A Commutative Hypercomplex Algebra with Associated Function Theory

The argument principle is one of the basic theorems from the classical complex analysis and it's crucial role in complex analysis is very nicely underlined in the book of A. F. Beardon Complex Analysis – the Argument Principle in Analysis and Topology, John Wiley and Sons, 1979. We shall generalize a result of T. Radó and P.V. Reichelderfer from Continuous Transformations in Analysis, Springer-Verlag, Berlin, Göttingen, Heidelberg, 1955, p. 399, and a recent paper of P. Duren, w. Hengartner and R. S. LaugesenAmer. Math. Monthly 103(5) (1996), 411-415, showing that the argument principle is an entirely topological theorem (because it holds for continuous, open, discrete mappings in the plane) and hence this theorem belongs more to topology than to complex analysis. We also present as an application a generalization of Rouché's theorem.

A number of results concerning infinite series are presented in this chapter. The material spans convergence tests for series of constants, uniform convergence for series of functions, and power series. It is not the intention here to give a comprehensive account of this theory, but rather to review material that is relevant to later chapters. This chapter is intended to make the rest of the book more self contained and remind the reader of some basic results. It is assumed that the reader has a background in real analysis of a single variable and elementary complex analysis; consequently, no attempt is made to prove these results or provide examples. Proofs and examples can be found in textbooks on real analysis such as Apostol (Mathematical Analysis, Addison-Wesley, 2nd ed., 1974), Fulks (Advanced Calculus, 3rd ed., Wiley, 1978), and Little et al. (Real Analysis via Sequences and Series, Springer-Verlag, 2015) and complex analysis (Ahlfors, Complex Analysis, McGraw-Hill, 2nd ed., 1966; Brown and Churchill, Complex Variables and Applications, McGraw-Hill, 7th ed., 2004; Priestley, Introduction to Complex Analysis, Rev. ed., Oxford University Press, 1990). The exception is in the final section on double sequences and series. This material is often not part of courses on real or complex analysis. Here we provide proofs, examples and exercises.

Students of beginning algebra are quickly expected to solve linear equations. The solution procedures are generally abstract, involving the manipulation of numbers and algebraic symbols. Many students, even after completing a year of algebra, do not understand variables, equations, and solving equations (cf. Carpenter et al. [1982]). One way to help students learn to solve equations is to use physical objects, diagrams, and then symbols to represent equations. (Bruner [1964, 1967] calls such representations enactive (concrete), iconic (pictorial), and symbolic.) Although solving equations symbolically is essential, many students can benefit from working with physical problems that can also be symbolized mathematically. This article describes one way for students to learn to solve certain linear equations using pan balances, diagrams, and then symbols.

A comprehensive analysis of the full-bridge constant-frequency LCC-type parallel resonant converter (LCC-PRC) is presented. Owing to operation under constant frequency, the filter designs are simplified and utilisation of magnetic components are improved. The LCC-PRC takes on the desirable characteristics of the pure series and the pure parallel converter, thus removing the main disadvantages. A useful analytic technique, based on classical AC complex analysis, is introduced for designing the LCC-PRC. By using a proper transformation on the state variable, the converter is analysed by means of a two-dimensional state-plane diagram, which shows that the converter possesses three operation modes (I, II, III). It is shown that operating the converter in mode II provides the desirable higher efficiency for a wide load range. A CPU (8031) is used to control the phase-shift time. Its control characteristic is very accurate and the cost is low. Finally, a constant-frequency controlled full-bridge LCC-type parallel resonant converter (LCC-PRC) using the CPU (8031) control is implemented.

From classical complex analysis it is well-known that the index of the Riemann-Hilbert problem is connected with the winding number. How to generalize the winding number to a rotational number or rotation is explained in [nf] and also in [krza]. How to get a connection between the index and the rotation can be found in [dei]. For matrices of singular integral operators a rotation can be defined which is connected with the index of this system of singular integral operators. Because Clifford analysis is a higher dimensional analog to complex analysis there should be some analogous behavior of singular integrals. In fact, we will show that the index of singular integral operators with paravector-valued symbols in Clifford algebras can be connected with the so-called wrapping number which is the higher dimensional analog to the winding number. The proof is rather elementary.

Одним з напрямкiв наукових дослiджень Ю. В. Козаченка є рiвняння математичної фiзики з випадковими факторами. Цi фактори можуть мати рiзну природу: випадковi початковi умови, випадковi крайовi умови, випадкова права частина, випадковi коефiцiєнти i т. д. Умови та оцiнки збiжностi за ймовiрнiстю випадкових рядiв знаходять широке застосування при розв’язаннi задач математичної фiзики з випадковими умовами. Фiзичнi постановки таких задач розглядав Кампе де Фер’є . Вiн розглядав крайову задачу для рiвняння коливання струни з випадковими початковими умовами. У роботах В. В. Булдигiна показано, що вимога, щоб майже всi реалiзацiї випадкової початкової функцiї задовольняли умови, при яких є розв’язуваною детермiнована задача, значно звужує клас випадкових умов, за яких розв’язок iснує в класичному розумiннi. Є багато робiт, в яких вивчались задачi математичної фiзики з випадковими умовами, якi базуються на дослiдженнi збiжностi за ймовiрностю в функцiональних просторах послiдовностi випадкових функцiй, що апроксимують розв’язки крайових задач. Зауважимо, що у бiльшостi з цих робiт, для знаходження умов рiвномiрної збiжностi випадкових рядiв застосовується метод, що ґрунтується на iдеї Ж.Канаха. Булдигiним В.В. та Козаченком Ю.В. був запропонований метод, який дозволяє обґрунтовувати застосування методу Фур’є до задач математичної фiзики у багатовимiрному випадку. Метод, що ґрунтується на iдеї Кахана для цього випадку не пiдходить. У роботах Козаченка Ю.В. та його учнiв дослiджувалися рiвняння гiперпболiчного та параболiчного типiв математичної фiзики з випадковими факторами. Зокрема, вивчалися властивостi класичних та узагальнених розв’язкiв таких задач, було обґрунтувано застосування методу Фур’є, знайдено оцiнок для розподiлу супремуму розв’язкiв, та побудовано моделi розв’язкiв деяких задач, що наближають розв’язок iз заданою надiйнiстю та точнiстю в рiвномiрнiй метрицi. Всi цi результати мають не лише теоретичне, але й практичне застосування для подальшого вивчення та розвинення теорiї гiперболiчних i параболiчних рiвнянь математичної фiзики з випадковими факторами. Крiм того, цi результати дозволяють моделювати розв’язки крайових задач математичної фiзики iз заданою надiйнiстю та точнiстю в рiвномiрнiй метрицi, що може застосовуватися в наукових дослiдженнях в галузi радiотехнiки, фiзики, геофiзики, фiнансової математики, математичної економiки, в технiчних науках та в механiцi, зокрема, де використовуються методи комп’ютерного моделювання випадкових процесiв.

The concept of analysis of geodetic monitoring results solely from a geometric point of view is recognized as an obsolete approach. A complex analysis of geodetic measurements (geometric approach) and the structure stress-strain analysis (mechanical approach) allows obtaining the whole picture of any engineering structure displacements. The detailed scheme of the structural analysis of geospatial monitoring results of the long-span roof structures has been given in the presented paper. The results of the geospatial monitoring of the large warehouse have been chosen as a study subject. The structure's roof consists of planar trusses, the main objects of external loads combined with dead loads. According to the complex analysis procedure, the trusses were analyzed using the method of joints with the determination of partial member forces. At the next step, these forces were leveraged in the following order member force–member deformation–node displacement. To obtain the actual displacements of the truss nodes, one has to account for the vertical displacements of the leaning points where the truss touched the column. That step is also being accomplished using the method of joints. Having the actual node displacements, one may compare them with geodetic monitoring results. The comparison results generally allow us to reveal the places with unacceptable displacements and estimate whether they are determined with the necessary accuracy. In this particular case, the final node's displacements were yielded as an output of combined analysis, both geometric and mechanical. That, in turn, lets to acquire the deformation process's genuine parameters. The study results have shown the high efficiency of the presented research methodology.

The theoretical part of this monograph examines the distribution of the spectrum of operator polynomials, focusing on quadratic operator polynomials with discrete spectra. The second part is devoted to applications. Standard spectral problems in Hilbert spaces are of the form A- I for an operator A, and self-adjoint operators are of particular interest and importance, both theoretically and in terms of applications. A characteristic feature of self-adjoint operators is that their spectra are real, and many spectral problems in theoretical physics and engineering can be described by using them. However, a large class of problems, in particular vibration problems with boundary conditions depending on the spectral parameter, are represented by operator polynomials that are quadratic in the eigenvalue parameter and whose coefficients are self-adjoint operators. The spectra of such operator polynomials are in general no more real, but still exhibit certain patterns. The distribution of these spectra is the main focus of the present volume. For some classes of quadratic operator polynomials, inverse problems are also considered. The connection between the spectra of such quadratic operator polynomials and generalized Hermite-Biehler functions is discussed in detail. Many applications are thoroughly investigated, such as the Regge problem and damped vibrations of smooth strings, Stieltjes strings, beams, star graphs of strings and quantum graphs. Some chapters summarize advanced background material, which is supplemented with detailed proofs. With regard to the reader s background knowledge, only the basic properties of operators in Hilbert spaces and well-known results from complex analysis are assumed

The study of economic maps has consistently attracted researchers due to their rich dynamics and practical relevance. A deeper understanding of these systems enables the development of more effective control strategies. In this work, we examine the influence of the fractional order υ with the Caputo fractional difference on an economic market map. The primary contribution is the comprehensive analysis of how both commensurate and incommensurate fractional orders affect the stability and complexity of the map. Numerical investigations, including phase portraits, largest Lyapunov exponents, and bifurcation analysis, reveal that the system undergoes a cascade of period-doubling bifurcations before transitioning into chaos. To further characterize the dynamics, complexity is evaluated using the 0–1 test and C0 complexity, both confirming chaotic behavior. Furthermore, two-dimensional control schemes are introduced and theoretically validated to both stabilize the chaotic economic market map and achieve synchronization with a combined response map. The theoretical and numerical results are validated through MATLAB 2016 simulations.

Self-consistent amplitude profile of ducted VLF transmitter signal due to resonant interaction with energetic electrons in the magnetosphere

In this paper, we use four-dimensional quaternionic algebra to describe space-time geodesics in curvature form. The transformation relations of a quaternionic variables are established with the help of basis-transformations of quaternion algebra. We deduce the quaternionic covariant derivative that explains how the quaternion components vary with scalar and vector fields. The quaternionic metric tensor and the geodesic equation are also discussed to describe the quaternionic line element in curved space-time. We examine a quaternionic metric tensor equation for the Riemannian Christoffel curvature tensor. We present the quaternionic Einstein’s field-like equation, which indicates that quaternionic matter and geometry are equivalent. Relevance of the work:In recent decades, hypercomplex algebra, viz., quaternion and octonions, has been widely used to explain various branches of physics. In this way, we have investigated quaternionic transformations and field equations in curved space-time. The present novel work will help to explain the characteristics of the curved space-time universe in terms of quaternion algebra. It can also be used to describe quaternionic gravitational waves, the black hole formulation, and so on.

Science sets itself apart from other paths to truth by recognizing that even its greate practitioners sometimes err.

From the stereographic projection the ancient Greeks used to chart the heavens to modern circle-packing techniques that can chart the inner reaches of the human brain, conformal maps have played a central role in mathematical analysis and applications for over two millennia. It seems a shame, then, that undergraduate mathematics students rarely learn about them, the more so since a basic knowledge of calculus and some facility with trigonometry are enough to make a significant start. In this paper, we use single-variable calculus to establish the conformality of the stereographic projection and the Mercator world map. Along the way, the complex exponential function arises in a natural way that establishes it as a conformal mapping—without using any complex analysis. Furthermore, the exponential function provides a link between the symmetric structures of the two conformal maps. We conclude by indicating how one can construct conformal maps of surfaces of revolution other than the sphere.

Chapter 2 Aspects of Mathematics and Mechanics

Algorithm for Navier-Stokes and continuity equations used to derive curvilinear coordinate system