Daviau showed the equivalence of matrix Dirac theory, formulated within a spinor bundle (S_x C _x^4 ), to a Clifford algebraic formulation within space Clifford algebra (C ( R ^3 , ) M _ 2 ( C ) P ) Pauli algebra (matrices) ≃ ℍ ⨁ ℍ ≃ biquaternions. We will show, that Daviau's map θ: ( : C ^4 M _ 2 ( C ) ) is an isomorphism. It is shown that Hestenes' and Parra's formulations are equivalent to Daviau's Clifford algebra formulation, which uses outer automorphisms. The connection between different formulations is quite remarkable, since it connects the left and right action on the Pauli algebra itself viewed as a bi-module with the left (resp. right) action of the enveloping algebra (P^ P P^T on P ). The isomorphism established in this article and given by Daviau's map does clearly show that right and left actions are of similar type. This should be compared with attempts of Hestenes, Daviau, and others to interprete the right action as the iso-spin freedom.
A further genuine and important approach to the spinor-tensor transition was developed starting probably with Crawford by P. Lounesto, @cite_6 and references there. He investigated the question, how a spinor field can be reconstructed from known tensor densities. The major characterization is derived, using Fierz-Kofink identities, from elements called Boomerangs --because they are able to come back to the spinorial picture. Lounesto's result is a characterization of spinors based on multi-vector relations which unveils a new unknown type of spinor.
@cite_6
2198155329 2022437913 2073656615 1009009209
Patch-based low-rank models have shown effective in exploiting spatial redundancy of natural images especially for the application of image denoising. However, two-dimensional low-rank model can not fully exploit the spatio-temporal correlation in larger data sets such as multispectral images and 3D MRIs. In this work, we propose a novel low-rank tensor approximation framework with Laplacian Scale Mixture (LSM) modeling for multi-frame image denoising. First, similar 3D patches are grouped to form a tensor of d-order and high-order Singular Value Decomposition (HOSVD) is applied to the grouped tensor. Then the task of multiframe image denoising is formulated as a Maximum A Posterior (MAP) estimation problem with the LSM prior for tensor coefficients. Both unknown sparse coefficients and hidden LSM parameters can be efficiently estimated by the method of alternating optimization. Specifically, we have derived closed-form solutions for both subproblems. Experimental results on spectral and dynamic MRI images show that the proposed algorithm can better preserve the sharpness of important image structures and outperform several existing state-of-the-art multiframe denoising methods (e.g., BM4D and tensor dictionary learning). The symmetric tensor spherical harmonics (STSH’s) on the N‐sphere (SN), which are defined as the totally symmetric, traceless, and divergence‐free tensor eigenfunctions of the Laplace–Beltrami (LB) operator on SN, are studied. Specifically, their construction is shown recursively starting from the lower‐dimensional ones. The symmetric traceless tensors induced by STSH’s are introduced. These play a crucial role in the recursive construction of STSH’s. The normalization factors for STSH’s are determined by using their transformation properties under SO(N+1). Then the symmetric, traceless, and divergence‐free tensor eigenfunctions of the LB operator in the N‐dimensional de Sitter space‐time which are obtained by the analytic continuation of the STSH’s on SN are studied. Specifically, the allowed eigenvalues of the LB operator under the restriction of unitarity are determined. Our analysis gives a group‐theoretical explanation of the forbidden mass range observed earlier for the spin‐2 field theory in de Sit... In the present paper we analyze a class of tensor-structured preconditioners for the multidimensional second-order elliptic operators in ℝ d , d≥2. For equations in a bounded domain, the construction is based on the rank-R tensor-product approximation of the elliptic resolvent ℬ R ≈(ℒ−λ I)−1, where ℒ is the sum of univariate elliptic operators. We prove the explicit estimate on the tensor rank R that ensures the spectral equivalence. For equations in an unbounded domain, one can utilize the tensor-structured approximation of Green’s kernel for the shifted Laplacian in ℝ d , which is well developed in the case of nonoscillatory potentials. For the oscillating kernels e −i κ‖x‖ ‖x‖, x∈ℝ d , κ∈ℝ+, we give constructive proof of the rank-O(κ) separable approximation. This leads to the tensor representation for the discretized 3D Helmholtz kernel on an n×n×n grid that requires only O(κ |log e|2 n) reals for storage. Such representations can be applied to both the 3D volume and boundary calculations with sublinear cost O(n 2), even in the case κ=O(n). Abstract After a very brief recollection of how my scientific collaboration with Ugo started, in this talk I will present some recent results obtained with localization: the deformed gauge theory partition function Z ( τ → | q ) and the expectation value of circular Wilson loops W on a squashed four-sphere will be computed. The partition function is deformed by turning on τ J tr Φ J interactions with Φ the N = 2 superfield. For the N = 4 theory SUSY gauge theory exact formulae for Z and W in terms of an underlying U ( N ) interacting matrix model can be derived thus replacing the free Gaussian model describing the undeformed N = 4 theory. These results will be then compared with those obtained with the dual CFT according to the AGT correspondence. The interactions introduced previously are in fact related to the insertions of commuting integrals of motion in the four-point CFT correlator and the chiral correlators are expressed as τ -derivatives of the gauge theory partition function on a finite Ω -background.