Cauchy-Riemann Research Articles

Geometric quantization on CR manifolds

Let [Formula: see text] be a compact connected orientable Cauchy–Riemann (CR) manifold with the action of a compact Lie group [Formula: see text]. Under natural pseudoconvexity assumptions we show that the CR Guillemin–Sternberg map is an isomorphism at the level of Sobolev spaces of CR functions, modulo a finite-dimensional subspace. As application we study this map for holomorphic line bundles which are positive near the inverse image of [Formula: see text] by the momentum map. We also show that “quantization commutes with reduction” for Sasakian manifolds.

On the Connection Between the Fueter–Sce–Qian Theorem and the Generalized CK-Extension

The Fueter–Sce–Qian theorem provides a way of inducing axial monogenic functions in $$\mathbb {R}^{m+1}$$ from holomorphic intrinsic functions of one complex variable. This result was initially proved by Fueter and Sce for the cases where the dimension m is odd using pointwise differentiation, while the extension to the cases where m is even was proved by Qian using the corresponding Fourier multipliers. In this paper, we provide an alternative description of the Fueter–Sce–Qian theorem in terms of the generalized CK-extension. The latter characterizes axial null solutions of the Cauchy–Riemann operator in $$\mathbb {R}^{m+1}$$ in terms of their restrictions to the real line. This leads to a one-to-one correspondence between the space of axially monogenic functions in $$\mathbb {R}^{m+1}$$ and the space of analytic functions of one real variable. We provide explicit expressions for the Fueter–Sce–Qian map in terms of the generalized CK-extension for both cases, m even and m odd. These expressions allow for a plane wave decomposition of the Fueter–Sce–Qian map or, more in particular, a factorization of this mapping in terms of the dual Radon transform. In turn, this decomposition provides a new possibility for extending the Coherent State Transform (CST) to Clifford Analysis. In particular, we construct an axial CST defined through the Fueter–Sce–Qian mapping, and show how it is related to the axial and slice CST’s already studied in the literature.

A bicomplex (ϑ,φ)-weighted fractional Borel-Pompeiu type formula

The purpose of this paper is to establish a Borel-Pompeiu type formula induced from a fractional bicomplex (ϑ,φ)-weighted Cauchy-Riemann operator, where the weights are two hyperbolic orthogonal bicomplex functions and the fractionality is understood in the sense of Riemann-Liouville and Caputo approaches.

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.

Well hydraulics in dual-zone heterogeneous wedge-shaped aquifers under an ambient flow from stream-wise varying boundary heads

• Closed-form analytical solution for flow net considering non-uniform ambient flow. • Generality of solution in choosing hydrological input parameters. • A framework for assessment of mass balance at aquifer and sub-aquifer scales. • Uncertainty analysis to quantify model response to randomness in input parameters. A steady-state analytical solution based on the method of Green’s function is presented for the analysis of hydraulic interference between pumping wells and ambient flow in a heterogeneous wedge-shaped aquifer. Two intersecting streams with spatially variable head form the boundaries of the wedge and establish non-uniform distribution of ambient flow across it. The dual-zone aquifer consists of two porous media of constant but contrasting transmissivities and refraction conditions are exactly satisfied along their common interface. The method of Green’s function affords an integral representation of boundary heads which is readily evaluated in terms of Lerch transcendent function. Integration of Cauchy-Riemann equations gives the associated stream function in closed-form which in turn completes analytical description of the flow net. A systematic way for the assessment of mass balance at both aquifer and sub-aquifer (well capture zone) scales is discussed which also helps identifying whether a certain reach of the boundary stream is losing or gaining water to or from the adjoining aquifer. The solution is general in the sense that the wedge angle, location and discharge of the wells, dual-zone transmissivities and slopes of boundary head variation can be arbitrary chosen. The solution encompasses the case of stream-barrier wedge as a subset. Hypothetical examples demonstrate the complex interaction between surface–subsurface waters showing formation of stagnation point on the capture zone envelopes. In each case, head profiles are in excellent agreement with numerical counterparts from adaptive finite element method. Finally, uncertainty analysis quantifies the extent to which stochastic nature of dual-zone transmissivities and boundary head slopes (model inputs) affects hydraulic head at specific observation points (model output).

Towards a general [formula omitted]-resolvent equation and Riesz projectors

The Fueter-Sce-Qian mapping theorem is a two steps procedure to extend holomorphic functions of one complex variable to quaternionic or Clifford algebra-valued functions in the kernel of a suitable generalized Cauchy-Riemann operator. Using the Cauchy formula of slice monogenic functions it is possible to write the Fueter-Sce-Qian extension theorem in integral form and to define the F-functional calculus for n-tuples of commuting operators. This functional calculus is defined on the S-spectrum and generates a monogenic functional calculus in the spirit of McIntosh and collaborators. One of the main goals of this paper is to show that the F-functional calculus generates the Riesz projectors. The existence of such projectors is obtained via the F-resolvent equation which was previously known only in the quaternionic setting and also its existence was under question. In this paper we prove the F-resolvent equation in the Clifford algebra setting. It is much more complicated than the one in the quaternionic case since it contains various pieces, however it still allows to nicely define the Riesz projectors.

Cauchy-Riemann operators and local slice analysis over real alternative algebras

We prove some formulas relating Cauchy-Riemann operators defined on hypercomplex subspaces of an alternative *-algebra to a differential operator associated with the concept of slice-regularity and to the spherical Dirac operator. These results in particular allow to introduce a definition of locally slice-regular function and open the path for local slice analysis. Since Cauchy-Riemann operators factor the corresponding Laplacian operators, the proven formulas let us also obtain several results about the harmonicity and polyharmonicity properties of slice-regular functions.

Carleman’s Formula for Solutions of Generalized Cauchy–Riemann Systems in Multidimensional Spatial Domains

In this paper, we consider the restoration problem for solutions of the generalized Cauchy–Riemann system in a multidimensional spatial domain, using their values on a piece of the boundary of the domain, i.e., the Cauchy problem. We construct an approximate solution of this problem based on the Carleman matrix method.

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.

A note on the temporal and spatial attenuation of ocean waves

We study attenuation of free waves in the homogeneous ocean. Using the Cauchy–Riemann relations and then energy equation, we derive a novel equation that relates temporal and spatial decay to the energy dissipation in the fluid. In particular, for spatially damped waves we find that the attenuation coefficient is inversely proportional to the group velocity. We discuss the consequences of this for continental shelf waves where the group velocity may tend to zero in wave number space, and hence the spatial attenuation coefficient may become infinitely large. We show that generally for a travelling wave solution, the spatial damping will remain unchanged in a frame of reference moving with the group velocity, revealing the different physics behind temporal wave damping and spatial wave damping.

Complex Analysis and its Several Applications

Complex analysis is important for science because it extends analytical methods from real variables to complex variables and complex numbers. Also, complex number has two independent components, one variable will not change when the other is changing, that are particularly useful when two variables must be dealt with simultaneously. In this essay, we are going to talk about the history chronologically such as who first introduced the idea of complex number, who first discovered the rule of complex number, and why complex analysis is important. Also, the essay includes some basics about the complex variable and complex analysis. For example, the definition of complex number, Cauchy-Riemann Equations, and Cauchy Goursat theorem can help us to get further known of the complex analysis and solve some basic analytic problems.

Some Fundamental Results from Complex Analysis

This paper is going to introduce the most basic theory of analytic functions of one complex variable. It begins at the discussion of meaning of complex number and the historical development from the formula of cubic equation to the square root of negative number. In the middle section, which is divided in to four small parts. First part states the expression of complex number z, algebraic properties, and the relationship of each single complex number with whole complex plane. Second part concerns about several elementary functions of complex number. Next part relates to the derivative of complex number, such as the partial derivative and Cauchy-Riemann Equation to verdict that whether a function is analytic in the domain. Last part is the integral of complex functions, which concludes some important theorems with the proof, and examples. After the section of introduction of complex number, it comes to the final section which is talking about an identity involving complex number and the prove of it.

Geometry of hyperbolic Cauchy–Riemann singularities and KAM-like theory for holomorphic involutions

This article is concerned with the geometry of germs of real analytic surfaces in $$({\mathbb {C}}^2,0)$$ having an isolated Cauchy–Riemann (CR) singularity at the origin. These are perturbations of Bishop quadrics. There are two kinds of CR singularities stable under perturbation: elliptic and hyperbolic. Elliptic case was studied by Moser–Webster (Acta Math 150(3–4), 255–296, 1983) who showed that such a surface is locally, near the CR singularity, holomorphically equivalent to normal form from which lots of geometric features can be read off. In this article we focus on perturbations of hyperbolic quadrics. As was shown by Moser and Webster (1983), such a surface can be transformed to a formal normal form by a formal change of coordinates that may not be holomorphic in any neighborhood of the origin. Given a non-degenerate real analytic surface M in $$({\mathbb {C}}^2,0)$$ having a hyperbolic CR singularity at the origin, we prove the existence of a non-constant Whitney smooth family of connected holomorphic curves intersecting M along holomorphic hyperbolas. This is the very first result concerning hyperbolic CR singularity not equivalent to quadrics. This is a consequence of a non-standard KAM-like theorem for pair of germs of holomorphic involutions $$\{\tau _1,\tau _2\}$$ at the origin, a common fixed point. We show that such a pair has large amount of invariant analytic sets biholomorphic to $$\{z_1z_2=const\}$$ (which is not a torus) in a neighborhood of the origin, and that they are conjugate to restrictions of linear maps on such invariant sets.

Boundary invariants and the closed range property for [formula omitted

This paper provides a connection between two distinct branches of research in CR geometry – namely, analytic and geometric conditions that suffice to establish the closed range of the Cauchy-Riemann operator and CR invariants on CR manifolds. Specifically, we work on not necessarily pseudoconvex domains Ω⊂Cn and define third and fourth order CR invariants on bΩ and show that these invariants provide enough information to establish closed range for the ∂¯-Laplacian in L(0,q)2(Ω) for a given, fixed q. The closed range estimates follow from our previously defined weak Z(q) condition. We also develop powerful linear algebra machinery to translate the information from the invariants into information about the Levi form and its eigenvalues. We conclude with several examples that demonstrate the usefulness and ease of use of the new conditions.

Extendability and the $$\overline{\partial }$$ operator on the Hartogs triangle

In this paper it is shown that the Hartogs triangle $${\mathbf{T}}$$ in $${\mathbf{C}}^2$$ is a uniform domain. This implies that the Hartogs triangle is a Sobolev extension domain. Furthermore, the weak and strong maximal extensions of the Cauchy-Riemann operator agree on the Hartogs triangle. These results have numerous applications. Among other things, they are used to study the Dolbeault cohomology groups with Sobolev coefficients on the complement of $${\mathbf{T}}$$ .

Schwarz problem in a ring domain*

In this article, we discuss the solutions of Schwarz problem in a ring domain. Firstly we introduce the solutions of inhomogeneous Cauchy-Riemann equation. Afterwards we extend the discussions to higher-order equations. The relevant solvability conditions are obtained. The last part is reserved for the solutions of higher-order linear equations in .

Some Results About the Structure of Primitivity Domains for Linear Partial Differential Operators with Constant Coefficients

Let G(D) be a linear partial differential operator on {mathbb {R}}^n, with constant coefficients. Moreover let Omega subset {mathbb {R}}^n be open and Fin L^1_{text {loc}} (Omega , {mathbb {C}}^N). Then any set of the form Af,F:={x∈Ω|(G(D)f)(x)=F(x)},withf∈Wlocg,1(Ω,Ck)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} A_{f,F}:= \\{ x\\in \\Omega \\, \\vert \\, (G(D)f)(x)=F(x)\\}, \\text { with }f\\in W^{g,1}_{\\text {loc}}(\\Omega , {\\mathbb {C}}^k) \\end{aligned}$$\\end{document}is said to be a G-primitivity domain of F. We provide some results about the structure of G-primitivity domains of F at the points of the (suitably defined) G-nonintegrability set of F. A Lusin type theorem for G(D) is also provided. Finally, we give applications to the Maxwell type system and to the multivariate Cauchy-Riemann system.

Riemann boundary value problems for multiplicative doubly quasi-periodic meta-analytic functions

This paper investigates multiplicative doubly quasi-periodic Riemann boundary value problems for null-solutions to a class of polynomially Cauchy-Riemann equations, i.e. meta-analytic functions. To begin with, the decomposition theorem for multiplicative doubly quasi-periodic meta-analytic functions is constructed. Then the growth order of multiplicative doubly quasi-periodic meta-analytic functions at origin is discussed. Finally, Riemann boundary value problems for multiplicative doubly quasi-periodic analytic functions and for multiplicative doubly quasi-periodic meta-analytic functions are investigated, respectively. The explicit integral expressions of their solutions and conditions of solvability are attained.