How do I align subscripts? When doing the following, the subscript are not on an line (becasue of the ^T): \documentclass[10pt,a4paper]{article} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb}  \begin{document}          $$X_1^TX_1$$      \end{document}
When i put the code $(f_{\mathcal{X}},f_{\mathcal{X}}^{\partial})$ I get the following:  But I want the subscripts to be aligned. How can I do this?
Why does Ubuntu 12.10 not detect my Win7 partition? I'm trying to install Ubuntu 12.10 alongside Windows 7 on my ASUS laptop and I'm new to all of this.   I'm using  the Ubuntu 12.10 64-bit ISO burned onto a CD.  The first time I tried to install it, Ubuntu said it did not detect any other OS, which meant I only had 4 options:   Erase disk and install Ubuntu (I don't want to do this) Encrypt new Ubuntu. Use LVM. Something else.   If I choose the Something else option, it brings me to the partition menu and says that I have 1 disk with free space of 750516Mb (750Gb), but that's not true because I have Windows 7.  I've already read through several questions and people have mentioned that you could only have 4 primary partitions. So I got hold of partition magic and removed my D: and E:, leaving me with primary C: partition and System reserved partition and a lot of unallocated space.  I restarted the laptop several times and booted the CD again and I got exactly the same as I did previously.  How do fix this problem and install Ubuntu alongside Windows 7?  EDIT 1  Here are my results for   sudo fdisk -l; echo; sudo parted -l; echo; mount; echo; df -h   WARNING: GPT (GUID Partition Table) detected on '/dev/sda'! The util fdisk doesn't support GPT. Use GNU Parted.   Disk /dev/sda: 750.2 GB, 750156374016 bytes 255 heads, 63 sectors/track, 91201 cylinders, total 1465149168 sectors Units = sectors of 1 * 512 = 512 bytes Sector size (logical/physical): 512 bytes / 4096 bytes I/O size (minimum/optimal): 4096 bytes / 4096 bytes Disk identifier: 0x90050371     Device Boot      Start         End      Blocks   Id  System /dev/sda1   *        2048      206847      102400    7  HPFS/NTFS/exFAT /dev/sda2          206848   146368214    73080683+   7  HPFS/NTFS/exFAT  Warning: /dev/sda contains GPT signatures, indicating that it has a GPT table. However, it does not have a valid fake msdos partition table, as it should. Perhaps it was corrupted -- possibly by a program that doesn't understand GPT partition tables.  Or perhaps you deleted the GPT table, and are now using an msdos partition table.  Is this a GPT partition table? Yes/No?     After selecting YES  Yes/No? yes                                                                Error: The backup GPT table is corrupt, but the primary appears OK, so that will be used. OK/Cancel? What happens to the semiconductor in an LED when it is given voltage that it cannot handle? How can I verify a Google authentication API access token? How can I verify a Google authentication access token?  I need to somehow query Google and ask: Is [given access token] valid for the [example@example.com] Google account?  Short version: It's clear how an access token supplied through the  can be used to then request data from a range of Google services. It is not clear how to check if a given access token is valid for a given Google account. I'd like to know how.  Long version: I'm developing an API that uses token-based authentication. A token will be returned upon provision of a valid username+password or upon provision of a third-party token from any one of N verifiable services.  One of the third-party services will be Google, allowing a user to authenticate against my service using their Google account. This will later be extended to include Yahoo accounts, trusted OpenID providers and so on.  Schematic example of Google-based access:    The 'API' entity is under my full control. The 'public interface' entity is any web- or desktop-based app. Some public interfaces are under my control, others will not be and others still I may never even know about.  Therefore I cannot trust the token supplied to the API in step 3. This will be supplied along with the corresponding Google account email address.  I need to somehow query Google and ask: Is this access token valid for example@example.com?  In this case, example@example.com is the Google account unique identifier - the email address someone uses to log in to their Google account. This cannot be assumed to be a Gmail address - someone can have a Google account without having a Gmail account.  The Google documentation clearly states how, with an access token, data can be retrieved from a number of Google services. Nothing seems to state how you can check if a given access token is valid in the first place.  Update The token is valid for N Google services. I can't try a token against a Google service as means of verifying it as I won't know which subset of all Google's services a given user actually uses.  Furthermore, I'll never be using the Google authentication access token to access any Google services, merely as a means of verifying a supposed Google user actually is who they say they are. If there is another way of doing this I'm happy to try. How to reliably keep an SSH tunnel open? I use an SSH tunnel from work to go around various idotic firewalls (it's ok with my boss :)). The problem is, after a while the ssh connection usually hangs, and the tunnel is broken.  If I could at least monitor the tunnel automatically, I could restart the tunnel when it hangs, but I haven't even figured a way of doing that.  Bonus points for the one who can tell me how to prevent my ssh connection from hanging, of course! Let $G$ be a finite group of order $n\ge2$. Which of the following statements are true? a. There always exists an injective homomorphism from $G$ into $S_n$. b. There always exists an injective homomorphism from $G$ into $S_m$ for some $m\lt n$. c. There always exists an injective homomorphism from $G$ into $GL_n(R)$.  I think b. is right (don't know how to prove though). I also think that c. is false. But not sure. Is there an easy way to calculate the residue at one of the poles of a rational expression of the form $\frac{1}{1+z^n}$?  I end up having to add a bunch of polar-form complex numbers in the denominator which I have no idea how to do except to convert them to rectangular coordinates, which becomes really messy.  I feel like there ought to be simple pattern/rule for such a canonical expression. Calculating $\int_{- \infty}^{\infty} \frac{\sin x \,dx}{x+i} $ I'm having trouble calculating the integral  $$\int_{- \infty}^\infty \frac{\sin x}{x+i}\,dx $$  using residue calculus. I've previously encountered expressions of the form   $$\int_{- \infty}^\infty f(x) \sin x \,dx $$  where you would consider $f(z)e^{iz}$ on an appropriate contour (half circle), do away with the part of the contour that wasn't on the real axis by letting the radius go to infinity, then recover the imaginary part of the answer to get back the sine. However here, I can't replace the sine with $e^{iz}$ in my complex function because $$\operatorname{Im} \frac{e^{ix}}{x+i} \neq \frac{\sin x}{x+i}\cdots$$  How to remedy this? I'm not sure if substituting $\sin x = \frac{1}{2i}(e^{ix}-e^{-ix})$ and solving two integrals is how this problem is meant to be solved, although I'm 99% sure it would work I need to find all functions $f:\mathbb R \rightarrow \mathbb R$ which are continuous and satisfy $f(x+y)=f(x)+f(y)$ I need to find all functions $f:\mathbb R \rightarrow \mathbb R$ such that $f(x+y)=f(x)+f(y)$. I know that there are other questions that are asking the same thing, but I'm trying to figure this out by myself as best as possible. Here is how I started out:  Try out some cases:  $x=0:$ $$f(0+y)=f(0)+f(y) \iff f(y)=f(0)+f(y) \iff 0=f(0) $$ The same result is for when $y=0$  $x=-y:$ $$f(-y+y)=f(-y)+f(y) \iff f(0)=f(-y)+f(y) \iff 0=f(-y)+f(y)\iff \quad f(-y)=-f(y)$$ I want to extend the result of setting $x=-y$ to numbers other that $-1$, perhaps all real numbers or all rational numbers. I got a little help from reading other solutions on the next part:  Let $q=1+1+1+...+1$. Then  $$f(qx)=f((1+1+...+1)x)=f(x+x+...+x)=f(x)+f(x)+...+f(x)=qf(x)$$ I understood this part, but I don't understand why this helps me find all the functions that satisfy the requirement that $f(x+y)=f(x)+f(y)$, but here is how I went on:  Thus  $$f(qx)=qf(x)$$ and it should follow that $$f \bigg (\frac {1}{q} x\bigg)= \frac{1}{q}f(x)$$ where $q\not =0$, then it further follows that  $$f \bigg (\frac {p}{q} x\bigg)= \frac{p}{q}f(x)$$ where $\frac{p}{q}$ is rational, and lastly it further follows that  $$f (ax)= af(x)$$ where $a$ is real. Thus functions of the form $f(ax)$ where $a$ is real satisfies the requirement of  $f(x+y)=f(x)+f(y)$.  I don't know how much of what I did is correct\incorrect, and any help would be greatly appreciated. Also is there any way that I can say that functions of the form $f(ax)$ where $a$ is real are the only functions that satisfy the requirement of  $f(x+y)=f(x)+f(y)$? Or do other solutions exist?  Again, thanks a lot for any help! (Hints would be appreciated, I'll really try to understand the hints!) Is there an MD5 Fixed Point where md5(x) == x? Ready-to-Use Database models example Where can I find ready-to-use database models ?  I don't need a database with data in it, but only schemas (UML diagrams). Perhaps something like the , but much more complex and real world. Why do statisticians like "$n-1$" instead of "$n$"? What is the command to open the shutdown dialog? I would like to use a gesture to shutdown/restart my computer and I need to know the command that launches the shutdown dialog: How to fix Eclipse with OpenJDK 9 Guidelines for customizing biblatex styles How to convert Local time to UTC time in Delphi? and how to convert it back from UTC to local time? Every element of a group has order $2$. Why, intuitively, is it abelian? I know that for every $n\in\mathbb{N}$, $n\ge 1$, there exists $p(x)\in\mathbb{F}_p[x]$ s.t. $\deg p(x)=n$ and $p(x)$ is irreducible over $\mathbb{F}_p$.     I am interested in counting how many such $p(x)$ there exist (that is, given $n\in\mathbb{N}$, $n\ge 1$, how many irreducible polynomials of degree $n$ exist over $\mathbb{F}_p$).   I don't have a counting strategy and I don't expect a closed formula, but maybe we can find something like "there exist $X$ irreducible polynomials of degree $n$ where $X$ is the number of...".  What are your thoughts ? Enable copy and paste on UITextField without making it editable How to change the background color only for the current page? Prove the inequality $\,\frac{1}{\sqrt{1}+ \sqrt{3}} +\frac{1}{\sqrt{5}+ \sqrt{7} }+\ldots+\frac{1}{\sqrt{9997}+\sqrt{9999}}\gt 24$ Prove the inequality $$\frac{1}{\sqrt{1}+ \sqrt{3}}  +\frac{1}{\sqrt{5}+ \sqrt{7} }+......... +\frac{1}{\sqrt{9997}+\sqrt{9999}} &gt; 24$$   My work:   Rationalizing the denominator gives   $$\frac{\sqrt{3}-1}{2}  +\frac{\sqrt{7}-\sqrt{5}}{2}+......+\frac{\sqrt{9999}-\sqrt{9997}}{2} .$$  Now by taking two as common and separating the positive and negative terms gives  $$\frac{1}{2} [ \{\sqrt{3} +\sqrt{7}+\dots +\sqrt{9999}\} - \{1+\sqrt{5} +\dots+\sqrt{9997}\}].$$  Can we do like this please suggest. Thanks.