Simplicial Homology of Random Configurations

Given a Poisson process on a d-dimensional torus, its random geometric simplicial complex is the complex whose vertices are the points of the Poisson process and simplices are given by the C̆ech complex associated to the coverage of each point. By means of Malliavin calculus, we compute explicitly the three first-order moments of the number of k-simplices, and provide a way to compute higher-order moments. Then we derive the mean and the variance of the Euler characteristic. Using the Stein method, we estimate the speed of convergence of the number of occurrences of any connected subcomplex as it converges towards the Gaussian law when the intensity of the Poisson point process tends to infinity. We use a concentration inequality for Poisson processes to find bounds for the tail distribution of the Betti number of first order and the Euler characteristic in such simplicial complexes.

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First passage time for the state of exact chemical equilibrium

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc19800777 ◽

1980 ◽

Vol 45 (3) ◽

pp. 777-782 ◽

Cited By ~ 1

Author(s):

Milan Šolc

Keyword(s):

Chemical Equilibrium ◽

First Passage Time ◽

Passage Time ◽

The State ◽

Recurrence Time ◽

First Passage ◽

First Order ◽

The Mean ◽

Passage Times ◽

Number Of Particles

The establishment of chemical equilibrium in a system with a reversible first order reaction is characterized in terms of the distribution of first passage times for the state of exact chemical equilibrium. The mean first passage time of this state is a linear function of the logarithm of the total number of particles in the system. The equilibrium fluctuations of composition in the system are characterized by the distribution of the recurrence times for the state of exact chemical equilibrium. The mean recurrence time is inversely proportional to the square root of the total number of particles in the system.

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The Structure and Photochromism of 2,4,4,6-Tetraphenyl-4H-thiopyran

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc19921326 ◽

1992 ◽

Vol 57 (6) ◽

pp. 1326-1334 ◽

Cited By ~ 11

Author(s):

Jaroslav Vojtěchovský ◽

Jindřich Hašek ◽

Stanislav Nešpůrek ◽

Mojmír Adamec

Keyword(s):

Solid State ◽

Uv Light ◽

Order Reaction ◽

Boat Conformation ◽

X Rays ◽

Exponential Time ◽

First Order ◽

Double Plane ◽

The Mean ◽

Independent Molecules

2,4,4,6-Tetraphenyl-4H-thiopyran, C29H22S, orthorhombic, Pna21, a = 17.980(4), b = 6.956(2), c = 34.562(11) Å, V = 4323(2) Å3, Z = 8, Dx = 1.237 g cm-3, F(000) = 1696, λ(CuKα) = 1.54184 A, μ = 1.372 mm-2, T = 294 K. The final R was 0.050 for the unique set of 3103 observed reflections. The central 4H-thiopyran ring forms a boat conformation for both symmetrically independent molecules with average boat angles 4.4(3) and 6.8(3)° at S and C(sp3), respectively. The mean planes of phenyls at the position 2 and 6 are turned from the double plane of 4H-thiopyran by 42.5(5) and 35.8(3)°, respectively. The investigated material undergoes a photochromic change in the solid state after irradiation with UV light or X-rays. The maximum of the new absorption band is situated at 564 nm. The non-exponential time dependence of photochromic bleaching is analysed in terms of a dispersive first-order reaction.

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A Bimodal Extension of the Exponential Distribution with Applications in Risk Theory

Symmetry ◽

10.3390/sym13040679 ◽

2021 ◽

Vol 13 (4) ◽

pp. 679

Author(s):

Jimmy Reyes ◽

Emilio Gómez-Déniz ◽

Héctor W. Gómez ◽

Enrique Calderín-Ojeda

Keyword(s):

Exponential Distribution ◽

Hazard Rate ◽

Rate Function ◽

Estimation Methods ◽

Inverse Gaussian ◽

New Family ◽

Tail Distribution ◽

Zero Values ◽

The Mean ◽

Positive Support

There are some generalizations of the classical exponential distribution in the statistical literature that have proven to be helpful in numerous scenarios. Some of these distributions are the families of distributions that were proposed by Marshall and Olkin and Gupta. The disadvantage of these models is the impossibility of fitting data of a bimodal nature of incorporating covariates in the model in a simple way. Some empirical datasets with positive support, such as losses in insurance portfolios, show an excess of zero values and bimodality. For these cases, classical distributions, such as exponential, gamma, Weibull, or inverse Gaussian, to name a few, are unable to explain data of this nature. This paper attempts to fill this gap in the literature by introducing a family of distributions that can be unimodal or bimodal and nests the exponential distribution. Some of its more relevant properties, including moments, kurtosis, Fisher’s asymmetric coefficient, and several estimation methods, are illustrated. Different results that are related to finance and insurance, such as hazard rate function, limited expected value, and the integrated tail distribution, among other measures, are derived. Because of the simplicity of the mean of this distribution, a regression model is also derived. Finally, examples that are based on actuarial data are used to compare this new family with the exponential distribution.

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Diffraction Measurements and Equilibrium Parameters

Advances in Physical Chemistry ◽

10.1155/2011/864714 ◽

2011 ◽

Vol 2011 ◽

pp. 1-14 ◽

Cited By ~ 8

Author(s):

Victor A. Sipachev

Keyword(s):

Low Frequency ◽

Mean Square ◽

Structural Studies ◽

Free Rotation ◽

Theory Analysis ◽

First Order ◽

Internuclear Distances ◽

The Mean ◽

Asymmetry Parameters ◽

First Order Perturbation

Structural studies are largely performed without taking into account vibrational effects or with incorrectly taking them into account. The paper presents a first-order perturbation theory analysis of the problem. It is shown that vibrational effects introduce errors on the order of 0.02 Å or larger (sometimes, up to 0.1-0.2 Å) into the results of diffraction measurements. Methods for calculating the mean rotational constants, mean-square vibrational amplitudes, vibrational corrections to internuclear distances, and asymmetry parameters are described. Problems related to low-frequency motions, including torsional motions that transform into free rotation at low excitation levels, are discussed. The algorithms described are implemented in the program available from the author (free).

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Identification and isotropy characterization of deformed random fields through excursion sets

Advances in Applied Probability ◽

10.1017/apr.2018.32 ◽

2018 ◽

Vol 50 (3) ◽

pp. 706-725

Author(s):

Julie Fournier

Keyword(s):

Random Field ◽

Euler Characteristic ◽

Random Fields ◽

Weak Form ◽

Invariance Property ◽

Basic Domain ◽

Excursion Sets ◽

The Mean ◽

Isotropic Random

Abstract A deterministic application θ:ℝ2→ℝ2 deforms bijectively and regularly the plane and allows the construction of a deformed random field X∘θ:ℝ2→ℝ from a regular, stationary, and isotropic random field X:ℝ2→ℝ. The deformed field X∘θ is, in general, not isotropic (and not even stationary), however, we provide an explicit characterization of the deformations θ that preserve the isotropy. Further assuming that X is Gaussian, we introduce a weak form of isotropy of the field X∘θ, defined by an invariance property of the mean Euler characteristic of some of its excursion sets. We prove that deformed fields satisfying this property are strictly isotropic. In addition, we are able to identify θ, assuming that the mean Euler characteristic of excursion sets of X∘θ over some basic domain is known.

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Unified uncertainty analysis by the mean value first order saddlepoint approximation

Structural and Multidisciplinary Optimization ◽

10.1007/s00158-012-0794-4 ◽

2012 ◽

Vol 46 (6) ◽

pp. 803-812 ◽

Cited By ~ 44

Author(s):

Ning-Cong Xiao ◽

Hong-Zhong Huang ◽

Zhonglai Wang ◽

Yu Liu ◽

Xiao-Ling Zhang

Keyword(s):

Uncertainty Analysis ◽

Saddlepoint Approximation ◽

Mean Value ◽

First Order ◽

The Mean

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Open books for Boothby-Wang bundles, fibered Dehn twists and the mean Euler characteristic

Journal of Symplectic Geometry ◽

10.4310/jsg.2014.v12.n2.a6 ◽

2014 ◽

Vol 12 (2) ◽

pp. 379-426 ◽

Cited By ~ 7

Author(s):

River Chiang ◽

Fan Ding ◽

Otto van Koert

Keyword(s):

Euler Characteristic ◽

Open Books ◽

Dehn Twists ◽

The Mean

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Stochastic Model of the n-Stage Reversible First-Order Reaction: Relation between the Time of First Passage to the Most Probable Microstate and the Mean Equilibrium Fluctuations Lifetime

Zeitschrift für Physikalische Chemie ◽

10.1524/zpch.2002.216.7.869 ◽

2002 ◽

Vol 216 (7) ◽

Cited By ~ 4

Author(s):

M. Solc

Keyword(s):

Stochastic Model ◽

Order Reaction ◽

First Order Reaction ◽

First Passage ◽

Equilibrium Fluctuations ◽

First Order ◽

N Stage ◽

The Mean

Using the stochastic model of

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The Second-Order Master Equation

The Master Equation and the Convergence Problem in Mean Field Games ◽

10.23943/princeton/9780691190716.003.0005 ◽

2019 ◽

pp. 128-158

Author(s):

Pierre Cardaliaguet ◽

François Delarue ◽

Jean-Michel Lasry ◽

Pierre-Louis Lions

Keyword(s):

Master Equation ◽

Mean Field ◽

Second Order ◽

Well Posedness ◽

Mean Field Game ◽

First Order ◽

Common Noise ◽

System P ◽

The Mean

This chapter investigates the second-order master equation with common noise, which requires the well-posedness of the mean field game (MFG) system. It also defines and analyzes the solution of the master equation. The chapter explains the forward component of the MFG system that is recognized as the characteristics of the master equation. The regularity of the solution of the master equation is explored through the tangent process that solves the linearized MFG system. It also analyzes first-order differentiability and second-order differentiability in the direction of the measure on the same model as for the first-order derivatives. This chapter concludes with further description of the derivation of the master equation and well-posedness of the stochastic MFG system.

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