Chris Cheung's Memo

Do you need R library for unconstrained minimization of high-dimensional functions? If so, you may be interested in my project RDistParNelderMead for faster and more memory efficient unconstrained high-dimensional minimization: https://github.com/chrischeungnf/RDistParNelderMead Any comment is highly appreciated.

Convert pdf to png (one page of pdf, one png): pdftoppm -png [pdf file name] [png file name prefix] From https://askubuntu.com/a/50180 Convert all jpg in one directory to pdf (one jpg, one page of pdf): convert *.jpg [Output pdf name] Combine multiple pdf to one pdf: pdftk [First pdf name] ... [Last pdf name] cat output [Output pdf name]

Numbering equations in align environment In align environment, all equations are numbered by default, use \nonumber to make equation on the same line not numbered. Example: Code: \documentclass{article} \usepackage{amsmath} \begin{document} \begin{align} \frac{w^* +\delta V_u^1}{r + \delta} &= V_u^{0} \nonumber \\ w^* &= r V_u^1 - \frac{b(r + \delta)}{1 + r} \end{align} \end{document} Result: In align* environment, all equations are not numbered by default, how about we want to make one equation numbered? Is there any analog of \nonumber? From Ian Thompson at https://tex.stackexchange.com/questions/42726/align-but-show-one-equation-number-at-the-end , we can add the following command before the document begins: \newcommand\numberthis{\addtocounter{equation}{1}\tag{\theequation}} % From Ian Thompson at https://tex.stackexchange.com/questions/42726/align-but-show-one-equation-number-at-the-end Then, we can use \numberthis to make equation on the same line numbered in ...

This is a summary of discussion with Alecos Papadopoulos at http://economics.stackexchange.com/questions/16118/stability-of-steady-state-equilibrium-for-overlapping-generation-model as well as from what I learnt and thought in my graduate macroeconomics class. Draft version. You are welcome to go to http://economics.stackexchange.com/questions/16118/stability-of-steady-state-equilibrium-for-overlapping-generation-model to comment and answer. In draft edition of Daron Acemoglu's Introduction to Modern Economic Growth (2007), proposition 9.4 which states that: In the overlapping-generations model with two-period lived households, Cobb-Douglas technology and CRRA preferences, there exists a unique steady-state equilibrium with the capital-labor ratio k* given by (9.15) and as long as $\theta \geq 1$ , this steady-state equilibrium is globally stable for all k (0) > 0. has a typo. In 2009 edition of the textbook, proposition 9.4 is corrected to: In the overlappin...