scholarly journals Lorenzen and Constructive Mathematics

2021 ◽  
pp. 47-61
Author(s):  
Thierry Coquand
Keyword(s):  
Mathematical Logic ◽  
Measure Theory ◽  
Constructive Mathematics ◽  
Short Survey ◽  
Recent Developments ◽  
Constructive Algebra

Abstract The goal of this paper is to present a short survey of some of Lorenzen’s contributions to constructive mathematics, and its influence on recent developments in mathematical logic and constructive algebra. We also present some work in measure theory which uses these contributions in an essential way.

Mathematical Logic: Proof Theory, Constructive Mathematics

2008 ◽  
pp. 907-952
Author(s):  
Samuel Buss ◽  
Helmut Schwichtenberg ◽  
Ulrich Kohlenbach
Keyword(s):  
Mathematical Logic ◽  
Proof Theory ◽  
Constructive Mathematics

Mathematical Logic: Proof Theory, Constructive Mathematics

2011 ◽  
Vol 8 (4) ◽  
pp. 2963-3002 ◽  
Author(s):  
Samuel Buss ◽  
Ulrich Kohlenbach ◽  
Michael Rathjen
Keyword(s):  
Mathematical Logic ◽  
Proof Theory ◽  
Constructive Mathematics

Mathematical Logic: Proof Theory, Type Theory and Constructive Mathematics

2005 ◽  
pp. 779-813
Author(s):  
Helmut Schwichtenberg ◽  
Vladimir Keilis-Borok ◽  
Samuel Buss
Keyword(s):  
Mathematical Logic ◽  
Type Theory ◽  
Proof Theory ◽  
Constructive Mathematics

scholarly journals Premium Calculation by Transforming the Layer Premium Density

1996 ◽  
Vol 26 (1) ◽  
pp. 71-92 ◽  
Author(s):  
Shaun Wang
Keyword(s):  
Distribution Function ◽  
Decision Theory ◽  
Stochastic Dominance ◽  
Measure Theory ◽  
Additive Measure ◽  
Proportional Hazard ◽  
Economic Decision ◽  
Recent Developments ◽  
Premium Calculation ◽  
Economic Decision Theory

AbstractThis paper examines a class of premium functionals which are (i) comonotonic additive and (ii) stochastic dominance preservative. The representation for this class is a transformation of the decumulative distribution function. It has close connections with the recent developments in economic decision theory and non-additive measure theory. Among a few elementary members of this class, the proportional hazard transform seems to stand out as being most plausible for actuaries.

scholarly journals Statistics with Imprecise Probabilities—A Short Survey

2021 ◽  
pp. 67-79
Author(s):  
Thomas Augustin
Keyword(s):  
Probabilistic Models ◽  
Statistical Modelling ◽  
Imprecise Probabilities ◽  
Paradigmatic Shift ◽  
Short Survey ◽  
Recent Developments ◽  
Data Imprecision

AbstractThis chapter aims at surveying and highlighting in an introductory way some challenges and big opportunities a paradigmatic shift to imprecise probabilities could induce in statistical modelling. Working with an informal understanding of imprecise probabilities, we discuss the concepts of model imprecision and data imprecision as the two main types of imprecision in statistical modelling. Then we provide a short survey of some major developments, methodological questions and applications of imprecise probabilistic models under model imprecision in the context of different inference schools and summarize some recent developments in the area of data imprecision.

scholarly journals Неограниченные сходимости в конвергентных векторных решетках

2018 ◽  
Author(s):  
Y.A.M. Dabboorasad ◽  
E.Yu. Emelyanov
Keyword(s):  
Risk Measures ◽  
Measure Theory ◽  
Variational Calculus ◽  
Mathematical Finance ◽  
Norm Convergence ◽  
Order Convergence ◽  
Vector Lattices ◽  
Almost Everywhere ◽  
Recent Developments ◽  
Compactness Arguments

Various convergences in vector lattices were historically a subject of deep investigation which stems from the begining of the 20th century in works of Riesz, Kantorovich, Nakano, Vulikh, Zanen, and many other mathematicians. The study of the unbounded order convergence had been initiated by Nakano in late 40th in connection with Birkhoff's ergodic theorem. The idea of Nakano was to define the almost everywhere convergence in terms of lattice operations without the direct use of measure theory. Many years later it was recognised that the unbounded order convergence is also rathe useful in probability theory. Since then, the idea of investigating of convergences by using their unbounded versions, have been exploited in several papers. For instance, unbounded convergences in vector lattices have attracted attention of many researchers in order to find new approaches to various problems of functional analysis, operator theory, variational calculus, theory of risk measures in mathematical finance, stochastic processes, etc. Some of those unbounded convergences, like unbounded norm convergence, unbounded multi-norm convergence, unbounded $\tau$-convergence are topological. Others are not topological in general, for example: the unbounded order convergence, the unbounded relative uniform convergence, various unbounded convergences in lattice-normed lattices, etc. Topological convergences are, as usual, more flexible for an investigation due to the compactness arguments, etc. The non-topological convergences are more complicated in genelal, as it can be seen on an example of the a.e-convergence. In the present paper we present recent developments in convergence vector lattices with emphasis on related unbounded convergences. Special attention is paid to the case of convergence in lattice multi pseudo normed vector lattices that generalizes most of cases which were discussed in the literature in the last 5 years.

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