O-Regular Mappings on C(C): A Structured Operator–Theoretic Framework

1. Some problems arising in the study of value distribution of functions of two complex variables. One of the objectives of modern analysis consists in the generalization of methods of the theory of functions of one complex variable in such a way that the procedures in the revised form can be applied in other fields, in particular, in the theory of functions of several complex variables, in the theory of partial differential equations, in differential geometry, etc. In this way one can hope to obtain in time a unified theory of various chapters of analysis. The method of the kernel function is one of the tools of this kind. In particular, this method permits us to develop some chapters of the theory of analytic and meromorphic functions f(z1, * * *, zn) of the class j2(I2n), various chapters in the theory of pseudo-conformal transformations (i.e., of transformations of the domains V2n by n analytic functions of n complex variables) etc. On the other hand, it is of considerable interest to generalize other chapters of the theory of functions of one variable, at first to the case of several complex variables. In particular, the study of value distribution of entire and meromorphic functions represents a topic of great interest. Generalizing the classical results about the zeros of a polynomial, Hadamard and Borel established a connection between the value distribution of a function and its growth. A further step of basic importance has been made by Nevanlinna and Ahlfors, who showed not only that the results of Hadamard and Borel in a sharper form can be obtained by using potential-theoretical and topological methods, but found in this way important new relations, and opened a new field in the modern theory of functions. As one passes from the theory of functions of one variable to the case of two and more variables(2), the question of the value distribution becomes more complex and many directions arise which can be considered as a generalization of the above mentioned chapter of the theory of functions of one variable. However, the formulation of the problems which can be answered, and generalization of methods used in one variable, is not immediate. A function of two complex variables may vanish in a domain on a segment

In this paper we deal with monogenic and k-monogenic Clifford-valued automorphic forms in two and several vector variables. We focus on the construction of several families of Eisenstein and Poincare series within the setting of this function theory. The functions that we construct provide non-trivial examples for generalizations of the classical Hilbert modular forms within the framework of real and complex Clifford analysis.

We introduce a generalized Cauchy integral formula for special classes of functions, holomorphic in several complex variables. These functions also satisfy a generalized Cachy-Riemann equation. The integral formula is then applied to demonstrate that these classes of holomorphic functions are Fréchet modules over complex Clifford algebras, and to provide maximal domains of convergence for Taylor expansions, with homogeneous terms grouped together, of real analytic solutions to Maxwell's equations in vacuo.

We define a new modified basis which is an association of two bases, e 1 and 2 . We give an expression of the form = 0 + 0 , where 0 is a real number and 0 is a complex number on three-dimensional real skew field. And we research the properties of regular functions with values in ternary field and reduced quaternions by Clifford analysis.

The theory of analytic functions in one and several complex variables has been founded by Cauchy, Riemann and Weierstraß and belongs to the most beautiful mathematical creations of modern times. We recommend the textbooks of Behnke-Sommer (1955), Grauert-Fritzsche (1964, 1974), Hurwitz-Courant (1964) and Vekua (1963). The investigations of analytic functions with respect to their differentiable properties will be founded on the integral theorems from Chapter 1 and with respect to their topological properties will be based on the winding number from Chapter 3. We additionally obtain a direct approach to the solutions of the inhomogeneous Cauchy-Riemann differential equations in this chapter. In the last section we investigate the discontinuous behavior of Cauchy’s integral across the boundary.

The aim of this article is to extend the theory of several complex variables to the non-commutative realm. Some basic results, such as the Bochner-Martinelli formula, the existence theorem of the solutions to the non-homogeneous Cauchy-Riemann equations, and the Hartogs theorem, are generalized from complex analysis in several variables to Clifford analysis in several paravector variables. In particular, the Bochner-Martinelli formula in several paravector variables unifies the corresponding formulas in the theory of one complex variable, several complex variables, and several quaternionic variables with suitable modifications.

We describe along the guidelines of Kohn (Quantitative Estimates for Global Regularity. Analysis and Geometry in Several Complex Variables, pp. 97–128. Trend Math. Birkhauser, Boston, 1999), the constant \({\mathcal {E}}_s\) which is needed to control the commutator of a totally real vector field \(T_{{\mathcal {E}}}\) with \(\bar{\partial }^*\) in order to have \(H^s\) a-priori estimates for the Bergman projection \(B_k, k\ge q-1\), on a smooth \(q\)-convex domain \(D\subset \subset {\mathbb {C}}^{n}\). This statement, not explicit in Kohn (Quantitative Estimates for Global Regularity. Analysis and Geometry in Several Complex Variables, pp. 97–128. Trend Math. Birkhauser, Boston, 1999), yields regularity of \(B_k\) in specific Sobolev degree \(s\). Next, we refine the pseudodifferential calculus at the boundary in order to relate, for a defining function \(r\) of \(D\), the operators \((T^+)^{-\frac{\delta }{2}}\) and \((-r)^{\frac{\delta }{2}}\). We are thus able to extend to general degree \(k\ge 0\) of \(B_k\), the conclusion of (Quantitative Estimates for Global Regularity. Analysis and Geometry in Several Complex Variables, pp. 97–128. Trend Math. Birkhauser, Boston, 1999) which only holds for \(q=1\) and \(k=0\): if for the Diederich–Fornaess index \(\delta \) of \(D\), we have \((1-\delta )^{\frac{1}{2}}\le {\mathcal {E}}_s\), then \(B_k\) is \(H^s\)-regular.

In the classical theory of several complex variables, holomorphic mappings are just n-tuples of holomorphic functions in m variables, with arbitrary n andm, and no relations between these functions are assumed. Some 30 years ago John Ryan introduced complex, or complexified, Clifford analysis which is, in a sense, the study of certain classes of holomorphic mappings where the components are not independent, and instead obey the relations generated by the Cauchy- Riemann and Dirac-type operators. In this paper, we take a closer look at this theory emphasizing some additional properties that holomorphic mappings satisfy in this context. Our attention is mostly restricted to the case of low dimensions where it is possible to identify new and interesting properties and to single out the special role played by bicomplex analysis.

The theory of real functions is a relatively young chapter of mathematical analysis. In fact, in the 19th century, the expression “function theory” was applied to mean the theory of complex-valued analytic functions of one or more complex variables. The first investigations that initiated the part of analysis called today “theory of real functions” or briefly “real analysis” were constructions of various real-valued functions of a real variable whose characteristic properties are very far from those of analytic functions; as a typical example, we can mention the construction, due to Karl Weierstrass, of a real-valued function continuous in an interval but not differentiable at any point of this interval.

This study is to determine the location of the difficulty of complex variable courses. Complex variable courses require a strong understanding, high concentration, good learning readiness, reasoning power and higher order logic in understanding direct objects. The direct objects referred to are Facts, Concepts, Skills, and Principles. In addition to direct objects in obtaining maximum results, indirect objects are needed. Indirect objects that are meant are tools used in the form of media, support and encouragement from others in obtaining good results. The results of the mid-semester grades for the 2014/2015 academic year were about 13% of students who got good grades and the rest were not good. In fact, this complex variable course is the main course of several courses. Another fact is that in the 2019/2020 semester VII, there are 35% in the poor category and 50% sufficient and the rest are good and very good. From 35% of students with poor learning outcomes took remedial, 1 person graduated with sufficient marks, while the rest did not pass. The specific objectives of the research are 1) to determine the mastery in complex variable courses, 2) the location of the difficulty and difficulty factors of complex variables, 3) to know the mastery and difficulty of indirect objects. This type of research approach is descriptive qualitative. Research wants to describe and describe how the actual event. The research was conducted in the Complex Variable Analysis course, Semester VII, implementation October-January and a total of 12 subjects. Techniques for obtaining data from observations, interviews and documentation. The analysis technique, presenting data, reducing and drawing conclusions. Conclusion,The difficulty of students lies in the facts and concepts that exist in the direct object. In the interview session, 71.42% said that the difficulty factor was the mastery of facts and concepts. In online implementation, students experience percentage difficulties when writing symbols and operating the media used. This is a new problem for mathematics students in achieving good learning outcomes with objective assessment standards.

Analytical function theory approach to the heat transfer problem of a cylinder in cross-flow at small Péclet numbers

The classical mechanics of large molecules and fullerenes is studied. The approach is based on the model of collective motion of these objects. The mixed Lagrangian (material) and Eulerian (space) description of motion is used. In particular, the Green and Cauchy deformation tensors are geometrically defined. The important issue is the group-theoretical approach to describing the affine deformations of the body. The Hamiltonian description of motion based on the Poisson brackets methodology is used. The Lagrange and Hamilton approaches allow us to formulate the mechanics in the canonical form. The method of discretization in analytical continuum theory and in classical dynamics of large molecules and fullerenes enable us to formulate their dynamics in terms of the polynomial expansions of configurations. Another approach is based on the theory of analytical functions and on their approximations by finite-order polynomials. We concentrate on the extremely simplified model of affine deformations or on their higher-order polynomial perturbations.

The theory of analytic functions of several complex variables enjoyed a period of remarkable development in the middle part of the twentieth century. After initial successes by Poincare and others in the late 19th and early 20th centuries, the theory encountered obstacles that prevented it from growing quickly into an analogue of the theory for functions of one complex variable. Beginning in the 1930s, initially through the work of Oka, then H. Cartan, and continuing with the work of Grauert, Remmert, and others, new tools were introduced into the theory of several complex variables that resolved many of the open problems and fundamentally changed the landscape of the subject. These tools included a central role for sheaf theory and increased uses of topology and algebra. The book by Gunning and Rossi was the first of the modern era of the theory of several complex variables, which is distinguished by the use of these methods. The intention of Gunning and Rossi's book is to provide an extensive introduction to the Oka-Cartan theory and some of its applications, and to the general theory of analytic spaces. Fundamental concepts and techniques are discussed as early as possible. The first chapter covers material suitable for a one-semester graduate course, presenting many of the central problems and techniques, often in special cases. The later chapters give more detailed expositions of sheaf theory for analytic functions and the theory of complex analytic spaces. Since its original publication, this book has become a classic resource for the modern approach to functions of several complex variables and the theory of analytic spaces.

The theory of analytical functions is widely used not only in fundamental research. It is a universal tool that allows you to formalize physical phenomena, facilitate the analysis of complex systems, solve complex physical problems, increasing the accuracy and efficiency of the results. With the help of analytical functions, many problems are reduced to simpler formulas that are easily interpreted. It also becomes possible to obtain analytical solutions without numerical methods, and the Cauchy-Riemann equations allow you to simultaneously describe several physical characteristics. In particular, in electrodynamics and field theory, analytical functions are used to describe potentials and fields, and the conformal mappings they perform are used to determine the distribution of the electric field in complex geometric regions, for example, around an electrode in the form of a corner or near sharp edges. In this article, an analytical function called the complex potential is used to determine the characteristics of the electrostatic field of flat and cylindrical capacitors. In particular, by studying the complex potential of the electrostatic field of flat and cylindrical capacitors, families of equipotentials and flux lines were obtained for them, the direction of the electrostatic field strength vector E was determined, and the capacitance C was calculated for these types of capacitors. It is indicated that the electrostatic field strength vector E of a flat capacitor is constant and directed normal to the surface of the capacitor plates, and in the case of a cylindrical capacitor, the vectors E and D (electrostatic induction vector) are directed radially from the inner cylinder (r = a) to the outer (r = b), if the inner one has a positive charge).

Analytic Function Theory and Differential Equations