arXiv research

A major issue in harmonic analysis is to capture the phase dependence of frequency representations, which carries important signal properties. It seems that convolutional neural networks have found a way. Over time-series and images, convolutional networks often learn a first layer of filters which are well localized i…

The covariance of a stationary process $X$ is diagonalized by a Fourier transform. It does not take into account the complex Fourier phase and defines Gaussian maximum entropy models. We introduce a general family of phase harmonic covariance moments, which rely on complex phases to capture non-Gaussian properties. The…

Study finds weak solutions for complex map flows with optimal lifespan.

A new model for generating point processes with complex geometries.

The paper analyzes MACD using operator theory.

Motivated by the need for accurate frequency information, a novel algorithm for estimating the fundamental frequency and its rate of change in three-phase power systems is developed. This is achieved through two stages of Kalman filtering. In the first stage a quaternion extended Kalman filter, which provides a unified…

Study harmonic surfaces in 3D space, proving superposition principle.

Symmetry operators of twistor spinors and harmonic spinors can be constructed from conformal Killing-Yano forms. Transformation operators relating twistors to harmonic spinors are found in terms of potential forms. These constructions are generalized to gauged twistor spinors and gauged harmonic spinors. The operators …

We prove that harmonic morphisms preserve the Jacobi operator along harmonic maps. We apply this result to prove infinitesimal and local rigidity (in the sense of Toth) of harmonic morphisms to a sphere.

In this paper we explore the functional correlation approach to operational risk. We consider networks with heterogeneous a-priori conditional and unconditional failure probability. In the limit of sparse connectivity, self-consistent expressions for the dynamical evolution of order parameters are obtained. Under equil…

Paper studies Laplace operator estimates in harmonic map heat flows.

Study invariant operators and vanishing theorems in CR geometry.

Study differential operators on non-compact harmonic manifolds, finding conditions for radial fundamental solutions and dense heat-semigroups.

New tan-concavity property for Lagrangian phase operators helps in studying dHYM metrics.

Revisits Gaussian process model with spherical harmonics for scalable deep learning.

Study left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.

We establish dimension-independent estimates related to heat operators e^{tL} on manifolds. We first develop a very general contractivity result for Markov kernels which can be applied to diffusion semigroups. Second, we develop estimates on the norm behavior of harmonic and non-negative subharmonic functions. We apply…

Harmonic gauge simplifies geometric analysis of Riemannian metrics.

New findings on compact manifolds with specific curvature properties.

We first present the natural definitions of the horizontal differential, the divergence (as an adjoint operator), and a $p$-harmonic form on a Finsler manifold. Next, we prove a Hodge-type theorem for a Finsler manifold in the sense that a horizontal $p$-form is harmonic if and only if the horizontal Laplacian vanishes…

We consider several differential operators on compact almost-complex, almost-Hermitian and almost-Kähler manifolds. We discuss Hodge Theory for these operators and a possible cohomological interpretation. We compare the associated spaces of harmonic forms and cohomologies with the classical de Rham, Dolbeault, Bott-Che…

Using geometric quantization procedure, the quantization of algebra of observables for physical system with Ricci-flat phase space is obtained. In the classical case the appointed physical system is reduced to harmonic oscillator when the one real parameter is vanished.

Scattering theory for harmonic one-forms on Riemann surfaces.

We use the density function of a harmonic space to obtain estimates for the eigenvalues of the Jacobi operator; when these estimates are sharp, then the harmonic space is a symmetric Osserman space.

The study proves conditions for complete Riemannian manifolds to be Einstein.

The study refines stability results for Yang-Mills fields and harmonic maps.

Discretizes diffusions and harmonic functions on covering spaces.

Estimates for harmonic forms on a 3-Torus, proving their existence.

Study $L^{2}$-harmonic forms on almost Kähler manifolds, extending vanishing theorems.

Study on harmonic spinors on specific Lie groups.

DPI quantifies phase differences in 1D and multidimensional signals using Riesz transform.

Harmonic morphisms and p-harmonic functions constructed on symmetric spaces.

Study on metrics maximizing eigenvalues of Paneitz operator on 4-manifolds.

This paper studies the space of $L ^2$ harmonic forms and $L ^2$ harmonic spinors on Taub-bolt, a Ricci-flat Riemannian 4-manifold of ALF type. We prove that the space of harmonic square-integrable 2-forms on Taub-bolt is 2-dimensional and construct a basis. We explicitly find a 2-parameter family of $L ^2$ zero mod…

Constructs explicit harmonic functions and morphisms on complex and quaternionic Grassmannians.

Develops calculus for QFB metrics, proving Fredholm properties and decay of harmonic forms.

Quantifies uniqueness of conformal-harmonic maps on 4-manifolds.

New approach to heat flow for half-harmonic maps, related to minimal surfaces.

We develop the notion of renormalized energy in CR geometry, for maps from a strictly pseudoconvex pseudohermitian manifold to a Riemannian manifold. This energy is a CR invariant functional, whose critical points, which we call CR-harmonic maps, satisfy a CR covariant subelliptic partial differential equation. The cor…

The paper constructs new non-trivial harmonic maps into higher-dimensional target manifolds.

Study fourth-order geometric flow of shape operator for co-dimension one immersions.

The paper focuses on various properties and applications of the homotopy operator, which occurs in the Poincaré lemma. In the first part, an abstract operator calculus is constructed, where the exterior derivative is an abstract derivative and the homotopy operator plays the role of an abstract integral. This operator …

We show that for a suitable class of ``Dirac-like'' operators there holds a Gluing Theorem for connected sums. More precisely, if $M_1$ and $M_2$ are closed Riemannian manifolds of dimension $n\ge 3$ together with such operators, then the connected sum $M_1 # M_2$ can be given a Riemannian metric such that the spectrum…

The concept of harmonic metallic structure on a metallic pseudo-Riemannian manifold is introduced. In the case of compact manifolds we prove that harmonicity of a metallic structure $J$, with $J^2=pJ+qI$ and $p^2+4q\neq 0$, is equivalent to $dJ=0$. Conditions for a harmonic metallic structure to be preserved by harmoni…

Geometric phases describe how in a continuous-time dynamical system the displacement of a variable (called phase variable) can be related to other variables (shape variables) undergoing a cyclic motion, according to an area rule. The aim of this paper is to show that geometric phases can exist also for discrete-time sy…

Study on critical Lagrangian phase singularities in mean curvature flow.

Sharp Veronese rigidity theorem for submanifolds of unit ball.

This paper explains a mechanism called phase collapse that improves image classification accuracy.