Beltrami Operator Research Articles

Beltrami fields exhibit knots and chaos almost surely

Abstract In this paper, we show that, with probability $1$ , a random Beltrami field exhibits chaotic regions that coexist with invariant tori of complicated topologies. The motivation to consider this question, which arises in the study of stationary Euler flows in dimension 3, is V.I. Arnold’s 1965 speculation that a typical Beltrami field exhibits the same complexity as the restriction to an energy hypersurface of a generic Hamiltonian system with two degrees of freedom. The proof hinges on the obtention of asymptotic bounds for the number of horseshoes, zeros and knotted invariant tori and periodic trajectories that a Gaussian random Beltrami field exhibits, which we obtain through a nontrivial extension of the Nazarov–Sodin theory for Gaussian random monochromatic waves and the application of different tools from the theory of dynamical systems, including Kolmogorov–Arnold–Moser (KAM) theory, Melnikov analysis and hyperbolicity. Our results hold both in the case of Beltrami fields on ${\mathbb {R}}^3$ and of high-frequency Beltrami fields on the 3-torus.

Contact structures and Beltrami fields on the torus and the sphere

Contact structures and Beltrami fields on the torus and the sphere

Neumann boundary value problem for the Beltrami equation in a ring domain

Neumann boundary value problem for the Beltrami equation in a ring domain

The Dirichlet problem in an unbounded cone-like domain for second order elliptic quasilinear equations with variable nonlinearity exponent

In this paper we consider the Dirichlet problem for quasi-linear second-order elliptic equation with the m ( x ) -Laplacian and the strong nonlinearity on the right side in an unbounded cone-like domain. We study the behavior of weak solutions to the problem at infinity and we find the sharp exponent of the solution decreasing rate. We show that the exponent is related to the least eigenvalue of the eigenvalue problem for the Laplace–Beltrami operator on the unit sphere.

Oscillating singular integral operators on compact Lie groups revisited

Fefferman (Acta Math 24:9–36, 1970, Theorem 2') has proved the weak (1,1) boundedness for a class of oscillating singular integrals that includes the oscillating spectral multipliers of the Euclidean Laplacian Delta , namely, operators of the form 0.2Tθ(-Δ):=(1-Δ)-nθ4ei(1-Δ)θ2,0≤θ<1.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} T_{\ heta }(-\\Delta ):= (1-\\Delta )^{-\\frac{n\ heta }{4}}e^{i (1-\\Delta )^{\\frac{\ heta }{2}}},\\,0\\le \ heta <1. \\end{aligned}$$\\end{document}The aim of this work is to extend Fefferman’s result to oscillating singular integrals on any arbitrary compact Lie group. We also consider applications to oscillating spectral multipliers of the Laplace–Beltrami operator. The proof of our main theorem illustrates the delicate relationship between the condition on the kernel of the operator, its Fourier transform (defined in terms of the representation theory of the group) and the microlocal/geometric properties of the group.

On the Generalised Transfer Operators of the Farey Map with Complex Temperature

We consider the problem of showing that 1 is an eigenvalue for a family of generalised transfer operators of the Farey map. This is an important problem in the thermodynamic formalism approach to dynamical systems, which in this particular case is related to the spectral theory of the modular surface via the Selberg Zeta function and the theory of dynamical zeta functions of maps. After briefly recalling these connections, we show that the problem can be formulated for operators on an appropriate Hilbert space and translated into a linear algebra problem for infinite matrices. This formulation gives a new way to study numerically the spectrum of the Laplace–Beltrami operator and the properties of the Selberg Zeta function for the modular surface.

Homogenization of iterated singular integrals with applications to random quasiconformal maps

We study homogenization of iterated randomized singular integrals and homeomorphic solutions to the Beltrami differential equation with a random Beltrami coefficient. More precisely, let (F_j)_{j \geq 1} be a sequence of normalized homeomorphic solutions to the planar Beltrami equation \partial_{\overline z} F_j (z)=\mu_j(z,\omega) \partial_{z} F_j(z) , where the random dilatation satisfies |\mu_j|\leq k&lt;1 and has locally periodic statistics, for example of the type \mu_j (z,\omega)=\phi(z) \sum_{n\in\mathbb{Z}^2} g(2^j z-n,X_{n}(\omega)), where g(z,\omega) decays rapidly in z , the random variables X_{n} are i.i.d., and \phi \in C^\infty_0 . We establish the almost sure and local uniform convergence as j \to \infty of the maps F_j to a deterministic quasiconformal limit F_\infty . This result is obtained as an application of our main theorem, which deals with homogenization of iterated randomized singular integrals. As a special case of our theorem, let T_1,\ldots , T_{m} be translation and dilation invariant singular integrals on \mathbb{R}^d , and consider a d -dimensional version of \mu_j , e.g., as defined above or within a more general setting, see Definition 3.4 below. We then prove that there is a deterministic function f such that almost surely, \mu_j T_{m}\mu_j\ldots T_1\mu_j\to f as j\to\infty weakly in L^p , for 1 &lt; p &lt; \infty .

On the Existence of Solutions of Quasilinear Beltrami Equations with Two Characteristics

On the Existence of Solutions of Quasilinear Beltrami Equations with Two Characteristics

Jacobi partial waves for a set of 3D noncentral rational scatterers

The common tool of choice for basis expansions for the scattering problem with 3D quantum systems remains the spherical harmonics as eigenfunctions of the Laplace–Beltrami operator on the sphere, with approximations for deviations made around the usually dominant s-wave spherically symmetric state. However, with the growing number of technologically accessible nonspherically symmetric geometries of cold atomic and molecular systems, there is a need to explore as orthonormal bases for partial wave analysis the larger class of weighted Jacobi polynomials, subsuming the spherical harmonics. In particular, the lowest angular state for this bigger class is a toroid instead of a spherical s-orbital. This allows analytic treatment of a wider array of rational angular-dependent potentials which can describe rings and systems with topological constraints such as monopoles. Here, we present exact analytic solutions for the quantum scattering problem by partial wave analysis using the weighted Jacobi polynomials as an expanded basis. We obtain the scattering amplitude, differential and total cross-sections for exactly solvable 3D potentials included in the Smorodinsky-Winternitz noncentral systems with dynamical symmetries. Moreover, this procedure also solves the quantum scattering problem from a novel series of rational trigonometric forms of anisotropic potentials including double ring-shaped configurations.

BMO and Dirichlet Problem for Degenerate Beltrami Equation

Following Bojarski and Vekua, we have studied the Dirichlet problem \( \underset{z\to \zeta }{\lim}\operatorname{Re}\ f(z)=\varphi \left(\zeta \right) \) as z → ζ, z ∈ D, ζ ∈ ∂D, with continuous boundary data φ(ζ) in bounded domains D of the complex plane ℂ, where f satisfies the degenerate Beltrami equation \( {f}_{\overline{z}}=\mu (z){f}_z,\left|\mu (z)\right|<1 \), a.e. in D. Assuming that D is an arbitrary simply connected domain, we have established, in terms of μ, the BMO and FMO criteria, as well as a number of other integral criteria, on the existence and representation of regular discrete open solutions to the stated above problem. We have also proven similar theorems on the existence of multivalued solutions to the problem with single-valued real parts in an arbitrary bounded domain D with no boundary component degenerated to a single point. Finally, we have given a similar solvability and representation results concerning the Dirichlet problem in such domains for the degenerate A-harmonic equation associated with the Beltrami equation.

Fast simulation for Gaussian random fields on compact Riemannian manifolds

We develop a fully discrete approximation scheme to fast simulate centered Gaussian random fields (GRFs) over compact Riemannian manifolds. These GRFs can be either anisotropic or isotropic ones. The scheme is proposed by the truncation of a series expansion depending solely on the decay of the angular power spectrum, which is independent of the chosen space and time discretization. The pth moment convergence for the scheme is proved and the error bound with its associated rate of convergence is determined. Three simulation case studies are conducted by applying the Laplacian operator to find the required surface harmonics for a torus endowed with a Riemannian metric and by applying the ellipsoidal Beltrami and Laplacian operators to find the required surface ellipsoidal harmonics respectively for anisotropic and isotropic GRFs over ellipsoids endowed with two different Riemannian metrics.

Harmonic Beltrami Signature: A Novel 2D Shape Representation for Object Classification

There is a growing interest in shape analysis in recent years. We present a novel shape signature for 2D bounded simply-connected domains, named the Harmonic Beltrami signature (HBS). The proposed signature is based on the harmonic extension of the conformal welding map of a unit circle and its Beltrami coefficient. We show that there is a one-to-one correspondence between the quotient space of HBS and the space of 2D simply-connected shapes up to a translation, rotation and scaling. With a suitable normalization, each equivalence class in the quotient space of HBS is associated to a unique representative. It gets rid of the conformal ambiguity. As such, each shape is associated to a unique HBS. Conversely, the associated shape of a HBS can be reconstructed based on quasiconformal Teichmuller theories, which is uniquely determined up to a translation, rotation and scaling. The HBS is thus an effective fingerprint to represent a 2D shape. The robustness of HBS is studied both theoretically and experimentally. With the HBS, simple metric, such as L2, can be used to measure geometric dissimilarity between shapes. Experiments have been carried out to classify shapes in different classes using HBS. Results show good classification performance, which demonstrate the efficacy of our proposed shape signature.

NOTES ON CONFORMAL PERTURBATION OF HEAT KERNELS

Let [Formula: see text] be a smooth n-dimensional Riemannian manifold for [Formula: see text]. Consider the conformal perturbation [Formula: see text] where [Formula: see text] is a smooth bounded positive function on [Formula: see text]. Denote by [Formula: see text] the Laplace–Beltrami operator of manifold [Formula: see text]. In this paper, we derive the upper bounds of the heat kernels for [Formula: see text] with [Formula: see text]. Moreover, we also investigate the gradient estimates of the heat kernel for [Formula: see text].

Quadratic Sparse Domination and Weighted Estimates for Non-integral Square Functions

We prove a quadratic sparse domination result for general non-integral square functions S. That is, for p_0 in [1,2) and q_0 in (2,infty ], we prove an estimate of the form where q_{0}^{*} is the Hölder conjugate of q_{0}/2, M is the underlying doubling space and {mathcal {S}} is a sparse collection of cubes on M. Our result will cover both square functions associated with divergence form elliptic operators and those associated with the Laplace–Beltrami operator. This sparse domination allows us to derive optimal norm estimates in the weighted space L^{p}(w).

Computability and Beltrami fields in Euclidean space

In this article, we pursue our investigation of the connections between the theory of computation and hydrodynamics. We prove the existence of stationary solutions of the Euler equations in Euclidean space, of Beltrami type, that can simulate a universal Turing machine. In particular, these solutions possess undecidable trajectories. Heretofore, the known Turing complete constructions of steady Euler flows in dimension 3 or higher were not associated to a prescribed metric. Our solutions do not have finite energy, and their construction makes crucial use of the non-compactness of R3, however they can be employed to show that an arbitrary tape-bounded Turing machine can be robustly simulated by a Beltrami flow on T3 (with the standard flat metric). This shows that there exist steady solutions to the Euler equations on the flat torus exhibiting dynamical phenomena of (robust) computational complexity as high as desired. We also quantify the energetic cost for a Beltrami field on T3 to simulate a tape-bounded Turing machine, thus providing additional support for the space-bounded Church-Turing thesis. Another implication of our construction is that a Gaussian random Beltrami field on Euclidean space exhibits arbitrarily high computational complexity with probability 1. Finally, our proof also yields Turing complete flows and diffeomorphisms on S2 with zero topological entropy, thus disclosing a certain degree of independence within different hierarchies of complexity.

Laplacian and Brownian motion on positive definite matrices, revisited

We present explicit expression for the Laplace–Beltrami operator on the symmetric space of positive definite matrices by using the Iwasawa decomposition and study the corresponding diffusion process, the Brownian motion. We show that some inductive arguments in the dimensions work.

Alignment of highly resolved time-dependent experimental and simulated crash test data

We investigate for car and component crash tests the comparison of highly resolved experimental data with corresponding simulation data. Due to recent advances for optical measurement systems, one can nowadays obtain surface measurement data from a real crash experiment with high resolution in space and time. These advances call for new data processing methods that allow an alignment of this experimental data with numerical simulation results. We propose an approach based on a data representation stemming from a discrete Laplace–Beltrami operator, which allows such an alignment as well as a joint visual comparative analysis of both data sources. The method enables the identification of the best corresponding simulation among several numerical results, which also allows inferring physical quantities that cannot be measured in experiments. We evaluate the procedure on synthetic and real experimental data from two different crashworthiness setups.

Numerical approximation of the square phase-field crystal dynamics on the three-dimensional objects

In the natural world, the phase field transition on the surface of a three-dimensional (3D) object is common. The square phase-field crystal (SPFC) equation is an effective model for describing the phase transition from liquid to solid with lattice structure. This study aims to present an efficient and practical computational scheme to simulate the SPFC dynamics on arbitrary surfaces. The surface is represented by the zero level-set of a singed distance function. The closest point method and appropriate pseudo-Neumann boundary condition are used to make the numerical solution to be constant along the normal direction to the arbitrary surface. In this sense, the calculations are transformed into a 3D discrete narrow computational band domain embedding the arbitrary surface and the finite difference method (FDM) is adopted to discretize the governing equation in 3D space. To maintain second-order spatial accuracy, the standard seven-point discrete Laplacian operator is applied to replace the discrete surface Laplace–Beltrami operator. We treat the nonlinear term in a totally explicit form and utilize the stabilization technique to suppress the stiffness of the numerical scheme. Therefore, we avoid solving any nonlinear equation or linear equation with variable coefficients. In each time step, we only need to update the linear system with constant coefficients. The proposed numerical scheme is highly efficient and its implementation is simple. Various numerical experiments demonstrate that the proposed numerical scheme not only achieves the expected accuracy but also works well for the simulations of phase transition and crystallization on surfaces with smooth and sharp shapes.

Product formulas and convolutions for Laplace–Beltrami operators on product spaces: beyond the trivial case

Product formulas and convolutions for Laplace–Beltrami operators on product spaces: beyond the trivial case

Generalized Helical Hypersurfaces Having Time-like Axis in Minkowski Spacetime

In this paper, the generalized helical hypersurfaces x=x(u,v,w) with a time-like axis in Minkowski spacetime E14 are considered. The first and the second fundamental form matrices, the Gauss map, and the shape operator matrix of x are calculated. Moreover, the curvatures of the generalized helical hypersurface x are obtained by using the Cayley–Hamilton theorem. The umbilical conditions for the curvatures of x are given. Finally, the Laplace–Beltrami operator of the generalized helical hypersurface with a time-like axis is presented in E14.