I have found a few different definitions of the term &quot;norm&quot; on a vector space $V$. Some of them require nonnegativity (i.e. the domain is $[0, \infty)$), while others don’t. Is there a way to derive nonnegativity from a definition like the following? A norm is a mapping $\|\cdot\|\colon V \to \mathbb{R}$ that satisfies: (N1) $\| k v\| = | k| \cdot\| v\| $ (N2) $\| v + w\|\leq \| v\| + \| w\|$ (N3) $\| v\| = 0 \iff v = 0$ This definition is taken from &quot;Lineare Algebra&quot; by Beutelspacher, but I have found similar definitions elsewhere.
Question:   Prove that a norm is always non-negative.   Assume that the norm is negative.   Define: $f: \mathbb{R}^{n}\rightarrow \mathbb{R}^{-}_{0}$  $v \mapsto \left ( \vec{v} \right )f=\left \| \vec{v} \right \|$  $\left ( \vec{u}+\vec{v} \right )f=\left \| \vec{u}+\vec{v} \right \| \in \mathbb{R}^{-}_{0}$ and  $\left ( \vec{u} \right )f=\left \| \vec{u} \right \|,\left ( \vec{v} \right )f=\left \| \vec{v} \right \|=\left \| \vec{v} \right \| \in \mathbb{R}^{-}_{0}$  At this point I am unable to progress further.  Any help is appreciated.
Invisible software updates list in Software Updater Is there a way to set my current location manually on Windows 8? Is there a way to set my current location manually on Windows 8? I don't have a GPS module, but the device is always located at the same place. The automatic detection via network sets a place 200 kilometers from here.  I searched for a setting that allows me to set the location by hand, but I didn't find one. Have red shifted photons lost energy and where did it go? Overnight charges for brokers holding stocks? I'm trying to learn about stock markets. I eventually want to invest a small amount over a long period. Yet, I notice on a lot of broker sites that they tend to charge an overnight fee for any stocks held over night. I presume this is every night thereafter?  I just wondered because it seems like a wall when you want to let your stock price grow, and an overnight charge eats into this. or, is this only the case for "CFD" stocks? I'm confused.  Any help?  Cheers Without appealing to choice, can we prove that if $X$ is well-orderable, then so too is $2^X$? Without appealing to the axiom of choice, it can be shown that (Proposition:) if $X$ is well-orderable, then $2^X$ is totally-orderable.  Question: can we show the stronger result that if $X$ is well-orderable, then so too is $2^X$?  Proof of Proposition. Pick any well-ordering of $X$. Then the lexicographic order totally orders $2^X$.  More explicitly: for any two $f,g \in 2^X$, define $f &lt; g$ iff   there exists $x \in X$ such that $f(x) \neq g(x)$, and if $x \in X$ is minimal such that $f(x) \neq g(x)$, then $f(x)=0$ and $g(x)=1$.   It can be shown that $&lt;$ totally-orders $2^X$. The set of functions which map convergent series to convergent series Suppose $f$ is some real function with the above property, i.e.  if $\sum\limits_{n = 0}^\infty {x_n}$ converges, then $\sum\limits_{n = 0}^\infty {f(x_n)}$ also converges.   My question is: can anything interesting be said regarding the behavior of such a function close to $0$, other than the fact that $f(0)=0$? Convergence of the series $\sum_n \frac{n^{\log(n)}}{\log(n)^n}$ I have a rough outline of a proof. Is this correct?  $$\sum_{n \geq 2} \frac{n^{\log(n)}}{\log(n)^n} \to a \in \mathbb{R} \iff  \sum_{n \geq 2} 2^n \frac{(2^n)^{\log(2^n)}}{\log(2^n)^{2^n}} \to b \in \mathbb{R}$$ via dyadic criterion (or Cauchy condensation test).  Analyzing the right-hand side:  $$\sum_{n \geq 2} 2^n \frac{(2^n)^{\log(2^n)}}{\log(2^n)^{2^n}} = \sum_{n \geq 2} 2^n \frac{(2^n)^{n\log2}}{(n\log2)^{2^n}} = \sum_{n \geq 2} \frac{(2^n)^{n\log2 + 1}}{(n\log2)^{2^n}} = \sum_{n \geq 2} \frac{2^{n^2\log2 + n}}{(n\log2)^{2^n}} = (\ast). $$  We know for some sufficiently large $N$, $n \log 2 \ge 2$ for all $n\ge N$, so we can bound the series above by:  $$(\ast) \leq C_N + \sum_{n \geq N} \frac{2^{n^2\log2 + n}}{2^{2^n}} = \sum_{n \geq N} 2^{n^2\log2 + n - 2^n} = (\ast\ast),$$  where $C_N$ is the finite sum of the original series until $N$. And once again, for sufficiently large $M$, $m^2\log2+m - 2^m \leq -m^2$ for all $m \ge M$. Hence  $$ (\ast\ast) \leq C_{\max(N,M)} + \sum_{n \geq \max(N,M)} 2^{-n^2} &lt; \infty.$$  We then use the $p$-test to conclude that the sequence does converge. I would like to work out the Fourier transform of the Gaussian function   $$f(x) = \exp \left(-n^2(x-m)^2 \right)$$  It seems likely that I will need to use differentiation and the shift rule at some point, but I can't seem to get the calculation to work. Does anyone have any advice?  By the way, I am using   $$\mbox{FT}(f)(k)=\int_{-\infty}^\infty f(x)e^{-ikx}\,\mathrm d x$$  as my definition of a Fourier transform. Many thanks. Definitions of "linearity" across branches of mathematics or levels of math education Linearity is a ubiquitous concept in mathematics; however, each branch of mathematics appears to have its own definition of what a linear map (function, functional, functor, transformation, form, operator, or whatsoever) is. These different definitions can lead to confusion, especially when people apply conflicting definitions of linearity to the same mathematical object (just  to figure out what I mean).  One of my favorite examples with potential for conflict is the natural logarithm $\ln : \mathbb{R}_{&gt;0} \to \mathbb{R}$, which the majority of people, as I could learn during hot disputations not only  but also with colleagues in my field, regard as clearly non-linear, even though it fully complies with the definition of a linear map from the vector space $\mathbb{R}_{&gt;0}$ to the vector space $\mathbb{R}$ (cf. Definition 4 below). A similar conflict arises analogously for the exponential function $\exp :\mathbb{R} \to \mathbb{R}_{&gt;0}$.  I do not want to initiate yet another discussion here, about what the right and what the wrong definition of linearity is! Rather, ...  ... I would like to compile a list of existing definitions, to get a better overview of the different uses of the term "linearity" across the different branches of mathematics and levels of education.    Let be begin with the following four, which are certainly far from constituting a complete list. What about bilinear or multilinear forms, linear morphisms and linear functors in category theory, linear metrics, or other concepts of linearity? I am looking forward to see your answers.  At school I was taught that     Definition 1:   A function $f$ is linear if it is of the form $f(x) = ax + b$.   Later I learned that $f(x) = ax + b$ is affine, rather than linear, because     Definition 2:   A function $f$ is linear if it is of the form $f(x) = ax$.   From real analysis and systems theory, I know the following     Definition 3:   A map $f$ is linear if         $f(x + y) = f(x) + f(y)$   $f(a \cdot x) = a \cdot f(x)$.      This definition also applies to the expected value in statistics, as        $\mathrm{E}(X + Y) = \mathrm{E}(X) + \mathrm{E}(Y)$   $\mathrm{E}(a \cdot X) = a \cdot \mathrm{E}(X)$.      In a linear algebra course, I learned for the first time that we must not forget to take into account that the vector-space operations of the domain and the codomain of a map $f$ may be dissimilar. In fact, if $(V, \oplus, \odot)$ and $(W, \boxplus, \boxdot)$ are two $\mathbb{K}$-vector spaces, then     Definition 4:   A map $f : V \to W$ is linear if         $f(x \oplus y) = f(x) \boxplus f(y)$   $f(a \odot x) = a \boxdot f(x)$.         Notice that I deliberately used distinct symbols for the operations in both vector spaces, to emphasize how this definition differs from Definition 3.   Correct me if I am wrong, but this definition should also apply to maps between two modules.    Updates: How to search for email of a particular date from years ago in Gmail There are similar questions and several answers. I tried to follow the methods provided in the answers.  They did not work. So I am going to ask it again with more specifics.  I only knew the date of the email in the in-box in 2012 and the content of it but did not know the exact wording of the subject. Is it possible to find this email? Ubuntu 19.10 change desktop environment I've installed another desktop environment which I would like to try. I am not sure how to switch to it.  Is there still a button on the log in screen that will switch desktop environments? This same question for previous versions of ubuntu yields screenshots of this button. But I cannot find any button on the log in screen that does this.  So, in ubuntu 19.10, does there exist a button to switch desktop environments? If so, where is it? If not, what is an alternative method to switch desktop environments after I've already installed one?  ██████@████-linux:~$ ls /usr/share/xsessions gnome-flashback-compiz.desktop    ubuntu.desktop gnome-flashback-metacity.desktop   The gear icon does not show up for me, as the answers indicate it should.    Solved. I was not logged out properly. I locked the screen thinking that was logging out. Logging out properly fixed the issue. How does negation affect the use and understanding of "or" and "and" I'm trying to make more sense of how negation effects how a sentence is parsed and understood if and's and or's are used within them.  Pop quiz: You are trapped on a bus with a bomb going 50 MPH. You have a radio... (okay enough Keanu Reeves references) :-)  Let's say the bomb expert says "Don't cut the red or green wires." I think that means "Do not cut the wires colored red. Do not cut the wires colored green."  Now, let's say the bomb expert said on the radio "Do not cut the red and green wires", instead. Do I take that to mean:     "Do not cut the wires colored red. Do not cut the wires colored green."   OR     "Do not cut the wires whose insulation has red and green stripes." ?   OR     "It's okay to cut the wires colored red or the wires colored green, as long I don't cut one of each color."   Which meaning should I take if the radio breaks and I can't ask follow up questions? Limit of a stochastic integral Here is a minimal example that showcases my problem:  \documentclass{article}  \usepackage{amsthm} \usepackage{hyperref} \usepackage{cleveref}  \newtheorem{thm}{Theorem} \newtheorem{lem}[thm]{Lemma}  \begin{document}  \begin{lem} Lemma \end{lem}  \end{document}   The warning I get is the following:  [1pdfTeX warning (ext4): destination with the same identifier (name{thm.1}) has been already used, duplicate ignored  \AtBegShi@Output ...ipout \box \AtBeginShipoutBox                                                    \fi \fi  l.16 \end{document}                    {&lt;install dir&gt;/MiKTeX/2.9/pdftex/config/pdftex.map}]   I know that it is just a warning and that, my document will still work perfectly fine with regards to cross-referencing and hyperlinking, but I would still like to know why I am getting this warning and if anything can be done to fix it without removing functionality of the included packages.   Bonus question: Whats going on behind the scenes when this warning (or any generic 'destination with the same identifier (name{XXX.YYY}) has been already used, duplicate ignored') is produced, as this is not limited to theorem environments.  From log file:  Package: amsthm 2004/08/06 v2.20 Package: hyperref 2011/04/17 v6.82g Hypertext links for LaTeX Package: cleveref.sty 2011/03/22 v0.17.9 Intelligent cross-referencing Package cleveref Info: `hyperref' support loaded on input line 2157. Package cleveref Info: `amsthm' support loaded on input line 2300. What are the different endings in Dishonored? I have the CKeditor plugin installed by itself (i.e. not using WYSIWYG module.) Then, it uses a CKEditor library located in sites/all/libraries/ckeditor. This works fine. I'd like to add more plugins to it; namely, the footnotes plugin. (For the moment, please refrain from "just download a new library from ckeditor.com and replace it"; there were other issues/considerations involved.)  Now, I see this everywhere when I try to look for a solution  config.extraPlugins = 'footnotes';   I tried adding that to admin/config/content/ckeditor/edit// -> Advanced Options -> Custom JavaScript configuration, but it white screens the editor.  I tried adding it to the config.js in sites/all/libraries/ckeditor like below, but it had no effect.  CKEDITOR.editorConfig = function( config ) {   config.extraPlugins = 'footnotes'; };   I can't really add it to a ckeditor.config.js file, dump that file in my theme, and choose "Yes" for "Load ckeditor.config.js from the theme path", since (for content creation/editing) it uses the admin theme which is different from the theme we're developing.  So, I'm at a loss. The only other thing I can think of is to use a hook in the module we have for the site, but I'm not sure if that's write/what hook would be called.  Is it just me, or does it seem like clear documentation on how to manage CKEditor in Drupal is severely lacking? Any help would be appreciated.  UPDATE: After some testing, it looks like the "Custom JavaScript configuration" is supposed to work, but it is possibly broken for CKEditor 4.4+ I read something (can't seem to find it now) that suggested the plugin/widget import was broken on 4.4+, so I changed to 4.3 and the extraPlugins worked. Now, I wasn't able to get footnotes working (which is what kicked this whole thing off), but some other plugins that previously failed during the same steps now work on 4.3. So, there's that. I'm wondering now if it's something with footnotes itself. Note: this post is different to , which just cites part from somewhere else, without any further analysis and thinking. Furthermore, none of the answers to that post answers my question - the situations where &quot;will (going to) be doing&quot; is more appropriate.   I will be participating in IELTS test next Tuesday afternoon.  In an online class, the teacher says the following at the beginning of the lesson.  I will be using NotePod (pointed out by red rectangle) to write relevant points we talk about today.   I guess &quot;will be using&quot; is more appropriate than &quot;will use&quot; there, right?  A  explains  This &quot;tense&quot; is used for reporting a scheduled activity with special focus on its timing— as if one is standing in the future and experiencing the flow of the time. The focus is on the temporary, ongoing, or repetitive nature of the activity.   I understand the grammar rules of those verb tenses and the difference between this group  I am going to drive home tomorrow.   I am going to be driving home tomorrow.  The former is just convey my activity in the near future while the latter probably makes an implication that &quot;don't disturb me as I am driving in that time&quot; However, in the following situations, I don't see the difference between 2 verb tenses.  Welcome back to this class. Today's lesson is for the IELTS students who are going to be taking the IELTS test. it's for everybody to be honest with you. this lesson is useful for anybody who needs to do some writing.   who are going to take the IELTS test ... ref to subsection number only is there a way to reference the subsection part only? Example:  \documentclass{article} \begin{document}  \section{This is my section} For further information see subsection \ref{subsecthree} within this section. \subsection{One} \subsection{Two} \subsection{Three}\label{subsecthree}  \end{document}   Usually gives (without the stars):  1 This is my section For further information see subsection **1.3** within this section.   But I want:  1 This is my section For further information see subsection **3** within this section.   Is that possible?  [Edit] To make it more clear: I am looking for a command that works like \ref, but prints (the arabic number of) the value of the subsection counter at the position of the \label.  ======================= == Solution in LaTeX ==  There is another solution below. This is a different solution I came up with  (I cannot post an answer to my own question, so I put it here):  \documentclass{article}  \newcommand{\labelsubseccounter}[1]{     \renewcommand\thesubsection{\arabic{subsection}}     \addtocounter{subsection}{-1}     \refstepcounter{subsection}     \label{#1}     \renewcommand\thesubsection{\thesection.\arabic{subsection}} }  \begin{document}  \section{This is my section} For further information see subsection \ref{subsecthree} within this section,  which is called \ref{onepointthree}, not \ref{onepointfour}. \subsection{One} \subsection{Two} \subsection{Three}\label{onepointthree}\labelsubseccounter{subsecthree} \subsection{Four}\label{onepointfour}  \end{document}   (I would like to insert the compiled code, but don't know how, sorry) Directing graph such that any outdegree would be at most 2 Let $G=(V,E)$ be a graph. Suppose that for every subgraph $G'=(V',E') , G' \subset G$, we have $|E'| \le 2|V'|$. Show it's possible to direct $G$ such that any outdegree would be at most $2$.   I tried proving it by induction. I've realized that the minimum degree at any subgraph of $G$ is at most 4. I tried removing the vertex $v_0$ with the minimal degree at a certain subgraph from the original graph, claiming that the graph from the remaining vertexes can be directed as required. How can I include the vertex $v_0$ in the new directed graph such that any outdegree would still be at most 2? Thanks I'm trying to get images in my document in two different ways:   On top of the entire page, centered On top of a single column   I tried it with both twocols and multicols. I ruled out twocols because it forces a new page every time you turn it on, because I need the abstract to be wide and the rest of the cover page to be two columns.  Here are some examples of both points (screenshot from a downloaded PDF):   I want to achieve both, but somehow I can only solve the first point with:  \end{multicols}  \begin{figure}   \centering     \includegraphics[width=0.5\textwidth]{figure}   \caption{Dataflow pipeline of the edge detector}   \label{fig:workflowedge} \end{figure}  \begin{multicols}{2}   Is it awkward to turn off multicols, then turn it back on again?  For my second point, I've tried everything, but almost always, the image doesn't show up at all! The only way I can get it to show at all is by specifying the H placement specifier, but that places the image inside the text where I defined it, not on top of the column.  This is my best try, but it hides the image and breaks all references to ??:  \begin{figure}[t!]   \centering     \includegraphics[width=0.25\textwidth]{figure}   \caption{Dataflow pipeline of the edge detector}   \label{fig:workflowedge} \end{figure}   and I'm getting the warning:  Package multicol Warning: Floats and marginpars not allowed inside `multicols' environment!.   How can I make a figure a non-float?  I also tried the figure* environment, but that just creates an image centered on top of the page, like my first point.