Riesz potentials of fractal measures μ in metric spaces and their inverses are introduced . They define self–adjoint operators in the Hilbert space L2(μ) and the former are shown to be compact. In the Euclidean case the corresponding spectral asymptotics are derived with Besov space methods. The inverses of the Riesz potentials are fractal pseudodifferential operators. For the order two operato...

This paper proposes an unsupervised method for material classification of raw printed circuit boards (PCBs) based on the dimension-reduced spectral information. First, a spectral imaging system is constructed for acquiring precise spectral reflectance images. Only 3-dimensional spectral features are then extracted from the original 31-dimensional spectral image data by PCA for segmentation prop...

We extend the localization techniques of Bloch to simplicial spaces. As applications, we give an extension of Bloch’s localization theorem for the higher Chow groups to schemes of finite type over a regular scheme of dimension one (including mixed characteristic) and, relying on a fundamental result of Friedlander-Suslin, we globalize the Bloch-Lichtenbaum spectral sequence to give a spectral s...

A thermodynamical treatment of a massless scalar field (a photon) confined to a fractal spatial manifold leads to an equation of state relating pressure to internal energy, PV(s) = U/d(s), where d(s) is the spectral dimension and V(s) defines the "spectral volume." For regular manifolds, V(s) coincides with the usual geometric spatial volume, but on a fractal this is not necessarily the case. T...

Recent auditory physiological evidence points to a modulation frequency dimension in the auditory cortex. This dimension exists jointly with the tonotopic acoustic frequency dimension. Thus, audition can be considered as a relatively slowly-varying two-dimensional representation, the “modulation spectrum,” where the first dimension is the well-known acoustic frequency and the second dimension i...

Recent auditory physiological evidence points to a modulation frequency dimension in the auditory cortex. This dimension exists jointly with the tonotopic acoustic frequency dimension. Thus, audition can be considered as a relatively slowly-varying two-dimensional representation, the " modulation spectrum, " where the first dimension is the well-known acoustic frequency and the second dimension...

An imaging mass spectrometer is an analytical instrument that can determine the spatial distribution of chemical compounds on complex surfaces. The output of the device is a multi-spectral datacube; a three-dimensional (3D) dataset in which the xy-dimension represents the surface position and the z-dimension represents the mass spectral distribution. Analysts try to discover correlations in spe...

In this paper, by combining modular forms and characteristic forms, we obtain general anomaly cancellation formulas of any dimension. For .4kC 2/–dimensional manifolds, our results include the gravitational anomaly cancellation formulas of Alvarez-Gaumé and Witten in dimensions 2, 6 and 10 [1] as special cases. In dimension 4k C 1 , we derive anomaly cancellation formulas for index gerbes. In d...

In this paper we calculated the spectral dimension of loop quantum gravity (LQG) using the scaling property of the area operator spectrum on spin-network states and using the scaling property of the volume and length operators on Gaussian states. We obtained that the spectral dimension of the spatial section runs from 1.5 to 3, and under particular assumptions from 2 to 3 across a 1.5 phase whe...

With the number of channels in the hundreds instead of in the tens hyperspectral imagery possesses much richer spectral information than multispectral imagery .The increased dimensionality of such hyperspectral data provides a challenge to the current technique for analyzing data. Conventional classification methods may not be used without dimension reduction preprocessing. Supervised classific...