Riemann Equations Research Articles

Elliptic bootstrapping and the nonlinear Cauchy–Riemann equation

The goal of this paper is to deduce a nonlinear elliptic regularity result from a linear one. In particular, elliptic bootstrapping is a powerful method to determine the regularity of a solution to a partial differential equation. We apply elliptic bootstrapping and linear elliptic regularity to the nonlinear Cauchy–Riemann equation. In doing so, we generalize the fundamental analytic result that holomorphic functions are automatically smooth. In particular, we show that, under certain conditions, the same is true for so-called J-holomorphic functions. We conclude by discussing how this nonlinear regularity result relates to ideas in symplectic geometry.

On a Statement of the Boundary Value Problem for a Generalized Cauchy–Riemann Equation with Nonisolated Singularities in a Lower-Order Coefficient

On a Statement of the Boundary Value Problem for a Generalized Cauchy–Riemann Equation with Nonisolated Singularities in a Lower-Order Coefficient

Mathematics-Informed Neural Network for Acoustic Scattering by Parallel Plates

This paper proposes a mathematics-informed neural network (MINN) approach for resolving the long-term challenge in wave scattering modeling. The central innovation lies in integrating Cauchy–Riemann equations into machine learning architectures. By incorporating Cauchy integrals and boundary conditions, the neural network successfully learns to numerically produce matrix kernel factorization for Wiener–Hopf analytical models. To validate and demonstrate the approach, a benchmark case of wave scattering from parallel hard–soft plates is studied by comparing the machine learning results with the available analytical solutions. The proposed MINN approach could provide a new route to extensively enhance the theoretical modeling capability for several wave scattering and fluid mechanics problems. The code can be found at https://github.com/lscapku/MINN.

Hyperholomorphicity by Proposing the Corresponding Cauchy–Riemann Equation in the Extended Quaternion Field

In algebra, the sedenions, an extension of the octonion system, form a 16-dimensional noncommutative and nonassociative algebra over the real numbers. It can be expressed as two octonions, and a function and differential operator can be defined to treat the sedenion, expressed as two octonions, as a variable. By configuring elements using the structure of complex numbers, the characteristics of octonions, the stage before expansion, can be utilized. The basis of a sedenion can be simplified and used for calculations. We propose a corresponding Cauchy–Riemann equation by defining a regular function for two octonions with a complex structure. Based on this, the integration theorem of regular functions with a sedenion of the complex structure is given. The relationship between regular functions and holomorphy is presented, presenting the basis of function theory for a sedenion of the complex structure.

The Hilbert problem in a half-plane for generalized analytic functions with a singular point on the real axis

This article analyzes the inhomogeneous Hilbert boundary value problem for an upper half-plane with the finite index and boundary condition on the real axis for one generalized Cauchy–Riemann equation with a singular point on the real axis. A structural formula was obtained for the general solution of this equation under restrictions leading to an infinite index of the logarithmic order of the accompanying Hilbert boundary value problem for analytic functions. This formula and the solvability results of the Hilbert problem in the theory of analytic functions were applied to solve the set boundary value problem.

SONAR target classification with complex-valued neural networks

Popular acoustic signal processing techniques analyze acoustic color, which is a magnitude representation of a frequency spectrum. This paradigm allows for easy visualization of results and simpler models at the cost of throwing out phase information. Analysis and modeling of complex-valued data does have inherent difficulty from the nature of complex numbers. Optimization on the complex field requires alternate partial derivative definitions to circumvent consequences of the Cauchy–Riemann equations regarding holomorphic functions. To make use of phase information, we demonstrate classifier model optimization with complex-valued parameters on data with both magnitude and phase components and show how complex neural networks yield marked improvements over similarly shaped real-valued networks in both classification accuracy and generalization ability. We apply these techniques to small target SONAR with simulated Lamb wave resonance signals for hollow spheres, differentiating between different material classes. [Research funded by the DoD Navy (NEEC) Grant No. N001742010016.]

Riemann–Hilbert Problems for Axially Symmetric Null-Solutions to Iterated Generalised Cauchy–Riemann Equations in $$\mathbb {R}^{n+1}$$

Riemann–Hilbert Problems for Axially Symmetric Null-Solutions to Iterated Generalised Cauchy–Riemann Equations in $$\mathbb {R}^{n+1}$$

IDENTIFICATION OF SEPARATION CURVES OF STRONGLY CONTRASTING ENVIRONMENTS USING COMPLEX ANALYSIS METHODS

When modeling mass transfer processes (for example, filtration) in porous media, there may be cases of the existence of highly permeable layers, which are separated from the corresponding studied parts by some curves that must be found (identified) in the process of solving the problem. When constructing a mathematical model of the corresponding physical process, we will consider a highly permeable medium as "ideally (theoretically infinitely) permeable." In this case, the desired curve can be considered an equipotential line. This work considers the stationary process of fluid movement in a homogeneous horizontal layer of infinitely large dimensions - a soil massif, which is limited by infinite sections of curves, in particular by the desired curve of theoretical water resistance and the horizontal axis, on which the local speed of movement is known. On the basis of the methods of conformal reflections and total images, an approach to the identification of the curve of the separation of media is proposed. The constructed algorithm is modified for solving nonlinear inverse boundary value problems on quasi-conformal mappings of curvilinear polygonal regions bounded by uncertain streamlines and equipotential lines. The feature of the method proposed is that the summary representation formulae allow to explicitly write the solution to the localized linear (main) part of the obtained system of equations. The unknown coefficients are determined here by solving a non-linear system generated solely by the boundary conditions and numerical analogues of the Cauchy–Riemann equations.

An Atomic Representation for Hardy Classes of Solutions to Nonhomogeneous Cauchy–Riemann Equations

An Atomic Representation for Hardy Classes of Solutions to Nonhomogeneous Cauchy–Riemann Equations

Developing Geometric Reasoning of the Relationship of the Cauchy Riemann Equations and Differentiation

Developing Geometric Reasoning of the Relationship of the Cauchy Riemann Equations and Differentiation

Pre-twisted calculus and differential equations

In this paper we introduce the φA-differentiability for functions f:U⊂Rk→Rn, where U is an open set, A is the linear space Rn endowed with a unital associative commutative algebra product, and φ:U⊂Rk→A is a differentiable function in the usual sense. We call it pre-twisted differentiability. With respect to the φA-differentiability we introduce: (a) a type Cauchy–Riemann equations, which serve as φA-differentiability criteria, (b) a Cauchy-integral theorem, and (c) φA-differential equations, which can be used to solve linear and nonlinear ODE systems. It has recently been shown that the φA-differentiable functions define a complete solutions for the PDEs of the form Auxx+Buxy+Cuyy=0, which is used in this paper for solving the corresponding Cauchy problems. Furthermore, solutions of φA-differential equations define solutions for linear and nonlinear PDE systems.

On a Generalization of Monge–Ampère Equations and Monge–Ampère Systems

In the present paper, we discuss Monge-Ampère equations from the viewpoint of differential geometry. It is known that a Monge–Ampère equation corresponds to a special exterior differential system on a 1-jet space. In this paper, we generalize Monge–Ampère equations and prove that a (k+1)st order generalized Monge–Ampère equation corresponds to a special exterior differential system on a k-jet space and that its solution naturally corresponds to an integral manifold of the corresponding exterior differential system. Moreover, we show that the Korteweg-de Vries (KdV) equation and the Cauchy–Riemann equations are examples of our equations.

Generalized Cauchy–Riemann equations in non-identity bases with application to the algebrizability of vector fields

Abstract We complete the work done by James A. Ward in the mid-twentieth century on a system of partial differential equations that defines an algebra 𝔸 {\mathbb{A}} for which this system is the generalized Cauchy–Riemann equations for the derivative introduced by Sheffers at the end of the nineteenth century with respect to 𝔸 {\mathbb{A}} , which is also known as the Lorch derivative with respect to 𝔸 {\mathbb{A}} , and recently simply called 𝔸 {\mathbb{A}} -differentiability. We get a characterization of finite-dimensional algebras, which are associative commutative with unity.

A NOVEL CONCEPT FOR FINDING THE FUNDAMENTAL RELATIONS BETWEEN STREAM FUNCTION AND VELOCITY POTENTIAL IN REAL NUMBERS IN TWO-DIMENSIONAL FLUID MOTIONS

In this paper, the author has presented the fundamental relations between stream function or current function,  and velocity potential or velocity function, φ which are ∂φ/∂x= ∂/∂y and ∂φ/∂y= – ∂/∂x where x,y,φ(x_(, ) y),  (x_(, ) y) are all real in two-dimensional fluid motions using real variables only whereas these relations had been established by using complex variables by Cauchy – Riemann which are known as Cauchy – Riemann equations in fluid dynamics.

Schwarz Boundary Value Problems for Polyanalytic Equation in a Sector Ring

In this article, we first give a modified Schwarz–Pompeiu formula in a general sector ring with angle \(\vartheta =\frac{\pi }{\alpha },\ \alpha \ge 1/2\) by proper conformal mappings, and obtain the solution of the Schwarz problem for the Cauchy–Riemann equation in explicit forms. Furthermore, we construct some integral operators and discuss their properties in detail. Finally, by virtue of these operators, Schwarz problems for an inhomogeneous polyanalytic equation and for a generalized polyanalytic equation are investigated, respectively.

Three boundary value problems of the Cauchy–Riemann equation in the Eclipse domain

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.

COMPLEX BOUNDARY VALUE PROBLEMS FOR THE CAUCHY–RIEMANN OPERATOR ON A TRIANGLE

There are three over-determined boundary value problems for the inhomogeneous Cauchy–Riemann equation, the Dirichlet problem, the Neumann problem and the Robin problem. These problems have found their significant importance and applications in diverse fields of science, for example, applied mathematics, physics, engineering and medicine. In this paper, we explicitly investigate the solvability of these three basic boundary value problems for the Cauchy–Riemann operator on an isosceles orthogonal triangle. The solvability conditions are explicitly obtained. The plane parqueting reflection principle is used. Cauchy–Pompeiu-type formulae for the triangle are established. The boundary behavior of a related integral operator of the Schwarz type is studied in detail. Then, the boundary values of solutions at the corner points are found.

A new class of exact solutions of von Karman’s equation in the nonlinear theory of gas dynamics

Abstract In this paper, by using invariants of Riemann and general solutions of Euler–Poisson–Darboux–Riemann equations, a new class of exact solutions of von Karman’s equation in the nonlinear theory of gas dynamics is given.

A Riemann–Hilbert Type Problem for the Generalized Cauchy–Riemann Equation with Supersingular Points on a Half-Plane

For an equation with the Cauchy–Riemann operator having strong isolated point singularities in the lowest coefficient, an integral representation of the solution is found in the class of bounded functions at infinity, and a Riemann–Hilbert type problem on a half-plane is investigated. Specialists are well aware about the important role played in applications by I.N. Vekua’s theory of generalized analytic functions of an equation in the case when the coefficients and the right-hand side of the generalized Cauchy–Riemann system belong to the class of summable functions. It is deeply linked with many areas of analysis, geometry and mechanics, including quasi-conformal maps, surface theory, shell theory, and gas dynamics. In particular, it is widely used in modeling transonic gas flows, states of momentless stressed equilibrium of convex shells, and many other processes. The Vekua–Pompeiu integral plays a key role in this theory. When the lowest coefficient has strong singularities, some researchers believe that the Vekua–Pompeiu integral, which is main research engine in the theory of generalized analytic functions, cannot be applied. By using the Vekua–Pompeiu integral, we have succeeded in constructing solutions of the Cauchy–Riemann system the lowest coefficient of which contains strong isolated point singularities in an infinite domain. A Riemann–Hilbert type problem on a half-plane is investigated for the equation under consideration by using an explicit integral representation.

Investigation of the two-cut phase region in the complex cubic ensemble of random matrices

We investigate the phase diagram of the complex cubic unitary ensemble of random matrices with the potential V(M)=−13M3+tM, where t is a complex parameter. As proven in our previous paper [Bleher et al., J. Stat. Phys. 166, 784–827 (2017)], the whole phase space of the model, t∈C, is partitioned into two phase regions, Oone−cut and Otwo−cut, such that in Oone−cut the equilibrium measure is supported by one Jordan arc (cut) and in Otwo−cut by two cuts. The regions Oone−cut and Otwo−cut are separated by critical curves, which can be calculated in terms of critical trajectories of an auxiliary quadratic differential. In Bleher et al. [J. Stat. Phys. 166, 784–827 (2017)], the one-cut phase region was investigated in detail. In the present paper, we investigate the two-cut region. We prove that in the two-cut region, the endpoints of the cuts are analytic functions of the real and imaginary parts of the parameter t, but not of the parameter t itself (so that the Cauchy–Riemann equations are violated for the endpoints). We also obtain the semiclassical asymptotics of the orthogonal polynomials associated with the ensemble of random matrices and their recurrence coefficients. The proofs are based on the Riemann–Hilbert approach to semiclassical asymptotics of the orthogonal polynomials and the theory of S-curves and quadratic differentials.