I have an equation with this shape, where $k,t_1,r \in \Bbb N$:  $$2^{(x+1)^2}=k+t_1(x^2+r)$$  And I noticed that I can find $t_1$ in terms of $x$ as follows:     $$ \frac{{\rm d} }{{\rm d}x} 2^{(x+1)^2} = \frac{{\rm d} }{{\rm d}x} \left( k+t_1(x^2+r) \right)$$   And then I would continue as follows:  $$ 2(x+1)\cdot2^{(x+1)^2}\cdot\ln(2) = t_12x$$  $$ (x+1)\cdot2^{(x+1)^2}\cdot\ln(2) = t_1x$$  So finally:  $$ t_1=\frac{(x+1)\cdot2^{(x+1)^2}\cdot\ln(2)}{x}$$     Is it possible to use a derivative in such case? I think that if both functions $2^{(x+1)^2}$ and $t_1(x^2+r)$ are equivalent and the derivative exists it might be possible, but I am not very sure about the validity of that step. Thank you!
Suppose we have $x^2+4x+3=0$  We differentiate both sides to get $2x+4=0$.  So $x=-2$  But $(-2)^2+4(-2)+3=0$   $4-8+3=-1$ is not equal to $0$.  Which tells that we cannot differentiate equalities. But in many proofs, like in integral calculus, differentiating both sides occurs. So, where is the fallacy?
Prove that the function : $f(x) = x$ when $x$ is rational and $f(x) = 1 - x$ when $x$ is irrational is continuous at $x = \frac{1}{2}$  To prove the continuity it is necessary to prove that for every $\epsilon$ there exists a $\delta$ such that $|f(x) - f(\frac{1}{2})| &lt; \epsilon$ whenever $|x -c| &lt; \delta$ $- (1)$ Now if we choose $a  € (\frac{1}{2} - \delta, \frac{1}{2} + \delta)$ and also $a $ is rational and choose $\epsilon &lt; |a - \frac{1}{2}|$ then condition in $(1)$ is not satisfied. How to move from here? How to prove $\sum\limits_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$? Other than the general inductive method,how could we show that $$\sum_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$$  Apart from induction, I tried with  to check the validity, but I can't think of an easy (manual) alternative.  Please suggest an intuitive/easy method. Is the "Final Release" a "Development Release"? Update  appear to have been updated recently (as of writing this the last update appears to have taken place on 3rd of June) to include the following statement:  The -d switch is necessary to upgrade from Ubuntu 18.04 LTS as upgrades have not yet been enabled and will only be enabled after the first point release of 20.04 LTS.  Also note that .  TLDR (as of 15 Aug 2020):   is the canonical tracking document for the first Focal Fossa point-release (20.04.1). It’s a live document. The Ubuntu release team will be updating it as we work on releasing 20.04.1. […] Upgrades from 18.04 to 20.04.1 are still disabled as we are working through a few upgrade blockers   Edit The answers (so far) do not answer the question. The question is pretty simple. Ubuntu.com points to a &quot;FinalRelease&quot;, which turns out to be a &quot;Development Release&quot;, see evidence below. I'm not asking how and when to upgrade. I'm pointing out what seems to me to be an apparent inconcistency (in semantics, linguistics, logic) in the way Ubuntu names or labels its releases. From an answer to the question I'd except someone to pick up and acknowledge this inconsistency. An answer could start with a &quot;Yes, ...&quot; or a &quot;No, ...&quot;, for starters. (Moreover, a good answer could distinguish between the release being in dev stage and/or the upgrade process (e.g. from LTS to LTS) being in dev stage, if appropriate.) This question, as I see it, is not a duplicate. Futher, I think it is a question which deserves a non-patronizing answer. Thanks.   indicates that today (23rd April 2020) the new Ubuntu 20.04 LTS is to be released.  See &quot;condensed screenshot&quot; here:  Indeed, it seems that that's what happened, see  and :  However, when following the instructions to upgrade (from 18.04 LTS to 20.04 LTS) found  and elsewhere (edit: actually , too!), even prior to the July point release, I'm a bit confused to see that the instruction is to use update-manager -d which also points to the &quot;Development Release&quot; (see screenshot below) rather than the &quot;FinalRelease&quot; (see screenshot and link above). Is the &quot;Final Release&quot; (which I'm after) the/a &quot;Development Release&quot;? Why isn't it called something like &quot;Production Release&quot; or &quot;Final Release&quot;? When will it no longer be referred to as &quot;Development Release&quot;? In July? I'm confused :( Programmatically open new pages on Tabs Why aren't Christoffel symbols tensors? - asked from a fibre bundle perspective Is my Geoserver "Direct WMS Integration" working? At one point, I decided to make friends with the low-lying spectrum of QCD. By this I do not mean the symmetry numbers (the "quark content"), but the actual dynamics, some insight.  The pions are the sloshing of the up-down condensate, and the other pseudoscalars by extending to strangeness. Their couplings are by soft-particle theorems. The eta-prime is their frustrated friend, weighed down by the instanton fluid. The rho and omega are the gauge fields for flavor SU(2), and A1(1260) gauges the axial SU(2), and they have KaluzaKlein-like echoes at higher energies, these can decay into the appropriate "charged" hadrons with couplings that depend on the flavor symmetry multiplet. The proton and the neutron are the topological defects. That accounts for everything up to 1300 but a few scalars and the b1.  There are scalars starting at around 1300 MeV which are probably some combination of glue-condensate sloshing around and quark-condensate sloshing around, some kind of sound in the vacuum glue. Their mass is large, their lifetime is not that big, they have sharp decay properties.  On the other hand, there is nothing in AdS/QCD which should correspond to the sigma/f0(600), or (what seems to be) its strange counterpart f0(980). While looking around, I found this discussion: . The literature that it pointed to suggests that the sigma is a very unstable bound state of pions (or, if you like, tetraquarks).   gives strong evidence for an actual pole;  gives a more cursory review. The location of the pole is far away from the real axis, the width is larger than the mass by 20% or so, and the mass is about 400MeV. The authors though are confident that it is real because they tell me that the interpolation the interactions of pions is safe in this region because their goldstone properties dominate the interactions. I want to believe it, but how can you be sure?  I know this particle was controversial. I want to understand what kind of picture this is giving. The dispersion subtraction process is hard for me to visualize in terms of effective fields, and the result is saying that there is an unstable bound state.  Is there a physical picture of the sigma which is more field theoretical, perhaps even just an effective potential for pions? Did anyone who convinced himself of the reality of the sigma have a way of understanding the bound state properties? Is there an analog unstable bound state for other goldstone bosons? Any insight would be welcome. Every non-empty subset of the integers which is bounded above has a largest element. I was reading a proof about every non-empty subset of the integers which is bounded above has a largest element, but i have troubles in one step. Here is the proof:  Since $S$ is a non-empty subset of $\mathbb{Z}$(hence $\mathbb{R}$) which is bounded above, by the supremum property $sup S$ exists. Let $w=supS$, then we have to show that $w \in S$, so we suppose $w \notin S$, then we have $w-1&lt;w$, then exists $m\in S$ such that $w-1&lt;m&lt;w$(here is my problem, why can we take $m\in S$ such that happens? we are in $\mathbb{Z}$, so that does not happens). Then he takes $n\in S$ such that $w-1&lt;m&lt;n&lt;w$. The inequality $n&lt;w$ implies that $-w&lt;-n$, and $w-1&lt;m$, if we add up both inequalities we have $-1&lt;m-n$ implies $n-m&lt;1$, then $0&lt;n-m&lt;1$ which is a contradiction because there are no integers between $0$ and $1$ Can anybody explains me if that step is right?? How can I prove this if there is a problem with that step? Thanks! How can I do a "broad" search for flights? Several airline and trip planning sites will allow you to check "I am flexible with my dates" or a similar option.  Is there any way to also search more broadly for location?  For example, if you wanted to leave from YYZ or YKF and arrive at any airport in California, can a search for such flights be done somehow?  I ask because when I plan plane trips, I want the lowest price.  I'm OK with leaving a day or two later if I save $200 (same with the return trip).  There are 3 airports within 80 km of me, I'll leave from the one with the cheapest flight. And I'm also OK with changing the trip to spend time exploring San Francisco instead of lying on the beach near San Diego, and that sort of thing.  Normally this would take dozens of searches, but I'd like to see it all at once. I have 1-band GeoTIFF file. The color interpretation for band is gray. I want a color table in order to transform my Col.tiff to Col.jpeg or Col.png.  I tried putting a color table in this tiff via a .vrt:  gdal_translate -of VRT  Col.tiff Col.vrt   then adding in the .vrt:  &lt;ColorInterp&gt;Palette&lt;/ColorInterp&gt; &lt;ColorTable&gt;    &lt;Entry c1="0" c2="0" c3="255" c4="240"/&gt;    &lt;Entry c1="0" c2="0" c3="254" c4="240"/&gt;    &lt;Entry c1="0" c2="0" c3="253" c4="240"/&gt;    &lt;Entry c1="0" c2="0" c3="252" c4="240"/&gt; &lt;/ColorTable&gt;   And translate again :  gdal_translate Col.vrt Col.tiff   But it answered:  Warning 1: Unable to export color table to GeoTIFF file.  Color tables can only be written to 1 band or 2 bands Byte or UInt16 GeoTIFF files.   How can I do it?  For information, there is gdalinfo for the Band:  Band 1 Block=55x18 Type=Float64, ColorInterp=Gray   Description = Something   Metadata:     GRIB_COMMENT=Something[p]     GRIB_ELEMENT=TMP     GRIB_FORECAST_SECONDS=0 sec     GRIB_REF_TIME=1401926400 sec UTC     GRIB_SHORT_NAME=SMTHG     GRIB_UNIT=[p]     GRIB_VALID_TIME=1401926400 sec UTC Probability that a quadratic polynomial with random coefficients has real roots The following is a homework question for which I am asking guidance.     Let $A$, $B$, $C$ be independent random variables uniformly distributed between $(0,1)$. What is the probability that the polynomial $Ax^2 + Bx + C$ has real roots?   That means I need $P(B^2 -4AC \geq 0$). I've tried calling $X=B^2 -4AC$ and finding $1-F_X(0)$, where $F$ is the cumulative distribution function.  I have two problems with this approach. First, I'm having trouble determining the product of two uniform random variables. We haven't been taught anything like this in class, and couldn't find anything like it on Sheldon Ross' Introduction to Probability Models.  Second, this strategy just seems wrong, because it involves so many steps and subjects we haven't seen in class. Even if I calculate the product of $A$ and $C$, I'll still have to square $B$, multiply $AC$ by four and then subtract those results. It's too much for a homework question. I'm hoping there might be an easier way. Limit $\lim_{n\to\infty}\left(1-\frac{1}{3}\right)^2\left(1-\frac{1}{6}\right)^2\cdots\left(1-\frac{1}{\frac{n(n+1)}{2}}\right)^2$ How can I find the following limit :$$\lim_{n\to\infty}\left(1-\frac{1}{3}\right)^2\left(1-\frac{1}{6}\right)^2\left(1-\frac{1}{10}\right)^2\cdots\left(1-\frac{1}{\frac{n(n+1)}{2}}\right)^2$$There are four options $(a)\frac1{3}(b)\frac1{9}(c)\frac1{81}(d)0$ Thanks! C++ equivalent of java's instanceof What is the preferred method to achieve the C++ equivalent of java's instanceof? ViewExpiredException in Firefox and IE after migrating to Jboss 7.0 AS I'm using the official Stack Exchange mobile app for iOS or Android, and  My app crashed! It doesn't look good on my new smartphone. It won't run after I updated my device. I noticed another bug. I have an idea for a new feature which is currently not in the app.  How do I report bugs with or request features for the iOS and Android apps? Also:  I've used the built-in functionality of the app to report a bug or submit a feature request. Did I do the right thing? The app prompted me to post on Meta after an error occurred. Should I post about it here? I'm wondering why the latest version is from 2017. I've heard rumors that the apps are no longer supported; is that true? I can't find the app on my device's app store anymore! Where did it go?     We get a lot of bug reports from users about the mobile apps, which are unsupported, and at the moment, there is no centralized Q&amp;A pair for those to be closed as duplicates of (the relevant answers are spread across multiple unrelated questions). This question is an FAQ proposal, to serve as a directly-related duplicate target for those questions. Duplicate explanation should encourage improvement of original How to tell Google that I have changed my website URLs? Avoiding jumping frames in beamer Are all n-ary operators simply compositions of binary operators? Take for example $A \times B \cdot C$ = $(A \times B) \cdot C$ where $A, B, C$ are 3-component real vectors.  We can define a 3-nary operator $\times - \cdot$ that is a composition of the two common binary operators $\times$ and $\cdot$.  The same thing happens with most functions (operators) - the way we calculate them is by doing smaller binary problems and adding together.  Every time I try to come up with an $(n &gt; 2)$-ary operator my mind automatically looks for binary operators.  So, the question is, do there exist operators (of some weird kind in some branch of math) that cannot be decomposed into 2-ary and 1-ary operators?  Thanks. My awk program to change spaces into tabs doesn't work How to write a shell script which uses awk to read in the data file students.txt and output the data in the tabbed format as shown:  Surname Forename MSc Stream Date of Birth Smith John IT 15.01.1986 Taylor Susan IT 04.05.1987 Thomas Steve MIT 19.04.1986    Do not worry if tabbed columns don’t line up. The distance between each of (Surname, Forename, MSc Stream and Date of Birth) column is one tab.   Question:  Why this bellow code doesn't work for me?  awk 'BEGIN {IFS=" "} {OFS="\t"} {print $1,$2,$3,$4}' students.txt