Riemann Operator Research Articles

Cauchy–Riemann operator in Cayley–Dickson–Clifford analysis

Cauchy–Riemann operator in Cayley–Dickson–Clifford analysis

Duality of fractional derivatives: On a hybrid and non-hybrid inclusion problem

Abstract The main goal of this paper is to investigate a newly proposed hybrid and hybrid inclusion problem consisting of fractional differential problems involving two different fractional derivatives of order μ, Caputo and Liouville–Riemann operators, with multi-order mixed Riemann–Liouville integro-derivative conditions. Although α is between one and two, we need three boundary value conditions to find the integral equation. The study investigates the results of existence for hybrid, hybrid inclusion, and non-hybrid inclusion problems by employing several analytical approaches, including Dhage’s technique, α - ψ {\alpha-\psi} -contractive mappings, fixed points, and endpoints of the product operators. To further illustrate our findings, we present three examples.

Integral Formulas for Slice Cauchy–Riemann Operator and Applications

Integral Formulas for Slice Cauchy–Riemann Operator and Applications

Characterization of hybrid quantum eigenstates in systems with mixed classical phase space.

Generic low-dimensional Hamiltonian systems feature a structured, mixed classical phase space. The traditional Percival classification of quantum spectra into regular states supported by quasi-integrable regions and irregular states supported by quasichaotic regions turns out to be insufficient to capture the richness of the Hilbert space. Berry's conjecture and the eigenstate thermalization hypothesis are not applicable and quantum effects such as tunneling, scarring, and localization do not obey the standard paradigms. We demonstrate these statements for a prototype Bose-Hubbard model. We highlight the hybridization of chaotic and regular regions from opposing perspectives of ergodicity and localization.

Some estimates for elliptic systems generalizing the Bitsadze system of equations

This article explores an elliptic system of n equations where the main part is the Bitsadze operator (the square of the Cauchy–Riemann operator) and the lower term is the product of a given matrix function by the conjugate of the desired vector function. The system was analyzed in the Banach space of vector functions that are bounded and uniformly H¨older continuous in the entire complex plane. It was revealed that the problem of solving the system in the specified space may not be Noetherian. An example of a homogeneous system with an infinite number of linearly independent solutions was given. As is known, for many classes of elliptic systems, the Noetherianity of boundary value problems in a compact domain is equivalent to the presence of a priori estimates in the corresponding spaces. In this regard, it is important to study the issues related to the establishment of a priori estimates for the system under consideration in the above space. In the case of coefficients weakly oscillating at infinity, necessary and sufficient conditions for the validity of the a priori estimate were found. These conditions were written out in the language of the spectrum of limit matrices formed by the partial limits of the coefficient matrix at infinity. Specific examples were provided to illustrate how the limit matrices are constructed and what the above conditions look like.

Generalized Partial-Slice Monogenic Functions: A Synthesis of Two Function Theories

In this paper, we review the notion of generalized partial-slice monogenic functions that was introduced by the authors in Xu and Sabadini (Generalized partial-slice monogenic functions, arXiv:2309.03698, 2023). The class of these functions includes both the theory of monogenic functions and of slice monogenic functions over Clifford algebras and it is obtained via a synthesis operator which combines a generalized Cauchy–Riemann operator with an operator acting on slices. Besides recalling the fundamental features, we provide a notion of ∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$*$$\\end{document}-product based on the CK-extension and discuss the smoothness of generalized partial-slice functions.

Eigenmodes of fractal drums: A numerical student experiment

“Can one hear the shape of a drum?” was a question posed (and made famous) by mathematician Mark Kac in the mid-1960s. It addresses whether a deeper connection exists between the resonance modes (eigenmodes) of a drum and its shape. Here, we propose a numerical experiment, suitable for advanced undergraduate physics students, on the calculation of the eigenmodes of a square Koch fractal drum, for which experimental results do exist. This exercise is designed to develop the students' understanding of the vibrations of fractal drums, their eigenmodes, and potentially their integrated density of states. The students calculate the lowest order eigenmodes of the fractal drum, visualize these modes, and study their symmetry properties. As an extension, the students may investigate the integrated density of states of the fractal drum and compare their findings to the Weyl–Berry conjecture.

Fluctuations of eigenvector overlaps and the Berry conjecture for Wigner matrices

Fluctuations of eigenvector overlaps and the Berry conjecture for Wigner matrices

Complete Hilbert-Space Ergodicity in Quantum Dynamics of Generalized Fibonacci Drives.

Ergodicity of quantum dynamics is often defined through statistical properties of energy eigenstates, as exemplified by Berry's conjecture in single-particle quantum chaos and the eigenstate thermalization hypothesis in many-body settings. In this work, we investigate whether quantum systems can exhibit a stronger form of ergodicity, wherein any time-evolved state uniformly visits the entire Hilbert space over time. We call such a phenomenon complete Hilbert-space ergodicity (CHSE), which is more akin to the intuitive notion of ergodicity as an inherently dynamical concept. CHSE cannot hold for time-independent or even time-periodic Hamiltonian dynamics, owing to the existence of (quasi)energy eigenstates which precludes exploration of the full Hilbert space. However, we find that there exists a family of aperiodic, yet deterministic drives with minimal symbolic complexity-generated by the Fibonacci word and its generalizations-for which CHSE can be proven to occur. Our results provide a basis for understanding thermalization in general time-dependent quantum systems.

Cauchy Formulae and Hardy Spaces in Discrete Octonionic Analysis

In this paper, we continue the development of a fundament of discrete octonionic analysis that is associated to the discrete first order Cauchy–Riemann operator acting on octonions. In particular, we establish a discrete octonionic version of the Borel–Pompeiu formula and of Cauchy’s integral formula. The latter then is exploited to introduce a discrete monogenic octonionic Cauchy transform. This tool in hand allows us to introduce discrete octonionic Hardy spaces for upper and lower half-space together with Plemelj projection formulae.

Properties of a polyanalytic functional calculus on the S‐spectrum

AbstractThe Fueter mapping theorem gives a constructive way to extend holomorphic functions of one complex variable to monogenic functions, that is, null solutions of the generalized Cauchy–Riemann operator in , denoted by . This theorem is divided in two steps. In the first step, a holomorphic function is extended to a slice hyperholomorphic function. The Cauchy formula for this type of functions is the starting point of the S‐functional calculus. In the second step, a monogenic function is obtained by applying the Laplace operator in four real variables, namely, Δ, to a slice hyperholomorphic function. The polyanalytic functional calculus, that we study in this paper, is based on the factorization of . Instead of applying directly the Laplace operator to a slice hyperholomorphic function, we apply first the operator and we get a polyanalytic function of order 2, that is, a function that belongs to the kernel of . We can represent this type of functions in an integral form and then we can define the polyanalytic functional calculus on S‐spectrum. The main goal of this paper is to show the principal properties of this functional calculus. In particular, we study a resolvent equation suitable for proving a product rule and generate the Riesz projectors.

BGG Sequences with Weak Regularity and Applications

We investigate some Bernstein–Gelfand–Gelfand complexes consisting of Sobolev spaces on bounded Lipschitz domains in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^{n}$$\\end{document}. In particular, we compute the cohomology of the conformal deformation complex and the conformal Hessian complex in the Sobolev setting. The machinery does not require algebraic injectivity/surjectivity conditions between the input spaces, and allows multiple input complexes. As applications, we establish a conformal Korn inequality in two space dimensions with the Cauchy–Riemann operator and an additional third-order operator with a background in Möbius geometry. We show that the linear Cosserat elasticity model is a Hodge–Laplacian problem of a twisted de Rham complex. From this cohomological perspective, we propose potential generalizations of continuum models with microstructures.

The Fueter-Sce mapping and the Clifford–Appell polynomials

AbstractThe Fueter-Sce theorem provides a procedure to obtain axially monogenic functions, which are in the kernel of generalized Cauchy–Riemann operator in ${\mathbb{R}}^{n+1}$. This result is obtained by using two operators. The first one is the slice operator, which extends holomorphic functions of one complex variable to slice monogenic functions in $ \mathbb{R}^{n+1}$. The second one is a suitable power of the Laplace operator in n + 1 variables. Another way to get axially monogenic functions is the generalized Cauchy–Kovalevskaya (CK) extension. This characterizes axial monogenic functions by their restriction to the real line. In this paper, using the connection between the Fueter-Sce map and the generalized CK-extension, we explicitly compute the actions $\Delta_{\mathbb{R}^{n+1}}^{\frac{n-1}{2}} x^k$, where $x \in \mathbb{R}^{n+1}$. The expressions obtained is related to a well-known class of Clifford–Appell polynomials. These are the building blocks to write a Taylor series for axially monogenic functions. By using the connections between the Fueter-Sce map and the generalized CK extension, we characterize the range and the kernel of the Fueter-Sce map. Furthermore, we focus on studying the Clifford–Appell–Fock space and the Clifford–Appell–Hardy space. Finally, using the polyanalytic Fueter-Sce theorems, we obtain a new family of polyanalytic monogenic polynomials, which extends to higher dimensions the Clifford–Appell polynomials.

Comparative analysis of new approximate analytical method and Mohand variational transform method for the solution of wave-like equations with variable coefficients

In a current article, two novel analytical approaches are compared for analytical analysis of wave like equations with variable coefficients, which describe the evolution of stochastic phenomena. One is the New Approximate Analytical Method which based on Caputo–Riemann operator with simple decomposition procedure. This new method directly provides a fractional order series form solution which fastly converge to exact form solution for integer order. The second is Mohand Variational Iteration Transform Method which base on iteration procedure with Mohand Transform. The Mohand Variational Iteration Transform Method provides series form solution without using any decomposition, discretization, and He’s polynomial. The solution in series form converges directly to the exact solution for integer orders. The comparative analysis validates that Mohand Variational Iterative Method has less computational work and simple procedure without using any decomposition, discretization, and He’s polynomial procedure as compared to New Approximate Analytical Method.

Reconstruction of the Cauchy–Riemann Operator by Complex Integration Operators along Circles

Reconstruction of the Cauchy–Riemann Operator by Complex Integration Operators along Circles

On a Spectral Problem for the Cauchy–Riemann Operator with Boundary Conditions of the Bitsadze–Samarskii Type

On a Spectral Problem for the Cauchy–Riemann Operator with Boundary Conditions of the Bitsadze–Samarskii Type

Lie analysis and laws of conservation for the two-dimensional model of Newell–Whitehead–Segel regarding the Riemann operator fractional scheme in a time-independent variable

The framework of fractional partial differential models is the first-rate hyperlink between mathematics and applied physics. This article project intends to utilize outcomes for the time-fractional Newell–Whitehead–Segel model in 2 D -space including conservation laws, Lie point symmetry analysis, and series solutions. We introduce a particular fractional model that is free of the type of approximate methods, whilst, environmental flow and magnetohydrodynamics processes are considered to be the main real-world phenomena treated with such a model. Herein, the method of power series is exercised to provide an analytical solution to the current governing model. The idea and undertaking of the method lie in the assumption that the solution is a power series of coefficients that are determined by a recurrence relation obtained by substituting the series solution in the considered model. Also, the Riemann approach is utilized as a total derivative. Simulation effects are systematically demonstrated through a chain of check cases. Strong proofs with some related plots are performed to confirm the accuracy and fitness of the model version and the presented approach. Moreover, the laws of conservation depend on the existence of a Lagrangian of the fractional Newell–Whitehead–Segel model utilized. Ultimately, a few related comments and future proposals are epitomized. HIGHLIGHT The Lie point symmetries of the time-fractional Newell–Whitehead–Segel equation in two-dimensional space is utilized and derived. The technique of the power series is applied to conclude the explicit solutions for the time-fractional Newell–Whitehead–Segel equation in two-dimensional space for the first time. The conservation laws for the time-fractional Newell–Whitehead–Segel equation in two-dimensional space are built using a novel conservation theorem. Several graphical countenances were utilized to award a visual performance of the obtained solutions.

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.

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.

Interval Fejér-Type Inequalities for Left and Right-λ-Preinvex Functions in Interval-Valued Settings

For left and right λ-preinvex interval-valued functions (left and right λ-preinvex IVFs) in interval-valued Riemann operator settings, we create Hermite–Hadamard (H-H) type inequalities in the current study. Additionally, we create Hermite–Hadamard–Fejér (H-H-Fejér)-type inequalities for preinvex functions of the left and right interval-valued type under some mild conditions. Moreover, some exceptional new and classical cases are also obtained. Some useful examples are also presented to prove the validity of the results.