattering diagram asymptotics. [ABSTRACT FROM PUBLISHER]

We use bilinear transformations to map points z = cos(α)+isin(α) on the unit circle in the complex plane into points x on the real line. Given any density function g(α) on the interval (-π, π), we show how a corresponding density function f(x) on (-∞, ∞) is induced. When α is uniformly distributed on (-π, π), we show that x has a Cauchy distribution in (-∞, ∞). When g(α) = K[SUBn] (1+cos(α))[SUPn], we show that x has a t-distribution in (-∞, ∞). [ABSTRACT FROM AUTHOR]

Given the n complex coefficients of a degree n - 1 complex polynomial, we wish to evaluate the polynomial at a large number m ≥ n of points on the complex plane. This problem is required by many algebraic computations and so is considered in most basic algorithm texts (e.g., [A. V. Aho, J. E. Hopcroft, and J. D. Ullman, The Design and Analysis of Computer Algorithms, Addison-Wesley, 1974]). We assume an arithmetic model of computation, where on each step we can execute an arithmetic operation, which is computed exactly. All previous exact algorithms [C. M. Fiduccia, Proceedings 4th Annual ACM Symposium on Theory of Computing, 1972, pp. 88-93; H. T. Kung, Fast Evaluation and Interpolation, Carnegie-Mellon, 1973; A. B. Borodin and I. Munro, The Computational Complexity of Algebraic and Numerical Problems, American Elsevier, 1975; V. Pan, A. Sadikou, E. Landowne, and O. Tiga, Comput. Math. Appl., 25 (1993), pp. 25-30] cost at least work &Ohm;(log[SUP2] n) per point, and previously, there were no known approximation algorithms for complex polynomial evaluation within the unit circle with work bounds better than the fastest known exact algorithms. There are known approximation algorithms [V. Rokhlin, J. Complexity, 4 (1988), pp. 12-32; V. Y. Pan, J. H. Reif, and S. R. Tate, in Proceedings 32nd Annual IEEE Symposium on Foundations of Computer Science, 1992, pp. 703-713] for polynomial evaluation at real points, but these do not extend to evaluation at general points on the complex plane. We provide approximation algorithms for complex polynomial evaluation that cost, in many cases, near constant amortized work per point. Let k = log(|P|/∊ ), where |P| is the sum of the moduli of the coefficients of the input polynomial P(z). Let &Ptilde;(z[SUBj]) be an &espi;-approx of P(z) if upper bounds the modulus of the error of the approximation &Ptilde;(z[SUBj]) at each evaluation point z[SUBj], that is. [ABSTRACT FROM AUTHOR]

This paper presents an efficient numerical method for solving nonlinear mixed Volterra- integral equations in the complex plane. The method is based on the fixed point iteration procedure. In each iteration of this method, periodic quasi-wavelets are used as basis functions to approximate the solution. Also, using the Banach fixed point theorem, some results concerning the error analysis are obtained. Finally, some numerical examples show the implementation and accuracy of this method. [ABSTRACT FROM AUTHOR]