Cauchy-Riemann Conditions Research Articles

Physics-Informed Holomorphic Neural Networks (PIHNNs): Solving 2D linear elasticity problems

We propose physics-informed holomorphic neural networks (PIHNNs) as a method to solve boundary value problems where the solution can be represented via holomorphic functions. Specifically, we consider the case of plane linear elasticity and, by leveraging the Kolosov–Muskhelishvili representation of the solution in terms of holomorphic potentials, we train a complex-valued neural network to fulfill stress and displacement boundary conditions while automatically satisfying the governing equations. This is achieved by designing the network to return only approximations that inherently satisfy the Cauchy-Riemann conditions through specific choices of layers and activation functions. To ensure generality, we provide a universal approximation theorem guaranteeing that, under basic assumptions, the proposed holomorphic neural networks can approximate any holomorphic function. Furthermore, we suggest a new tailored weight initialization technique to mitigate the issue of vanishing/exploding gradients. Compared to the standard PINN approach, noteworthy benefits of the proposed method for the linear elasticity problem include a more efficient training, as evaluations are needed solely on the boundary of the domain, lower memory requirements, due to the reduced number of training points, and C∞ regularity of the learned solution. Several benchmark examples are used to verify the correctness of the obtained PIHNN approximations, the substantial benefits over traditional PINNs, and the possibility to deal with non-trivial, multiply-connected geometries via a domain-decomposition strategy.

On Grothendieck-type duality for spaces of holomorphic functions of several variables

We describe the strong dual space $({\mathcal O} (D))^*$ of the space ${\mathcal O} (D)$ of holomorphic functions of several complex variables in a bounded domain $D$ with Lipschitz boundary and connected complement (as usual, ${\mathcal O} (D)$ is endowed with the topology of local uniform convergence in $D$). We identify the dual space with the closed subspace of the space of harmonic functions on the closed set ${\mathbb C}^n\setminus D$, $n>1$, whose elements vanish at the point at infinity and satisfy the Cauchy-Riemann tangential conditions on $\partial D$. In particular, we generalize classical Grothendieck-Köthe-Sebastião e Silva duality for holomorphic functions of one variable to the multivariate situation. We prove that the duality we produce holds if and only if the space ${\mathcal O} (D)\cap H^1 (D)$ of Sobolev-class holomorphic functions in $D$ is dense in ${\mathcal O} (D)$. Bibliography: 35 titles.

A new HEAP-inspired method for embedded power flows and path-related holomorphic properties

Power flow problems can be robustly solved by holomorphic embedding (HE) methods which require the holomorphic property of solution functions near a reference state and also along a path bridging this state and the target state, for determining the effectiveness of series expansion and physical realizability (PRL), respectively, where PRL means the complex conjugacy of solution values at the target state. Existing studies generally assume holomorphic properties beforehand or assert them from a convergence of series expansion or implicit function theorem at separate points, however, which can hardly be applied to confirm the holomorphic property along a path and then PRL (because a path has infinitely many points and cannot be numerically inspected point-by-point). To this end, a new approach PHVACR is enlightened by HEAP and employs an expansion in the rectangular form and also Cauchy–Riemann (CR) conditions to numerically verify holomorphic properties around a path in an arc-by-arc manner. Here, ‘HE’ is short for holomorphic embedding, ‘AP’ in HEAP indicates an arc-length parametrization, and ‘PHVA’ in PHVACR refers to the path-related holomorphy validation with ‘CR’ representing the Cauchy–Riemann conditions. To explain the importance of PHVACR and its performance, counter-intuitive examples are discussed to explain the limitations of existing approaches, and large problems with 10,000+ buses are also simulated to examine the effectiveness and scalability of PHVACR which are compared with other HE methods.

Integral theorems in finite-dimensional commutative algebra

For monogenic (continuous and differentiable in the sense of G\^ateaux) functions given in special real subspaces of an arbitrary finite-dimensional commutative associative algebra over the complex field and taking values in this algebra, we establish basic properties analogous to properties of holomorphic functions of a complex variable. Methods for proving results are based on a representation of monogenic functions via holomorphic functions of complex variables that allows to establish analogues of Cauchy-Riemann conditions and the continuity of G\^ateaux derivatives of all orders for monogenic functions. In such a way, analogues of a number of classical theorems of complex analysis (the Cauchy integral theorem for a curvilinear integral, the Cauchy integral formula, the Morera theorem, the Taylor theorem) are proved and different equivalent definitions for the mentioned monogenic functions are established. An analogue of the Cauchy theorem for an integral over non piecewise smooth surfaces is proved.

Simulation of asymmetrical rotational flow in the flow part of a well jet pump

In order to improve the mathematical model of the working process of the above-bit ejection system for the conditions of its asymmetric rotation in the well, a method of modeling the circulating injected flow using a hydrodynamic vortex function, the center of which is shifted relative to the axis of the mixing chamber of the jet pump, has been developed. In the process of modeling the rotating asymmetric movement of the injected flow, the concept of circulation of the vector of the translational velocity of the liquid along a closed circuit is used. With the help of the components of the complex potential of the hydrodynamic circulation function, a graphical interpretation of the equipotential lines and flow lines of the spatial vortex with one-sided and two-sided displacement of the coordinate center was obtained. Using analytical relations for the velocity potential and stream function, the components of the circulation velocity are determined and the analyticity of the function of the complex potential with a shifted vortex center is proven. The fulfillment of the Cauchy-Riemann conditions allowed us to present the resulting velocity of the circulating current with the coordinate center shifted along the vertical axis in the form of a partial derivative of the complex potential. In contrast to the complex potential for a symmetric vortex, the ratio for determining the circulation speed in case of misalignment of the working nozzle and the mixing chamber of the jet pump contains an additional component in the form of the displacement of the vortex center.In the process of analyzing the results of using the proposed mathematical model, it was established that the resulting speed of the circulation current and the relative displacement of the spatial vortex center are directly proportional. The conducted studies proved that the ratio of the velocities of the symmetric and asymmetric circulation flow is a function of the displacement of the center of the vortex flow coordinates and varies from 1 to zero. The magnitude of the relative speed of the circulation flow is inversely proportional to the displacement of the coordinate center and the distance to the mixing chamber of the jet pump. The developed mathematical model can be used to predict the influence of the rotation of the ejection system on its pressure characteristic in the case of radial relative displacement of the working nozzle and the mixing chamber of the jet pump.

ONE CLASS OF SMOOTH BOUNDED SOLUTIONS TO THE CAUCHY PROBLEM FOR A THREE-DIMENSIONAL FILTRATION MODEL WITH DARCY'S LAW

In the filtration theory there are numerous approaches to solving three-dimensional problems of fluid motion in a porous medium. Basically, the solutions of such problems are obtained by numerical methods. The question of finding an analytical solution of precisely three-dimensional problems of fluid motion is open. The first results on the use of the apparatus of four-dimensional mathematics for solving the three-dimensional model of the Navier-Stokes equations by the analytical method were obtained by the Kazakh mathematician Professor M.M. Abenov. After the author of this article with other researchers proved the theorem on the existence of a solution to the Cauchy problem for a three-dimensional model of filtration theory. This paper is devoted to the study of a three-dimensional model of the filtration theory in one of the spaces of four-dimensional numbers. The purpose of this article is to obtain an analytical solution of the three-dimensional Cauchy problem for the mathematical model of linear filtration model by the method of four-dimensional regular functions.  In this study, a class of infinitely differentiable and bounded functions of the initial conditions of the Cauchy problem, satisfying the Cauchy-Riemann condition, with five degrees of freedom for a specific four-dimensional function is found, and also a class of infinitely differentiable and bounded solutions of this problem is found that satisfy the linear Darcy law.

Substantiation of the method of modeling circulating flows during the rotation of the overhead jet pump

The scheme of theoretical analysis of the influence of rotational motion on the working process of the near-bit jet pump is chosen, the methods of its realization and limits of use are determined. On the basis of the potential flows theory with the application of elementary hydrodynamic functions of a complex variable, basics for the development of analytical models of the near-bit ejector pump operation for conditions of its rotation in a well have been worked out. Components of a mixed flow are modeled as functions of the vortex flow and parallel jet flow. Application of certain hydrodynamic functions of a complex variable has allowed to combine partial outcomes of differential equations of elementary streams motions and to determine the structure of the mixed flow function. The equation of the mixed flow motion is given in the form of a complex potential, components of which are equal to Laplace’s equation and Cauchy-Riemann Conditions, and determine the kinematics of irrotational flow. Possibility of application of the complex variable functions’ theory for velocities profiles modeling and determination of limits for dissociation of mixed flows by rotation of the near-bit jet pump in a well has been shown. The list of hydro-dynamic parameters, which should be taken into consideration by modeling of the ejector system working process, has been determined, and the structure of initial equations, that characterize kinematic picture of velocities field in the chamber of the ejector pump mixing has been established. Limit conditions for determination of a complex potential of mixed flows and application limits of the proposed method of circulation flows modeling have been stated. The class of hydrodynamic problems has been established, to which one may relate the problem of finding the complex potential of mixed flows in a chamber for mixing of an ejector pump by its rotation in a well.

Functions in one space of four-dimensional numbers

For the first time, the theory of functions of four-dimensional numbers with commutative product was described in works of Abenov M.M., in which the mathematical apparatus was defined, algebraic operations and their properties were determined, functions of four-dimensional numbers, their limits, continuity and differentiability were found. The continuation was the joint work of Abenov M.M. and Gabbasov M.B., where similar anisotropic four-dimensional spaces (withthe notation M2-M7) were defined, which are also commutative with zero divisors. This work is devoted to the study of functions of a four-dimensional variable, definitions and analysis of fourdimensional functions, their properties, as well as the regularity of functions. The purpose of this work is to analyze the definition of functions of four-dimensional variables of the space M5, as well as theorems on the continuity and existence of differentiability of functions of four-dimensionalvariables. This work is descriptive for comparing the spaces of four-dimensional numbers M5 and M3. In the article, theorems on the continuity and differentiability of functions of four-dimensional variables and their properties are proved, and the Cauchy-Riemann conditions are found. The form of trigonometric, exponential, logarithmic, exponential and power functions of four-dimensional variables is determined and the regularity of functions of M5 space is proved. Key words: four-dimensional function, continuity, differentiability, regular function, Cauchy-Riemann condition.

On one physical interpretation of generalized conditions Cauchy-Riemann

The message shows that in variables, the generalized Cauchy-Riemann conditions (CRC) coincide with the Maxwell system for the electromagnetic field, and the possible existence of magnetic charges and currents is assumed. This makes it possible to give a physical interpretation to all 8 components of the solution of generalized СRС. When the generalized СRС is taken as a basis, two scalar fields necessarily appear. They are interpreted as some fields of spatially distributed electric and magnetic charges and currents. The introduction of the attached generalized СRC allows us to give a method for determining the potentials of the electromagnetic field under the assumptions made and to determine the gauge transformations more General than the well-known ones. Of course, having made certain restrictions, all the results coincide with those accepted in classical electrodynamics. The author points out that he does not make any physical statements and all constructions are purely mathematical in nature.

New approach to the generation of orthogonal body-fitted meshes and its application to non-steady plane-strain ideal plastic flow

For rigid-perfect plastic solids, ideal plastic flow happens when all material elements follow minimum work paths and the two principal stretch lines are materially embedded. To describe its kinematics effectively, Lagrange convective orthogonal curvilinear system has to be constructed. Therefore, a numerical procedure for generating orthogonal body-fitted meshes has been developed and applied to non-steady plane-strain ideal plastic flow. The mapping between the Cartesian system and the orthogonal system is based on Laplace equations and Cauchy–Riemann condition. The numerical procedure takes into consideration of all the prescribed boundary curves instead of one as the boundary conditions, which has been adopted in earlier studies when using conventional characteristic line method. After each iteration step, the obtained boundary points are adjusted towards the orthogonality conditions until error tolerance requirement is met. For demonstration purpose, the numerical procedure is applied to the optimization of a bulk part under forging. The obtained orthogonal body-fitted meshes are in good agreement with previous methods. Then the optimal initial scale factor is also calculated based on this method and compared with earlier studies. Finally, the evolution of the intermediate shapes and frictional external boundary conditions is also calculated when there are no elastic dead zones.

Плоска задача механіки суцільного середовища в полярних координатах з використанням аргумент функцій комплексного змінного

The general approaches to the solution of the plane problem of continuum mechanics, which have been successfully tested in the theory of plasticity, elasticity, dynamic problems of the theory of elasticity, are considered. Based on the argument function method and the method of a complex variable, new approaches to the determination of components of the stress tensor in polar coordinates have been developed. The equilibrium equation systems were used to solve the flat problem. A fundamental substitution is suggested. Use of a trigonometric substitution that connects integral characteristics of a stressed state with components of a stress tensor is demonstrated. Argument functions of basic variables are introduced. When substituting into differential equations, operators are formed, which are characterized by these argument functions and that perform a role of special search regulators. As a result of this, dependencies of existence of solutions in a form of the invariant Cauchy–Riemann conditions and Laplace’s equations are determined. The result obtained is conveniently applied for simplification, allowing linearization of boundary conditions. The solution uses generalized relations in the differential form for specific functions - functions of harmonic type. The trigonometric shape of the shearing stress distribution diagram is actually confirmed by theoretical and experimental data. The solutions that determine not the functions themselves, but the conditions of their existence using Cauchy–Riemann differential conditions are obtained. The solution is a more general case with the feature that is represented not by the product of functions, each of which is determined by one coordinate, but by the product of different functions simultaneously dependent on two coordinates. Comparison of the obtained results with the solutions of other authors shows that the presented solution after simple transformations can be simplified and consider the obtained solution as more generalized.

Some algebraic properties of complex Segre quaternoins

This paper deals with the basic properties the algebra of Segre quaternions over the field of complex numbers. We study idempotents, ideals, matrix representation and the Peirce decomposition of this algebra. We also investigate the structure of zeros of a polynomial in Segre complex quaternions by reducing it to the system of four polynomial equations in the complex field. In addition, Cauchy-Riemann type conditions are obtained for the differentiability of a function on the complex Segre quaternionic algebra.

Stochastic optimization on complex variables and pure-state quantum tomography

Real-valued functions of complex arguments violate the Cauchy-Riemann conditions and, consequently, do not have Taylor series expansion. Therefore, optimization methods based on derivatives cannot be directly applied to this class of functions. This is circumvented by mapping the problem to the field of the real numbers by considering real and imaginary parts of the complex arguments as the new independent variables. We introduce a stochastic optimization method that works within the field of the complex numbers. This has two advantages: Equations on complex arguments are simpler and easy to analyze and the use of the complex structure leads to performance improvements. The method produces a sequence of estimates that converges asymptotically in mean to the optimizer. Each estimate is generated by evaluating the target function at two different randomly chosen points. Thereby, the method allows the optimization of functions with unknown parameters. Furthermore, the method exhibits a large performance enhancement. This is demonstrated by comparing its performance with other algorithms in the case of quantum tomography of pure states. The method provides solutions which can be two orders of magnitude closer to the true minima or achieve similar results as other methods but with three orders of magnitude less resources.

Application of Three-Dimensional Quasi-Conformal Mappings to Grid Construction

Two-dimensional conformal mappings are a powerful and elegant tool for solving many mathematical and physical problems. The conformal mapping method is suitable for constructing two-dimensional grids. The quasi-conformal mappings constructed in this paper naturally generalize the application of conformal mappings to grid construction in the three-dimensional case. For a steady irrotational flow of an ideal incompressible fluid, in addition to the velocity potential, two stream functions are introduced. Generalized Cauchy–Riemann conditions from which three-dimensional quasi-conformal mappings follow are presented. The mappings constructed can be represented as a sequence of two-dimensional conformal mappings. Examples of grid construction using the theory of quasi-conformal mappings are given. The best proof of these results is their visualization.

About vortex equations of two dimensional flows

A method to obtain a time-independent vortex solution of a nonlinear differential equation describing two-dimensional flow is investigated. In the usual way, starting from the Navier–Stokes equation the vortex equation is derived by taking a curl operation. After rearranging the equation of the vortex, we get a continuity equation or a divergence-free equation: \(\partial _1V_1+\partial _2V_2=0\). Additional irrotationality of \(V_1\) and \(V_2\) leads us to the Cauchy–Riemann condition satisfied by a newly introduced stream function \(\Psi\) and velocity potential \(\Phi\). As a result, if we know \(V_1\) and \(V_2\) or a combination of two, the differential equation is mapped to a lower-order partial differential equation. This differential equation is the one satisfied by the stream function \(\psi\) where the vorticity vector \(\omega\) is given by \(-(\partial _1^2+\partial _2^2) \psi\). A simple solution is discussed for the two different limits of viscosity.

Extending coherent state transforms to Clifford analysis

Segal-Bargmann coherent state transforms can be viewed as unitary maps from L2 spaces of functions (or sections of an appropriate line bundle) on a manifold X to spaces of square integrable holomorphic functions (or sections) on Xℂ. It is natural to consider higher dimensional extensions of X based on Clifford algebras as they could be useful in studying quantum systems with internal, discrete, degrees of freedom corresponding to nonzero spins. Notice that the extensions of X based on the Grassmann algebra appear naturally in the study of supersymmetric quantum mechanics. In Clifford analysis, the zero mass Dirac equation provides a natural generalization of the Cauchy-Riemann conditions of complex analysis and leads to monogenic functions. For the simplest but already quite interesting case of X = ℝ, we introduce two extensions of the Segal-Bargmann coherent state transform from L2(ℝ, dx) ⊗ ℝm to Hilbert spaces of slice monogenic and axial monogenic functions and study their properties. These two transforms are related by the dual Radon transform. Representation theoretic and quantum mechanical aspects of the new representations are studied.

Algebraic First Integrals of the Polynomial Systems Satisfying the Cauchy–Riemann Conditions

We consider polynomial planar systems of ordinary differential equations that can be written in the form \(\dot{z}={\mathcal {P}}(z), \ z\in {\mathbb {C}}.\) We describe the classes of such systems that have rational first integrals and study some properties of these integrals.

GEOMETRICAL QUATERNIONIC COUPLING FOR THREE DIMENSIONAL WAVE EQUATIONS

The present work has the scope to show the relationship between four three-dimensional waves. This fact will be made in the form of coupling, us- ing for it the Cauchy-Riemann conditions for quaternionic functions (1), through certain Laplace's equation in (2). The coupling will relate those functions that determine the wave as well as their respective propagation speeds.

Riesz potential versus fractional Laplacian

This paper starts by introducing the Grünwald–Letnikov derivative, the Riesz potential and the problem of generalizing the Laplacian. Based on these ideas, the generalizations of the Laplacian for 1D and 2D cases are studied. It is presented as a fractional version of the Cauchy–Riemann conditions and, finally, it is discussed with the n-dimensional Laplacian.

Adaptive Dynamic Programming for a Class of Complex-Valued Nonlinear Systems

In this brief, an optimal control scheme based on adaptive dynamic programming (ADP) is developed to solve infinite-horizon optimal control problems of continuous-time complex-valued nonlinear systems. A new performance index function is established on the basis of complex-valued state and control. Using system transformations, the complex-valued system is transformed into a real-valued one, which overcomes Cauchy-Riemann conditions effectively. With the transformed system and the performance index function, a new ADP method is developed to obtain the optimal control law by using neural networks. A compensation controller is developed to compensate the approximation errors of neural networks. Stability properties of the nonlinear system are analyzed and convergence properties of the weights for neural networks are presented. Finally, simulation results demonstrate the performance of the developed optimal control scheme for complex-valued nonlinear systems.