2014 316 pages
Series: Textbooks in Mathematics
978-1-49-870265-2
30th November 2014
Analysis with Ultrasmall Numbers presents an intuitive treatment of mathematics using ultrasmall numbers. With this modern approach to infinitesimals, proofs become simpler and more focused on the combinatorial heart of arguments, unlike traditional treatments that use epsilon?delta methods. Students can fully prove fundamental results, such as the Extreme Value Theorem, from the axioms immediately, without needing to master notions of supremum or compactness.
The book is suitable for a calculus course at the undergraduate or high school level or for self-study with an emphasis on nonstandard methods. The first part of the text offers material for an elementary calculus course while the second part covers more advanced calculus topics.
The text provides straightforward definitions of basic concepts, enabling students to form good intuition and actually prove things by themselves. It does not require any additional "black boxes" once the initial axioms have been presented. The text also includes numerous exercises throughout and at the end of each chapter.
Elementary Analysis
Basic concepts
Continuity and limits
Differentiability
Elementary integration
Higher Analysis
Basic concepts revisited
Derivatives
Sequences and series
Topology of Real Numbers
Differential Equations
Integration
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2015 189 pages
Series: Chapman & Hall/CRC Monographs on Statistics & Applied Probability
978-1-43-986939-0
22nd March 2015
Many observational studies in epidemiology and other disciplines face inherent limitations in study design and data quality, such as selection bias, unobserved variables, and poorly measured variables. Accessible to statisticians and researchers from various disciplines, this book presents an overview of Bayesian inference in partially identified models. It includes many examples to illustrate the methods and provides R code for their implementation on the book’s website. The author also addresses a number of open questions to stimulate further research in this area.
Introduction
What Are Partially Identified Models and Why Are They Important?
What Is for and against Us?
Some Simple Examples of PIMs
Evaluating Inference
The Tell-Tale Signature of Partial Identification
The Structure of Posterior Distributions in PIMs
Frequentist Properties of Bayesian estimators in PIMs
Interval Estimation
Study Design
Posterior Computation
PIM versus Identified/Misspecified Model
Sensitivity Analysis
Further Examples
This book contains original research papers by leading experts in the fields of probability theory, stochastic analysis, potential theory and mathematical physics. There is also a historical account on Masatoshi Fukushima's contribution to mathematics, as well as authoritative surveys on the state of the art in the field.
Professor Fukushima's Work:
The Mathematical Work of Masatoshi Fukushima ? An Essay (Zhen-Qing Chen, Niels Jacob, Masayoshi Takeda and Toshihiro Uemura)
Bibliography of Masatoshi Fukushima
Contributions:
Quasi Regular Dirichlet Forms and the Stochastic Quantization Problem (Sergio Albeverio, Zhi-Ming Ma and Michael Rockner)
Comparison of Quenched and Annealed Invariance Principles for Random Conductance Model: Part II (Martin Barlow, Krzysztof Burdzy and Adam Timar)
Some Historical Aspects of Error Calculus by Dirichlet Forms (Nicolas Bouleau)
Stein's Method, Malliavin Calculus, Dirichlet Forms and the Fourth Moment Theorem (Louis H Y Chen and Guillaume Poly)
Progress on Hardy-Type Inequalities (Mu-Fa Chen)
Functional Inequalities for Pure-Jump Dirichlet Forms (Xin Chen, Feng-Yu Wang and Jian Wang)
Additive Functionals and Push Forward Measures Under Veretennikov's Flow (Shizan Fang and Andrey Pilipenko)
On a Result of D W Stroock (Patrick J Fitzsimmons)
Consistent Risk Measures and a Non-Linear Extension of Backwards Martingale Convergence (Hans Follmer and Irina Penner)
Unavoidable Collections of Balls for Processes with Isotropic Unimodal Green Function (Wolfhard Hansen)
Functions of Locally Bounded Variation on Wiener Spaces (Masanori Hino)
A Dirichlet Space on Ends of Tree and Superposition of Nodewise Given Dirichlet Forms with Tier Linkage (Hiroshi Kaneko)
Dirichlet Forms in Quantum Theory (Witold Karwowski and Ludwig Streit)
On a Stability of Heat Kernel Estimates under Generalized Non-Local Feynman-Kac Perturbations for Stable-Like Processes (Daehong Kim and Kazuhiro Kuwae)
Martin Boundary for Some Symmetric Levy Processes (Panki Kim, Renming Song and Zoran Vondra?ek)
Level Statistics of One-Dimensional Schrodinger Operators with Random Decaying Potential (Shinichi Kotani and Fumihiko Nakano)
Perturbation of the Loop Measure (Yves Le Jan and Jay Rosen)
Regular Subspaces of Dirichlet Forms (Liping Li and Jiangang Ying)
Quasi-Regular Semi-Dirichlet Forms and Beyond (Zhi-Ming Ma, Wei Sun and Li-Fei Wang)
Large Deviation Estimates for Controlled Semi-Martingales (Hideo Nagai)
A Comparison Theorem for Backward SPDEs with Jumps (Bernt Oksendal, Agnes Sulem and Tusheng Zhang)
On a Construction of a Space-Time Diffusion Process with Boundary Condition (Yoichi Oshima)
Lower Bounded Semi-Dirichlet Forms Associated with Levy Type Operators (Rene L Schilling and Jian Wang)
Ultracontractivity for Non-Symmetric Markovian Semigroups (Ichiro Shigekawa)
Metric Measure Spaces with Variable Ricci Bounds and Couplings of Brownian Motions (Karl-Theodor Sturm)
Intrinsic Ultracontractivity and Semi-Small Perturbation for Skew Product Diffusion Operators (Matsuyo Tomisaki)
Readership: Researchers in probability, stochastic analysis and mathematical physics.