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2. Formalization of the Equivalence among Completeness Theorems of Real Number in Coq
- Author
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Wensheng Yu and Yaoshun Fu
- Subjects
formalization ,analysis ,General Mathematics ,0102 computer and information sciences ,real number theory ,01 natural sciences ,Formal proof ,Compactness theorem ,Computer Science (miscellaneous) ,Coq ,Dedekind cut ,Gödel's completeness theorem ,0101 mathematics ,Engineering (miscellaneous) ,Mathematics ,Real number ,Fundamental theorem ,lcsh:Mathematics ,010102 general mathematics ,Monotone convergence theorem ,lcsh:QA1-939 ,Algebra ,Automated theorem proving ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,010201 computation theory & mathematics ,completeness theorems - Abstract
The formalization of mathematics based on theorem prover becomes increasingly important in mathematics and computer science, and, particularly, formalizing fundamental mathematical theories becomes especially essential. In this paper, we describe the formalization in Coq of eight very representative completeness theorems of real numbers. These theorems include the Dedekind fundamental theorem, Supremum theorem, Monotone convergence theorem, Nested interval theorem, Finite cover theorem, Accumulation point theorem, Sequential compactness theorem, and Cauchy completeness theorem. We formalize the real number theory strictly following Landau&rsquo, s Foundations of Analysis where the Dedekind fundamental theorem can be proved. We extend this system and complete the related notions and properties for finiteness and sequence. We prove these theorems in turn from Dedekind fundamental theorem, and finally prove the Dedekind fundamental theorem by the Cauchy completeness theorem. The full details of formal proof are checked by the proof assistant Coq, which embodies the characteristics of reliability and interactivity. This work can lay the foundation for many applications, especially in calculus and topology.
- Published
- 2021
3. A Compound Poisson Perspective of Ewens–Pitman Sampling Model
- Author
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Stefano Favaro and Emanuele Dolera
- Subjects
Class (set theory) ,General Mathematics ,Negative binomial distribution ,MathematicsofComputing_GENERAL ,Poisson distribution ,01 natural sciences ,log-series compound poisson sampling model ,Combinatorics ,010104 statistics & probability ,symbols.namesake ,Computer Science (miscellaneous) ,QA1-939 ,0101 mathematics ,Ewens–Pitman sampling model ,Engineering (miscellaneous) ,Mathematics ,wright distribution function ,Conjecture ,Berry–Esseen type theorem ,Exchangeable random partitions ,Log-series compound poisson sampling model ,Mittag–Leffler distribution function ,Negative binomial compound poisson sampling model ,Pitman’s α-diversity ,Wright distribution function ,010102 general mathematics ,Sampling (statistics) ,Extension (predicate logic) ,exchangeable random partitions ,negative binomial compound poisson sampling model ,Distribution (mathematics) ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,symbols ,Poisson sampling - Abstract
The Ewens–Pitman sampling model (EP-SM) is a distribution for random partitions of the set {1,…,n}, with n∈N, which is indexed by real parameters α and θ such that either α∈[0,1) and θ>, −α, or α<, 0 and θ=−mα for some m∈N. For α=0, the EP-SM is reduced to the Ewens sampling model (E-SM), which admits a well-known compound Poisson perspective in terms of the log-series compound Poisson sampling model (LS-CPSM). In this paper, we consider a generalisation of the LS-CPSM, referred to as the negative Binomial compound Poisson sampling model (NB-CPSM), and we show that it leads to an extension of the compound Poisson perspective of the E-SM to the more general EP-SM for either α∈(0,1), or α<, 0. The interplay between the NB-CPSM and the EP-SM is then applied to the study of the large n asymptotic behaviour of the number of blocks in the corresponding random partitions—leading to a new proof of Pitman’s α diversity. We discuss the proposed results and conjecture that analogous compound Poisson representations may hold for the class of α-stable Poisson–Kingman sampling models—of which the EP-SM is a noteworthy special case.
- Published
- 2021
- Full Text
- View/download PDF
4. A Study of Seven Asymmetric Kernels for the Estimation of Cumulative Distribution Functions
- Author
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Pierre Lafaye de Micheaux and Frédéric Ouimet
- Subjects
Mean squared error ,General Mathematics ,Weibull kernel ,Asymptotic distribution ,Mathematics - Statistics Theory ,inverse Gamma kernel ,Statistics Theory (math.ST) ,01 natural sciences ,Inverse Gaussian distribution ,010104 statistics & probability ,symbols.namesake ,asymptotic statistics ,0502 economics and business ,Birnbaum–Saunders kernel ,FOS: Mathematics ,Computer Science (miscellaneous) ,QA1-939 ,Applied mathematics ,Statistics::Methodology ,asymmetric kernels ,0101 mathematics ,Engineering (miscellaneous) ,050205 econometrics ,Mathematics ,Weibull distribution ,Gamma kernel ,reciprocal inverse Gaussian kernel ,Cumulative distribution function ,Probability (math.PR) ,05 social sciences ,Estimator ,62G05, 60F05, 62G20 ,nonparametric statistics ,Kernel method ,Kernel (statistics) ,symbols ,LogNormal kernel ,Mathematics - Probability ,inverse Gaussian kernel - Abstract
In Mombeni et al. (2019), Birnbaum-Saunders and Weibull kernel estimators were introduced for the estimation of cumulative distribution functions (c.d.f.s) supported on the half-line $[0,\infty)$. They were the first authors to use asymmetric kernels in the context of c.d.f. estimation. Their estimators were shown to perform better numerically than traditional methods such as the basic kernel method and the boundary modified version from Tenreiro (2013). In the present paper, we complement their study by introducing five new asymmetric kernel c.d.f. estimators, namely the Gamma, inverse Gamma, lognormal, inverse Gaussian and reciprocal inverse Gaussian kernel c.d.f. estimators. For these five new estimators, we prove the asymptotic normality and we find asymptotic expressions for the following quantities: bias, variance, mean squared error and mean integrated squared error. A numerical study then compares the performance of the five new c.d.f. estimators against traditional methods and the Birnbaum-Saunders and Weibull kernel c.d.f. estimators from Mombeni et al. (2019). By using the same experimental design, we show that the lognormal and Birnbaum-Saunders kernel c.d.f. estimators perform the best overall, while the other asymmetric kernel estimators are sometimes better but always at least competitive against the boundary kernel method., 38 pages, 2 tables, 9 figures
- Published
- 2021
- Full Text
- View/download PDF
5. A Coupling between Integral Equations and On-Surface Radiation Conditions for Diffraction Problems by Non Convex Scatterers
- Author
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Xavier Antoine, Chokri Chniti, Saleh Mousa Alzahrani, Department of Mathematics, Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)
- Subjects
[SPI.ACOU]Engineering Sciences [physics]/Acoustics [physics.class-ph] ,Coupling ,Surface (mathematics) ,Physics ,Diffraction ,Field (physics) ,on-surface radiation condition ,Scattering ,General Mathematics ,Operator (physics) ,Mathematical analysis ,Regular polygon ,010103 numerical & computational mathematics ,[INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation ,01 natural sciences ,Integral equation ,010101 applied mathematics ,integral equation ,QA1-939 ,Computer Science (miscellaneous) ,0101 mathematics ,acoustics ,Engineering (miscellaneous) ,Mathematics ,[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] - Abstract
The aim of this paper is to introduce an orignal coupling procedure between surface integral equation formulations and on-surface radiation condition (OSRC) methods for solving two-dimensional scattering problems for non convex structures. The key point is that the use of the OSRC introduces a sparse block in the surface operator representation of the wave field while the integral part leads to an improved accuracy of the OSRC method in the non convex part of the scattering structure. The procedure is given for both the Dirichlet and Neumann scattering problems. Some numerical simulations show the improvement induced by the coupling method.
- Published
- 2021
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