How do computers calculate the log of a value?
What algorithm is used by computers to calculate logarithms?
Let $A$, $B$ be two reduced $k$-algebras. Then if an element of the form $$\sum a_{i}\otimes b_{j}$$ is nilpotent, we can compose it with any $k$-homomorphism $f$ from $A$ to $k$ to get a homomorphism from $A\otimes B$ to $B$. This map's image is $$\sum f(a_{i})\otimes b_{j}$$must be nilpotent as well. So if we assume $b_{j}$ is linearly independent in the first place, we may reason that $f(a_{i})=0$. This showed that $a_{i}$ are in the Jacobson radical of $A$. However, we only know $\mathrm{Nil}(A)=0$, and it is not clear to me how to pass from the Jacobson radical to the nilradical without using some commutative algebra machinery.   Here is what I think that might work. If $A$ is of finite type, then by Nullstellensatz we can assert the above is true. But if $A$ is not of finite type, then I can write $A=k[x_{1},...,x_{n}][X]$, where $n$ is the transcendence degree of $k(A)$ and $k[X]$ is finite over $k$. Then I should be able to pass from $k[X]$ to $A$ by inductive argument on the transcendental degree of $k(A)$, since prime ideals are maximal in a UFD (I am working with $k[x]$ by introducing $[x]$).   The proof I have for the case of finite type is not satisfying. Given $I\subset k[X]$, using Nullstellensatz one can show that $$I_{X}V_{X}[I]=\sqrt{I}$$ where $V_{X}[I]$ is the set of all points in $X$ on which $I$ vanishes, and $I_{X}(V_{X}[I])$ be the ideal in $k[X]$ which vanishes on $V_{X}[I]$. Granting this, let $I=(a_{i})$, then we want to show $I=0$, which is $V_{X}[I]=X$. This follows since if $U\subset W$, then $V_{X}[U]\supset V_{X}[W]$. Since $V_{X}[I]\supset V_{X}[m_{x}]$ for all maximal ideal corresponding to points, it must be $X$. This proof is unnecessarily winding by using the ideal-variety correspondence. Ideally I should be able to construct a pure algebraic proof.   The thing not clear to me is whether the above strategy really works. Namely I am not sure if I can really induct on the transcendental degree of $k(A)$. To make matters simple assume we showed the statement holds for $A$ with transdental degree $&lt;n$. For $A$ with transdental degree $n$ we can thus write it as $k[A'][x]$. Then any maximal ideal $m'$ of $k[A'][x]$ must correspond to a maximal ideal $m=m'\cap k[A']$ in $k[A']$. But it seems not possible to show that any prime ideal is maximal. Indeed considering the simple example of $k[x,y]$, where $(x)$ is a prime ideal but not maximal since $k[x,y]/(x)=k[y]$ is not a field. The problem is in the inducting process the base ring is not neccessarily a field, and so this strategy cannot work.   I am thinking maybe I can show $m'=m[f(x)]$ with $f(x)$ some irreducible polynomial in $k[A'][x]$. Because we know $k[A'][x]$ is noetherian by Hilbert's basis theorem, so in particular $m'$ must have a finite generating set. And we can reduce the number of irreducible polynomial to 1 since $m'$ is a maximal ideal. But I still do not know how to use this to finish the proof. What is the difference between /sbin/nologin and /bin/false? I have often heard it recommended that a user account should be disabled by setting its shell to /bin/false. But, on my existing Linux systems, I see that a great number of existing accounts (all of them service accounts) have a shell of /sbin/nologin instead.  I see from the man page that /sbin/nologin prints a message to the user saying the account is disabled, and then exits. Presumably /bin/false would not print anything.  I also see that /sbin/nologin is listed in /etc/shells, while /bin/false is not.  The man page says that FTP will disable access for users with a shell not listed in /etc/shells and implies that other programs may do the same. Does that mean that somebody could FTP in with an account that has /sbin/nologin as its shell?  What is the difference here? Which one of these should I use to disable a user account, and in what circumstances? What other effects does a listing in /etc/shells have? Let $G$ be a group. Show that a nonempty subset $H$ is a subgroup of $G$ if any only if $ab^{-1} \in H $ for any $a,b \in H$.  The forward direction is quite easy. Suppose $H$ is a subgroup. Then by closure, $ab \in H$ for any $a,b \in H$. Every element has an inverse. Hence, if $b \in H $, then $b^{-1} \in H$. Hence, by closure again, $ab^{-1} \in H$.   Backward direction, suppose $ab^{-1} \in H $ for any $a,b \in H$. Let $a=b$. Then we have $bb^{-1}=1_H\in H$. Let $a=1_H$ , we have $1_Hb^{-1}=b^{-1} \in H$. I don't know how to show closure.  Reference: Fraleigh p. 58  Question 5.45 in A First Course in Abstract Algebra I've installed phpmyadmin sudo apt-get install phpmyadmin. Of course, I have also mysql and a web server running properly.  Now how can I reach it? AWS S3, Paperclip missing required :bucket option The server version of Ubuntu does not have the  command. How can I add a  (PPA) to the server without this command? How to check if a string contains a substring in Bash I have a string in Bash: string=&quot;My string&quot;  How can I test if it contains another string? if [ $string ?? 'foo' ]; then   echo &quot;It's there!&quot; fi  Where ?? is my unknown operator. Do I use echo and grep? if echo &quot;$string&quot; | grep 'foo'; then   echo &quot;It's there!&quot; fi  That looks a bit clumsy. Formatting the union of sets In the example  $\cup^{i=1}_{\infty}F_i$   How do you get the i=1 directly below the union symbol and the infinity directly above? 19.10: Deactivate Alt Key From Triggering Applications' Menu Bar How can I stop the Alt key from triggering the menu bar of applications in Ubuntu 19.10?  I know that there are these , , and , as well as  but those solutions are no longer applicable. I did try searching for a similar option to the one in the answer of those questions but with no success. Beryllium Vacuum Sphere Boat/Aircraft Is it possible to make a solid rigid evacuated "balloon" out of Beryllium or other elements or alloys?  The critical buckling pressure at which an evacuated sphere is given as $$ P_1=\frac{2E\cdot\left(h/R\right)^2}{\sqrt{3\left(1-\nu^2\right)}} $$ where $E$ is , $h$ the shell's thickness, $R$ the radius to the midsurface shell, and $\nu$ is .  Of course this is theoretical and experimental and literature values for submersibles hover around 1/4 of the theoretical pressure.  Beryllium has the following properties:   Density=1.85 g/cm$^3$ Young's Modulus=287*10$^9$ Pa Poisson Ratio=0.032   Calculating the upward force, buoyancy at 101.3 kPa atmospheric pressure and Radius of 10m gives 4.24*10$^8$ N. The weight of this sphere with shell thickness 1mm has 2.32*10$^7$ N downward force of Beryllium material, so it will theoretically float at ground level.   Calculating the critical buckling stress of this beryllium sphere is 331.57 kPa, which is 3.27 times the atmospheric pressure of 101.3 kPa, so theoretically it won't buckle. But in experimentation done in earlier experiments as stated before, due to construction/material non-ideal situations, the critical buckling formula predicts about 4 times stronger than actual, so it would need about 400 kPa, not 331 kPa as Beryllium sphere with this specification would hold. Increasing the strength capability also increases weight and therefore lowers it's ability to float, but Beryllium is the closest element with this situation I have calculated that could work if ideal (I have only been using periodic table elements and household materials)  Is it possible with an alloy of elements? Using the equations for buckling and buoyancy, the following is needed. What alloys/materials come to your mind?  Material considerations:   Material with larger Poisson ratio Material with larger Young's Modulus Less dense material Construction (is it possible)   Material considerations affect on lift   Thickness can be larger, increases strength by the squared. But also increases weight. If a material with lower density with similar structural properties as Beryllium is used, it would help. Radius can be lower for same reason, but also lowers buoyancy (volume)   My other designs (don't worry about these just yet, just if you are curious is a vacuum sphere with a pressurized "outer tube" with spokes in tension (carbon fiber nanotubes maybe) to keep the 2 spheres in position. Another is a spherical truss out of carbon fiber nanotubes with a light membrane around. The truss prevents the outer membrane from collapsing due to the atmospheric pressure  The "point" of this if you ask, as to why not use a traditional balloon is it can be used as a boat by filling the sphere with air, and then used as a aircraft at will by evacuating the sphere. An alternative mode of transportation for sea vessels etc. Spring Data JPA Update @Query not updating? How to make wget faster or multithreading? I've just upgraded my computer hardware(cpu + motherboard + graphic card + memory + hard disk), so that install a new OS is needed. I tried to download debian-6.0.6-amd64-netinst.iso with wget command but the speed is so slow that I could not bear. 4Kb/s ~ 17 Kb/s, slow like a running turtle, or even more slower if I use Chrome.  I've read the help information of wget, it seems like there are no options could make it more faster.  Is there anyway to make wget faster? Or is it possible to make it multi-threading download?  PS: my bandwidth is 4M. I use this command:  wget -c url  http://hammurabi.acc.umu.se/debian-cd/6.0.6/amd64/iso-cd/debian-6.0.6-amd64-netinst.iso Is it true that if $A$ is a nonnegative primitive matrix, then $AA^T$ is also primitive? Obviously $A^T$ is primitive but in general product of primitive matrices is not primitive.  Any hint? I play several games on Steam and I love the achievement system. It would be great to get an overview of all the achievements ever gained in Steam games. Does Steam support this? I haven't found such an overview.  If such a feature does not exist, I would already be happy with an easy way to calculate the sum of all my achievements automatically.  Edit: I stumbled upon an interesting website called  today. Unfortunately it only works for XBox-Achievements. Does anyone know of a similar service for Steam? This would solve my question right away. Implementing my own solution based on the links posted in the answers didn't work because the APIs are not sufficient for my needs. Remove duplicates from a List in C# Anyone have a quick method for de-duplicating a generic List in C#? A theorem about Cesàro mean, related to Stolz-Cesàro theorem Original Title: Tauberian theorems and Cesàro sum  Theorem (Landau-Hardy, From Rudin's Principle of Mathematical Analysis Exercise 3.14)  $\newcommand\abs[1]{\left\lvert#1\right\rvert}$ If $\{s_n\}$ is a complex sequence, define its arithmetic means $\sigma_n$ by $$\sigma_n=\frac{s_0+s_1+\dotsb+s_n}{n+1}\qquad(n=0,1,2,\dotsc)$$ Put $a_n=s_n-s_{n-1}$ for $n\ge1$. Assume $M&lt;+\infty$ and $\abs{na_n}\le M$ for all $n$, and $\lim_{n\to\infty}\sigma_n=\sigma$, then $\lim_{n\to\infty}s_n=\sigma$.  The outline of the proof  If $m&lt;n$, then $$s_n-\sigma_n=\frac{m+1}{n-m}(\sigma_n-\sigma_m)+\frac1{n-m}\sum_{k=m+1}^n(s_n-s_k)\tag{*}$$ Notice that $\abs{s_n-s_k}\le(n-m-1)M\,/\,(m+2)$, fix $\epsilon&gt;0$ and associate with each $n$ the integer $m$ that satisfies $$m\le\frac{n-\epsilon}{1+\epsilon}&lt;m+1$$ Then $(m+1)\,/\,(n-m)\le1/\epsilon$ and $\abs{s_n-s_k}&lt;M\epsilon$. Hence $$\limsup_{n\to\infty}\,\abs{s_n-\sigma}\le M\epsilon$$  Questions and thoughts  It seems that the equation (*) comes out strangely. I wonder how to discover such kind of strange identities. So is there any observation, even deeper, to look through that equation?  Thanks! describes how to get the changelog for a given package. However, it's the whole changelog.  What I'd like is a way to see what will be changed in whatever I'm about to upgrade. For example, I was imagining something like:  $ sudo apt-get upgrade --show-changelogs Reading package lists... Done Building dependency tree        Reading state information... Done Calculating upgrade... Done The following packages will be upgraded:   foo     -- adds the bar feature for better snazziness     -- removes the deprecated baz feature  1 upgraded, 0 newly installed, 0 to remove and 0 not upgraded. Need to get 8,864 kB of archives. After this operation, 285 kB disk space will be freed. Do you want to continue [Y/n]?    Unfortunately, I don't see anything like that in the man pages for apt-get. Does this or something similar exist?  UPDATE: I've written a  talking about apt-listchanges and Debian changelogs in a little more depth that incorporates some of the feedback in this question. $\int_0^{\infty } \frac{\log (x)}{e^x+1} \, dx = -\frac{1}{2} \log ^2(2)$ How to show? Check if convergent in the end points of the convergence interval I have to find the radius of convergence for $\displaystyle\sum_{n=0}^{\infty}\frac{n}{n+1}x^n$, for x$\in$R. I think it is $r^{-1}=\lim_{n\to\infty}\frac{a_{n+1}}{a_n}=\lim_{n\to\infty}\frac{ \frac{n+1}{n+2} }{ \frac{n}{n+1}}=1$ so r=1. Then I have to find out if $\displaystyle\sum_{n=0}^{\infty}\frac{n}{n+1}x^n$ is convergent in  the end points of the convergence interval. I think I have to check for $|r|$. But we can't use ratio test (L=1) or integral test(not decreasing) for $\displaystyle\sum_{n=0}^{\infty}\frac{n}{n+1}$. How can I then check if it's convergent in the end points of the convergence interval? Triangle Inequality question with fractions Given $a, b \in \mathbb{R}$, prove that   $$\frac{|a + b|}{1 + |a + b|} \le \frac{|a|}{1 + |a|} + \frac{|b|}{1+|b|}$$  When does equality hold?  The only useful thing I could get (using the triangle inequality) is:  $$\begin{align}1 + \left|\left(a + \frac{ab}{2}\right) + \left(b - \frac{ab}{2}\right)\right| &amp;\le 1 + \left|a + \frac{ab}{2}\right| + \left|b - \frac{ab}{2}\right|\\ &amp;\le 1 + |a| + |b| + |ab|\\ &amp;=(1 + |a|)(1 + |b|)\end{align}$$  But it seems to be inapplicable! Any ideas?