Curvature Measures and Random Sets, I

1984 ◽  
Vol 119 (1) ◽  
pp. 327-339 ◽  
Author(s):  
M. Zähle
1996 ◽  
Vol 28 (02) ◽  
pp. 335-336
Author(s):  
Kiên Kiêu ◽  
Marianne Mora

Random measures are commonly used to describe geometrical properties of random sets. Examples are given by the counting measure associated with a point process, and the curvature measures associated with a random set with a smooth boundary. We consider a random measure with an invariant distribution under the action of a standard transformation group (translatioris, rigid motions, translations along a given direction and so on). In the framework of the theory of invariant measure decomposition, the reduced moments of the random measure are obtained by decomposing the related moment measures.


1996 ◽  
Vol 28 (2) ◽  
pp. 335-336 ◽  
Author(s):  
Kiên Kiêu ◽  
Marianne Mora

Random measures are commonly used to describe geometrical properties of random sets. Examples are given by the counting measure associated with a point process, and the curvature measures associated with a random set with a smooth boundary. We consider a random measure with an invariant distribution under the action of a standard transformation group (translatioris, rigid motions, translations along a given direction and so on). In the framework of the theory of invariant measure decomposition, the reduced moments of the random measure are obtained by decomposing the related moment measures.


2020 ◽  
pp. 108128652097275
Author(s):  
Miroslav Šilhavý

The paper presents a coordinate-free analysis of deformation measures for shells modeled as 2D surfaces. These measures are represented by second-order tensors. As is well-known, two types are needed in general: the surface strain measure (deformations in tangential directions), and the bending strain measure (warping). Our approach first determines the 3D strain tensor E of a shear deformation of a 3D shell-like body and then linearizes E in two smallness parameters: the displacement and the distance of a point from the middle surface. The linearized expression is an affine function of the signed distance from the middle surface: the absolute term is the surface strain measure and the coefficient of the linear term is the bending strain measure. The main result of the paper determines these two tensors explicitly for general shear deformations and for the subcase of Kirchhoff-Love deformations. The derived surface strain measures are the classical ones: Naghdi’s surface strain measure generally and its well-known particular case for the Kirchhoff-Love deformations. With the bending strain measures comes a surprise: they are different from the traditional ones. For shear deformations our analysis provides a new tensor [Formula: see text], which is different from the widely used Naghdi’s bending strain tensor [Formula: see text]. In the particular case of Kirchhoff–Love deformations, the tensor [Formula: see text] reduces to a tensor [Formula: see text] introduced earlier by Anicic and Léger (Formulation bidimensionnelle exacte du modéle de coque 3D de Kirchhoff–Love. C R Acad Sci Paris I 1999; 329: 741–746). Again, [Formula: see text] is different from Koiter’s bending strain tensor [Formula: see text] (frequently used in this context). AMS 2010 classification: 74B99


Author(s):  
Aniket Bhattacharya ◽  
Vineet Jha ◽  
Khushboo Singhal ◽  
Mahar Fatima ◽  
Dayanidhi Singh ◽  
...  

Abstract Alu repeats contribute to phylogenetic novelties in conserved regulatory networks in primates. Our study highlights how exonized Alus could nucleate large-scale mRNA-miRNA interactions. Using a functional genomics approach, we characterize a transcript isoform of an orphan gene, CYP20A1 (CYP20A1_Alu-LT) that has exonization of 23 Alus in its 3’UTR. CYP20A1_Alu-LT, confirmed by 3’RACE, is an outlier in length (9 kb 3’UTR) and widely expressed. Using publically available datasets, we demonstrate its expression in higher primates and presence in single nucleus RNA-seq of 15928 human cortical neurons. miRanda predicts ∼4700 miRNA recognition elements (MREs) for ∼1000 miRNAs, primarily originated within these 3’UTR-Alus. CYP20A1_Alu-LT could be a potential multi-miRNA sponge as it harbors ≥10 MREs for 140 miRNAs and has cytosolic localization. We further tested whether expression of CYP20A1_Alu-LT correlates with mRNAs harboring similar MRE targets. RNA-seq with conjoint miRNA-seq analysis was done in primary human neurons where we observed CYP20A1_Alu-LT to be downregulated during heat shock response and upregulated in HIV1-Tat treatment. 380 genes were positively correlated with its expression (significantly downregulated in heat shock and upregulated in Tat) and they harbored MREs for nine expressed miRNAs which were also enriched in CYP20A1_Alu-LT. MREs were significantly enriched in these 380 genes compared to random sets of differentially expressed genes (p = 8.134e-12). Gene ontology suggested involvement of these genes in neuronal development and hemostasis pathways thus proposing a novel component of Alu-miRNA mediated transcriptional modulation that could govern specific physiological outcomes in higher primates.


Author(s):  
Andreas Bernig ◽  
Dmitry Faifman ◽  
Gil Solanes

AbstractThe recently introduced Lipschitz–Killing curvature measures on pseudo-Riemannian manifolds satisfy a Weyl principle, i.e. are invariant under isometric embeddings. We show that they are uniquely characterized by this property. We apply this characterization to prove a Künneth-type formula for Lipschitz–Killing curvature measures, and to classify the invariant generalized valuations and curvature measures on all isotropic pseudo-Riemannian space forms.


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