arXiv research

The paper proves rigidity theorems for area widths of Riemannian manifolds.

Estimates area and spectrum of stable minimal surfaces in Euclidean and hyperbolic spaces.

The geodesic length spectrum of a complete, finite volume, hyperbolic 3-orbifold M is a fundamental invariant of the topology of M via Mostow-Prasad Rigidity. Motivated by this, the second author and Reid defined a two-dimensional analogue of the geodesic length spectrum given by the multiset of isometry types of total…

Study estimates hypersurface areas in curved spaces, with applications to spectrum bounds.

We analyze the frequency spectrum of quantum neural networks using algebraic methods and prove maximality results.

Study counts minimal Lagrangians in hyperbolic surfaces with precise growth rate.

Quantum theory of curved tetrahedrons yields quantum group intertwiners.

In this paper, we prove the existence of the free boundary minimal hypersurface of least area in compact manifolds with boundary. Such hypersurface can be viewed as the ground state of the volume spectrum introduced by Gromov. Moreover, we characterize the orientation and Morse index of them.

Study geometric rigidity of surfaces in negative curvature manifolds.

Consider a smooth closed surface $M$ of fixed genus $\geqslant 2$ with a hyperbolic metric $σ$ of total area $A$. In this article, we study the behavior of geometric and dynamical characteristics (e.g., diameter, Laplace spectrum, Gaussian curvature and entropies) of nonpositively curved smooth metrics with total area …

Paper proves ellipticity of certain Reeb orbits and estimates ECH spectrum on lens spaces.

We describe the asymptotic behavior of minimal area submanifolds in product spacetimes of an asymptotically hyperbolic space times a compact internal manifold. In particular, we find that unlike the case of a minimal area submanifold just in an asymptotically hyperbolic space, the internal part of the boundary submanif…

This paper studies the problem of spectrum shortage in an unmanned aerial vehicle (UAV) network during critical missions such as wildfire monitoring, search and rescue, and disaster monitoring. Such applications involve a high demand for high-throughput data transmissions such as real-time video-, image-, and voice- st…

For a given bounded domain $Ω\subset {\Bbb R}^n$ with smooth boundary, we explicitly calculate the first two coefficients of the asymptotic expansion of the heat trace associated with the Stokes operator as $t\to 0^+$. These coefficients (i.e., heat invariants) provide precise information for the volume of the domain $…

The volume spectrum of fiber bundles is bounded by the product of the base's volume spectrum and the fiber's volume.

This article develops a statistical test for the null hypothesis of strict stationarity of a discrete time stochastic process in the frequency domain. When the null hypothesis is true, the second order cumulant spectrum is zero at all the discrete Fourier frequency pairs in the principal domain. The test uses a window …

For a bounded domain $Ω\subset {\Bbb R}^n$ with smooth boundary, we explicitly calculate the first two coefficients of the asymptotic expansion of the trace of the strongly continuous semigroup associated with the Navier-Lamé operator on $Ω$ as $t\to 0^+$. These coefficients (i.e., spectral invariants) provide precise …

We uniquely and explicitly reconstruct the instantaneous intrinsic metric of the Kerr-Newman Event Horizon from the spectrum of its Laplacian. In the process we find that the angular momentum parameter, radius, area; and in the uncharged case, mass, can be written in terms of these eigenvalues. In the uncharged case th…

Study calculates spectra of minimal hypersurfaces in hyperbolic space.

Study calculates geometric invariants from Navier-Lamé spectrum.

We apply topological methods to study the smallest non-zero number $λ_1$ in the spectrum of the Laplacian on finite area hyperbolic surfaces. For closed hyperbolic surfaces of genus two we show that the set $\{S \in {\mathcal{M}_2}: {λ_1}(S) > 1/4 \}$ is unbounded and disconnects the moduli space ${\mathcal{M}_2}$.

Kernel method is a very powerful tool in machine learning. The trick of kernel has been effectively and extensively applied in many areas of machine learning, such as support vector machine (SVM) and kernel principal component analysis (kernel PCA). Kernel trick is to define a kernel function which relies on the inner-…

In this paper, we explicitly construct large classes of incommensurable hyperbolic knot complements with the same volume and the same initial (complex) length spectrum. Furthermore, we show that these knot complements are the only knot complements in their respective commensurabiltiy classes by analyzing their cusp sha…

Study on Dirac operator spectrum on shrinking surfaces with cusps.

In this paper we examine the relationship between the length spectrum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M. In particular we analyze the extent to which the geometry of M is determined by the closed geodesics coming from finite area totally geodesic surfaces. Using a variety of tech…

We prove lower Dirac eigenvalue bounds for closed surfaces with a spin structure whose Arf invariant equals 1. Besides the area only one geometric quantity enters in these estimates, the spin-cut-diameter which depends on the choice of spin structure. It can be expressed in terms of various distances on the surfaces or…

Stable solutions found for a specific physics model.

Weak lensing maps contain information beyond two-point statistics on small scales. Much recent work has tried to extract this information through a range of different observables or via nonlinear transformations of the lensing field. Here we train and apply a 2D convolutional neural network to simulated noiseless lensi…

Large batch size training of Neural Networks has been shown to incur accuracy loss when trained with the current methods. The exact underlying reasons for this are still not completely understood. Here, we study large batch size training through the lens of the Hessian operator and robust optimization. In particular, w…

We analyze the cumulative distribution of total personal income of USA counties, and gross domestic product of Brazilian, German and United Kingdom counties, and also of world countries. We verify that generalized exponential distributions, related to nonextensive statistical mechanics, describe almost the whole spectr…

Sharp comparison theorems for 3D manifolds with scalar curvature bound.

Spectrum management and resource allocation (RA) problems are challenging and critical in a vast number of research areas such as wireless communications and computer networks. The traditional approaches for solving such problems usually consume time and memory, especially for large size problems. Recently different ma…

We study the SL(2,R)-infimal lengths of simple closed curves on half-translation surfaces. Our main result is a characterization of Veech surfaces in terms of these lengths. We also revisit the "no small virtual triangles" theorem of Smillie and Weiss and establish the following dichotomy: the virtual triangle area spe…

Recent developments link Steklov eigenvalues to manifold geometry.

Study on Dirac operator spectrum on hyperbolic surfaces with shrinking geodesics.

The spectroscopy measurement is one of main pathways for exploring and understanding the nature. Today, it seems that racing artificial intelligence will remould its styles. The algorithms contained in huge neural networks are capable of substituting many of expensive and complex components of spectrum instruments. In …

A theorem of J. Hersch (1970) states that for any smooth metric on $S^2$, with total area equal to $4π$, the first nonzero eigenvalue of the Laplace operator acting on functions is less than or equal to 2 (this being the value for the standard round metric). For metrics invariant under the standard $S^1$-action on $S^2…

In 1995 the author, Jones, and Segal introduced the notion of "Floer homotopy theory". The proposal was to attach a (stable) homotopy type to the geometric data given in a version of Floer homology. More to the point, the question was asked, "When is the Floer homology isomorphic to the (singular) homology of a natural…

Bird sounds possess distinctive spectral structure which may exhibit small shifts in spectrum depending on the bird species and environmental conditions. In this paper, we propose using convolutional recurrent neural networks on the task of automated bird audio detection in real-life environments. In the proposed metho…

We consider the Laplacian in a domain squeezed between two parallel hypersurfaces in Euclidean spaces of any dimension, subject to Dirichlet boundary conditions on one of the hypersurfaces and Neumann boundary conditions on the other. We derive two-term asymptotics for eigenvalues in the limit when the distance between…

We define a new spectrum for compact length spaces and Riemannian manifolds called the "covering spectrum" which roughly measures the size of the one dimensional holes in the space. More specifically, the covering spectrum is a set of real numbers $δ>0$ which identify the distinct $δ$ covers of the space. We investigat…

The subject of this paper is the relationship among the marked length spectrum, the length spectrum, the Laplace spectrum on functions, and the Laplace spectrum on forms on Riemannian nilmanifolds. In particular, we show that for a large class of three-step nilmanifolds, if a pair of nilmanifolds in this class has the …

The subject of this paper is the relationship among the marked length spectrum, the length spectrum, the Laplace spectrum on functions, and the Laplace spectrum on forms on Riemannian nilmanifolds. In particular, we show that for a large class of three-step nilmanifolds, if a pair of nilmanifolds in this class has the …

Study on null-torsion holomorphic curves in 6-sphere, focusing on their second variation.

Let $M$ be a compact connected manifold of dimension $n$ endowed with a conformal class $C$ of Riemannian metrics of volume one. For any integer $k\geq0$, we consider the conformal invariant $λ_k ^c (C)$ defined as the supremum of the $k$-th eigenvalue $λ_k (g)$ of the Laplace-Beltrami operator $Δ_g$, where $g$ runs ov…

Study the energy spectrum of metrics on surfaces and its relation to simple length spectrum.

Study on $k$-surfaces in negatively curved 3-manifolds, focusing on energy and entropy.

Study shows spectrum properties for specific Hadamard manifolds.