Large-$N$ Limit of Crossing Probabilities, Discontinuity, and Asymptotic Behavior of Threshold Values in Mandelbrot's Fractal Percolation Process

Let R be a commutative Noetherian Nagata ring, let M be a non-zero finitely generated R-module, and let I be an ideal of R such that height MI > 0. In this paper, there is a definition of the integral closure Na for any submodule N of M extending Rees' definition for the case of a domain. As the main results, it is shown that the operation N → Na on the set of submodules N of M is a semi-prime operation, and for any submodule N of M, the sequences Ass R M/(InN)a and Ass R (InM)a/(InN)a(n=1,2,…) of associated prime ideals are increasing and ultimately constant for large n.

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Random fractals and their intersection with winning sets

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004121000360 ◽

2021 ◽

pp. 1-30

Author(s):

YIFTACH DAYAN

Keyword(s):

Hausdorff Dimension ◽

General Result ◽

Countable Collection ◽

Percolation Process ◽

Fractal Percolation ◽

Random Fractals ◽

Affine Hyperplane ◽

Badly Approximable

Abstract We show that fractal percolation sets in $\mathbb{R}^{d}$ almost surely intersect every hyperplane absolutely winning (HAW) set with full Hausdorff dimension. In particular, if $E\subset\mathbb{R}^{d}$ is a realisation of a fractal percolation process, then almost surely (conditioned on $E\neq\emptyset$ ), for every countable collection $\left(f_{i}\right)_{i\in\mathbb{N}}$ of $C^{1}$ diffeomorphisms of $\mathbb{R}^{d}$ , $\dim_{H}\left(E\cap\left(\bigcap_{i\in\mathbb{N}}f_{i}\left(\text{BA}_{d}\right)\right)\right)=\dim_{H}\left(E\right)$ , where $\text{BA}_{d}$ is the set of badly approximable vectors in $\mathbb{R}^{d}$ . We show this by proving that E almost surely contains hyperplane diffuse subsets which are Ahlfors-regular with dimensions arbitrarily close to $\dim_{H}\left(E\right)$ . We achieve this by analysing Galton–Watson trees and showing that they almost surely contain appropriate subtrees whose projections to $\mathbb{R}^{d}$ yield the aforementioned subsets of E. This method allows us to obtain a more general result by projecting the Galton–Watson trees against any similarity IFS whose attractor is not contained in a single affine hyperplane. Thus our general result relates to a broader class of random fractals than fractal percolation.

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Stability and Asymptotic Behavior of a Regime-Switching SIRS Model with Beddington–DeAngelis Incidence Rate

Mathematical Problems in Engineering ◽

10.1155/2020/7181939 ◽

2020 ◽

Vol 2020 ◽

pp. 1-12

Author(s):

Shan Wang ◽

Youhua Peng ◽

Feng Wang

Keyword(s):

Asymptotic Behavior ◽

Incidence Rate ◽

Regime Switching ◽

Invariant Set ◽

Stochastic Noise ◽

Threshold Values ◽

Unique Positive Solution ◽

Stochastic Asymptotic Stability ◽

Sirs Model

A regime-switching SIRS model with Beddington–DeAngelis incidence rate is studied in this paper. First of all, the property that the model we discuss has a unique positive solution is proved and the invariant set is presented. Secondly, by constructing appropriate Lyapunov functionals, global stochastic asymptotic stability of the model under certain conditions is proved. Then, we leave for studying the asymptotic behavior of the model by presenting threshold values and some other conditions for determining disease extinction and persistence. The results show that stochastic noise can inhibit the disease and the behavior will have different phenomena owing to the role of regime-switching. Finally, some examples are given and numerical simulations are presented to confirm our conclusions.

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Geometric functionals of fractal percolation

Advances in Applied Probability ◽

10.1017/apr.2020.33 ◽

2020 ◽

Vol 52 (4) ◽

pp. 1085-1126

Author(s):

Michael A. Klatt ◽

Steffen Winter

Keyword(s):

Topological Phase ◽

Explicit Formulas ◽

Percolation Process ◽

Intrinsic Volumes ◽

Fractal Percolation ◽

Fractal Models ◽

Topological Phase Transition ◽

Finite Approximations ◽

Percolation Thresholds ◽

Transition Points

AbstractFractal percolation exhibits a dramatic topological phase transition, changing abruptly from a dust-like set to a system-spanning cluster. The transition points are unknown and difficult to estimate. In many classical percolation models the percolation thresholds have been approximated well using additive geometric functionals, known as intrinsic volumes. Motivated by the question of whether a similar approach is possible for fractal models, we introduce corresponding geometric functionals for the fractal percolation process F. They arise as limits of expected functionals of finite approximations of F. We establish the existence of these limit functionals and obtain explicit formulas for them as well as for their finite approximations.

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Desynchronosis markers in risk assessment of metabolic syndrome development

Medical alphabet ◽

10.33667/2078-5631-2019-1-4(379)-21-26 ◽

2019 ◽

Vol 1 (4) ◽

pp. 21-26 ◽

Cited By ~ 1

Author(s):

O. V. Bobko ◽

O. V. Tikhomirova ◽

N. N. Zybina ◽

O. A. Klitsenko

Keyword(s):

Risk Assessment ◽

Metabolic Syndrome ◽

Threshold Values ◽

Fat Tissue ◽

Diurnal Dynamics ◽

Laboratory Markers ◽

Dyscirculatory Encephalopathy ◽

Melatonin Production ◽

Pai 1 ◽

Logistic Regression Equation

The objective of the study is to show significance of desynchronosis laboratory markers in risk assessment of metabolic syndrome (MS) development. Materials and Methods. There were examined 98 men, aged 43-88, diagnosed with dyscirculatory encephalopathy showing one and more risk factors for development of cardiovascular diseases. They were divided into 2 groups according to the international guidelines of 2009: with MS (n = 61) and without MS (n = 37). Parameters of fats, glucose metabolism, inflammatory mediators, fat tissue metabolism markers, melatonin metabolite excretion (6-sulfatoxymelatonin) were defined in blood serum and urine. Results. The article presents data on changes in leptin, adiponectin, PAI-1, testosterone production and 6-sulfatoxymela-tonin excretion in patients with MS. There are calculated threshold values of these markers definitely increasing MS risk and logistic regression equation which allows assessing MS risk for an individual patient. Conclusion. Detected disorders of melatonin synthesis diurnal dynamics in patients with MS and interconnection between melatonin production and adiponectin, leptin, PAI-1, testosterone synthesis allow considering these parameters as desynchronosis markers significant for MS development.

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