BKK bound – an “elementary proof”

Some comments of Jonathan Korman made me realize that it would have been good to include in How Many Zeroes? an “elementary” proof of the Bernstein-Kushnirenko formula via Hilbert polynomials. This approach only yields a “weak” version of the bound since it applies only to the case that the number of solutions is finite. Nevertheless,… Continue reading BKK bound – an “elementary proof”

Notes (part 2): S. Bosch, U Güntzer and R. Remmert, Non-Archimedean Analysis

\(\newcommand{\ff}{\mathbb{F}} \DeclareMathOperator{\ord}{ord} \newcommand{\qq}{\mathbb{Q}} \newcommand{\rr}{\mathbb{R}} \newcommand{\rnneg}{\rr_{\geq 0}} \newcommand{\rpos}{\rr_{> 0}} \newcommand{\zz}{\mathbb{Z}} \newcommand{\znneg}{\zz_{\geq 0}} \)Given a normed (abelian) group \(A,\) the following are some important subsets of \(\prod_{i=1}^\infty A\): It is clear that \(A^{(\infty)} \subseteq c(A) \subseteq b(A).\) If \(A\) is a ring/algebra over a field/module over a ring, each of these sets inherits the structure from \(A.\)… Continue reading Notes (part 2): S. Bosch, U Güntzer and R. Remmert, Non-Archimedean Analysis

Notes (part 1): S. Bosch, U Güntzer and R. Remmert, Non-Archimedean Analysis

Semi-normed and normed groups \(\newcommand{\mmm}{\mathfrak{m}} \newcommand{\qqq}{\mathfrak{q}} \newcommand{\rr}{\mathbb{R}} \newcommand{\rnneg}{\rr_{\geq 0}} \newcommand{\rpos}{\rr_{> 0}} \newcommand{\zz}{\mathbb{Z}} \newcommand{\znneg}{\zz_{\geq 0}} \)All groups are going to abelian with the group operation denoted by “+”. A filtration, a generalization of valuation, is a function \(\nu\) from a group \(G\) to \(\rr \cup \{\infty\}\) such that Filtrations are in one-to-one correspondence with ultrametric functions… Continue reading Notes (part 1): S. Bosch, U Güntzer and R. Remmert, Non-Archimedean Analysis

Notes: Jonathan D. Cryer and Kung-Sik Chan, Time Series Analysis with Applications in R

This post is to record my notes on the first book on time series that I am reading seriously, namely the 2008 edition of Time Series Analysis with Applications in R by Jonathan D. Cryer and Kung-Sik Chan. Chapter 2. Fundamental Concepts The model for a time series is a stochastic process, which is a… Continue reading Notes: Jonathan D. Cryer and Kung-Sik Chan, Time Series Analysis with Applications in R