scholarly journals On Random Disc Polygons in Smooth Convex Discs

2014 ◽  
Vol 46 (04) ◽  
pp. 899-918 ◽  
Author(s):  
F. Fodor ◽  
P. Kevei ◽  
V. Vígh
Keyword(s):  
Smooth Convex ◽  
Convex Disc ◽  
Asymptotic Formulae ◽  
Planar Convex ◽  
Area And Perimeter ◽  
Spindle Convexity

In this paper we generalize some of the classical results of Rényi and Sulanke (1963), (1964) in the context of spindle convexity. A planar convex discSis spindle convex if it is the intersection of congruent closed circular discs. The intersection of finitely many congruent closed circular discs is called a disc polygon. We prove asymptotic formulae for the expectation of the number of vertices, missed area, and perimeter difference of uniform random disc polygons contained in a sufficiently smooth spindle convex disc.

scholarly journals On Random Disc Polygons in Smooth Convex Discs

2014 ◽  
Vol 46 (4) ◽  
pp. 899-918 ◽  
Author(s):  
F. Fodor ◽  
P. Kevei ◽  
V. Vígh
Keyword(s):  
Smooth Convex ◽  
Convex Disc ◽  
Asymptotic Formulae ◽  
Planar Convex ◽  
Area And Perimeter ◽  
Spindle Convexity

In this paper we generalize some of the classical results of Rényi and Sulanke (1963), (1964) in the context of spindle convexity. A planar convex disc S is spindle convex if it is the intersection of congruent closed circular discs. The intersection of finitely many congruent closed circular discs is called a disc polygon. We prove asymptotic formulae for the expectation of the number of vertices, missed area, and perimeter difference of uniform random disc polygons contained in a sufficiently smooth spindle convex disc.

scholarly journals Variance estimates for random disc-polygons in smooth convex discs

2018 ◽  
Vol 55 (4) ◽  
pp. 1143-1157 ◽  
Author(s):  
Ferenc Fodor ◽  
Viktor Vígh
Keyword(s):  
Probability Model ◽  
Upper Bounds ◽  
Smooth Convex ◽  
Convex Disc ◽  
Variance Estimates

Abstract In this paper we prove asymptotic upper bounds on the variance of the number of vertices and the missed area of inscribed random disc-polygons in smooth convex discs whose boundary is C+2. We also consider a circumscribed variant of this probability model in which the convex disc is approximated by the intersection of random circles.

scholarly journals All-Trans Retinoic Acid Increases DRP1 Levels and Promotes Mitochondrial Fission

Cells ◽  
2021 ◽  
Vol 10 (5) ◽  
pp. 1202
Author(s):  
Bojjibabu Chidipi ◽  
Syed Islamuddin Shah ◽  
Michelle Reiser ◽  
Manasa Kanithi ◽  
Amanda Garces ◽  
...  
Keyword(s):  
Retinoic Acid ◽  
Mitochondrial Fission ◽  
Normal Function ◽  
Mitochondrial Network ◽  
Protein Levels ◽  
Trans Retinoic Acid ◽  
All Trans Retinoic Acid ◽  
Area And Perimeter

In the heart, mitochondrial homeostasis is critical for sustaining normal function and optimal responses to metabolic and environmental stressors. Mitochondrial fusion and fission are thought to be necessary for maintaining a robust population of mitochondria, and disruptions in mitochondrial fission and/or fusion can lead to cellular dysfunction. The dynamin-related protein (DRP1) is an important mediator of mitochondrial fission. In this study, we investigated the direct effects of the micronutrient retinoid all-trans retinoic acid (ATRA) on the mitochondrial structure in vivo and in vitro using Western blot, confocal, and transmission electron microscopy, as well as mitochondrial network quantification using stochastic modeling. Our results showed that ATRA increases DRP1 protein levels, increases the localization of DRP1 to mitochondria in isolated mitochondrial preparations. Our results also suggested that ATRA remodels the mitochondrial ultrastructure where the mitochondrial area and perimeter were decreased and the circularity was increased. Microscopically, mitochondrial network remodeling is driven by an increased rate of fission over fusion events in ATRA, as suggested by our numerical modeling. In conclusion, ATRA results in a pharmacologically mediated increase in the DRP1 protein. It also results in the modulation of cardiac mitochondria by promoting fission events, altering the mitochondrial network, and modifying the ultrastructure of mitochondria in the heart.

Asymptotic behavior of repeated eigenvalues of perturbed self-adjoint elliptic operators

2021 ◽  
pp. 1-16
Author(s):  
Alexander Dabrowski
Keyword(s):  
General Type ◽  
Differential Operators ◽  
Laplacian Operator ◽  
Variational Characterization ◽  
Repeated Eigenvalues ◽  
Direct Extension ◽  
Asymptotic Formulae ◽  
Elliptic Differential Operators ◽  
Domain Perturbations ◽  
Boundary Deformation

A variational characterization for the shift of eigenvalues caused by a general type of perturbation is derived for second order self-adjoint elliptic differential operators. This result allows the direct extension of asymptotic formulae from simple eigenvalues to repeated ones. Some examples of particular interest are presented theoretically and numerically for the Laplacian operator for the following domain perturbations: excision of a small hole, local change of conductivity, small boundary deformation.

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