Is trace of regular representation in Lie group a delta function? My major is physics. I need to use some tools in group theory, but I am really  confused by the trace in compact infinite groups. The following is my question:  In discrete group theory, the irreducible representation of identity group element $e$ is always an identity matrix. So the trace of $e$ under  regular representation is the order of the group : $\chi(e) = |G|$. I hope to get  similar result for Lie group. For example, SO(2) has infinite number of irreducible representations, $D^{(m)}$, where $m=0,\pm 1, \pm 2,\cdots $. All of them are 1-dimensional $D^{(m)} = e^{im\phi}$. Here $\phi$ is the rotation angle. For element $R(\phi)$ in SO(2), the trace in $m$th irreducible representation should be  $\chi(R(\phi))=e^{im\phi}$, if we sum all these traces up, we get the trace in regular representation. So what is the trace of $e$ in the regular representation ? It seems to be infinite.
Is trace of regular representation in Lie group a delta function? My major is physics. I need to use some tools in group theory, but I am really  confused by the trace in compact infinite groups. The following is my question:  In discrete group theory, the irreducible representation of identity group element $e$ is always an identity matrix. So the trace of $e$ under  regular representation is the order of the group : $\chi(e) = |G|$. I hope to get  similar result for Lie group. For example, SO(2) has infinite number of irreducible representations, $D^{(m)}$, where $m=0,\pm 1, \pm 2,\cdots $. All of them are 1-dimensional $D^{(m)} = e^{im\phi}$. Here $\phi$ is the rotation angle. For element $R(\phi)$ in SO(2), the trace in $m$th irreducible representation should be  $\chi(R(\phi))=e^{im\phi}$, if we sum all these traces up, we get the trace in regular representation. So what is the trace of $e$ in the regular representation ? It seems to be infinite.
Why do some devices have 200V on the case? I recently got a desktop LED lamp, and I feel a "strange" slight vibration when touching the metal case. I found that other devices also have high voltage on their case.  What is the reason/trade-off to design the PSU that way? Are "real" transformers inefficient? Even if it's harmless, why is the high voltage connected to the metal case, and not just through insulated wires?    When I turn the lamp on, the setup in the picture measures the power line (220V) 214Vpp (59Vrms). When I also touch the metal case, the voltage drops down to around 30V.  AC adaptor: How can I hibernate or sleep my Ubuntu 10.10 desktop and have it "wake up" the next day?  I have seen software that can do this on windows, so it can't be hard on Ubuntu! Change to default start folder for Windows command prompt My new Windows XP work computer's command prompt automatically brings up "H:\&gt;" when I open it.  Is there a way for me to change that safely?  Specifically, to "C:\foo\bar\Something Else"?  I found the method of changing the Autorun value through Google, but it came with a warning that it "might affect the functionality of batch scripts."  Bonus note: I was originally getting at this issue in , but I didn't ask very directly.  The answer I ended up accepting was really good, so I wanted to leave it there. The exact phrase from the book of Swan (the context is choosing the words permit or allow) -      The passive structure with it is only possible with permit   Having said that...     It is not allowed to smoke in the kitchen - incorrect     It is not permitted to smoke in the kitchen - correct   I feel absolutely fine for the incorrect sentence.   Your inputs please. Make me understand this. My earliest PhD graduation date is 10 months away. Funding permits me to stay 1–3 months longer, so the final date is still uncertain.  It is never too early to make contacts for possible post-docs. But how long in advance is too early to formally apply to post-doc job openings or funding opportunities? Depending on the bureaucracy, the path from applying to starting can probably be anything from a week up to a year — someone I know is applying with  which had the opening advertised in August, application deadline in October, will have interviews in December/January, and the commencement of work only in September, 13 months after the initial job advertisement. My personal guess is that the long time between interviews and starting is to create a level playing field between those who do and those who don't need to go through a visa application procedure.  If I'm too early in applying for a post-doc, a hosting institution might prefer someone who can start earlier and where the graduation date is more certain. For example, in my case the graduation date is still uncertain, so there is a risk in accepting even if they are willing to wait. Project funding might also require the money to be spent rather soon.  If I wait too long, there might be a considerable period where I'm between jobs. Having a long (>6 months) period between jobs is a disadvantage for me. Maybe it's also a disadvantage for a hosting institution (I'm not sure).  How do I find the balance? For example, considering the duration of funding and visa applications, does it make sense to apply e.g. ? Or is it in any case virtually inevitable to have at least several months before my PhD graduation and the start of a post-doc? Possible Duplicate:          I'm working on a website that displays distance to various locations to the nearest 1/10 mile. One of the developers (a non-native speaker) asked me an interesting question: If the distance is 1.0 mile(s), is that singular or plural? Clearly, anything other that 1.0 is plural, but is it:     1.0 miles away   or     1.0 mile away   The first option sounds more correct to my ear, but the voice of my elementary school language arts tells me that the latter option is correct.  Edit: Added the word 'away' to the example. It doesn't change the question, but might provide clarity for the answer. Also, regarding numbers between 0 and 1, after more thought, I'm not sure that the answer is as clear as I originally thought. See . I have always recommended pressing ALT+F2 and typing for example gksu nautilus or gksudo nautilus both are equivalent; to run graphical programs with root privileges.  However, I have just installed raring on a test system and this no longer works.  gksu and gksudo are not installed as part of a default raring install; though they are in the repositories. In 12.04 at least gksudo was just a symbolic link to gksu  The reasons I have always done this are because while it is often safe to use sudo under some circumstances it can cause problems with graphical applications because gksu has you running the program as root while with sudo you are running as yourself but with elevated privileges.  This can mean you inadvertently change the owner of critical system files.  This can cause problems.  This reasoning is explained better here:         Why was command removed from a fresh install and what should we be using instead? How do I fix a "Unknown configuration key `foreign-architecture' found in your `dpkg' configuration files." error? How do I bulkify queries for directly or indirectly related records? What is Chromatic Aberration? What is Chromatic Aberration? Is it a physical part of the lens, or just an optical illusion? How to customize title page in beamer so that it looks like How can I modify title page so that it looks like it is in  video. Please seek to 16 seconds and watch title page. I do not have permission to post images. I liked this theme and I want to make a beamer theme that looks similar. I tried with beamercolorbox but I failed to position the box to the bottom.      Edit: Here is my minimum working example  \documentclass[xcolor=dvipsnames]{beamer} \defbeamertemplate*{title page}{customized}[1][] { \usebeamerfont{subtitle} \usebeamercolor[fg]{subtitle}  {\flushright \setbeamercolor{author}{bg=white,fg=Red} \usebeamerfont{author} \insertauthor\par \usebeamerfont{institute}\insertinstitute\par }  \vspace{0.2in}  \setbeamercolor{postit}{fg=white,bg=YellowOrange}  \begin{beamercolorbox}[sep=1em,wd=1.062\textwidth,ht=3cm,dp=3cm,right]{postit} \usebeamerfont{title} {\huge \inserttitle}\par \usebeamerfont{subtitle} {\insertsubtitle} \end{beamercolorbox} } \usecolortheme[named=YellowOrange]{structure} \useoutertheme{infolines} \setbeamertemplate{navigation symbols}{}  \institute{Harvard College Professor and\\ Johnstone Family Professor in Psychology, Harvard University}  \title{SAY WHAT?} \subtitle{Linguistics as a Window to\\ Understanding the Brain} \author{Steven Pinker}  \begin{document} { \setbeamertemplate{footline}{} \begin{frame} \titlepage \end{frame} } \end{document} Count number of increasing functions, nondecreasing functions $f: \{1, 2, 3, \ldots, n\} \to \{1, 2, 3, \ldots, m\}$, with $m \geq n$. I stumbled upon a question given like:  Let $m$ and $n$ be two integers such that $m \geq n \geq 1$.   Count the number of functions $$f: \{1, 2, · · · , n\} \to \{1, 2, · · · , m\}$$ of the following two types:  (a) strictly increasing; i.e., whenever $x &lt; y, f(x) &lt; f(y)$, and  (b) non-decreasing; i.e., whenever $x &lt; y, f(x) \leq f(y)$.  I tried in the following way.  For (a). We can say $f(n)&gt;f(n-1)$ AND $f(n)&gt;f(n-2) \ldots f(n)&gt;f(1)$ (total $n-1$ elements)  Similarly for $f(n-1)&gt;f(n-2), f(n-1)&gt;f(n-3) \ldots f(n-1)&gt;f(1)$ (total $n-2$ elements)...  and $f(2)&gt;1$  Like this $$(n-1) + (n-2) + \ldots + (1) = \frac{n(n-1)}{2}$$  Is this correct?  For (b). Since there's equality, it will be $$n + (n-1) + ... (1) = \frac{n(n+1)}{2}$$  Is my approach correct? Any help is appreciated. Thanks in advance. Could there be a body of water the size of a planet? How can I compute the integral $\int_{0}^{\infty} \frac{dt}{1+t^4}$? Total combinations from number of unique combination? How can you find count of the total combinations possible if you have the count of unique combinations? Suppose I need to form a sum of $3$ using $\{1,2\}$ Unique combn. $(1+1+1)$, $(1+2)$ hence '$2$' unique but total is '$3$' i.e, $(1+1+1)$, $(1+2)$, $(2+1)$ I know the elements used to make the combination Note: Number of unique combinations (ie. 2 here) is known. How can I find the total number of combination given the unique combinations? I have trouble comparing 2 double in Excel VBA  suppose that I have the following code  Dim a as double Dim b as double a = 0.15 b = 0.01   After a few manipulations on b, b is now equal to 0.6  however the imprecision related to the double data type gives me headache because  if a = b then  //this will never trigger end if   Do you know how I can remove the trailing imprecision on the double type? Vertical spacing for two consecutive figure environments Why do universal constants have the values they do? This is meant to be a generic question of the type that we get repeatedly on this site, in different versions:        Why do universal constants have the values they do? Can we predict their values theoretically? Do they change over time? How would the world be different if a particular constant had a different value? How to end paragraph text in an inverted pyramid shape? In many old books, the final paragraphs of chapters end in an inverted pyramid shape, like this:    I think this is beautiful, so I would like to imitate this for my book. I've found the code for a triangle in this post   \documentclass{article} \usepackage{microtype,shapepar}  \def\triangleupshape{% {0}% {0}b{0}\\% {8.66}t{-5}{10}\\% {17.32}t{-10}{20}\\% {17.32}e{0}% }      \begin{document} \shapepar{\triangleupshape}A ipsum dolor sit amet, consectetuer    adipiscing elit, sed diam nonummy nibh euismod tincidunt ut laoreet dolore magna aliquam erat volutpat. Ut wisi enim ad minim veniam, quis nostrud exerci tation ullamcorper suscipit lobortis nisl ut aliquip ex ea commodo consequat. Duis autem vel eum iriure dolor in hendrerit in vulputate velit esse molestie consequat, vel illum dolore eu feugiat nulla facilisis at vero eros et accumsan et iusto odio dignissim qui blandit praesent luptatum zzril delenit augue duis dolore te feugait nulla facilisi. Lorem ipsum dolor sit amet, consectetuer adipiscing elit, sed diam nonummy nibh euismod tincidunt ut laoreet dolore magna aliquam erat volutpat. \par \end{document}   Which results in this:    But I'd like it to be inverted and the first part of the paragraph need to be full line width. Only the last few lines should end in an inverted pyramid like in the example above.  Is there a way to do this? Thanks in advance for your help. Show the title character limit when writing a title I just ran into this while writing .  [I originally wrote:     you need a minimum of X reputation to comment on questions that are not your own, and on answers that are not to your own questions   But had to shorten it to:     you need a minimum of X reputation to comment]   First, it didn't tell me that I'd gone over the character limit for the title until after I had already tabbed out of the title field.  Second, it didn't tell me how much I'd gone over or anything. I had to copy-paste the text I'd written into my text editor in order to get some real-time feedback on the exact character count, to help figure out how to best shorten it.