Explicit bounds on twin primes and Brun's Constant.

2007 ◽  
Author(s):  
Dominic. Klyve
Keyword(s):  
Twin Primes ◽  
Explicit Bounds ◽  
Brun's Constant

scholarly journals Normal approximation for mixtures of normal distributions and the evolution of phenotypic traits

2021 ◽  
Vol 53 (1) ◽  
pp. 162-188
Author(s):  
Krzysztof Bartoszek ◽  
Torkel Erhardsson
Keyword(s):  
Normal Distribution ◽  
Phenotypic Trait ◽  
Phenotypic Traits ◽  
Normal Distributions ◽  
Average Value ◽  
Conditional Moments ◽  
Mixture Of Normal Distributions ◽  
Explicit Bounds ◽  
Mixtures Of Normal Distributions ◽  
Parameter Values

AbstractExplicit bounds are given for the Kolmogorov and Wasserstein distances between a mixture of normal distributions, by which we mean that the conditional distribution given some $\sigma$ -algebra is normal, and a normal distribution with properly chosen parameter values. The bounds depend only on the first two moments of the first two conditional moments given the $\sigma$ -algebra. The proof is based on Stein’s method. As an application, we consider the Yule–Ornstein–Uhlenbeck model, used in the field of phylogenetic comparative methods. We obtain bounds for both distances between the distribution of the average value of a phenotypic trait over n related species, and a normal distribution. The bounds imply and extend earlier limit theorems by Bartoszek and Sagitov.

scholarly journals New Gronwall-Bellman Type Inequalities and Applications in the Analysis for Solutions to Fractional Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Bin Zheng ◽  
Qinghua Feng
Keyword(s):  
Quantitative Analysis ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Continuous Dependence ◽  
Initial Value ◽  
Liouville Fractional Derivative ◽  
Explicit Bounds ◽  
Fractional Differential ◽  
Differential And Integral Equations

Some new Gronwall-Bellman type inequalities are presented in this paper. Based on these inequalities, new explicit bounds for the related unknown functions are derived. The inequalities established can also be used as a handy tool in the research of qualitative as well as quantitative analysis for solutions to some fractional differential equations defined in the sense of the modified Riemann-Liouville fractional derivative. For illustrating the validity of the results established, we present some applications for them, in which the boundedness, uniqueness, and continuous dependence on the initial value for the solutions to some certain fractional differential and integral equations are investigated.

scholarly journals Some Nonlinear Delay Volterra–Fredholm Type Dynamic Integral Inequalities on Time Scales

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Yazhou Tian ◽  
A. A. El-Deeb ◽  
Fanwei Meng
Keyword(s):  
Time Scales ◽  
Qualitative Theory ◽  
Integral Inequalities ◽  
Dynamic Equations ◽  
Fredholm Type ◽  
Explicit Bounds ◽  
Theoretical Results ◽  
Nonlinear Delay

We are devoted to studying a class of nonlinear delay Volterra–Fredholm type dynamic integral inequalities on time scales, which can provide explicit bounds on unknown functions. The obtained results can be utilized to investigate the qualitative theory of nonlinear delay Volterra–Fredholm type dynamic equations. An example is also presented to illustrate the theoretical results.

scholarly journals Compound Poisson approximation for the distribution of extremes

2002 ◽  
Vol 34 (1) ◽  
pp. 223-240 ◽  
Author(s):  
A. D. Barbour ◽  
S. Y. Novak ◽  
A. Xia
Keyword(s):  
Point Processes ◽  
Poisson Approximation ◽  
Value Theory ◽  
Wasserstein Metric ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Limiting Behaviour ◽  
Explicit Bounds ◽  
Empirical Point ◽  
Poisson Cluster

Empirical point processes of exceedances play an important role in extreme value theory, and their limiting behaviour has been extensively studied. Here, we provide explicit bounds on the accuracy of approximating an exceedance process by a compound Poisson or Poisson cluster process, in terms of a Wasserstein metric that is generally more suitable for the purpose than the total variation metric. The bounds only involve properties of the finite, empirical sequence that is under consideration, and not of any limiting process. The argument uses Bernstein blocks and Lindeberg's method of compositions.

scholarly journals On nonlinear finite difference inequalities in two independent variables

2002 ◽  
Vol 33 (1) ◽  
pp. 57-66
Author(s):  
B. G. Pachpatte
Keyword(s):  
Finite Difference ◽  
Difference Equations ◽  
Qualitative Theory ◽  
Independent Variables ◽  
Finite Difference Equations ◽  
Explicit Bounds ◽  
Two Independent Variables

The aim of the present paper is to establish some new finite difference inequalities involving functions of two independent variables which provide explicit bounds on unknown functions. The inequalities given here can be used as tools in the qualitative theory of certain partial finite difference equations.

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