Beltrami Operator Research Articles

A Meshfree Collocation Scheme for Surface Differential Operators on Point Clouds

We present a meshfree collocation scheme to discretize intrinsic surface differential operators over scalar fields on smooth curved surfaces with given normal vectors and a non-intersecting tubular neighborhood. The method is based on discretization-corrected particle strength exchange (DC-PSE), which generalizes finite difference methods to meshfree point clouds. The proposed Surface DC-PSE method is derived from an embedding theorem, but we analytically reduce the operator kernels along surface normals to obtain a purely intrinsic computational scheme over surface point clouds. We benchmark Surface DC-PSE by discretizing the Laplace–Beltrami operator on a circle and a sphere, and we present convergence results for both explicit and implicit solvers. We then showcase the algorithm on the problem of computing Gauss and mean curvature of an ellipsoid and of the Stanford Bunny by approximating the intrinsic divergence of the normal vector field. Finally, we compare Surface DC-PSE with surface finite elements (SFEM) and diffuse-interface finite elements (DI FEM) in a validation case.

Finite Lipschitzness of regular solutions to nonlinear Beltrami equation

A sufficient condition providing a local finite Lipschitzness for the regular solutions of Sobolev class W loc 1 , 1 to a nonlinear Beltrami equation introduced in [Golberg A, Salimov R. Nonlinear Beltrami equation. Complex Var Elliptic Equ. 2020;65(1):6–21] is established. Some illustrating examples and preliminary results are also given.

Canonical momenta in digitized Su(2) lattice gauge theory: definition and free theory

Hamiltonian simulations of quantum systems require a finite-dimensional representation of the operators acting on the Hilbert space mathcal {H}. Here we give a prescription for gauge links and canonical momenta of an SU(2) gauge theory, such that the matrix representation of the former is diagonal in mathcal {H}. This is achieved by discretising the sphere S_3 isomorphic to SU(2) and the corresponding directional derivatives. We show that the fundamental commutation relations are fulfilled up to discretisation artefacts. Moreover, we directly construct the Casimir operator corresponding to the Laplace–Beltrami operator on S_3 and show that the spectrum of the free theory is reproduced again up to discretisation effects. Qualitatively, these results do not depend on the specific discretisation of SU(2), but the actual convergence rates do.

Differential Geometry of the Family of Helical Hypersurfaces with a Light-like Axis in Minkowski Spacetime L4

We investigate the class of helical hypersurfaces parametrized by x=x(u,v,w), characterized by a light-like axis in Minkowski spacetime L4. We determine the matrices that represent the fundamental forms, Gauss map, and shape operator of x. Furthermore, employing the Cayley–Hamilton theorem, we compute the curvatures associated with x. We explore the conditions under which the curvatures of x possess the property of being umbilical. Moreover, we provide the Laplace–Beltrami operator for the family of helical hypersurfaces with a light-like axis in L4.

Twisted Surfaces Family in Euclidean 3-Space

In this study, we examine a distinct set of twisted surfaces in the three-dimensional Euclidean space E3. Our focus lies in the investigation of the differential geometry of this surface family, including the determination of their curvatures. Furthermore, we establish the essential conditions for minimal surfaces within this framework. Additionally, we calculate the Laplace−Beltrami operator for this particular surface family and provide an illustrative example.

Conoid Surfaces Family in Minkowski 3-Space

This study focuses on exploring a distinct family of conoid surfaces in the three-dimensional Minkowski space L3. Our main objective is to delve into the differential geometry of this family, analyzing its curvatures in detail. Furthermore, we establish the essential conditions for achieving minimality within this specific framework. Additionally, we calculate the Laplace−Beltrami operator for this family of surfaces and illustrate our findings through an example.

Conoid Surfaces Family in Three-Dimensional Euclidean Space

In this research, we investigate a specific family of conoid surfaces within the three-dimensional Euclidean space E3. We consider the differential geometry of the family. We determine the curvatures of these particular surfaces. Moreover, we provide the necessary conditions for minimality within this framework. Additionally, we compute the Laplace−Beltrami operator for this family and present an example.

BRST BMS4 symmetry and its cocycles from horizontality conditions

The BRST structure of the extended Bondi-Metzner-Sachs symmetry group of asymptotically flat manifolds is investigated using the recently introduced framework of the Beltrami field parametrization of four-dimensional metrics. The latter identifies geometrically the two physical degrees of freedom of the graviton as fundamental fields. The graded BRST BMS4 nilpotent differential operator relies on four horizontality conditions giving a Lagrangian reformulation of the asymptotic BMS4 symmetry. A series of cocycles is found which indicate the possibility of anomalies for three-dimensional Lagrangian theories to be built in the null boundaries of asymptotically flat spaces from the principle of BRST BMS4 invariance.

Chaotic and integrable magnetic fields in one-dimensional hybrid Vlasov–Maxwell equilibria

The construction of kinetic equilibrium states is important for studying stability and wave propagation in collisionless plasmas. Thus, many studies over the past decades have been focused on calculating Vlasov–Maxwell equilibria using analytical and numerical methods. However, the problem of kinetic equilibrium of hybrid models is less studied, and self-consistent treatments often adopt restrictive assumptions ruling out cases with irregular and chaotic behaviour, although such behaviour is observed in spacecraft observations of space plasmas. In this paper, we develop a one-dimensional (1-D), quasineutral, hybrid Vlasov–Maxwell equilibrium model with kinetic ions and massless fluid electrons and derive associated solutions. The model allows for an electrostatic potential that is expressed in terms of the vector potential components through the quasineutrality condition. The equilibrium states are calculated upon solving an inhomogeneous Beltrami equation that determines the magnetic field, where the inhomogeneous term is the current density of the kinetic ions and the homogeneous term represents the electron current density. We show that the corresponding 1-D system is Hamiltonian, with position playing the role of time, and its trajectories have a regular, periodic behaviour for ion distribution functions that are symmetric in the two conserved particle canonical momenta. For asymmetric distribution functions, the system is nonintegrable, resulting in irregular and chaotic behaviour of the fields. The electron current density can modify the magnetic field phase space structure, inducing orbit trapping and the organization of orbits into large islands of stability. Thus, the electron contribution can be responsible for the emergence of localized electric field structures that induce ion trapping. We also provide a paradigm for the analytical construction of hybrid equilibria using a rotating two-dimensional harmonic oscillator Hamiltonian, enabling the calculation of analytic magnetic fields and the construction of the corresponding distribution functions in terms of Hermite polynomials.

Automatic landmark detection and registration of brain cortical surfaces via quasi-conformal geometry and convolutional neural networks

In medical imaging, surface registration is extensively used for performing systematic comparisons between anatomical structures, with a prime example being the highly convoluted brain cortical surfaces. To obtain a meaningful registration, a common approach is to identify prominent features on the surfaces and establish a low-distortion mapping between them with the feature correspondence encoded as landmark constraints. Prior registration works have primarily focused on using manually labeled landmarks and solving highly nonlinear optimization problems, which are time-consuming and hence hinder practical applications. In this work, we propose a novel framework for the automatic landmark detection and registration of brain cortical surfaces using quasi-conformal geometry and convolutional neural networks. We first develop a landmark detection network (LD-Net) that allows for the automatic extraction of landmark curves given two prescribed starting and ending points based on the surface geometry. We then utilize the detected landmarks and quasi-conformal theory for achieving the surface registration. Specifically, we develop a coefficient prediction network (CP-Net) for predicting the Beltrami coefficients associated with the desired landmark-based registration and a mapping network called the disk Beltrami solver network (DBS-Net) for generating quasi-conformal mappings from the predicted Beltrami coefficients, with the bijectivity guaranteed by quasi-conformal theory. Experimental results are presented to demonstrate the effectiveness of our proposed framework. Altogether, our work paves a new way for surface-based morphometry and medical shape analysis.

A stable radial basis function partition of unity method for solving convection–diffusion equations on surfaces

The radial basis function partition of unity (RBF-PU) method is used to solve various types of partial differential equations, since it is based on scattered node sets that can be easily implemented. In this paper, a framework for the RBF-PU method is proposed to solve the convection–diffusion equations on closed surfaces. The surface gradient and Laplace–Beltrami operators are discretized based on the RBF-PU method using the polyharmonic spline radial basis functions augmented with polynomials. Since the RBF-PU method is a local meshless method, an advantage is that the differential matrix obtained by discretizing the surface differential operator is sparse. Additionally, solving convection-dominated problems or transport equations directly using RBF-PU method can produce numerical oscillations. For this reason, we add hyperviscosity term to improve the stability of the RBF-PU method for solving the convection–diffusion equations on surfaces, and numerical experiments show that such a combination has higher-order convergence rates. Finally, we solve the discontinuous source problem on the torus and the Turing system with convection term on different surfaces to demonstrate the effectiveness of the proposed method.

Resonances and residue operators for pseudo-Riemannian hyperbolic spaces

For any pseudo-Riemannian hyperbolic space X over R,C,H or O, we show that the resolvent R(z)=(□−zId)−1 of the Laplace–Beltrami operator −□ on X can be extended meromorphically across the spectrum of □ as a family of operators Cc∞(X)→D′(X). Its poles are called resonances and we determine them explicitly in all cases. For each resonance, the image of the corresponding residue operator in D′(X) forms a representation of the isometry group of X, which we identify with a subrepresentation of a degenerate principal series. Our study includes in particular the case of even functions on de Sitter and Anti-de Sitter spaces.For Riemannian symmetric spaces analogous results were obtained by Miatello–Will and Hilgert–Pasquale. The main qualitative differences between the Riemannian and the non-Riemannian setting are that for non-Riemannian spaces the resolvent can have poles of order two, it can have a pole at the branching point of the covering to which R(z) extends, and the residue representations can be infinite-dimensional.

The Dirichlet problem for the Beltrami equations with sources

The paper is devoted to the study of the Dirichlet problem Re ω(z) → φ(ζ) as z → ζ, z ∈ D, ζ ∈ ∂D, with continuous boundary data φ : ∂D → ℝ for Beltrami equations $$ {\omega}_{\overline{z}} $$ = μ(z)ωz + σ(z), |μ(z)| < 1 a.e., with sources σ : D → ℂ in the case of locally uniform ellipticity. In this case, we have established a series of effective integral criteria of the BMO, FMO, Calderon-Zygmund, Lehto, and Orlicz types on the singularities of the equations at the boundary for the existence, representation, and regularity of solutions in arbitrary bounded domains D of the complex plane ℂ with no boundary component degenerated to a single point for sources σ in Lp (D), p > 2, with compact support in D. Moreover, we have proved the existence, representation, and regularity of weak solutions of the Dirichlet problem in such domains for the Poisson-type equation div[A(z)∇ u(z)] = g(z), whose source g ∈ Lp(D), p > 1, has compact support in D and whose matrix-valued coefficient A(z) guarantees its locally uniform ellipticity.

Chaos from Symmetry: Navier Stokes Equations, Beltrami Fields and the Universal Classifying Crystallographic Group

The core of this paper is the group-theoretical approach, initiated in 2015 by one of the present authors in collaboration with Alexander Sorin that brings into the classical field of mathematical fluid-mechanics a brand new vision, allowing for a more systematic classification and algorithmic construction of Beltrami flows on torii R3/Λ where Λ is a crystallographic lattice. Here this new hydro-theory based on the focal idea of a Universal Classifying Group UGΛ is revised, reorganized, improved and extended. In particular we construct the so far missing UGΛHex for the hexagonal lattice and we advocate that, mastering the cubic and hexagonal instances of this group, we can cover all cases. The relation between the classification of Beltrami Flows with that of contact structures is enlightened. The recent developments about the framework of b-manifolds are considered and it is shown that the choice of the allowed critical surfaces for the b-deformation of a Beltrami field seems to be strongly related to the group-theoretical structure of the latter. This opens new directions of investigations about a group theoretical classification of critical surfaces. Apart from that the most promising research direction opened by the present work streams from the fact that the Fourier series expansion of a generic Navier-Stokes solution can be regrouped into an infinite sum of contributions Wr, each associated with a spherical layer of quantized radius r in the momentum lattice. Each Wr is the superposition of a Beltrami field Wr+ plus an anti-Beltrami field Wr−. These latter have a priori exactly the same decomposition into irreps of the group UGΛ that are variously repeated on higher layers. This crucial property enables the construction of generic Fourier series with prescribed hidden symmetries as candidate solutions of the NS equations. Alternatively the Fourier series representation of known solutions can be analyzed from the point of view of such symmetries. As a further result of this research program a complete and versatile system of MATHEMATICA Codes named AlmafluidaNSPsystem has been constructed and is now available through the site of the Wolfram Community. The exact solutions presented in this paper have to be considered as an illustration of the new conceptions and ideas that have emerged and of what can be further done utilizing the computer codes as an instrument. The main message streaming from our constructions is that the more symmetric is the Beltrami Flow the highest is the probability of the on-set of chaotic trajectories.

Harnack Estimation for Nonlinear, Weighted, Heat-Type Equation along Geometric Flow and Applications

The method of gradient estimation for the heat-type equation using the Harnack quantity is a classical approach used for understanding the nature of the solution of these heat-type equations. Most of the studies in this field involve the Laplace–Beltrami operator, but in our case, we studied the weighted heat equation that involves weighted Laplacian. This produces a number of terms involving the weight function. Thus, in this article, we derive the Harnack estimate for a positive solution of a weighted nonlinear parabolic heat equation on a weighted Riemannian manifold evolving under a geometric flow. Applying this estimation, we derive the Li–Yau-type gradient estimation and Harnack-type inequality for the positive solution. A monotonicity formula for the entropy functional regarding the estimation is derived. We specify our results for various different flows. Our results generalize some works.

On Beltrami equations with inverse conditions and hydrodynamic normalization

We consider problems concerning the existence of solutions of the Beltrami equations and their convergence in the complex plane. We are mainly interested in the case when these solutions satisfy the so-called hydrodynamic normalization condition in the neighborhood of infinity. Under some conditions on dilatations of inverse mappings, we have established the existence of such solutions in the class of continuous Sobolev mappings. We have also obtained results on the locally uniform limit of a sequence of such solutions.

A new symmetric differential operator of normalized functions with applications in image processing

Recently, a symmetric differential operator (SDO) is attracted to studying in the field of mathematical analysis. A new formal of SDO is presented to generalize some well known differential operators in a complex domain. According to this formulation, we shall exam the boundedness and compactness of this operator in complex spaces, such as Hilbert space and Sobolev space. For this purpose, we suggest new norms to solve the fractional Beltrami equation in the open unit disk. This operator has ability to depict the analytic geometric representation of the solution of second order differential equation utilizing the concept of Schwarzian derivative in the open unit disk. Applications in imagings are given the sequel.

An extended tuned subdivision scheme with optimal convergence for isogeometric analysis

This paper presents an extended tuned subdivision scheme for quadrilateral meshes that recovers optimal convergence rates in isogeometric analysis while preserving high-quality limit surfaces. Convergence rates can be improved for subdivision surfaces through proper mask tuning with a lower subdominant eigenvalue λ. However, such a tuning might affect the quality of limit surfaces if the tuning parameters are restricted to one-ring refined control vertices only. In this work, we further relax Catmull–Clark subdivision rules for 2-ring refined vertices with additional degrees of freedom in tuning masks with limit surfaces towards curvature continuity at extraordinary positions. Curvatures are bounded near extraordinary positions of valences N≤7, and a relaxation approach is proposed to suppress the local curvature variation for N≥8. We compare the proposed tuned subdivision scheme with several state-of-the-art schemes in terms of both surface qualities and applications in isogeometric analysis. Laplace–Beltrami equations with prescribed test solutions are adopted for verification of analysis results, and the proposed scheme recovers optimal convergence rates in the L2-norm with considerably lower absolute solution error than other schemes. As to surface quality, the proposed scheme has significant improvements compared with other schemes with optimal convergence rates for isogeometric analysis. The proposed scheme produces tighter curvature bounds with comparable reflection lines to the prevalent Catmull–Clark subdivision.

Generalised spectral dimensions in non-perturbative quantum gravity

The seemingly universal phenomenon of scale-dependent effective dimensions in non-perturbative theories of quantum gravity has been shown to be a potential source of quantum gravity phenomenology. The scale-dependent effective dimension from quantum gravity has only been considered for scalar fields. It is, however, possible that the non-manifold like structures, that are expected to appear near the Planck scale, have an effective dimension that depends on the type of field under consideration. To investigate this question, we have studied the spectral dimension associated to the Laplace–Beltrami operator generalised to k-form fields on spatial slices of the non-perturbative model of quantum gravity known as causal dynamical triangulations. We have found that the two-form, tensor and dual scalar spectral dimensions exhibit a flow between two scales at which an effective dimension appears. However, the one-form and vector spectral dimensions show only a single effective dimension. The fact that the one-form and vector spectral dimension do not show a flow of the effective dimension can potentially be related to the absence of a dispersion relation for the electromagnetic field, but dynamically generated instead of as an assumption.

Robin boundary-value problem for the Beltrami equation

UDC 517.5 We investigate the unique solution of the Robin boundary-value problem for the Beltrami equation with constant coefficients in the unit disc by using a technique based on a singular integral operator defined on L p ( 𝔻 ) for all p &gt; 2.