Does anybody encounter this bug ? formula should return 0 but return 5,6843418860808E-14
Why is this simple Excel calculation not netting Zero?
I have a multi-line equation that I want to have matching sized parentheses. The advice I read online was to use \left( equation \right. \\ left. rest-of-equation \right) . However, this syntax leaves first half of the equation unaware of what is in the second half. For example, if you include a square root term after the break, the second parenthesis will be larger than the opening parenthesis. When you are using the aligned environment to split a long equation this can look somewhat offputting.    Compare the top equation to the bottom one. The parentheses match in size for the bottom equation, but the left hand parentheses for the top one is  noticeably smaller.  Top Code:   \begin{aligned}[t]   \gamma_{\text{u}} + \frac{1}{108}   \left(57\gamma_{\text{er}} \right. &amp;+  38\gamma_{\text{se}}\\ &amp; {}\pm  \left.\sqrt{2601 \gamma_{\text{er}}^2 + 3648  \gamma_{\text{er}}\gamma_{\text{se}} + 1444 \gamma_{\text{se}}^2}  \right)   \end{aligned}   Bottom Code:  \begin{aligned}[t] \gamma_{\text{u}} + \frac{1}{16} \left(9\gamma_{\text{er}}  +  6\gamma_{\text{se}} \pm  \sqrt{49 \gamma_{\text{er}}^2 + 76 \gamma_{\text{er}}\gamma_{\text{se}} + 36 \gamma_{\text{se}}^2} \right) \end{aligned} Naming threads and thread-pools of ExecutorService Why does locking the rear tires on a vehicle cause it to spin? are often used in motor-sports and stunt driving to turn around tight corners.  Essentially, the driver applies the handbrake as he enters the turn, in order to "kick out" the back end and slide the vehicle around the corner.  However, applying the handbrake hard enough to lock the rear wheels while traveling straight will cause the rear of the vehicle to "kick out".  Given enough speed, the vehicle will spin 180 degrees.  Why do sliding wheels on a car want to "get in front" of rolling wheels? Safest way to transport a laptop during flight 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? What's going on with "compact implies sequentially compact"? I've seen both counterexamples and proofs to &quot;compact implies sequentially compact&quot;, and I'm not sure what's going on. Apparently there are compact spaces which are not sequentially compact; quick googling and wikipedia checks will turn up examples floating around; they tend to be variants of $[0,1]^{[0,1]}$ with the product topology.  Here's a demonstration(?):  $[0,1]^{[0,1]}$ is compact by Tychonoff's theorem.  So, we demonstrate failure of sequential compactness. Choose a unique binary representation for each $x\in [0,1]$. For each $n\in\mathbb{N}$, let $f_n : [0,1]\to[0,1]$ (an element of $[0,1]^{[0,1]}$) be the function which maps each $x$ to its $n$-th place digit in its binary expansion. Let $f_{n_k}$ be a subsequence of this sequence. Let $x'\in [0,1]$ be such that the $n_{2m}$-th digit is $0$ and the $n_{2m+1}$-th digit is $1$, for all $m\in\mathbb{N}$. Then $f_{n_k} (x')$ does not converge (it alternates between $0$ and $1$), and hence $f_{n_k}$ cannot converge. Thus we have found a sequence in $[0,1]^{[0,1]}$ without any convergent subsequence and so it is not sequentially compact.  (Aside: this is apparently based off of an example in Steen's Counterexamples in Topology, according to  ) Nevertheless, there's also some proofs(?) floating around that compactness of a space implies sequential compactness, along these lines (this proof(?) is of the contrapositive):  Suppose $X$ is not sequentially compact. By definition, this means there is some sequence $(x_n)$ over $X$ with no convergent subsequence. If any $x\in X$ had for each of its neighborhood $U$ infinitely many $n$ for which $x_n \in U$, then we could define a convergent subsequence of $(x_n)$, contradicting our assumption.  (Presumably this is done by choosing for each neighborhood a sufficiently-large-indexed term in that neighborhood.) Thus, for each $x \in X$ we can select an open set $U_x$ such that $x\in U_x$ but with $x_n \in U_x$ for only finitely many $n$. The collection $\mathcal{U}=\{U_x : x\in X\}$ is clearly an open cover of $X$. If $\mathcal{U}$ had a finite subcover $\{U_1, \dots, U_k\}$ then the union $U_1 \cup \cdots \cup U_k$ would contain all of $X$ but only contain $x_n$ for finitely many $n$, which is impossible. Thus $X$ is not compact, since we have found an open cover without a finite subcover.  (Aside: this proof is essentially the same as the proof appearing in Rudin's Principles of Mathematical Analysis for Thm 2.37 that infinite subsets of compact spaces have limit points.) So, what's going on here?  It can't be that what both the counterexample and the proof are telling us is correct.  Is there some subtle (or more embarassingly for me -- glaring) flaw in the proof? What size difference is there between the PSP 1000, 2000 and 3000 for applying Skin Stickers? How to troubleshoot booting to a black screen on VirtualBox I was using 13.04 as a guest OS on my VirtualBox 14.2.18. Host is Windows 8. After upgrade to 13.10 Ubuntu does not start. I can see a black screen and blinking cursor. I am pretty new to Ubuntu. I suspect a problem with graphics drivers. I can access tty and all files (/home) are there. Say I had  and I used  on something, could I respond to Cloudshift bringing the creature back into play with  or something similar? Let $\Omega=\mathbb{N}$. For each $E\subset\Omega$ let $N_n(E)$ be the cardinality of the set $E\cap \{1,2,\ldots,n\}$. Define $C=\left\{E: \lim_{n\rightarrow \infty} \frac{N_n(E)}{n} \text{ exists}\right\}$. Show that $C$ isn't a field.  I already know that it isn't closed to finite union of non-disjoint set but I can't see why. I saw a post that somebody said &quot;the set of natural numbers whose first digit is 1 doesn't have this limit&quot;. I'm in trouble to see why this limit wouldn't exists in finite unions. Thanks in advance. How can I take pictures through a skyscraper, airplane, or train window? I find that I take a lot of pictures through skyscraper and vehicle windows (those that are impractical or impossible to open). There are all kinds of problems with taking photographs through windows: reflections from light inside, dirt on the window, etc. My technique mostly involves placing the lens directly on the window glass and shielding it from lights reflecting from above.   Are there any non-obvious gizmos / techniques for easily shooting through windows? I've heard that a polarizing filter might be used in this situation, but I don't carry one around most of the time. Besides, most of the pictures I am interested in taking are in low light situations (at sunset or in a moving train, etc). When I have these facts,   French coins were not found in the ruin. German coins were not found in the ruin.   which of these sentences are grammatically correct?  1.     No French and German coins were found in the ruin.   2.     No French or German coins were found in the ruin.   3.     No french nor German coins were found in the ruin.   I think 1 is correct but I'm not sure.  By the way, I can also say     Neither German nor French coins were found in the ruin.   Also, when you have an additional fact   Italian coins were not found in the ruin   which of these statements are grammatically correct?  1.     No French, German and Italian coins were found in the ruin.   2.     No French, German or Italian coins were found in the ruin.   3.     None of French, German and Italian coins were found in the ruin.   4.     None of French, German or Italian coins were found in the ruin.   Thanks, Why is rendered viewport faster than rendering individual frames? When I'm using rendered viewport it initially takes a while before rendering starts, but once it does, I can freely move around and rendering a new view happens almost instantaneously. When on the other hand I'm rendering a movie, for each frame Blender takes a significant amount of time before rendering even starts, usually doing the following:  Synchronizing object, Updating images, Updating meshes, Updating Scene BVH, Packing BVH,  Why does it do this? And is it possible to disable this to increase movie rendering speed? If $\,\gcd(a,b) = {\rm lcm}(a,b)\,$ then prove $a=b$ I'm using X11 forwarding over ssh to run Linux apps on my Windows box, and when the network drops it loses everything that was running. Is there anything similar to screen for X11? How do I save terminal output to a file? How do I save the output of a command to a file?  Is there a way without using any software? I would like to know how. Conditional typesetting / build Let's say I have .tex file which is a set of problems and solutions, of which I want to make two pdf files, one which includes the solutions and one which doesn't. so I need to build the .tex file under two different modes, and specify that parts of the document be ignored under one of the modes.  This is something common in programming context. e.g. in C++ i can code  #ifdef DEBUG   // do this additional stuff ... #endif   How can I do something equivalent in latex? Making 3 points stick on a plane / flag with wind I have a flag I want to waft in the wind. I imported the shape as an SVG from Adobe Illustrator.  I converted the mesh to a curve.  It wouldn't let me make some subdivisons but it still seems to work with the cloth property.    When I press play, you can see the structure of the flag stops.  How can I make those last 3 points intact? Also the fill colour seems to have disappeared, do you know why and how I can bring it back?    Thanks  Steve  File here: How do I grep for all non-ASCII characters? Reducing voltage with resistors This has always been a big want for me. I love using multi-monitor setups but only want a single laptop. Does the 2011 MBP Thunderbolt support chaining multiple monitors together? I know that you can add a monitor and say, a few hard drives, but can you connect it to 2 27" monitors and your flat screen TV (for example)? Thanks!