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Distribution (differential geometry)

About: Distribution (differential geometry) is a research topic. Over the lifetime, 911 publications have been published within this topic receiving 10149 citations.


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TL;DR: In this paper, the authors consider the Riemannian random wave model of Gaussian linear combinations of Laplace eigenfunctions on a general compact manifold and establish that the process properly rescaled and evaluated at an independently and uniformly chosen point X on the manifold, converges in distribution under the sole randomness of X towards an universal Gaussian field as the frequency tends to infinity.
Abstract: We consider the Riemannian random wave model of Gaussian linear combinations of Laplace eigenfunctions on a general compact Riemannian manifold. With probability one with respect to the Gaussian coefficients, we establish that, both for large band and monochro-matic models, the process properly rescaled and evaluated at an independently and uniformly chosen point X on the manifold, converges in distribution under the sole randomness of X towards an universal Gaussian field as the frequency tends to infinity. This result is reminiscent of Berry's conjecture and extends the celebrated central limit Theorem of Salem-Zygmund for trigonometric polynomials series to the more general framework of compact Riemannian manifolds. We then deduce from the above convergence the almost-sure asymptotics of the nodal volume associated with the random wave. To the best of our knowledge, these asymp-totics were only known in expectation and not in the almost sure sense due to the lack of sufficiently accurate variance estimates. This in particular addresses a question of S. Zelditch regarding the almost sure equidistribution of nodal lines.

6 citations

Patent
23 Sep 2005
TL;DR: A modular manifold assembly includes at least one barbed distribution connector disposed on a modular manifold body as discussed by the authors, which can be used to distribute fluid throughout a fluid distribution system, and can be assembled into various lengths as required to accommodate multiple different desired fluid distribution systems configurations.
Abstract: A modular manifold assembly includes at least one barbed distribution connector disposed on a modular manifold body. Individual modular manifold assemblies are assembled into various lengths as required to accommodate multiple different desired fluid distribution system configurations. The modular manifold body employs a tubular design with a D-shaped cross-section that allows ease of assembly. The at least one barbed distribution connector attaches to at least one distribution line to distribute fluid throughout a fluid distribution system.

6 citations

Posted Content
TL;DR: In this paper, the authors demonstrate how the local geometry of structures of nonholonomic nature, originated by Andrei Agrachev, works in the following two situations: rank 2 distributions of maximal class in R^n with non-zero generalized Wilczynski invariants and rank 2 distribution with additional structures such as affine control system with one input spanning these distributions, sub-pseudo Riemannian structures etc.
Abstract: We demonstrate how the novel approach to the local geometry of structures of nonholonomic nature, originated by Andrei Agrachev, works in the following two situations: rank 2 distributions of maximal class in R^n with non-zero generalized Wilczynski invariants and rank 2 distributions of maximal class in R^n with additional structures such as affine control system with one input spanning these distributions, sub-(pseudo)Riemannian structures etc. The common feature of these two situations is that each abnormal extremal (of the underlying rank 2 distribution) possesses a distinguished parametrization. This fact allows one to construct the canonical frame on a (2n-3)-dimensional bundle in both situations for arbitrary n greater than 4. The moduli spaces of the most symmetric models for both situations are described as well. The relation of our results to the divergence equivalence of Lagrangians of higher order is given

6 citations

Posted Content
TL;DR: A GAN-like method to transform a simple distribution to a data distribution in the latent space by solving only a minimization problem is proposed and results on the MNIST and the CelebA datasets validate the effectiveness of the proposed method.
Abstract: Variational Auto-Encoders enforce their learned intermediate latent-space data distribution to be a simple distribution, such as an isotropic Gaussian. However, this causes the posterior collapse problem and loses manifold structure which can be important for datasets such as facial images. A GAN can transform a simple distribution to a latent-space data distribution and thus preserve the manifold structure, but optimizing a GAN involves solving a Min-Max optimization problem, which is difficult and not well understood so far. Therefore, we propose a GAN-like method to transform a simple distribution to a data distribution in the latent space by solving only a minimization problem. This minimization problem comes from training a discriminator between a simple distribution and a latent-space data distribution. Then, we can explicitly formulate an Optimal Transport (OT) problem that computes the desired mapping between the two distributions. This means that we can transform a distribution without solving the difficult Min-Max optimization problem. Experimental results on an eight-Gaussian dataset show that the proposed OT can handle multi-cluster distributions. Results on the MNIST and the CelebA datasets validate the effectiveness of the proposed method.

6 citations

Patent
26 May 1992
TL;DR: A compact injection molding melt distribution manifold has integral or monolithic inlet and main portions (76,74) and both portions have a common electrical heating element (88) as mentioned in this paper.
Abstract: A compact injection molding melt distribution manifold (10) The manifold has integral or monolithic inlet and main portions (76,74) and both portions have a common electrical heating element (88) A melt passage (66) extends through the manifold from a common inlet (68) in the inlet portion (76) and branches in the main portion (74) to a number of spaced outlets leading to different gates (38)

6 citations


Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20221
202159
202067
201953
201843
201733