let $I \subseteq \mathbb{R}$ be an open interval. A function $\gamma : I \rightarrow \mathbb{R}^m $ is called a path and we write $\gamma (t)=(\gamma^{1}(t),\ldots,\gamma^{m}(t)) \in \mathbb{R}^m$. If all m components are differentiable functions from $I \ \mbox{to} \ \mathbb{R}^m$,then the path is a differentiable path. Then for every $t \in I$, we consider $\gamma'(t)=((\gamma^{1})'(t),\ldots,(\gamma^{m})'(t)) \in \mathbb{R}^m$ which is called the velocity vector of $\gamma$ at t.  Question: if $\gamma:I \rightarrow \mathbb{R}^m$ is a differentiable path and $||\gamma(t)||=1, \ \forall t \in I$. We fix a point $t_0 \in I$ and denote $\gamma(t_0):=\vec{a},\gamma'(t_0):=\vec{q}$ Prove that $$&lt;\vec{a},\vec{q}&gt;=0$$  My method is to use $0\leq |&lt;\vec{a},\vec{q}&gt;| \leq ||\vec{a}|| ||\vec{q}||=||\vec{q}||$, then by squeeze theorem, we can have the inner product equals zero. However, as you see, my method does not work.
How can I prove the following?  $\gamma (t)$ is unit speed, $\dot \gamma(t) \not= 0 \Rightarrow \ddot \gamma(t)$ is perpendicular to $\gamma(t)$   I don't really see where a problem would arise when $\dot \gamma(t)=0$ would cause such a problem
The equivalent electric field of a magnetic field Why separation of variables works in PDEs? Probably a basic issue, but still something that irks me and I can't find an answer online. I want to change the default folder for saving new shapes/processing outputs from &quot;last used folder&quot; to &quot;project folder&quot; or &quot;something in relation to project folder&quot; - is this possible? If so, how can I achieve this? Ubuntu desktop 18.04 LTS freezes with USB WiFi dongle OS: Ubuntu desktop 18.04 LTS Linux version 4.15.0-24-generic  USB WiFi: odroid WiFi Module 3 with Realtek RTL8188CUS-GR  syslog has only ^@  Jul  4 12:31:50 udbv-HP-ProBook sudo: pam_ecryptfs: pam_sm_authenticate: /home/udbv is already mounted ^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^$ Jul  4 12:33:30 udbv-HP-ProBook systemd-modules-load[396]: Inserted module 'ppdev' Jul  4 12:33:30 udbv-HP-ProBook systemd-modules-load[396]: Inserted module 'parport_pc' Jul  4 12:33:30 udbv-HP-ProBook systemd-cryptsetup[424]: crypt_init() failed: Block device required Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Starting Flush Journal to Persistent Storage... Jul  4 12:33:30 udbv-HP-ProBook swapon[415]: swapon: /swapfile: read swap header failed Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Started Create Static Device Nodes in /dev. Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Starting udev Kernel Device Manager... Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Started Set the console keyboard layout. Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Reached target Local File Systems (Pre). Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Mounting Mount unit for vlc... Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Mounting Mount unit for gnome-3-26-1604... Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Mounting Mount unit for gnome-calculator... Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Mounting Mount unit for pycharm-community... Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Mounting Mount unit for gnome-3-26-1604... Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Mounting Mount unit for skype, revision 41... Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Mounting Mount unit for gnome-calculator... Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Mounting Mount unit for gnome-calculator... Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Mounting Mount unit for skype, revision 38... Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Mounting Mount unit for pycharm-community... Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Mounting Mount unit for vlc... Jul  4 12:33:30 udbv-HP-ProBook kernel: [    0.000000] microcode: microcode updated early to revision 0x84, date = 2018-01-21   I tried following  but it didn't help.  Any ideas? Hill Cipher known plaintext attack I know a plaintext - ciphertext couple of length 6 for a hill cipher where its key is a [3x3] matrix.  Based on what I've read and learned, to attack and crack keys of [n x n], if we know a plaintext - ciphertext duo of length $n^2$ then we have our set of n equations with n variables, and this is generally solvable. However in my case there is only length of $6$ instead of $n^2 = 9$ .   My question is, how will I solve this problem and find the key? Or, since this is an obvious homework question, what should be my way of thinking? I cannot think of anything else but matrix multiplications and inverses but they do not help me at all. I am trying to fit a SVM to my data. My dataset contains 3 classes and I am performing 10 fold cross validation (in LibSVM):   ./svm-train -g 0.5 -c 10 -e 0.1 -v 10 training_data   The help thereby states:  -c cost : set the parameter C of C-SVC, epsilon-SVR, and nu-SVR (default 1)   For me, providing higher cost (C) values gives me higher accuracy. What does C in SVM actually mean? Why and when should I use higher/lower values (or the LibSVM given default value) of C? Diminishing returns / non-linear reputation from “great” answers Over time I realized a very annoying property of the Stack Overflow reputation system: Easy answers to popular question are rewarded with a lot more reputation points than actually great answers. In case anybody actually needs examples for this, here are two examples from :   gained me 420 rep, I invested 2 minutes writing it, and it only was a wild guess.   took me about an hour for the research and the math and gained me 40 rep.  Now I know that you are bound to get more rep for answers that are useful to more people, and I'd say that's fine. But the current situation is quite extreme: 30% of my reputation is from 5 of my 330 answers, and they are definitely far from being my best ones.  A final example that is not from my profile, but more extreme: Do you really think  is worth 40000 reputation?  My proposition to improve that situation would be to make the reputation gain from a single answer non-linear, maybe even have it capped at a certain point. I'm not sure on how that non-linear function should look exactly, and would leave good proposals for that to users with more experience than me. Convex function as supremum of countably many affine functions Let $f:\mathbb{R}^n\to \mathbb{R}$ be a convex function. It is well-known that (see e.g.~) that there exists a set $A$ such that  $$ f(x)=\sup_{a\in A}A(a)x+b(a), \quad \forall x\in \mathbb{R}^n, $$ where $A(a)\in\mathbb{R}^{n\times n}$ and $b(a)\in \mathbb{R}^n$.  I was wondering, is there any reference saying we can choose $A$ as a countable set? Show $GCD(a_1, a_2, a_3, \ldots , a_n)$ is the least positive integer that can be expressed in the form $a_1x_1+a_2x_2+ \ldots +a_nx_n$ Is there a way to see that $ \nabla_\mu g_{\nu \rho} = 0 $ without explicit computation, where $\nabla_\mu$ refers to the covariant derivative? In books, it is usually said that this is a consequence of the fact that parallel transport preserves dot product. How ? Ramsey Number Inequality: $R(\underbrace{3,3,...,3,3}_{k+1}) \le (k+1)(R(\underbrace{3,3,...3}_k)-1)+2$ I want to prove that:  $$R(\underbrace{3,3,...,3,3}_{k+1}) \le (k+1)(R(\underbrace{3,3,...3}_k)-1)+2$$  where R is a Ramsey number. In the LHS, there are $k+1$ $3$'s, and in the RHS, there are $k$ $3's$. I really have no clue how to start this proof. Any help is appreciated! I recently dismissed Dogmeat in favor of Codsworth, and I'm fairly sure I sent him to Sanctuary Hills. However, I've had a fairly detailed look around the place and I haven't found him yet. Does anyone know where he is? Change the URL in the browser without loading the new page using JavaScript How would I have a  action that may have some effects on the current page but would also change the URL in the browser so if the user hits reload or bookmark, then the new URL is used?    It would also be nice if the back button would reload the original URL.  I am trying to record JavaScript state in the URL. How to make a declaration from a command? There are two forms of setting the type style: the declarative method e.g, \itshape and the command form, e.g, \textit. The command form takes an argument whereas the declarative form does not --- it just applies itself to whatever comes afterwards.  I would like to develop a declarative form, say \UppercaseIt, of the command form \MakeUppercase{&lt;text&gt;}. I have read, in source2e.pdf on CTAN the code for implementing \textit, \itshape and \MakeUppercase and can't claim that I understand any of them. I am after something like {\UppercaseIt some text} instead of \MakeUppercase{some text}.  I only have a code outline of what I'm after, since I don't know how to do it:  \documentclass{article}  \newcommand{\UppercaseIt}{%   %... magic }  \begin{document}  \MakeUppercase{some text}% Yields SOME TEXT  {\UppercaseIt some text}% Should yield SOME TEXT  \end{document} To find tangents to given circle from a point outside it Find the combined equation of two tangents drawn from $P(x_1,y_1)$ to the circle $x^2+y^2 = a^2$. Point $P$ lies outside the circle. What is the easiest way to find and replace text in 20 text files in a directory? I have 20 text files with various extensions in a directory.  I want to replace "nnn" with "ooo" in all files.  I could import them into a project in Eclipse or Visual Studio to replace them all but that is overkill and would add other meta files etc.  What is a simple way to do this, I'm thinking: select all files, right click, choose some tool called (e.g. "QuickReplacer") and fill in find and replace, click and done.  Does anyone know of a tool like this? Why does the find command blow up in /run/? I used the following command to check python versions as given by . It does work except for one error, shown below. Can any one explain why?  $ sudo find / -type f -executable -iname 'python*' -exec file -i '{}' \; \     | awk -F: '/x-executable; charset=binary/ {print $1}' \     | xargs readlink -f | sort -u | xargs -I % sh -c 'echo -n "%: "; % -V'   Error:  find: ‘/run/user/1000/gvfs’: Permission denied Get a generic method without using GetMethods I want to get the method System.Linq.Queryable.OrderyBy&lt;T, TKey&gt;(the IQueryable&lt;T&gt; source, Expression&lt;Func&lt;T,TKey&gt;&gt; keySelector) method, but I keep coming up with nulls.  var type = typeof(T); var propertyInfo = type.GetProperty(group.PropertyName); var propertyType = propertyInfo.PropertyType;  var sorterType = typeof(Func&lt;,&gt;).MakeGenericType(type, propertyType); var expressionType = typeof(Expression&lt;&gt;).MakeGenericType(sorterType);  var queryType = typeof(IQueryable&lt;T&gt;);  var orderBy = typeof(System.Linq.Queryable).GetMethod("OrderBy", new[] { queryType, expressionType }); /// is always null.   Does anyone have any insight? I would prefer to not loop through the GetMethods result. Book recommendation on plane Euclidean geometry I consider myself relatively good at math, though I don't know it at a high level (yet). One of my problems is that I'm not very comfortable with geometry, unlike algebra, or to restate, I'm much more comfortable with algebra than geometry. I think that's mainly because my geometry education was sparse through the years, lacking in consistency etc. So I'd like to revise (and learn more) all at once, catching the basic axioms, understanding why such is such, etc. Essentially, a moderately rigorous textbook in plain Euclidean geometry (nothing fancy). Please don't say "The Elements" - I have browsed it at the bookstore, it is quite good, but not really what I'm looking for right now. I have a bunch of Spring beans which are picked up from the classpath via annotations, e.g.  @Repository("personDao") public class PersonDaoImpl extends AbstractDaoImpl implements PersonDao {     // Implementation omitted }   In the Spring XML file, there's a  defined:  &lt;bean id="propertyConfigurer"    class="org.springframework.beans.factory.config.PropertyPlaceholderConfigurer"&gt;     &lt;property name="location" value="/WEB-INF/app.properties" /&gt; &lt;/bean&gt;    I want to inject one of the properties from app.properites into the bean shown above. I can't simply do something like   &lt;bean class="com.example.PersonDaoImpl"&gt;     &lt;property name="maxResults" value="${results.max}"/&gt; &lt;/bean&gt;   Because PersonDaoImpl does not feature in the Spring XML file (it is picked up from the classpath via annotations). I've got as far as the following:  @Repository("personDao") public class PersonDaoImpl extends AbstractDaoImpl implements PersonDao {      @Resource(name = "propertyConfigurer")     protected void setProperties(PropertyPlaceholderConfigurer ppc) {     // Now how do I access results.max?      } }   But it's not clear to me how I access the property I'm interested in from ppc?