The title says it all but for some context:   In robotics the Jacobian maps a robot's joint velocities to the velocities at its end-point.   In some quadratic programming approaches (again in the robotics context) the Hessian is considered as $\mathbf{H}=J^TJ$. Note that in our field often $J\in\mathbb{R}^{m\times n}, n &gt; m$.  My question is, under what conditions $\mathbf{H}=J^TJ$ holds or is a valid approximation. I've seen papers that say this notion isn't valid in some cases.
I met this equation frequently in Guass-Newton optimizations. But I dont understand why the left and right side of the equation can be equal.  Lets say the Jacobian is $2$ by $2$   and Hessian is $$\begin{bmatrix}\frac{\partial^2f_1}{\partial^2 x_1 } &amp; \frac{\partial^2f_1}{\partial^2 x_2  } \\ \frac{\partial^2f_2}{\partial^2 x_1 } &amp; \frac{\partial^2f_2}{\partial^2 x_2  }\end{bmatrix}$$   But the right hand side of the equation is  $$J^TJ =\begin{bmatrix} \left( \frac{\partial f_1}{\partial x_1} \right)^2 + \left( \frac{\partial f_2}{\partial x_1} \right)^2  &amp;  \frac{\partial f_1}{\partial x_1}\frac{\partial f_1}{\partial x_2} +\frac{\partial f_2}{\partial x_1}\frac{\partial f_2}{\partial x_1}\\ \frac{\partial f_1}{\partial x_1}\frac{\partial f_1}{\partial x_2} +\frac{\partial f_2}{\partial x_1}\frac{\partial f_2}{\partial x_1} &amp;\left( \frac{\partial f_1}{\partial x_2} \right)^2 + \left( \frac{\partial f_2}{\partial x_2} \right)^2\end{bmatrix} $$  Why can these two be equal as presented in papers?
I just reinstalled my Ubuntu system with 11.10, but ran into a problem when I tried to install Gimp 2.7.  I always get unmet dependencies (it's german, so I can't tell you the exact english translation):     Needs gimp-data (&lt;= 2.6.11-z) but 2.7.3-2011082002~oo should be installed   I already read the follow post, but the answers didn't really help me.  When I type in     sudo apt-get install gimp=2.6   it just says that there is no version 2.6 found. I get the same with 2.7.  I really need Gimp. But installing Gimp 2.6 isn't possible as well.  When I remove the repos for 2.7, I still get the same error. How to best determine if a bike will fit without riding it Number of isosceles triangle. How to vertically center content with variable height within a div? What is the best way to vertically center the content of a div when the height of the content is variable.  In my particular case, the height of the container div is fixed, but it would be great if there were a solution that would work in cases where the container has a variable height as well. Also, I would love a solution with no, or very little use of CSS hacks and/or non-semantic markup. Can't transfer purchases from iPhone to iTunes in iOS 9 Url rewriting in Amazon S3 I have a webapp hosted in Amazon S3, the webapp is build with jQuery Mobile and I use changePage lo navigate between pages, I'm not using the hash operator(#) when I change the page.  I'm having problems when the user copy the url and paste it in a new tab  For example:    loads the page page1.html that is in amazon s3 with only the jquery page structure...  The behavior that I want is that load index.html and with js read the url, detect the page and load the correct page.  With Apache I solve this with URL Rewriting... but for Amazon S3 I didn't find any similar...  NOTE: I know that if I activate the hash operator (#) in my urls the problem is solved but I need clean urls without #. Story where the protagonist is turned off from an automated system of professional attribution and instant learning How do I use the Tooling API to create a new Apex Trigger? How do I use the Tooling API to create a new Apex Trigger (rather than update an existing one)?  According to the  I need to specify the ContentEntityId as an existing Trigger Id. But in my case I don't have an Id because I've trying to create a new sObject.  If I omit the ContentEntityId I get an error back in the SaveResult:     Status Code: INSUFFICIENT_ACCESS_ON_CROSS_REFERENCE_ENTITY   Message: "insufficient access rights on cross-reference id"   The following is more or less a test case in C# that attempts to use the SOAP tooling API to create the new trigger.  SforceService toolingService = //... Get instance of the Tooling API  MetadataContainer container = new MetadataContainer(); container.Name = "Test" + DateTime.Now.Ticks;  SaveResult[] saveResult = toolingService.create(new sObject[] { container }); string metaDataContainerId = saveResult[0].id;  ApexTriggerMember apexTriggerMember = new ApexTriggerMember(); apexTriggerMember.MetadataContainerId = metaDataContainerId; apexTriggerMember.Body = "trigger TestOpportunityBeforeUpdate on Opportunity (before update) {\n//Horray\n}";     // Adding this will successfuly update the existing trigger //apexTriggerMember.ContentEntityId = "01q70000000GuzP";  SaveResult[] triggerSaveResults = toolingService.create(new sObject[] { apexTriggerMember }); Assert.AreEqual(1, triggerSaveResults.Length); SaveResult triggerSaveResult = triggerSaveResults[0]; if (!triggerSaveResult.success) {     Assert.Fail(triggerSaveResult.errors[0].message); } Assert.IsTrue(triggerSaveResult.success);  ContainerAsyncRequest containerRequest = new ContainerAsyncRequest(); containerRequest.MetadataContainerId = metaDataContainerId; containerRequest.IsCheckOnly = false;  SaveResult[] containerResults = toolingService.create(new sObject[] { containerRequest }); The novel centers around a man living a comfortable existence in the pretty distant future.  To support himself, he is occasionally asked to enter a box which will copy his entire body, and send it out faster than the speed of light to some developing world, to work.  His copy on this world wakes up and realizes what a big mistake he's made.  The copy does grueling work, feels lonely, and eventually just dies of radiation sickness.  Back on earth, the man never realizes how bad it is out there, and keeps sending out more copies.  I think the story is probably 80s or earlier - not sure if it's a short story, or the beginning of a novel which I lost &amp; never finished. Making a recursive CTE like query, but for many records Computing irreducible components of algebraic set proof that topologist sine curve is not locally connected I know that this question has been asked before, but it is still not clear to me why the topologist sine curve is not locally connected.     The topologist's sine curve: Let $S$ denote the following subset of the plane. $$S = \{ (x, \sin(1/x)) \mid 0 &lt; x \le 1\}.$$ The set $\bar{S}$ is called the topologist's sine curve, which equals the union of $S$ and the vertical interval $0 \times [-1,1]$.   My attempt was:  Fix a point $p = (0, t)$ where $t &gt; 0$, and consider a neighborhood $N = B(p,\delta)\cap\bar S$, where $B(p,\delta)$ denotes a ball centered at $p$ with radius $\delta$.  Then, I want to show that every open subset of $N$ is disconnected, so there can't be connected neighborhood of $p$ contained in $N$.  But, I am having a trouble with establishing that every open subset of $N$ is disconnected.  How can I prove the claim? Testing a UPS by unplugging it from the wall Resistor in series We know that when a resistor is connected in series , the current flowing through that resistor will be constant.But how do we use a resistor to limit the current flowing through a circuit ,when resistor is connected in series with the circuit ? Is There Any Way to get the template Name By full URL for example  If the full url is  www.example.com/hello-world   I need a function that return the template name (page.php)  I know that WordPress matches every url with regular expression to determine the query and decide which template to include.  That's exactly what I need, a built in or (custom made) function that takes the URL and returns back the corresponding template What's the difference between TRUNCATE and DELETE in SQL What's the difference between TRUNCATE and DELETE in SQL?  If your answer is platform specific, please indicate that. ConTeXt: Flush right formula numbers in cases environment I would like to have the formula numbers of the cases environment flush right like regular formulas (not at a fixed distant from the cases like it is by default with the default setting numberdistance=2.5\emwidth). How can I do that?  \starttext  \startplaceformula     \startformula         \startcases             \NC x = 1 \NR[+]             \NC y = 1 \NR[+]         \stopcases     \stopformula \stopplaceformula  \stoptext     What I want (but with the left bracket):  \starttext  \startplaceformula     \startformula         \startalignment[n=1]             \NC x = 1 \NR[+]             \NC y = 1 \NR[+]         \stopalignment     \stopformula \stopplaceformula  \stoptext See my edit below, this is not totally a duplicate of   My reverse-search function in the CLI is broken and I don't know how to fix it back...  When I hit ctrl+r and search a command I did earlier, find it then press the right arrow key to use it, the output is broken. Here is an example:  # I reverse-searched the word "test" then pressed right arrow yoshiji@x1carbon:~/path/to/git/folder (master) $ch)`test': rake test #                                              ^^^^^^^^^^ weird output #                                                     ^ cursor right there   It also "shifts" my cursor to a different position. If I press enter, the right command is executed (rake test in this case) so it means the text displayed is not the same as what is actually executed.  I know this is most-likely related to my PS1 export in my ~/.bashrc, so here are the last lines of this file:  # Git branch in prompt. git_branch() {   branch=$(__git_ps1 2&gt; /dev/null)   if [ "$branch" = " (master)" ]; then      echo -e "\033[33m${branch}\033[00m" # yellow   elif [ "$branch" = "" ]; then      echo -e "\033[90m (no git)\033[00m" # red   else      echo -e "\033[32m${branch}\033[00m" # green   fi } PS1='\u@\h:\w$(git_branch) \$ '   This is something I wrote to displays the git branch in orange (if master branch) or else in green (or no git in gray).  Do you guys have any idea on what is causing this weird behavior?    Edit: This is a different problem from the  in the way that I am using echo -e "some string" to return a string having color+name of the git branch from the function git_branch(). Using echo -e does not escape the characters \[ and \] (try it yourself in your CLI: echo -e "\[\033[33m\]test").  Ideally, I would like to keep those conditional colors based on the branch's presence &amp; name (gray if no branch, orange if master, else green). Why is the even root of a number always positive? Let $n \in \mathbb N$ be a natural number and $a \in \mathbb R$ be a real number. The $n$-th root of the number $a$ is defined as follows:  Case I: $n$ is an odd number. In this case the $n^{\text{th}}$ root of $a$ is defined to be that number $b \in \mathbb R$ such that $b^n = a$.  Case II: $n$ is an even number. In this case the $n^{\text{th}}$ root of $a$ is defined to be that number $b \geq 0$ such that $b^n = a$.  Why is it that when $n$ is even, we only consider $b \geq 0$. For example, both $+2$ and $-2$ squared equal $4$, but when we say the square root of $4$ is $2$. Is there a reason for this? This question is from [Number Theory George E. Andrews 1-1 #3].     Prove that $$x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1}).$$   This problem is driving me crazy. $$x^n-y^n = (x-y)(x^{n-1}+x^{n-2}y+\dots +xy^{n-2}+y^{n-1)}$$  $(x^n-y^n)/(x-y) =$ the sum for the first $n$ numbers and then I added $(xy^{(n+1)-2}+y^{(n+1)-1})$ which should equal $(x^{n+1}-y^{n+1})/(x-y)$ but I can't figure it out    This is a similar problem in the book and I tried this method but it wasn't working out      $\quad$Thereom $\bf1$-$\bf2$: $\,\,\,\,$ If $\,x$ is any real number other than $1$, then $$\sum_{j=0}^{n-1}x^j=1+x+x^2+\ldots+x^{n-1}=\dfrac{x^n-1}{x-1}.$$    $\quad$Remark: $\displaystyle\sum_{j=0}^{n-1}A_j$ is shorthand for $A_0+A_1+A_2+\ldots+A_{n-1}.$     $\quad$Proof: Again we proceed by mathematical induction. If $n=1$ then $\displaystyle\sum_{j=0}^{1-1}x^j=x^0=1$ and $(x-1)/(x-1)=1$. Thus the theorem is true for $n=1$.  $\quad$ Assuming that $\displaystyle\sum_{j=0}^{k-1}x^j=(x^k-1)/(x-1)$, we find that $$ \eqalign{ \sum^{(k+1)-1}_ {j=0}x^j &amp; = \sum^{k-1}_ {j=0}x^j+x^k=\dfrac{x^k-1}{x-1}+x^k \\ &amp;= \dfrac{x^k-1+x^{k+1}-x^k}{x-1}\\ &amp;= \dfrac{x^{k+1}-1}{x-1}. }$$   Hence condition $(\rm ii)$ is fulfilled, and we have established the theorem.    $\quad$Corollary $\bf1$-$\bf1$: $\,\,$ If $\,m$ and $n$ are positive integers and if $m&gt;1$, then $n&lt;m^n.$