Example of nonconvex function with nonempty subgradient I understand subgradient is well defined for convex functions such as $f(x)=|x|$ but not for nonconvex functions. I don't have a good intuition how subgradient is defined for a nonconvex function. Can someone please provide a simple example that subgradient does exist for a nonconvex function? I tried to search but without success. Of course somebody have asked questions involving keywords nonconvex and subgradient, but that doesn't mean the same question.
Subgradients of non-convex functions In  (section 2.3), it is stated that:     A point $x^*$ is a minimizer of a function $f$ (not necessarily convex) if and only if $f$ is subdifferentiable at $x^*$ and $0 \in\partial f(x^*).$   Could anybody provide me with references for a proof of the above statement?   Is there a reference where we can learn more about subgradients of non-convex functions?  In Section 3. Calculus of subgradients of the above notes,many properties of subgradients are presented for convex functions. I would like to know which ones among these properties still hold for non-convex functions.  Thank you in advance!
(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). I am using QGIS and I created a vector grid on a crop field in order to cut it into plots. I want to save each grid cell into individual polygon shapefiles automatically.  Here is an example of a vector grid I would like to split into 117 shapefiles:    Eventually, I did it manually with the selection tool and "Save as" but it is really time-consuming when there are more than 100 plots to save.    Is there any way to do it faster and more automatically?  Each plot as an attribute named "Plot" that can discriminate against them (basically being id + 1). It's the attribute used here to label them. What should happen if a logo I made is similar to another? What word means what many people think 'ironic' means? 'Ironic' is often used to mean an unusual coincidence rather than its true meaning which is closer to sarcastic. That being said, is there a word that would be a good replacement for what many people seem to think it means? How to find the number of automorphisms on $\mathbb Z_2\times \mathbb Z_4$ Well, I was trying find out the number of group homomorphisms from $U(20)$ onto $U(15)$ which I reduce to the problem of finding the number of automorphisms for $\mathbb Z_2\times \mathbb Z_4$. But don't know how shall I proceed.   Any help will be appreciated. What is a "callback" in C and how are they implemented? From the reading that I have done, Core Audio relies heavily on callbacks (and C++, but that's another story).    I understand the concept (sort of) of setting up a function that is called by another function repeatedly to accomplish a task.  I just don't understand how they get set up and how they actually work.  Any examples would be appreciated. How to generate a random snowflake I recently moved into an in-law unit of a large old house.  In the kitchen, there are two pipes coming out of the wall (one protruding out a little more than the other).  I had assumed they were gas lines to connect my stove.  But since the stove only connects to one pipe, I'm now wondering if they're actually water lines. Can anyone explain how I can tell the difference? On the other side of the wall is a large closet with the air conditioning system, water heater, and what I think is the gas shutoff valve. Literature with Biber generates strange citations: firstnames appear erratically I had been using bibtex and then biblatex for quite some time. For the newest edition of my book I moved to biber. I happened t notice that firstnames are sometimes generated and sometimes not. Has this been observed by others?   I have tried to make a minimal example but without success. Using the following code, which shows all the commands related to biblatex I have in my preamble I could not get the firstnames to be shown. Even with the stylefile I made for this book (second line) the error did not occur.  I have a dropbox with the latex files for the book available. I can give access to these.   What do you propose?  \documentclass[fontsize=10pt,twoside]{scrbook} \usepackage{hormonbuch} \usepackage[%   style=authoryear%   ,natbib=true%   ,sorting=nyt,backend=biber,doi=false,isbn=false,url=false]{biblatex} \addbibresource{../Literatur/EndokrinologieKundeNourl.bib}  \renewcommand*{\bibpagespunct}{\addcolon} \DeclareFieldFormat{pages}{#1}  \renewrobustcmd*{\enquote}{}  \renewbibmacro{in:}{%       Herbert Voss auf http://tex.stackexchange.com/questions/10682/suppress-in-biblatex   \ifentrytype{article}{}{%   \printtext{\bibstring{in}\intitlepunct}}}  \renewbibmacro*{journal+issuetitle}{%   \usebibmacro{journal}%   \setunit*{\addspace}%   \iffieldundef{series}     {}     {\newunit      \printfield{series}%      \setunit{\addspace}}%   \usebibmacro{volume+number+eid}% %  \setunit{\addspace}%   \newunit}  \renewbibmacro*{volume+number+eid}{%   \printfield{volume}% %  \setunit{\addcolon\space}%   }% % \renewbibmacro*{note+pages}{%   \printfield{pages}%   \newunit} % \begin{document}  \citet{CRT+88,FDe+85} \citep{CRT+88,FDe+85} \cite{CRT+88,FDe+85}  \printbibliography  \end{document} By accident, I used rm on a file I didn't want to delete. Is there a way that I can get it back under Linux? Torsion elements of a group aren't necessarily a subgroup In a question from an exam in an undergraduate group theory course, we were asked to prove or disprove that the set of all Torsion elements of a group is necessarily a subgroup.  I knew that the set of Torsion elements is closed under the inverse operation, but was later told that it is not closed under multiplication, therefore disproving the claim. However, I couldn't find any examples of a group $G$ and two elements $a,b$ such that both $a$ and $b$ have a finite order, but $ab$ doesn't. I know that for this to happen $G$ must be an infinite and non-Abelian group, but still couldn't find a valid example.   What are some examples of groups/elements fulfilling the aforementioned property? Prove that f is one-to-one on D What are the advantages of stacking multiple LSTMs? What are the advantages, why would one use multiple LSTMs, stacked one side-by-side, in a deep-network? I am using a LSTM to represent a sequence of inputs as a single input. So once I have that single representation— why would I pass it through again?  I am asking this because I saw this in a natural-language generation program. Prove that any continuous map on $\mathbb{R}$ which is the identity on $\mathbb{Q}$ is the identity map, hence Aut$(\mathbb{R}/\mathbb{Q})= 1.$  proof: Suppose $\sigma$ is continuous on $\mathbb{R}$ which is an identity on $\mathbb{Q}$. let $a \in R $. Then since $\sigma$ is continuous, then by definition there is $\delta_1 &gt; 0$ such that $|a - b| &lt; \delta_1$.  And assume $|\sigma(a) -\sigma(b)| &lt; \epsilon/2$ .  Let $\delta $= min{$\epsilon/2, \delta_1$}.  And now let $b \in \mathbb{Q}$ such that $|a - b| &lt; \epsilon$. Then  $|\sigma(a) - a| = |\sigma(a)  - b + b - a| \leq |\sigma(a) - \sigma(b) | + |b - a| &lt; \epsilon/2 + |b-a|$.  Can someone please help me? I am not sure . Can I say $|b - a| &lt; \epsilon/2 $? Windows 7, any way to sort folders and files all mixed together? Is there a way to sort folders and files by their name all mixed together in Windows 7?  It would be easier to understand if you .  I want my folders and files arranged like the BEFORE situation of the first screenshot.  If there is no way to do that with Windows Explorer, is there any file management programs that I should try?  I've already tried Total Commander. I have a complex command that I'd like to make a shell/bash script of.  I can write it in terms of $1 easily:  foo $1 args -o $1.ext   I want to be able to pass multiple input names to the script. What's the right way to do it?    And, of course, I want to handle filenames with spaces in them. I want to make a generic class that accepts only serializable classes, can it be done with the where constraint?  The concept I'm looking for is this:  public class MyClass&lt;T&gt; where T : //[is serializable/has the serializable attribute] Ubuntu 13.10 Macbook Air Mavericks Parallels 9 Black Screen I can install Ubuntu 13.10 in parallels ok and then log in.  Problem is Ubuntu does an update and I get black screen / desktop such that I can't see any app windows that I open.  I've tried a few of the other suggestions and they don't work for me. How Do I Reset A Manual Installation Of Windows Subsystem For Linux? Related:   When installing a WSL distribution outside of the Microsoft Store (for example on a restricted network, see linked), many of the standard means of interacting with that distribution are not available.  In my example, when I go through Settings > Apps and Features, the distro is not shown in the list and the management features are not available.  If I get my manually installed linux instance into a bad state, what is the correct way to reset it and bring it back to a fresh install?  Edit:  Examples of a 'bad state' encountered thus far include a BSOD during an 'apt-get update', leading to an unrecoverable state.  Also, performing multiple configuration changes to troubleshoot a problem, removing or disabling services and installing other packages without keeping close track of what was changed, with a desire to "start fresh" to test if the solution arrived at does in fact work on its own.. Sophomore's dream: $\int_0^{1} x^{-x} \; dx = \sum_{n=1}^\infty n^{-n}$