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Ricci decomposition

About: Ricci decomposition is a research topic. Over the lifetime, 1972 publications have been published within this topic receiving 45295 citations.


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TL;DR: In this paper, it was shown that a complete non-compact manifold with non-negative Ricci curvature has at most linear volume growth and if there exists a nonconstant harmonic function, f, of polynomial growth of any given degree q, then the manifold splits isometrically, M = N x R.
Abstract: Lower bounds on Ricci curvature limit the volumes of sets and the existence of harmonic functions on Riemannian manifolds. In 1975, Shing Tung Yau proved that a complete noncompact manifold with nonnegative Ricci curvature has no nonconstant harmonic functions of sublinear growth. In the same paper, Yau used this result to prove that a complete noncompact manifold with nonnegative Ricci curvature has at least linear volume growth. In this paper, we prove the following theorem concerning harmonic functions on these manifolds. Theorem: Let M be a complete noncompact manifold with nonnegative Ricci curvature and at most linear volume growth. If there exists a nonconstant harmonic function, f, of polynomial growth of any given degree q, then the manifold splits isometrically, M= N x R.

22 citations

Journal ArticleDOI
TL;DR: A constructive algorithm that decomposes an arbitrary real tensor into a finite sum of orthonormal rank-1 outer products and allows, for the first time, a complete characterization of all tensors orthogonal with the original tensor.
Abstract: We propose a constructive algorithm that decomposes an arbitrary real tensor into a finite sum of orthonormal rank-1 outer products. The algorithm, called TTr1SVD, works by converting the tensor into a tensor-train rank-1 (TTr1) series via the singular value decomposition (SVD). TTr1SVD naturally generalizes the SVD to the tensor regime with properties such as uniqueness for a fixed order of indices, orthogonal rank-1 outer product terms, and easy truncation error quantification. Using an outer product column table it also allows, for the first time, a complete characterization of all tensors orthogonal with the original tensor. Incidentally, this leads to a strikingly simple constructive proof showing that the maximum rank of a real $2 \times 2 \times 2$ tensor over the real field is 3. We also derive a conversion of the TTr1 decomposition into a Tucker decomposition with a sparse core tensor. Numerical examples illustrate each of the favorable properties of the TTr1 decomposition.

22 citations

Journal ArticleDOI
TL;DR: In this article, the Ricci curvature of a Riemannian metric was studied as a differential operator acting on the space of metrics close to the standard metric of a rank-one noncompact symmetric space.
Abstract: We study the Ricci curvature of a Riemannian metric as a differential operator acting on the space of metrics close (in a weighted functional spaces topology) to the standard metric of a rank-one noncompact symmetric space. We prove that any symmetric bilinear field close enough to the standard may be realized as the Ricci curvature of a unique close metric if its decay rate at infinity (its weight) belongs to some precisely known interval. We also study what happens if the decay rate is too small or too large.

22 citations

Journal ArticleDOI
TL;DR: In this paper, the authors derived a Moser's parabolic Harnack inequality for the heat equation, which leads to upper and lower Gaussian bounds on the heat kernel.
Abstract: We study some function-theoretic properties on a complete smooth metric measure space $(M,g,e^{-f}dv)$ with Bakry-Emery Ricci curvature bounded from below. We derive a Moser's parabolic Harnack inequality for the $f$-heat equation, which leads to upper and lower Gaussian bounds on the $f$-heat kernel. We also prove $L^p$-Liouville theorems in terms of the lower bound of Bakry-Emery Ricci curvature and the bound of function $f$, which generalize the classical Ricci curvature case and the $N$-Bakry-Emery Ricci curvature case.

22 citations

Journal ArticleDOI
TL;DR: In this paper, a tensors which form bases of irreducible representations of the rotation group are used to express weakly-textured polycrystal polycrystals as a linear combination of an orthonormal set of basis tensors.
Abstract: Material tensors pertaining to polycrystalline aggregates should manifest also the influence of crystallographic texture on the material properties in question. In this paper we make use of tensors which form bases of irreducible representations of the rotation group and prove a representation theorem by which a given material tensor of a weakly-textured polycrystal is expressed as a linear combination of an orthonormal set of irreducible basis tensors, with the components given explicitly in terms of texture coefficients and a set of undetermined material parameters. Once the irreducible basis tensors that appear in the formula are determined, the representation formula, which is valid for all texture and crystal symmetries, will delineate quantitatively the effect of crystallographic texture on the material tensor in question. We present an integral formula and an orthonormalization process which serve as the basis for a procedure to determine explicitly the irreducible basis tensors required in the representation formula. For applications we determine a set of irreducible basis tensors for the elasticity tensor and a set for fourth-order tensors that define constitutive equations in incompressible elasticity and Hill’s quadratic yield functions in plasticity. We show that orientation averaging of a tensor can be done easily if we have in hand a set of irreducible basis tensors for the decomposition of the tensor in question. As illustration we derive a formula, which is valid for all texture and crystal symmetries, for the elasticity tensor under the Voigt model.

22 citations

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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202326
202241
20211
20203
20192
201810