Suppose $G$ is a group, $H\leq G$, and for all $g\in G$ we have $g^2\in H$. Is $H$ a normal subgroup of $G$?
Let $H$ be a subgroup of a group $G$ such that $x^2 \in H$, $\forall x\in G$. Prove that $H$ is a normal subgroup of $G$.    I have tried to using the definition but failed. Can someone help me please.
I would like to change a file extension from *.txt to *.text. I tried using the basename command, but I'm having trouble on changing more than one file.  Here's my code:  files=`ls -1 *.txt`  for x in $files do     mv $x "`basename $files .txt`.text" done   I'm getting this error:  basename: too many arguments Try basename --help' for more information Standard error of a ratio I have a linear regression and two estimates, say A and B and their standard errors. I need to find the standard error of a ratio A/B [or A/(1-B)].  I guess the main problem is that I don't know the correlation between A and B. I also guess that makes this unsolvable, since the correlation plays a major role. Is this correct?  If I knew it, what would be the way to calculate the standard error? Find $F$-related vector fields on $M\times N$, where $F(x)=(x, f(x))$ I am reading Lee's book on Differential geometry. In Chapter 4, Lee has this exercise.  Let $M$, $N$ be smooth manifolds, and let $f:M\to N$ be a smooth map. Define $F : M\to M \times N$ by $F(x) = (x, f(x) )$. Show that for every smooth vector field $V$ of $M$ , there is a smooth vector field on $W$ on $M \times N$ that is $F$-related to $V$.  I can understand that we must have $W_{(x,f(x))}=V_x\oplus Df(x)V_p$ for all $x\in M.$ The set $\{(x,f(x)):x\in M\}$ is a closed set in $M\times N.$ Therefore, if we can show that for all $(p,f(p))\in M\times N$ there is a neighbourhood $U_p$ and a smooth vector field on $U_p$ extending $V_x\oplus Df(x)V_x$ we are done by partition of unity. But I cannot show that. Can someone help me out? Single quotes vs. double quotes in Python According to the documentation, they're pretty much interchangeable.  Is there a stylistic reason to use one over the other? Allow us to vote to delete from the Android app Currently, the only way to vote to delete (and undelete, since you can't see deleted posts) in the Android app is to open the post in a browser, and then vote to delete from there.  Can we get delete vote functionality within the app, as well as the ability to delete our own posts?  Worth to mention: users , so it's possible in the API.  Related: How do I efficiently compute $a^b\bmod c$:  When $b$ is huge, for instance $5^{844325}\bmod 21$? When $b$ is less than $c$ but it would still be a lot of work to multiply $a$ by itself $b$ times, for instance $5^{69}\bmod 101$? When $(a,c)\ne1$, for instance $6^{103}\bmod 14$?  Are there any other tricks for evaluating exponents in modular arithmetic?  This is being asked in an effort to cut down on duplicates, see  and . I was wondering if there was any was I could right click inside any instance of a "Finder" window such that I have an option that says "Open Terminal Here". It would be really helpful. Copy existing property from e.g. Properties window to a custom panel So if I have this custom panel in the 3D view, is there a way I can take a property from for example the properties window and paste it onto my panel below the "text" prop, without creating a new property?  The reason I do not want to create a new property is because I want them to share all data, such as values, keyframes, drivers etc.        bl_info = {     "name": "",     "description": "",     "author": "",     "version": (0, 0, 1),     "blender": (2, 70, 0),     "location": "",     "warning": "",      "wiki_url": "",     "tracker_url": "",     "category": "" }  import bpy  from bpy.props import (StringProperty,                        BoolProperty,                        IntProperty,                        FloatProperty,                        FloatVectorProperty,                        EnumProperty,                        PointerProperty,                        ) from bpy.types import (Panel,                        Operator,                        AddonPreferences,                        PropertyGroup,                        )   class inouse_Settings(PropertyGroup):      my_bool = BoolProperty(         name="Enable or Disable",         description="A bool property",         default = False)     class UV_OT_my_panel(Panel):     bl_space_type = 'VIEW_3D'     bl_idname = "UV_OT_my_panel"     bl_label = "Custom"     bl_category = "Custom"     bl_region_type = 'TOOLS'      def draw(self, context):          scn = context.scene         mytool = scn.inhouse         layout = self.layout          col = layout.column(align=True)         row = col.row(align = True)         row.prop(mytool, "my_bool", text="Text")       def register():     bpy.utils.register_module(__name__)     bpy.types.Scene.inhouse = PointerProperty(type=inouse_Settings)  def unregister():     bpy.utils.unregister_module(__name__)     del bpy.types.Scene.inhouse  if __name__ == "__main__":     register() Divisibility Proof with Induction - Stuck on Induction Step I'm working on a problem that's given me the run around for about a weekend. The statement:  For all $m$ greater than or equal to $2$ and for all $n$ greater than or equal to $0$, $m - 1$ divides $m^n - 1$.  Using induction over $n$, the base step comes easy since $m^n - 1$ is $0$ when $n = 0$.  My induction hypothesis is letting $k \geq 0$ and assuming that $m - 1$ divides $m^k - 1$. In order to show that $m - 1$ divides $m^{k+1} - 1$, I obviously need to use the induction hypothesis. However, no matter where I try to use the fact that $m^k - 1 = (m-1)a$ for some $a$ in the integers, the expression $m^{k+1} - 1$ always becomes more difficult to get to $m^{k+1} - 1$ being equal to $(m-1)b$ for some $b$ in the integers.   In other words, I can't figure out any actually helpful way to apply the induction hypothesis with the goal of proving the next step.  Any help would be appreciated! Permalinks: custom post type -> custom taxonomy -> post I am having trouble working with WordPress rewrite rules, and could use some help.  I have a custom post type called _shows_.  All shows have a single custom taxonomy category _show-category_. A _show_ will never have more than one _show-category_.  I would like my urls to route in this manner:  www.mysite.com/shows/  =&gt;  archive-shows.php  www.mysite.com/shows/%category%/ =&gt; taxonomy-show-category.php  www.mysite.com/shows/%category%/%postname%/ =&gt; single-shows.php   So as a real world example, lets say we have a _show-category_ "Foo" and a _show_ post titled "Bar" that has "Foo" as it's _show-category_. I would expect my WordPress app to look like this:  www.mysite.com/shows/foo/ =&gt; shows all posts under the foo category www.mysite.com/shows/foo/bar =&gt; shows the indivual post   I try to avoid plugins when possible, but am open to any solution. For ridiculously small hard drives (8GB), what Ubuntu Based (although not necessarily official) distribution offers the smallest installed size?  Let me clarify: I still want a certain amount of personal file space, preferably a few gig. I also don't have a problem with performance, while it is mediocre, a 2.6GHz single-core and 1GB of ram suffice to run ubuntu fine, but i find that it  uses 6-7GB of space. I am aiming for closer to 4GB. Generating a separate pdf file of tables and figures from a latex file? Is it possible to generate a PDF of figures and tables of a paper from LaTeX itself?  I saw Journals and Conferences asking for a separate file of tables and figures.  In  question, questioner proposed a method that uses \usepackage{endfloat} to push all the figures and tables to the last pages and then use a PDF editor to cut the last pages of PDF. This is a solution but not a good one. LaTeX should be able to do this by its own.  What I have in mind is to compile the LaTeX file and it generates a complete PDF output and beside that a separate file that contains all the figures and tables of former PDF. Decision Tree in LaTeX with TikZ Grid line doesn't have to be included     \documentclass[10pt]{beamer}  \usetheme{default}  \usepackage{graphicx}  \usepackage{amssymb,amsmath}  \usepackage{tikz}  \usetikzlibrary{matrix}  \usepackage{adjustbox}  \usepackage{caption} \ usepackage{subcaption}   \setbeamercolor{title}{fg=black}  \setbeamercolor{subtitle}{fg=gray}  \setbeamercolor{footlinecolor}{fg=white,bg=green}  \setbeamertemplate{footline}[text line]{% \parbox{\linewidth}{\vspace*{-10pt} \textit{Where Wall Street Goes to School} \hfill \insertpagenumber} } \setbeamertemplate{navigation symbols}{} –  \begin{tikzpicture}[&gt;=stealth,sloped][scale=0.8]  \matrix (tree) [matrix of nodes, minimum size=0.25cm, column sep=2cm, row sep=1cm, ]  { &amp; $C_{i+1,j+1}$ \\ $C_{i,j}$ &amp;$C_{i+1,j}$ \\  &amp; $C_{i+1,j-1}$ \\   };  \draw[-&gt;] (tree-3-1) -- (tree-1-3) node [midway,above] {};  \draw[-&gt;] (tree-3-1) -- (tree-3-3) node [midway,above] {};  \draw[-&gt;] (tree-3-1) -- (tree-5-3) node [midway,below] {};  \end{tikzpicture}  \end{adjustbox} – I am using the Service Console in Lightning Experience, and using the Chatter Feed component on the Case layout to display updates to the Case.  If I use the "Log a Call" Quick Action from the Chatter publisher, these calls are displayed in the Chatter feed.  This is the desired behaviour.  However, if I insert a Log a Call record related to the Case any other way - e.g. using the Developer Console, or my CTI system's custom components - the call does not display in the feed.  If I query the record, it looks exactly the same as the record that is created with the Quick Action.  It just seems that the Chatter post is never made in this situation.  I have checked my Feed Tracking settings:   Task has Enable Feed Tracking and Subject checked Case has All Related Objects checked   How can I enable calls created with a custom component (part of a CTI system) to display in the case feed? How to install Chrome browser properly via command line? I am trying to install chrome browser via command line.  I tried this:  $ apt-cache search chrome browser   The results show that the proper term is "chrome-browser," so I tried that:  $ sudo apt-get install chrome-browser   And then "Y" for the Y/n question.  But the installation threw errors.  Does someone see anything wrong with the commands I issued? Order of calling base class constructor from derived class initialization list Passing multidimensional arrays as function arguments in C In C can I pass a multidimensional array to a function as a single argument when I don't know what the dimensions of the array are going to be?  Besides, my multidimensional array may contain types other than strings. Function returns None without return statement This is in very common use on the internet. I just read this:     "[The] hotel employee walks in, I say "uh, puppy" and she just NOPEd the [heck] out of the room."   This is often done by adding 'd to the word. Why using IN(...) when selecting on indexed fields, will kill the performance of SELECT query?