How many elements generate the cyclic group $\mathbb{Z}/p^r\mathbb{Z}$?
How many generator has a cyclic group of order n?
How to choose a lens for my first DSLR? I am considering buying my first DSLR.  How do I know whether I need a 18-200 lens, 17-85, or 18-55? Or something else? Value of $\frac{\sqrt{10+\sqrt{1}}+\sqrt{10+\sqrt{2}}+\cdots+\sqrt{10+\sqrt{99}} }{\sqrt{10-\sqrt{1}}+\sqrt{10-\sqrt{2}}+\cdots+\sqrt{10-\sqrt{99}}}$ Here is the question:     $$\frac{\sqrt{10+\sqrt{1}}+\sqrt{10+\sqrt{2}}+\cdots+\sqrt{10+\sqrt{99}} }{\sqrt{10-\sqrt{1}}+\sqrt{10-\sqrt{2}}+\cdots+\sqrt{10-\sqrt{99}}} = \;?$$   ()  I think we need to simplify it writing it in summation sign as you can see here:  $$\frac{\sum\limits_{n=1}^{99} \sqrt{10 + \sqrt{n}}}{\sum\limits_{n=1}^{99} \sqrt{10 - \sqrt{n}}}$$  or in Wolfram Alpha input in comments.  I can compute it too! It's easy to write a script for this kind of question. I need a way to solve it. How would you solve it on a piece of paper? No command 'bla' found, did you mean:? So, when a command is not found, by what means is the "did you mean:" list populated? What program finds these alternate commands?  What is the meaning of: "(main), (universe)..."? Can I change which program finds these? How to run scripts on start up? Cost increase and suspend During a recent game, my opponent Suspended two  in the same turn. Then, during my turn, I played a .    The way we resolved it, is that at the next turn, my opponent payed 1 for the second Rift Bolt. I'm not sure if this is correct, because the rule text for supspend says that when you remove the last suspend counter you cast the spell without paying its mana cost.  So, to me, it sounds like even if the cost was increased, one still does not need to pay for it.    What's the correct thing to do? If actually one needs to pay for the second suspended spell, what would happen if he can't? Is "gift" accepted as a verb? I have always argued adamantly, as long as the issue has been around, that gift should never be used as a verb. However, someone whose English knowledge I quite respect disagrees.  I’ve done some searching and I haven’t found a consensus; should gifting be shunned? All my properties are zero, and my array is zero, even though I have populated them with numbers and objects I'm really new to Mac OSX and UNIX based systems. I wanted to run a few Windows games on my Mac Mini so I started installing Wine and MacPorts. I think the installs are correct, but when I type sudo port install wine I see a not found error in my terminal. I tried other basic commands like say and clear and all of them return the same not found error. My research shows that my PATH might be incorrectly set, but it lacks steps I can implement. Specifically, I ran this command: echo $PATH and it returned:  /opt/local/bin:/opt/local/sbin:/opt/local/bin:/opt/local/sbin:/opt/local/bin  I'm a complete newbie to mac and have no idea where .bash_profile or any of those files are. My skills let me install Xcode and the Command Line tools for Xcode and I sense I need to level up on path management. At this point I need help managing my dot files and changing my path to fix these specific errors on Mountain Lion. What are my next steps? What is cool about generics, why use them? I thought I'd offer this softball to whomever would like to hit it out of the park. What are generics, what are the advantages of generics, why, where, how should I use them? Please keep it fairly basic. Thanks. What does a Terminator do after its mission is accomplished? A Terminator is essentially given a mission or an objective ("Go back in time and kill Sarah Connor" or something of the like).  What does a Terminator do once its mission is accomplished?  We see in Terminator 2: Judgment Day (the live-action movie) that the  (played by Arnold Terminator Schwarzenegger) completed its mission of protecting the Connors, and then sort-of just melted itself into nothingness (presumably to protect the world from discovering Terminators again).  But what would the  or the  (in Terminator 3) have done?   Are they given multiple directives ("Go kill Sarah Connor, then go kill Robert Baratheon, and then have a nice hot bath; you've earned it")?  This question is related, but not a dupe: How to derive $[x_i, F(\vec p)] = i \hbar \frac {\partial F(\vec p)}{\partial p_i}$ indicates that the following relation is &quot;easily shown&quot;: $[x_i, F(\vec p)] = i \hbar \frac {\partial F(\vec p)}{\partial p_i}$, however I'm having some trouble showing it.  I think I'm just messing up the multivariable Taylor expansion (of $F(\vec p)$).  Can one of you walk me through it or link me to site that will?  Thanks.  Edit: Here's what I get (without using $x=i\hbar \frac \partial {\partial p}$ which I haven't proven yet): $$F(\vec p) = F(\vec 0) + \sum^3_{j=1} \frac {\partial F(\vec 0)}{\partial p_j}p_j + \frac 12 \sum^3_{k=1} \sum^3_{j=1} \frac {\partial^2 F(\vec 0)}{\partial p_k \partial p_j} p_j p_k + \dots$$ so $$(x_iF(\vec p) - F(\vec p)x_i)\psi$$ $$= x_i[F(\vec 0)\psi -i\hbar \sum^3_{j=1} \frac {\partial F(\vec 0)}{\partial p_j}(\nabla \psi)_j + \hbar^2\frac 12 \sum^3_{k=1} \sum^3_{j=1} \frac {\partial^2 F(\vec 0)}{\partial p_k \partial p_j} (\nabla \psi)_j (\nabla \psi)_k + \dots]$$ $$- [F(\vec 0)x_i\psi -i\hbar \sum^3_{j=1} \frac {\partial F(\vec 0)}{\partial p_j}(\nabla x_i\psi)_j + \hbar^2\frac 12 \sum^3_{k=1} \sum^3_{j=1} \frac {\partial^2 F(\vec 0)}{\partial p_k \partial p_j} (\nabla x_i\psi)_j (\nabla x_i\psi)_k + \dots]$$ where $$(\nabla x_i\psi)_j=\frac {\partial x_i}{\partial x_j}\psi + x_i\frac {\partial \psi}{\partial x_j} = \delta_{ij}\psi + x_i\frac {\partial \psi}{\partial x_j}$$ and $$(\nabla x_i\psi)_j(\nabla x_i\psi)_k=(\delta_{ij}\psi + x_i\frac {\partial \psi}{\partial x_j})(\delta_{ik}\psi + x_i\frac {\partial \psi}{\partial x_k}) = \delta_{jk}\psi^2 + x_j \psi \frac {\partial \psi}{\partial x_k} + x_k \frac {\partial \psi}{\partial x_j} \psi + x_i^2 \frac {\partial^2 \psi}{\partial x_j \partial x_k}$$ Thus: $$(x_iF(\vec p) - F(\vec p)x_i)\psi$$ $$=  x_i[F(\vec 0)\psi -i\hbar \sum^3_{j=1} \frac {\partial F(\vec 0)}{\partial p_j}\frac {\partial \psi}{\partial x_j} + \hbar^2\frac 12 \sum^3_{k=1} \sum^3_{j=1} \frac {\partial^2 F(\vec 0)}{\partial p_k \partial p_j} \frac {\partial \psi}{\partial x_j} \frac {\partial \psi}{\partial x_k} + \dots]$$ $$- [F(\vec 0)x_i\psi -i\hbar \sum^3_{j=1} \frac {\partial F(\vec 0)}{\partial p_j}(\delta_{ij}\psi + x_i\frac {\partial \psi}{\partial x_j}) + \hbar^2\frac 12 \sum^3_{k=1} \sum^3_{j=1} \frac {\partial^2 F(\vec 0)}{\partial p_k \partial p_j} (\delta_{jk}\psi^2 + x_j \psi \frac {\partial \psi}{\partial x_k} + x_k \frac {\partial \psi}{\partial x_j} \psi + x_i^2 \frac {\partial^2 \psi}{\partial x_j \partial x_k}) + \dots]$$ From here it doesn't look like those higher order terms are all going to cancel out. How t show $f=g$ almost everywhere Chapter on the right side - How to turn it off? Placing items before \documentclass When looking in books or at sample code, I always see \documentclass appearing first. Is that absolutely necessary?  I have created some test code which seems to compile fine:  \def\SetClass{article} \documentclass{\SetClass} \begin{document}     This is some text. \end{document}   Is there any problem with placing other items, such as \input, \usepackage, or such details before the \documentclass? How to make sure one understands each and every detail while reading a textbook/research paper without leaving any gaps? I am a physics Master's student.  I have good grades, with CGPA above 9(/10)(However, all courses in my university are graded on a curve so the actual marks are significantly lower than 90%). This has been my marks throughout my undergraduate and graduate courses and I was happy because this was above average. Also, my summer projects in UG were usually computational and any difficulty I had in theoretical aspects I attributed it to my inexperience in advanced Physics.   However, recently I have started to work on pure theoretical project and while reading for the project I realized that I consistently miss some details from each topic. I understand the general idea, can re-derive most of the equations(with some effort) however when I discuss things with my professor or when I read further topics I realize that I have missed certain key-details. This is exactly similar to the case of me missing a few key questions in my exams and getting that 9 grade point instead of 10(or at-least I think so). I was not worried about this before but now I realize that there is some flaw in the way I study (but I do not know what).   Since I want to apply for Physics PhD I realized that I will need to have solid concepts and it would be best to correct my mistakes but I do not know how or even what could be the reason for my imperfect understanding.  Any direct suggestion would help or even certain general guidelines on what I can do.    Things I've tried recently -   1) I've started to written down almost everything I read. This is slightly laborious but is certainly helping. It also helps me to gloss over the things I read previously when I come back the next day.  2) I completely avoid smartphone when I'm library/study room so that I won't get distracted. Not sure if it helps me solve the above problem but certainly helps me study better in general.  3) I've tried to read the topics from several sources. The books I am reading is somewhat like a collection of research papers and hence does not have a exercises. So I though reading from various sources might help get better ideas but it's not really helping because most other sources are too high level as an introductory text. But this will probably help me in cases where it's practical to do this.  4) Somewhat similar to point (1)- I am trying reread, whenever I can, what I've done until that point. But not sure how much this will help me in filling the gaps. Polygamma function series: $\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2$ Applying the , I found: $$S=\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2\lt\dfrac{2}{3}\pi^2$$ where $\Psi^{(1)}(k)$ is the polygamma function. Is it known any sharper bound for the sum $S$? Thanks. Why does Venus spin in the opposite direction? Given: Law of Conservation of Angular Momentum.   Reverse spinning with dense atmosphere (92 times > Earth &amp; CO2 dominant sulphur based). Surface same degree of aging all over. Hypothetical large impact is not a sufficient answer.   Assuming any object large enough to alter a planets rotation or even orbit would likely destroy most of its shape, yet Venus has retained a spherical property with a seemingly flat, even terrain indicating no volcanoes,and few if any visible meteor impacts. It would be fragmented and dispersed for billions of years.  Even the question of what meteor, comet, asteroid composition could survive traveling that close to the sun's temperature, radiation, electromagnetic energy, solar flares, or gravity to equal a mass reactionary change as to alter it's spin. How do I pass parameter TO a Visualforce page from a link in an email? Here's what I am trying to do:  1) User creates record A.  2) Time-based workflow sends an email to the Record A creator based on a date field in the record.  3) The email template has a URL link.  4) The URL links to a Visualforce page that is basically a form with fields to be filled out (this Visualforce page is in a managed package) and a save button that creates a new User record.  5) When the User clicks the URL link in the email, the Visualforce page opens in a new window.  6) I would like to pass the values of the fields in record A to pre-fill the Visualforce page fields.   Here's my link:  &lt;a href="https://cs19.visual.force.com/apex/CloneThisUser?Desktop=true&amp;Id={!Pending_User__c.User_to_CloneId__c}&amp;formGroupInputSmall={!Pending_User__c.First_Name__c}   And here's the code from the relevant section of the Visualforce page:  &lt;div class="form-group form-group-sm"&gt;       &lt;!-- &lt;apex:outputLabel styleClass="col-sm-2 control-label" for="formGroupInputSmall"&gt;First Name&lt;/apex:outputLabel&gt; --&gt;       &lt;div class="col-xm-11 col-sm-12"&gt;           &lt;apex:inputField html-placeholder="First Name" styleClass="form-control FirstNameInput"  value="{!u.FirstName}" id="formGroupInputSmall"/&gt;       &lt;/div&gt;     &lt;/div&gt;   I have only included the first field I would like to pass the parameter to, in order to keep this post simple.   The link passes the User_to_Clone_Id__c just fine but does not fill in the "First Name" field with the value of the First_Name__c .  What am I doing incorrectly? Thanks. A man enters a comatose author's dreams to wake him up This is a short story, I think, but I might be wrong.  A man is recruited to enter an authors' dreams to try &amp; bring him out of a coma &amp; finds himself in a hackneyed barsoom type world where the coma victim is a muscle bound hero, it's a desert world with two suns, the damsel in distress is the typical scantily clad princess trope.  The "hero" basically spoils the authors' dreams by manifesting a pistol &amp; simply shooting the monsters dead at one point &amp; the author tries to write him out of the "story" by simply jumping forward several chapters &amp; is both surprised &amp; peeved to find him still there.  The author eventually comes out of his coma &amp; immediately assaults the man when he recognizes him from his dreams, he tries to strangle him if I remember right.  He enters the authors' dreams with some kind of experimental dream imaging tech, basically a two ECG sort of set up linked through a computer with him put to sleep with drugs I think. Evaluate:  $$\lim_{x\to{0}}\bigg(\frac{2}{x^3}.(\arcsin{x}-\arctan{x})\bigg)^{2/x^2}$$  I can just expand $\arcsin{x}$ and $\arctan{x}$ using their taylor expansions, but is there any other method?