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What is true for a ring with exactly two right ideals Visual Studio Solutions Folder as real Folders I have a Visual Studio Solution. Currently, it is an empty solution (=no projects) and I have added a few solution folders.  Solution Folders only seem to be "virtual folders", because they are not really created in the Filesystem and files inside solution folders are just sitting in the same folder as the .sln file.  Is there a setting that i've overlooked that tells Visual Studio to treat Solution Folders as "real" folders, that is to create them in the file system and move files into it when I move them inside the solution into one of those folders?  Edit: Thanks. Going to make a suggestion for VS2010 then :) Recovering data from a faulty disk When I plug in a removable 1TB USB disk in my laptop, the operations led keeps blinking for a while and then you can hear the servo mechanism resetting with a short "tack".  I'd like to recover the data.    I believe there might be some way to get the data through low level IOs.    What I believe is that the mounting process assumes the disk is OK and I should prevent this before using ddrescue or something similar.  Does anybody have a solution to this problem? Show that $(L^{p},\|\|_{p})$ is a Banach space. Show that $(L^{p},\|\|_{p})$ is a Banach space.  My approach: I prove the statement for $(L^{1},\|\|_{1})$, of the following way, first all, is easy show that $\|\|_{1}$ is a norm. So,  $(L^{1},\|\|_{1})$ is vector space. To show that is a Banach space, note that  Prop. 1: Let $(f_{n})_{n\in\mathbb{N}}\subset\mathcal{L}^{1}$ (where $\mathcal{L}^{1}=\mathcal{L}^{1}(X,\tau,\mu)$ is the space of all integrals function), such that $\sum_{i}{\|f_{n}\|}_{1}&lt;\infty$, then the sequences $\left(\sum_{n=1}^{N}{f_{n}}\right)_{n\in\mathbb{N}}$ converges almost everywhere to the integral function, that we called $\sum_{n\in\mathbb{N}}{f_{n}}$. Furthermore $$\sum_{n\in\mathbb{N}}{\int{f_{n}d\mu}}=\int{\sum_{n\in\mathbb{N}}{f_{n}d\mu}}\quad\lim_{N\to\infty}{\|\sum_{n=1}^{N}{f_{n}}-\sum_{n\in\mathbb{N}}{f_{n}}\|_{1}}=0$$ Prop. 2: Let $(E,\|\|)$ a normed vector space. Then $E$  is a Banach space if and only if for all $(e_{k})_{k\in\mathbb{N}}\subset E$ such that $\sum_{k\in\mathbb{N}}{\|e_{k}\|}&lt;\infty$, $\left(\sum_{k=1}^{n}{e_{k}}\right)_{n\in\mathbb{N}}$ converges in $E$  We take the proposition 2. Let $(f_{n})_{n\in\mathbb{N}}\subset L^{1}$ such that $\sum_{n}{\|\hat{f}_{n}\|_{1}}&lt;\infty$. Let $f_{n}\in\hat{f}_{n}$, then $\|f_{n}\|_{1}=\|\hat{f}_{n}\|_{1}$ and then $\sum_{n}{\|f_{n}\|_{1}}&lt;\infty$, by prop. 1, there exist $F\in\mathcal{L}^{1}$ such that $\|\sum_{n=1}^{N}{f_{n}-F\|_{1}}\to 0$, and then $\|\sum_{n=1}^{N}{\hat{f}_{n}-F\|_{1}}\to 0$. Finally note that $\hat{F}\in L^{1}$ is a Banach space. This was my answer for  $L^{1}$ space, but how I prove the general statement for $L^{p}$ (I want to do a similar response). Edit: I'm stuck in the problem, any idea or hint is appreciated.Thanks!! Calculate the area on a sphere of the intersection of two spherical caps I'm currently working on building a predictive model for a binary outcome on a dataset with ~300 variables and 800 observations. I've read much on this site about the problems associated with stepwise regression and why not to use it.  I've been reading into LASSO regression and its ability for feature selection and have been successful in implementing it with the use of the "caret" package and "glmnet".   I am able to extract the coefficient of the model with the optimal lambda and alpha from "caret"; however, I'm unfamiliar with how to interpret the coefficients.    Are the LASSO coefficients interpreted in the same method as logistic regression? Would it be appropriate to use the features selected from LASSO in logistic regression?   EDIT  Interpretation of the coefficients, as in the exponentiated coefficients from the LASSO regression as the log odds for a 1 unit change in the coefficient while holding all other coefficients constant. Supremum of a Continuous Function is Continuous Why does ltablex skip a number in captions? The below example presents two tables with an tabularx environment (taken from package ltablex) and a caption. After each tabularx-table, the table counter skips one, as if both the caption and the tabularx environment increase it, while the latter shouldn't.  \documentclass{report} \usepackage{ltablex} \begin{document}  \begin{table} \caption{Caption 1.} % "Table 1" \begin{tabularx}{\linewidth}{|c|} empty table \end{tabularx} \end{table}  \begin{table} \caption{Caption 2.} % "Table 3" \begin{tabularx}{\linewidth}{|c|} empty table \end{tabularx} \end{table}  \end{document}   My reputation doesn't allow me to upload a picture, but the captions are Table 1 and Table 3 respectively. If $f$ is continuous and $\,f\big(\frac{1}2(x+y)\big) \le \frac{1}{2}\big(\,f(x)+f(y)\big)$, then $f$ is convex Let $\,\,f :\mathbb R \to \mathbb R$ be a continuous function such that $$ f\Big(\dfrac{x+y}2\Big) \le \dfrac{1}{2}\big(\,f(x)+f(y)\big) ,\,\, \text{for all}\,\, x,y \in \mathbb R, $$ then  how do we prove that $f$ is convex that is $$ f\big(tx+(1-t)y\big)\le tf(x)+(1-t)f(y) , \forall x,y \in \mathbb R , t\in (0,1)? $$ I can prove it for dyadic rational's $t=\dfrac k {2^n}$ and then argue by continuity ; but I would like a more direct proof . Thanks in Advance How to ask for "Save Changes?" on Lighthing Component form when user tries to navigate away We have a Lightning Community and would like to prompt the user to "Save Changes" if they try to click one of the community's navigational components (home link, or other menu item) but they have unsaved form changes on a custom Lightning component.  Ideally, we'd have a window.confirm popup asking if they want to leave without saving their changes, and if they click Cancel, the URL navigation would be aborted.  Is this possible via custom Lighting Components and Lighting Communities, and if so, how? If not, are there any patterns to work around this? Find all primes $p>2$ for which $x^2+x+1$ is irreducible in $\mathbb{F}_p[x]$ Find all primes $p&gt;2$ for which $x^2+x+1$ is irreducible in $\mathbb{F}_p[x]$   Attempt. Since $x^2+x+1$ is of degree 2, it is reducible iff it has a root in $\mathbb{F}_p$. It has a root in $\mathbb{F}_p$ iff $x^3-1=(x-1)(x^2+x+1)$ has a root other than 1. The latter has a root other than $1$ iff $\mathbb{F}_p$ has elements of order $3$, which happens iff $3\mid |\mathbb{F}_p^{\times}|=p-1$, which happens iff $p\equiv 1\bmod 3$. So $x^2+x+1$ is irreducible iff $p\not \equiv 1\bmod3 $.   I am getting a different answer than my professor, who has $p\equiv 1\bmod 3$. If I am incorrect, were am I going wrong?  Update. Now I am unsure about my own reasoning. What if $1$ is a root of $x^2+x+1$? I don't know why I am considering roots of $x^2+x+1$ other than $1$ when $1$ can be a root. See how many duplicates of a post? Can I see how many duplicates there are of a post? I.e. how many posts have been labelled as duplicate because of this post? If so, how? And can I also get a list of these duplicated posts? I stumbled upon a  with the following paragraph:     MBIE has launched an investigation into claims Chinese galvanised   steel coil imports have been dumped on the local market, causing   "material injury" to New Zealand industry, and the case will end up on   the table of newly appointed Commerce Minister Jacqui Dean.   I thought that sentence was grammatically wrong because the author should put the word "that" (as a conjunction) right after the word "claim". I don't see why the author could drop the conjunction. Is this a prevailing phenomenon in American or Australian English? I am using XeTeX and have the following short document:  \documentclass[a4paper,12pt]{article} \usepackage{amsmath,amsthm} \usepackage[MnSymbol]{mathspec} \usepackage[no-sscript]{xltxtra} \defaultfontfeatures{Mapping=tex-text} \setallmainfonts{Minion Pro}  \begin{document}  $\phi, \varphi, \delta \ldots A \vee B$  \end{document}    Now no matter what font I am using the punctuation marks in math mode (commas, dots, \ldots etc.) come out in Computer Modern. Is there a way to change this? Show this equality (The factorial as an alternate sum with binomial coefficients). How to use siunitx and tabularx together? I'm using the tabular features of siunitx (for what it is worth: v1.3 since I'm on TexLive 2009) to align numbers in tables, eg.  \begin{tabular}{ p{1.5cm} S S S } bla &amp; 1.23 &amp; 4.5 &amp; 67.89 \tabularnewline \end{tabular}   and that works perfectly fine.  Now I would like to have this table span a specific width, or rather to have several tables to have the same width. For this I would normally use tabularx  \begin{tabularx}{\textwidth}{ p{1.5cm} X X X } bla &amp; 1.23 &amp; 4.5 &amp; 67.89 \tabularnewline \end{tabular}   Now, is there a way to combine the two? To have the number formatting and alignment features of siunitx inside an automatically resizing table? For as long as I can remember I have avoided using switch statement fall-through. Actually, I can't remember it ever entering my consciousness as a possible way to do things as it was drilled into my head early on that it was nothing more than a bug in the switch statement. However, today I ran across some code that uses it by design, which got me immediately wondering what everyone in the community thinks about switch statement fall-through. Is it something that a programming language should explicitly not allow (like C# does, though it supplies a workaround) or is it a feature of any language that is powerful enough to leave in the programmer's hands? Edit: I wasn't specific enough to what I meant by fall-through. I use this type a lot:     switch(m_loadAnimSubCt){         case 0:         case 1:             // Do something             break;         case 2:         case 3:         case 4:             // Do something             break;    }  However, I'm concerned about something like this.    switch(m_loadAnimSubCt){         case 0:         case 1:             // Do something, but fall through to the other cases             // after doing it.          case 2:         case 3:         case 4:             // Do something else.             break;    }  This way whenever the case is 0, 1 it will do everything in the switch statement. I've seen this by design and I just don't know if I agree that switch statements should be used this way. I think the first code example is very useful and safe. The second seems kind of dangerous. From injective map to continuous map How to show that topological groups are automatically Hausdorff? On page 146, James Munkres' textbook Topology(2ed),  Show that $G$ (a topological group) is Hausdorff. In fact, show that if $x \neq y$, there is a neighborhood $V$ of $e$ such that $V \cdot x$ and $V \cdot y$ are disjoint.  Noticeably, the definition of topological group in Munkres's textbook differs from that in .  A topological group $G$ is a group that is also a topological space satisfying the $T_1$ axiom, such that the map of $G \times G$ into $G$ sending $x \times y$ into $x \cdot y$ and the map of $G$ into $G$ sending $x$ into $x^{-1}$, are continuous maps. Prove $\frac{|a+b|}{1+|a+b|}<\frac{|a|}{1+|a|}+\frac{|b|}{1+|b|}$. Prove that for every $a, b\in R\setminus\{0\}$ is correct this inequality:  $$\frac{|a+b|}{1+|a+b|}&lt;\frac{|a|}{1+|a|}+\frac{|b|}{1+|b|}.$$