Good morning,  Can you give me a help to demonstrate this proposition:  $f$ is an even and continuous function on the interval $[-a,a], a&gt;0$. Demonstrate:  $$\int_{-a}^a \frac{f(x)}{1+e^x} \,dx= \int_0^a f(x) \,dx$$
I have a question:     Suppose $f$ is continuous and even on $[-a,a]$, $a&gt;0$ then prove that   $$\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$$   How can I do this? Don't know how to start.
I have come across a question about determining possible minimal polynomials for a rank one linear operator and I am wondering if I am using the correct proof method.  I think that the facts needed to solve this problem come from the section on Nilpotent operators from Hoffman and Kunze's "Linear Algebra".     Question: Let $V$ be a vector space of dimension $n$ over the field $F$ and consider a linear operator $T$ on $V$ such that $\mathrm{rank}(T) = 1$.  List all possible minimal polynomials for $T$.   Sketch of Proof:   If $T$ is nilpotent then the minimal polynomial of $T$ is $x^k$ for some $k\leq n$.  So suppose $T$ is not nilpotent, then we can argue that $T$ is diagonalizable based on the fact that $T$ must have one nonzero eigenvalue otherwise it would be nilpotent (I am leaving details of the proof of diagonalization but it is the observation that the characteristic space of the nonzero eigenvalue is the range of T and has dimension $1$).  Thus the minimal polynomial of $T$ is just a linear term $x-c$.    Did I make a mistake assuming that $T$ can have only one nonzero eigenvalue?  Thanks for your help A "fast" way for  computing $ \prod \limits_{i=1}^{45}(1+\tan i^\circ) $? Which  is the fastest paper-pencil approach to compute the product $$ \prod  \limits_{i=1}^{45}(1+\tan i^\circ) $$ Disable RAM stick in Windows 7 I have 2 8gb RAM sticks on my motherboard. One is faulty on some bytes and is giving me BSOD once in a while.  Is it possible to disable one of them without removing it physically? I want to keep it inside the motherboard as I dual boot with Xubuntu and I can block the fauly bits there, and as far as I know, it is impossible to block them in Windows. How do I solve this limit: $\lim _{x \to 0} \left(\frac{ \sin x}{x}\right)^{1/x}$? $$\lim _{x \rightarrow 0} \left(\frac{ \sin x}{x}\right)^{1/x}$$  I have spent an hour on the above limit and have no work to show. I tried using L'Hopital's Rule, but just kept going around in circles. Any help would be appreciated. Thank you. I'm Swedish, but living in Switzerland at the moment. Can I open a US bank account? Either when in the US on vacation or through the internet/phone?  There are lots of situations when I find myself in need of an american credit card or just a bank account. When ordering things online (most retailers refuse to accept my orders if I try to pay with my Swedish or Swiss credit card, even if I ship it to an US address... ) etc etc. Here's the question: Prove that $\lim_{n \to \infty} (\sqrt{n^2+n}-n) = \frac{1}{2}.$  Here's my attempt at a solution, but for some reason, the $N$ that I arrive at is incorrect (I ran a computer program to test my solution against some test cases, and it spits out an error). Can anyone spot the error for me?  $\left|\sqrt{n^2+n}-n-\frac{1}{2}\right| &lt; \epsilon$  $\Rightarrow \left|\frac{n}{\sqrt{n^2+n}+n} - \frac{1}{2}\right| &lt; \epsilon$  $\Rightarrow \frac{1}{2}-\frac{1}{\sqrt{1+\frac{1}{n}}+1} &lt; \epsilon$  $\Rightarrow \frac{1}{\sqrt{1+\frac{1}{n}}+1} &gt; \frac{1}{2} - \epsilon = \frac{1-2 \epsilon}{2}$  $\Rightarrow \frac{1}{\sqrt{1+\frac{1}{n}}} &gt; \frac{1-2 \epsilon}{2}$  $\Rightarrow \frac{1}{\sqrt{\frac{1}{n}}} &gt; \frac{1-2 \epsilon}{2}$  $\Rightarrow \sqrt{n} &gt; \frac{1-2 \epsilon}{2}$  $\Rightarrow n &gt; \frac{4 {\epsilon}^2-4 \epsilon +1}{4}$ Codomain of function complete implies function space complete Let $(X,d_X)$ and $(Y,d_Y)$ be two metric spaces.   Let $(C_b(X,Y), d_C)$ be a function space with the metric $$d_C(f,g):=\sup\limits_{x\in X} d_Y(f(x), g(x)).$$  Prove that if $(Y,d_Y)$ is complete then $(C_b(X,Y), d_C)$ is complete.  Unfortunately, I have no idea how to approach this problem. I can think of something like this:  Let $(y_k)_k\subset Y$ be a Cauchy sequence converging to $y\in Y$. Then there exists some $f\in C_b(X,Y)$ such that $f(x_k)=y_k$ for all $k\in \mathbb N$. Since $f$ is continuous, $\lim\limits_{k\to\infty} d_Y(f(x_k), y)=0$.  But how do I go from here? As far as I understand, what needs to be proved is that any Cauchy sequence of functions in $C_b(X,Y)$ converges in $C_b(X,Y)$, that is, that the limit is bounded and continuous. But how do we devise such a sequence of functions? What about lowering the edit character limit for characters in code snippets? , to correct a misspelled library enum in an answer. Since the edit was only one character, I also had to add a superfluous sentence to get over the character limit required to edit a post.  I've hit this problem before in other edits, always in code snippets. Since the edit limit exists to prevent trivial edits, and since there are many cases where one character code edits are not trivial (= to == etc), would it make sense to have a lower edit minimum inside code snippets? The minimum system requirements for 18.04 are "2 GHz dual core processor or better"; would a quad-core at 1.8 GHz meet (or approximate) this requirement? The link button in the post editor incorrectly escapes square brackets in inline code.  To reproduce:   Write the following MarkDown in the editor  text text `function[]` text text  Select `function[]`, press the link button (), and enter some link the result will be [`function\[\]`][1] which renders as function\[\]   This comes up very often in   when I'm linking to the documentation because the usual way to indicate that a symbol is a function in Mathematica is adding [] after it.  This is a minor thing, as it's easy to manually remove the backslashes, but it's somewhat annoying so I thought I'd mention it.    Edit by Mr.Wizard It is September 2013 and this still hasn't been fixed.  While "adding [] after it" is not presently the custom on Mathematica.SE this is still a bug and still needs to be fixed. pip: AttributeError: 'module' object has no attribute 'SSL_ST_INIT' Reprojecting LiDAR data with libLAS, error: "latitude or longitude exceeded limits"? I have a lidar file (file.las) with the following spatial reference:  Spatial Reference:            PROJCS["NAD83 / UTM zone 11N",     GEOGCS["NAD83",         DATUM["North_American_Datum_1983",             SPHEROID["GRS 1980",6378137,298.2572221010002,                 AUTHORITY["EPSG","7019"]],             AUTHORITY["EPSG","6269"]],         PRIMEM["Greenwich",0],         UNIT["degree",0.0174532925199433],         AUTHORITY["EPSG","4269"]],     PROJECTION["Transverse_Mercator"],     PARAMETER["latitude_of_origin",0],     PARAMETER["central_meridian",-117],     PARAMETER["scale_factor",0.9996],     PARAMETER["false_easting",500000],     PARAMETER["false_northing",0],     UNIT["metre",1,         AUTHORITY["EPSG","9001"]],     AUTHORITY["EPSG","26911"]]   Then I reproject it to WGS84 with las2las:  las2las --a_srs EPSG:26911 --t_srs EPSG:4326 -i file1.las -o output.las   And I get:   Spatial Reference:                GEOGCS["WGS 84",         DATUM["WGS_1984",             SPHEROID["WGS 84",6378137,298.257223563,                 AUTHORITY["EPSG","7030"]],             AUTHORITY["EPSG","6326"]],         PRIMEM["Greenwich",0],         UNIT["degree",0.0174532925199433],         AUTHORITY["EPSG","4326"]]   But when I try to revert the transformation: from WGS84 to NAD83/UTM 11N, I get an error:  las2las --a_srs EPSG:4326 --t_srs EPSG:26911 -i output.las -o wgs2utm.las ERROR 1: latitude or longitude exceeded limits error: Could not project point for ReprojectionTransform::latitude or longitude exceeded limits0.   Why I got this error, if I am just reversing the transformation (originally from NAD83 to WGS84, and then from WGS84 to NAD83)?. Do you get to use the additional attack from the Extra Attack feature as well when you Ready an Attack action?  One of my players, a monk, decided to Ready an action as an earth elemental was about to attack. The trigger was the elemental moving within five feet of him. When this occurred, he took his Attack action, and then proceeded to take the extra attack as well.  I was unsure at the time whether or not the extra attack would also happen, and I haven't had much luck figuring it out. Why are there directories called Local, LocalLow, and Roaming under \Users\\AppData? I have a workstation running Windows Server 2008 that's logged into an  domain, and I have a dual boot with Ubuntu Linux. When running Linux, I'd like to be able to use the same Thunderbird profile I use under Windows, so I pointed Thunderbird to use a profile I found under:  \Users\(myname)\AppData\Local\Thunderbird\Profiles   It turns out it was not the correct profile -- it had the right configuration, but the Inbox was an old version. I eventually found out the correct path was:  \Users\(myname)\AppData\Roaming\Thunderbird\Profiles   What's the rationale behind those different places to store application data? What is the difference between Fact and Truth? I'm curious about the difference between Fact and Truth. I was searching on the internet if I could find it. But still I'm confused about the exact meaning. I first read the forum discussion here  where an author has given two examples for each like below     A fact is a reality that cannot be logically disputed or rejected. If   I say "fire is hot," I don't care how great your reasoning skills are,   if you touch fire your skin will burn (and don't give me that "but   people can walk on hot coals!" bull. There's a difference between the   transfer of heat through conduction and training one's body to deal   with the agonizing pain of said conduction). Now when I say this, I am   not speaking a truth, I am speaking a fact. If you say "fire is not   hot," you are not lying, you are incorrect. Facts are concrete   realities that no amount of reasoning will change. When one   acknowledges a fact, they are doing just that. Facts are not   discovered, facts are not created, facts are simply acknowledged.      A truth on the other hand, is almost the opposite. Truths are those   things that are not simply acknowledged, but must be discovered, or   created. If I say "God exists," and I possess strong reasoning for the   affirmative of that statement, then God really does exist, that is a   reality. However, if another individual possesses strong reasoning for   the negative, and because of this reasoning they believe that God does   not exist, then that is also a reality. If we were to debate our   ideologies, and my reasoning appeared stronger than theirs, they may   choose to adopt my belief that God does exist. If they do, then the   existence of God is just as true as the nonexistence of God which they   believed a week ago. Truths, as opposed to fact, are much more fluid   and malleable than their empirical counterparts.   and followed by further discussion.  Then I found this . Article from above link says like below:     Facts are notes and lyrics on sheet music.  Truth is what the singer   gives to the listener when she’s brave enough to open up and sing from   her heart.   But still curious about the difference between both of them.  In our daily life, in general conversation, we generally use these both terms interchangeably. Then what is the difference? Are they synonym or have specific difference? No processing tools in QGIS 3.2 I can't find my bin folder. Where is it located? The other day I was trying to download something for Minecraft and it involved my bin folder. So I was looking for it and couldn't find it. I then thought that I might have accidentally deleted it, but didn't find it in the trash. I have no idea how to get the bin folder back with all the Minecraft binary files in them. Normal + Connected -> Uncountable Here's a solution (to the exercise "prove that if $X$ is a space with more than one element, and is normal and connected then $X$ is uncountable"):   by Urysohn's lemma, given $A$ and $B$, closed and disjoint in $X$, there exists a continuous function from $X$ into $[0,1]$ such that $f(A)={0}, f(B)={1}$.   If $X$ were countable, so it would be $f(X)\subset [0,1]$; choose $r\in (0,1)\setminus f(X)$, then $X=f^{-1}([0,r))\cup f^{-1}((r,1])$, so $X$ is disconnected.  But the full power of the lemma wasn't used, all that was used is that there exists a continuous function $X\longrightarrow [0,1]$, so the condition of being normal seems too much.  So my question is, what are the weakest conditions on $X$ for the existence of such continuous functions into $[0,1]$ (or $\mathbb{R}$, for that matter)? What is the difference between Early and Late Binding? What is the difference between early and late binding? I want to ask how to get id of element which was clicked by right clicking in open menu. I must know ID of this element to show only that history the element what was clicked fom open menu: