If $I$ and $J$ are ideals in a ring $R$ with 1 which are  co-maximal, i.e $I+J = R$, show that $I^m$ and $J^n$ are co-maximal for all $m,n$ in $\mathbb{N}$  Work done:     Should I proceed using    By taking $$K=I^m+J^m \neq R$$  Let $$S=\{I,J \in R \mid I^m+J^n \neq R \}$$ be the required partially order set ordered by inclusion then  $$I+J \subseteq I^n+J \subseteq I^n+J^m \subseteq \ldots $$  Then, I think this chain doesn't have an upper bound...as $I^m+J^n \neq R$ for all $ m,n \in \mathbb N$. Am I right??  Thanks for any help!!
Let $R$ be a commutative ring and $I_1, \dots, I_n$ pairwise comaximal ideals in $R$, i.e., $I_i + I_j = R$ for $i \neq j$. Why are the ideals $I_1^{n_1}, ... , I_r^{n_r}$ (for any $n_1,...,n_r \in\mathbb N$) also comaximal?
In (non-relativistic) classical physics, if the temperature of an object is proportional to the average kinetic energy ${1 \over 2} m\overline {v^{2}}$of its particles (or molecules), then shouldn't that temperature depend on the frame of reference - since $\overline {v^{2}}$ will be different in different frames?  (I.e. In the lab frame  $K_l = {1 \over 2} m\overline {v^{2}} $, but in a frame moving with velocity $u$ relative to the lab frame, $K_u =  {1 \over 2} m \overline {(v+u)^{2}}$). Profile tab hyperlink should match name change I have a list of cards for a deck I want to build or buy. What should I do to check whether that deck is legal in a particular format (Standard, for example)? How do I cherry pick packages from a PPA? Is there a way to cherry pick packages from a ppa repository you add? That is, to exclude some from updating while including others?  For example, I want just the chromium-browser package from  but not the others. So I don't want my sudo apt-get upgrade to have that ppa's other (besides chromium-browser) packages install over my existing ones, just that one.   I realize this is a license to shoot myself in the foot if I misread dependencies and exclude ones that are needed to ones that I included, but I know for a fact in this case that the other packages are not dependencies for that one.  Is this at all possible? Maybe there is some whitelist/blacklist file or something? number length limit in SOQL Where condition When I execute SOQL in Developer Console:  Select Id, Amount From Opportunity Where Amount &gt; 10000000000   An parsing error occurs:  Unknown error parsing query   Is there length limit of number on SOQL Where condition? Why is accuracy not the best measure for assessing classification models? How to reinstall kernels after accidental deletion with a separate boot partition and an encrypted LVM volume? My installation is a Ubuntu 12.04 64 bits (Classic "legacy" MBR - No  EFI/UEFI here) with two partition :   A separate boot partition An encrypted LUKS volume that contains the system Logical Volumes   I seems I accidentaly deleted all the current kernel files with:  sudo apt-get purge 'linux-image-3.8*'    I applied the updates with Ubuntu Software Update, rebooted and then: Ubuntu can't boot.  I tried to boot on a live DVD with a 14.04.1 Ubuntu and can still see my partitions (described later):  /dev/sda1 (boot) /dev/sda2 (Encrypted volume) /dev/sda5 (Logical Volume container)  What are the steps to recover/reinstall the kernel files and dependencies in order to cleanly reboot my installed system?  Thanks in advance Is it possible to upgrade a BGA 1224 CPU? boolean variable values in PHP to javascript implementation I am having a problem understanding displacement maps. I'm starting to think that I don't understand them at all.  Clarifying My Understanding  My understanding (which might be completely wrong), is that a displacement map is a gray-scale image whereby a value of 0.5 produces no displacement, >0.5 produces displacement in the positive direction, and &lt;0.5 produces displacement in the negative direction. This, however, does not seem to match my actual observations in Blender.  I am using a displacement map within Cycles, plugging it into the Displacement socket on the Material Output node (I am also using the new experimental adaptive subdivision and micro-vertex displacement). I note that a value of 0.5 actually produces a displacement in the positive direction, causing the whole mesh to jump up slightly, as well as creating the peaks and valleys that you would expect for >0.5 and &lt;0.5 values respectively.  This affects my entire understanding of displacement maps. Why is this happening? Why is a value of exactly 0.5 creating displacement? Further, how are you supposed to arrange and line up objects in your scene when, in rendered view, the displacement maps will push them out of position, even if those maps are only supposed to create displacement in specific areas.  The Problem in Practice  This is the problem I am having at the moment. I have a plane which is supposed to be a snowy field, and I am using a displacement map to give some depth to the snow. In rendered view, my plane jumps up significantly. This is what lead me to investigate my understanding of displacement maps, and to this question.  Additionally, what is the correct way to increase and decrease the strength of a displacement map? At the moment, I will run them through a Math node set to multiply, however, as you might imagine, this exaggerates the issue that I am complaining about above.  To be clear, however, I am not using a math node when experiencing the "jump up" for 0.5 values. This is not the source of my problem, but another aspect to it. What is the probability that $x_1+x_2+...+x_n \le n$? Given that $X_1, X_2...$ are mutually independent random variables. For each $i$ with $1\le i \le n$   the variable $X_i$ is equal to either $0$ or $n+1$ $E(X_i)$ = $1$   also.. if $X_i$ is equal to either $0$ or $n+1$, doesn't that mean that all $X_i$ need to be $0$?  Thanks What is meant by "object serialization"? Can you please explain it with some examples? I've gotten stuck at the following exercise;     There are m functions from a one-element set to the set {1, 2, …, m}.   How many functions are there from a two-element set to {1, 2, …, m}?   From a three-element set? Give a recurrence for the number of T(n) of   functions from an n-element set to {1, 2, …, m}. Solve the recurrence.   I'm not sure how to begin solving it? I'm wondering if there's a simple way to rewrite this in terms of $k$ and $x$, especially as a polynomial. It seems to me to crop up every so often, especially for $k=2$, when I integrate with trig substitution. But $k=2$ is not so bad, because I can use the double angle formula; it's the prospect of higher values of $k$ that motivates this question.   I think the law of sines may help? Or maybe even De Moivre's theorem, to find the length of the hypotenuse as the length of the angle changes, if we think of the right triangle drawn from $\arcsin(x)$ with side1 = $x$, hypotenuse = $1$, and side2 = $\sqrt{1 - x^2}$ as a complex number, though I'm not sure how that might work. Is color undefined I want to check whether a color has been previously defined, consider the below code which doesn't execute.  \documentclass{article}  \usepackage{xcolor} \newcommand{\colorprovide}[2]{\ifx\undefined{#1}\colorlet{#1}{#2}\fi}  \begin{document}     %Initial Definition     \colorprovide{mycolor}{red}     \textcolor{mycolor}{Test 1}      %Will overwrite     \colorlet{mycolor}{green}     \textcolor{mycolor}{Test 2}      %Should NOT overwrite as it exists     \colorprovide{mycolor}{red}     \textcolor{mycolor}{Test 3} \end{document}   In other words and analogous to \providecommand{...} not renewing an existing command if it exists, I need a macro which will will NOT redefine a color if the color has already been defined.  The above code throws an error as the logical checking part \ifx\undefined... is incorrect. Git keeps prompting me for a password What's the main difference between Intel processor generations? What are the main differences between each processor generation of Intel processors/chipsets? Is there a performance difference?  I've already seen processors from first to sixth generation, however I don't know/understand the exact differences. Configure proxy for APT? I need to configure proxy. There wasn't apt.conf file in etc/apt/ so I've created it using nano.   But I don't understand what I need to write there. I've written  http_proxy = http://lgn:pwd@192.168.1.254:8080/ there and then restarted networking. But when I try to apt-get update - it doesn't work.   Actually I don't understand what port and ip I should write in apt.conf file(there was example with 192.168.1.254 ip and 8080 port so I decided to use them). Value of $\frac{x^n -a^n}{ x-a}$? Let $$S=\sum_{n=1}^{\infty}x^{n^2}, \quad|x|&lt;1.$$  It is convergent and the sum is definitely less than $1/(1-x).$   Is there any closed form for $S$ ?