What is the mathematical rigorous proof that L1 regularization will give sparse solution?
Why does the Lasso provide Variable Selection?
Prove that if 51 positive integers between 1 and 100 are chosen, then one of them must divide another. Given $51$ integers $a_1, a_2, \ldots, a_{51}$ satisfying $1 \leq a_i \leq 100$. Prove that there exist $i$ and $j$ such that $a_i \mid a_j$.   I'm a little unsure how to approach this problem. Thanks for any help. Is it possible to install an earlier version of Skype on Windows 8? I want to writing the matrix on the arrow from X to Y 3x3 type, from Z to Y 3x2 type ; I want to do a search on StackOverflow for a few terms, but also by user, since I know which user referenced the item I'm looking for. I would like to replace the search submit button with a Font Awesome character like the Magnifier (Search Icon) character, and put Search the site inside the search box.  How can I achieve something like this? Do we actually get more sick (flu/cold) during winter? The word flu derives from the Italian phrase "influenza de freddo" meaning "influence of the cold".  Indeed it is that time of the year when my colleagues seem to have the flu/cold more often than not. I remember during university, winter lectures would be riddled with people coughing and sneezing - the so called "freshers flu"  Questions   Is there statistical evidence for this claim? If so, what are the causes/factors involved?   asks why our noses get runnier during cold whether, is this related? What is the word for when someone is overly nice and actually isn't nice at all? What is the word for when someone is overly nice and actually isn't nice at all because of the unnecessary added information/comments? For example, they will say things like "Your hair looks so good that wayyyy omg you should do it that way ALL the time! Looks way better than how you usually do it." And the word I'm looking for is not passive aggressive and it is not sarcasm. There is a word for these types of people. What is it? These people think they're being kind but actually blunt and rude. Another example would be for someone to say "hey you're beautiful, but maybe if you didn't wear harsh makeup you would look better" and they intend to sound like they're being nice when you know they don't like you and are saying in out of spite. Does $\lim_{n\to \infty} \int^{\infty}_{-\infty} f_n(x) dx\, = \int^{\infty}_{-\infty} f(x)\,dx\,$? Proving that these two sets are denumerable. (a) $S_k=\{A\subset\mathbb{N}: |A|=k\}$ for $k\in\mathbb{N}$  (b) $S = \bigcup_{k=1}^\infty S_k$   Work: For (a), I am not too sure about what approach I should use. I think finding a bijective function between the $S_k$ and $\mathbb{N}$ is hard to do . I also thought about using the Schroeder-Bernstein theorem and find injective functions $f:\mathbb{N}\rightarrow S_k$and $g:S_k\rightarrow\mathbb{N}$ in order to prove that $S_k$ and $\mathbb{N}$ are numerically equivalent. However, I have a hard time finding and proving these injective functions. I haven't really looked at (b) as I have yet to complete (a). What is the progression route in Terraria? Why is the de-facto standard sigmoid function, $\frac{1}{1+e^{-x}}$, so popular in (non-deep) neural-networks and logistic regression?   Why don't we use many of the other derivable functions, with faster computation time or slower decay (so vanishing gradient occurs less). Few examples are . One of my favorites with slow decay and fast calculation is $\frac{x}{1+|x|}$.  EDIT  The question is different to  as I'm only interested in the 'why' and only for the sigmoid. I often find  syntax very helpful, e.g. process substitution like in diff &lt;(sort file1) &lt;(sort file2).  Is it possible to use such Bash commands in a Makefile? I'm thinking of something like this:  file-differences:     diff &lt;(sort file1) &lt;(sort file2) &gt; $@   In my GNU Make 3.80 this will give an error since it uses the shell instead of bash to execute the commands. When I googled stomach flu, there were many questions and statements like      How do you get the stomach flu?    Or     Difference between a stomach flu and food poisoning    Or      Six facts about stomach flu   When do you use the, a or nothing? And which should I use for a sentence like "I got a stomach flu from someone" ? Evaluating $\int_{-\infty}^{\infty}\frac{1}{(x^2+b^2)^2}dx$ What we've gelost — why doesn't English use the prefix "ge-"? The Germanic languages that I'm familiar with all use a prefix similar to ge- on past participles:  German: Ich habe mir den Fuß gebrochen. Dutch: Ik heb mijn voet gebroken.  But English doesn't do this at all:  English: I've broken my foot.  Where did this prefix come from? Did English ever have the ge- past participle prefix? If so, why was it lost? 2 different passports with different surnames "In" and "on": How can I decide which one to use for vehicles? Examples:   In a car, van, etc. On a bus, boat, motorcycle, etc.   How can one decide which preposition to use? Is memorization the only way or is there a better way?  Note: People generally explain this by either distinguishing between open and closed vehicles or between large and small vehicles. However, the examples I've given defeat both explanations. "Give him a box that everyone knows what it contains." "Give him a box that everyone knows what it contains."  Is this correct English? It sounds wrong to me. This is a general, community wiki  question and answer pair intended to address any questions   concerning the reasons a site or specific site contents do not appear   in search engine results.      If your question was closed as a duplicate of this question and you   feel that the information provided here does not provide a sufficient   answer, please open a discussion on .   My site (or specific pages on my site) is not appearing in search engine results.  Why isn't my content indexed and what can I do about it? Equivalent definitions of convexity for $f\in\mathcal C^1(\mathbb R^n)$ I noticed many identical questions on this site relating to equivalent definitions of convexity, see for example , , , , , ,  and there might be more. So I wanted to create a &quot;big list&quot; of equivalent definitions of convex functions, I will start with some equivalent definitions and post my proofs in my answer below. Feel free to add any other equivalent definitions. This question can then serve as an abstract duplicate. Claim. Let $n\in\mathbb N$ and $f\in\mathcal C^1(\mathbb R^n)$. Then the following are equivalent:  $f$ is convex, i.e. $f(\lambda x_1 + (1-\lambda x_2))\le\lambda f(x_1)+(1-\lambda)f(x_2)$ for all $\lambda\in[0,1]$ and $x_1,x_2\in\mathbb R^n$. For all $x,y\in\mathbb R^n$, $\langle\nabla f(x)-\nabla f(y),x-y\rangle\geq0$. For every $x,y\in\mathbb R^n$, $f(y)\geq f(x)+\langle y-x,\nabla f(x)\rangle$.