A new metric formed from the old
If $d(x,y)$ is a metric, then $\frac{d(x,y)}{1 + d(x,y)}$ is also a metric
What kind of components are black blobs on a PCB? What should I do to avoid ear pain when flying? ART has better performance and better battery life on paper at least.  Should I consider switching to it on my moto g? What are the results I can expect? What precautions should I take? Is it easy to turn back to dalvik? String processing in windows batch files: How to pad value with leading zeros? equal size numerator and denominator I am trying to size the numerator and denominator of the below equation so that they are equal in size.  I have not had much luck suggestions are appreciated.  \begin{equation} \sum_{i=1}^{n} \frac{\frac{cf_{n}}{(1+i)^n}} {\displaystyle\sum_{i=1}^{n}\frac{cf_n}{(1+i)^n}} , n_{i} \end{equation} $\sum_{n=-\infty}^{\infty}\frac{1}{(u +n)^2}=\frac{\pi^2}{(\sin \pi u)^2}$ I've already see a  by Marko Riedel which I list it follows:     The standard way to treat these sums is to integrate   $$ f(z) = \frac{1}{(z+\alpha)^2} \pi \cot(\pi z)$$   along a contour consisting of a circle of radius $R$ and with $R$ going to infinity and hence being larger than $\alpha$, where the circle does not pass through the poles on the real axis.   Now along the semicircle in the upper half plane we have   $$|f(z)| \le \frac{1}{(R-|\alpha|)^2}\pi \left|\frac{e^{i\pi R\exp(i\theta)} + e^{-i\pi R\exp(i\theta)}} {e^{i\pi R\exp(i\theta)} - e^{-i\pi R\exp(i\theta)}}\right|= \frac{1}{(R-|\alpha|)^2} \pi  \left|\frac{e^{2i\pi R\exp(i\theta)}+1}{e^{2i\pi R\exp(i\theta)}-1}\right| &lt; \frac{1}{(R-|\alpha|)^2} \pi   \frac{1+e^{-2\pi R\sin(\theta)}e^{2i\pi R\cos(\theta)}} {1-e^{-2\pi R\sin(\theta)}e^{2i\pi R\cos(\theta)}}$$   This last term is clearly $O(1/R^2)$ as $R$ goes to infinity as the quotient of the two exponentials goes to one since $\exp(-R)$ vanishes and there is no singularity when $\theta = 0$ or $\theta = \pi$ as $R\cos\theta = \pm R$, which is not an integer by the assumption that the circle avoids the poles and hence cannot be one.     My question: It is clear that for each $z=Re^{i \theta}$ , $f(z)=O(1/R^2)$ ，but can this guarantee that the last term on RHS is uniform bounded ? Since $\theta$ varies on a compact interval $[0, \pi]$ , I want to show that for each $ \theta \in [0, \pi]$ , and a fixed positive number $M \gt 1 $ there exist an open ball contains $\theta$ and a fixed $R_{\theta}\gt 0$ , for every $R \ge R_{\theta}$ and $x \in $ the open ball $$\frac{1+e^{-2\pi R\sin(x)}e^{2i\pi R\cos(x)}} {1-e^{-2\pi R\sin(x)}e^{2i\pi R\cos(x)}} \le M$$  For $\theta \neq 0,\pi$ , it is easy to find the desired $R_{\theta}$ and the open ball , but if $\theta = 0$ or $ \theta = \pi$ , for every open ball containing $\theta$ , it behave erratically near $\theta$ , I have no idea how to deal with this . Highlighting pattern strings in a large text ... namely that if $n &gt; 1$ and $(n − 1)!\equiv−1\pmod{n}$, then $n$ is prime.  This is for a number theory class I'm in at Penn State. My idea is to follow accordingly, but I can't get it concretely..  Reason by contradiction, suppose that $n$ is not prime.   Then $n = ab$, where $1 &lt; a, b &lt; n$  In particular $(n-1)! = ai$ for some $i\in\Bbb{Z}$. $\\$ If $(n-1)! = nj - 1$ for some $j\in\Bbb{Z}$ then it follows that a probably doesn't divide $-1$? I flew a long range flight carrying my laptop, a quite heavy one, inside a neoprene briefcase in the passenger cabin. The recommendation by the aircrew was to place it in the overhead compartment or under the seat in front of me. However, during take-off and landing, the vibrations on the plane are quite strong and, as a result, it received minor damage.   Placing them in the overhead compartment might be equivalent to placing it on the floor under the seat, with the added issue of additional luggage in the compartment shifting.  I considered the alternative of holding the laptop by the handle while sitting, so it won't be any contact with the aircraft's floor. This is impracticable during cruise, but I thought it could be better during the take-off and landing segments. Another option I considered is to wrap it with completely with bubble wrap before boarding the plane.   Which is the safest way to transport a laptop during flight? Do not delete good duplicates! In the early days of C++ when it was bolted on top of C, you could not use NULL as it was defined as (void*)0. You could not assign NULL to any pointer other than void*, which made it kind of useless. Back in those days, it was accepted that you used 0 (zero) for null pointers.  To this day, I have continued to use zero as a null pointer but those around me insist on using NULL. I personally do not see any benefit to giving a name (NULL) to an existing value - and since I also like to test pointers as truth values:  if (p &amp;&amp; !q)   do_something();   then using zero makes more sense (as in if you use NULL, you cannot logically use p &amp;&amp; !q - you need to explicitly compare against NULL, unless you assume NULL is zero, in which case why use NULL).  Is there any objective reason to prefer zero over NULL (or vice versa), or is all just personal preference?  Edit: I should add (and meant to originally say) that with RAII and exceptions, I rarely use zero/NULL pointers, but sometimes you do need them still. Why is chi square used when creating a confidence interval for the variance? This is a very basic question. Why do we use a chi square distribution? What is the meaning of this distribution? Why is this the distribution used for creating a confidence interval for the variance?   Every place I google for an explanation just presents this a fact, explaining when to use chi, but not explaining why to use chi, and why it look the way it does.  Many thanks to anyone that can point me towards the right direction and that is - really understanding why I am using chi when I am creating a confidence interval for the variance. I have an ASUS K55VM and it came with DOS, and then Windows 7 was installed. I am planning to install Ubuntu on it. My computer had 4 partitions: 240GB, 230GB, 230GB, and 230GB, and I changed this to 5 partitions: 243GB, 230GB, 230GB, 150GB, and 80GB. I installed Ubuntu 12.04 to the 80GB partition through a USB flash drive. It installed fine, however the problem is during the Boot option in Windows Boot Manager. When I choose Ubuntu, this what is displayed:  Windows failed to start. A recent hardware or software change might be the cause. To fix the problem:   1. Insert your Windows installation disc and restart your computer.   2. Choose your language settings, and then click "Next,"   3. Click "Repair your computer." If you don't have this disc, contact your system administrator or computer manufacturer for assistance.    File: \ubuntu\winboot\wubildr.mbr   Status: 0xc000000e   Info: The selected entry could not be loaded because the application is missing or corrupt.   I tried to change the boot option for the OS and made Ubuntu as my first option. It worked, however Windows 7 does not show up in GRUB. In this fear I did the following:     Restarted my PC changed boot option 1 to Windows.   Formatted the drive to which Ubuntu was installed (80GB one) through Windows.   Restarted again.     The problem persists. Windows Boot Manager still shows Ubuntu, and when I choose this it gives two lines one like: grub rescue&gt;, but my Windows 7 is working perfectly fine.  Now I want a solution to remove Ubuntu from the Windows boot option and reinstall completely. Converting a system command to python for file search and delete Word for a temporary project name In software development, and I imagine in many other fields, we often make up temporary nicknames for new projects or features. These are not meant to be the real names for the end product and are often funny, playful, or sarcastic in nature. These are also not intended to be "codenames" that hide the nature of the project or feature. Is there a word for such nicknames? Subjunctive mood: 'was' usage after 'I wish' and 'if only' had declared the sentence below as follow: 1) STAR TURN, 2) BEST LINES.      I wish Bernard Cribbins was my grandad. Perhaps he would be willing to   adopt?     Rattigan: “If only that was possible.” The Doctor: “If only that were   possible. Conditional clause.”    Is 'was' after 'I wish', 'if only' and 'if' (in conditional clauses) colloquial?  Does it be used in formal speech and writing? A "non-believer" of C# was asking me what the purpose to extension methods was. I explained that you could then add new methods to objects that were already defined, especially when you don't own/control the source to the original object.  He brought up "Why not just add a method to your own class?" We've been going round and round (in a good way). My general response is that it is another tool in the toolbelt, and his response is it is a useless waste of a tool... but I thought I'd get a more "enlightened" answer.  What are some scenarios that you've used extension methods that you couldn't have (or shouldn't have) used a method added on to your own class? What does "rigorous proof" mean? I have heard several times that some mathematician has given another and more rigorous for an established theorem, but I don't know what does it really mean and what differences makes it to be more 'rigorous'. My understanding of a proof is that a proof is some explanation to convincing others that a statement is true. If some proof is not rigorous, does it mean that while the proof is acceptable for many but it is not for everybody? If so, the proof is not correct since it going to be rejected by some people and it is not a proof at all. Or, does rigorous mean it uses more advanced tools? If so, why not any alternative (but simpler) proof is also acceptable as rigorous (which is more 'valuable' as well since it's simpler!)?   Saying those limited insight of mine, my questions are:   $1-$ What does rigorous proof mean, in general definition?  $2-$ In order to understand your answers, as an example I am re-writing three of the proofs for the compactness of the torus. Which of the following proofs are most rigorous, 'normal' and least rigorous; and why so?    Theorem: The torus $T^2$ is compact.  First proof: The torus is homeomorphic to the product space $S^1 \times S^1$. The circle $S^1$ is compact. Therefore the torus is the product of two compact spaces, and thus it is compact.   Second proof: The torus $T$ is the subspace of $\mathbb{R}^3$ obtained by rotating a circle about the $z$-axis. The torus is closed since given any point in its complement, there is an open ball of sufficiently small radius centered at that point and disjoint from the torus. Also, the torus is bounded since it is contained in the ball of radius $4$ centered at the origin. Since the torus is closed and bounded in $\mathbb{R}^3$, it is compact.   Third proof: The torus is homeomorphic to a space obtained as a quotient  space of the square by identifying opposite edges. The square is a closed and  bounded subset of $\mathbb{R}^3$ and therefore is compact. Furthermore, a quotient map on the square is a continuous function, and the image of a compact space under a continuous function is compact. Therefore the torus is compact.   Thanks a lot for your guidance. CPT Template Not Showing - Getting 404 I'm developing a Custom Post Type plugin, which works fine in dev. Now that I've moved it to production, I'm getting a 404 when I try to display the template for the CPT.   So, two questions:  1- Is there some mechanism to determine the path WP took to get through the hierarchy?  2- Why wouldn't it just give me the single or index page?  I've dumped $wp_query and a stacktrace at the end of the 404. Here's part of what I found:  [query] =&gt; Array     (         [page] =&gt;          [pagename] =&gt; refletters/proximity     )  [request] =&gt; SELECT   wp_posts.* FROM wp_posts  WHERE 1=1  AND (wp_posts.ID = '0') AND wp_posts.post_type = 'page'  ORDER BY wp_posts.post_date DESC  [posts] =&gt; Array     (     )  [queried_object] =&gt;  [queried_object_id] =&gt; 0   And the stack trace:  2013-04-27 22:16:41 - 24.52.197.40 - TRACE - index.php - 404.php[39] -      wordpress/wp-content/themes/notoriousconsultant/404.php[39] - aaLog-&gt;logtrace     wordpress/wp-includes/template-loader.php[50] - include     wordpress/wp-blog-header.php[16] - require_once     wordpress/index.php[17] - require   Thanks for your help. Is it possible to run three versions of Google Chrome? How might I run 3+ versions of google chrome so I can log into to 3 different google accounts.  More specifically I am looking to have separate icons in the dock for the 3 separate running applications.  How might I run duplicate and or copy versions of the same application.  The following  by Mark works for a lot of applications but not for Google Chrome.  It appears the following might work .