Northwest vs Northwestern
Why do we say "Eastern Europe" but "East Asia"?
Shrink exFAT partition Is there any way to shrink an exFAT partition?   The windows disk manager has the "Shrink" option greyed out. How to retrieve and display images from a database in a JSP page? ArcMap Attribute Table not seen when second screen disconnected from Main Computer? I don't know what to do with this. Nothing works. I hope somebody can help me to find a decision $$n!\leq {\left(\frac{n+1}{2}\right)}^{n}$$ Quick way to subtract white outline from knotwork pattern in Illustrator? I am working with a pretty complex traced AI that I received, and I want to basically make it so that all the visually black stuff is shapes, and all the white outlines (invisible here) are removed. Most of the spaces when a line crosses over another is created by a white outline.   I know about expanding appearance and the pathfinder tool, but because there are so many lines that are crisscrossing, I haven't had an easy way of minusing the fronts away. I could expand appearance and sort every object carefully, but I feel like there may be a quicker way of doing this.  If I put it in Photoshop as a smart object in 'screen' mode, it looks right.. I could put that back in Illy and trace it, but I don't want to complicate the paths. Suppose John has 5 sweets. Is there any difference between the following two sentences?     Jack has 3 times as many sweets as John.      Jack has 3 times more sweets than John.   I prefer the first construction and would know unambiguously that Jack has 15 sweets in this case. However in the second construction I would be inclined to think that Jack has 20 sweets, since it seems to suggest 15 sweets in addition to the original 5. In how many ways can the letters of the word CHROMATIC be arranged? Allow people to answer protected questions if they have earned at least some amount of combined SE rep The purpose of protected questions is, according to the message presented on such a question, "to prevent 'thanks!', 'me too!', or spam answers by new users." The more verbose  agrees on the purpose. And the  says this is why protected questions exist:     Some questions are protected because they are expected to attract   either spam or users -- often new users -- who may mistake the site as   a traditional forum, posting "noisy" answers such as "Thank you" or   "This worked for me" or "I'm also having this problem".   Considering this, shouldn't a user be able to answer protected questions if they have gained at least 10 rep on any stack exchange site? If they've already demonstrated one site that they are familiar enough with the SE Q/A format to not be a drive-by useless answerer, shouldn't that confidence in them carry over to other SE sites?  Or maybe at some level of combined SE rep this restriction should be waived. I find it silly that at 15k+ combined network rep, I'm unable to answer a protected question on a SE site I've just started using. I'm getting "jumpy" loadings in rollapply PCA in R. Can I fix it? I have 10 years of daily returns data for 28 different currencies. I wish to extract the first principal component, but rather than operate PCA on the whole 10 years, I want to rollapply a 2 year window, because the currencies' behaviours evolve and so I wish to reflect this. However I have a major problem, that is that both the princomp() and prcomp() functions will often jump from positive to negative loadings in adjacent PCA analyses (ie 1 day apart). Have a look at the loading chart for the EUR currency:    Clearly I can't use this because adjacent loadings will jump from positive to negative, so my series which uses them will be erroneous. Now take a look at the absolute value of the EUR currency loading:    The problem is of course that I still cannot use this because you can see from the top chart that the loading does go from negative to positive and back at times, a characteristic which I need to preserve.   Is there any way I can get around this problem? Can I force the eigenvector orientation to always be the same in adjacent PCAs?   By the way this problem also occurs with the FactoMineR PCA() function. The code for the rollapply is here:  rollapply(retmat, windowl, function(x) summary(princomp(x))$loadings[, 1], by.column = FALSE, align = "right") -&gt; princomproll Practical way to implement prevention of IP Spoofing I am an undergraduate Computer Science student and was hoping to gain some knowledge of ways to help prevent IP spoofing but all the resources I have tried out elaborate this concept in a theoretical way. I want to try out my hands at one of the techniques like:      How do I simulate this whole situation in my own system were I myself am the attacker and I myself have to defend it? And once I have gained an understanding of it, then how do I start translating that into programming terms? canonical-livepatch, Doesn't seem to patch anything No border-spacing before first item and after last item Russian babel and chemformula Subcaption defaulting to capital letter labels? I seem to be having a unique problem. There are many questions asking about how to change subcaption to a label with capital letters but for me my program is defaulting to capital letters when I would prefer lowercase.  My preamble  \documentclass[12pt,reqno]{amsart} \setlength\parindent{0pt} %no auto indentation for new paragraphs \usepackage{geometry}  \geometry{a4paper}  \usepackage{natbib}  \usepackage[latin1]{inputenc} \usepackage{amsmath, amsthm, amssymb, calc, dsfont}    \usepackage{enumerate} \usepackage{subcaption, caption, graphicx} %figure packages \usepackage{authblk} %author affiliations \usepackage{setspace, lineno} %double spaced etc. options, line numbering  \usepackage{tikz} \usetikzlibrary{shapes,arrows,chains,automata,positioning}   My figure  \begin{figure}[h!]   \centering   \begin{subfigure}[]{0.48\textwidth}           \includegraphics{Figure1.pdf}           \caption{}           \label{FigPractice}    \end{subfigure}    \begin{subfigure}[]{0.48\textwidth}           \includegraphics{Figure2.pdf}           \caption{}           \label{PracticeAge}   \end{subfigure}   \caption{a caption} \end{figure}   I've also tried \renewcommand{\thesubfigure}{\alph{subfigure}} Getting opencv error in c++ Picking a $\delta$ for a convenient $\varepsilon$? I'm studying some proofs in a book from the library  and  unlike my own book, it doesn't prove by showing that a certain expression is striclty less than $\varepsilon$ (given an  appropiate $\delta$ and $\vert x-a\vert &lt; \delta$, etc). Instead, it shows that it's less than some expression involving $\varepsilon$.   E.g. Given $\varepsilon &gt; 0$, we can pick a $\delta &gt; 0$ so $|f(x) - g(a)| &lt;  \text{insert stuff} &lt; k \cdot \varepsilon$ when $|x-a| &lt; \delta$, and thus $f(x) - g(a) \to 0 $ when $x \rightarrow a$.  How does this make sense formally? (Intuitively, I am convinced.)  I don't have much experience with $\varepsilon$-$\delta$ -proofs, so I might be missing something obvious. Prove that $\int_0^1t^{p-1}(1-t)^{q-1}\,dt=\frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}$ for positive $p$ and $q$ I'm trying to prove that for $p,q&gt;0$, we have $$\int_0^1t^{p-1}(1-t)^{q-1}\,dt=\frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}.$$    The hint given suggests that we express $\Gamma(p)\Gamma(q)$ as a double integral, then do a change of variables, but I've been unable thus far to express it as a double integral.  Can anyone get me started or suggest an alternate approach?  Note: This wasn't actually given to me as the $\Gamma$ function, just as a function $f$ satisfying $$f(p)=\int_0^\infty e^{-t}t^{p-1}\,dt$$ for all $p&gt;0$, but I recognized that. This is in the context of an advanced calculus practice exam. Why KL divergence is non-negative? Why is KL divergence non-negative?  From the perspective of information theory, I have such an intuitive understanding:  Say there are two ensembles $A$ and $B$ which are composed of the same set of elements labeled by $x$. $p(x)$ and $q(x)$ are different probability distributions over ensemble $A$ and $B$ respectively.   From the perspective of information theory, $\log_{2}(P(x))$ is the least amount of bits that required for recording an element $x$ for ensemble $A$. So that the expectation $$\sum_{x \in  ensemble}-p(x)\ln(p(x))$$ can be interpreted as at least how many bits that we need for recording an element in $A$ on average.   Since this formula puts a lower bound on the bits that we need on average, so that for a different ensemble $B$ which brings about a different probability distribution $q(x)$, the bound that it gives for each element $x$ will surely not bit that is given by $p(x)$, which means taking the expectation, $$\sum_{x\in ensemble}-p(x)\ln(q(x))$$ this average length will surely be greater than the former one, which leads to $$\sum_{x\in ensemble }p(x)\frac{\ln(p(x))}{\ln(q(x))} &gt; 0$$  I don't put $\ge$ here since $p(x)$ and $q(x)$ are different.  This is my intuitive understanding, is there a purely mathematical way of proving KL divergence is non-negative? The problem can be stated as:  Given $p(x)$ and $q(x)$ are both positive over real line, and $\int_{-\infty}^{+\infty}p(x)dx = 1$, $\int_{-\infty}^{+\infty}q(x)dx = 1$.  Prove $$\int_{-\infty}^{+\infty}p(x)\ln\frac{p(x)}{q(x)}$$ is non-negative.  How can this be proved? Or can this be proved without extra conditions? How to obtain the sum of the following series? $\sum_{n=1}^\infty{\frac{n^2}{2^n}}$ It seems that I'm missing something about this.  First of all, the series is convergent: $\lim_{n\rightarrow\infty}\frac{2^{-n-1} (n+1)^2}{2^{-n} n^2}=\frac{1}{2}$ (ratio test)  What I tried to do is to find a limit of a partial sum $\lim_{n\rightarrow\infty}S_n$ as follows: $S_n=\frac{\frac{1}{6} n (n+1) (2 n+1)}{\frac{1-\left(\frac{1}{2}\right)^n}{2 \left(1-\frac{1}{2}\right)}}$. Still, the limit is $\infty$ and I'm clearly doing something wrong. /proc/PID/fd/X link number