Let $(G, *)$ be a group such that $(a * b)^3 = a^3 * b^3$ and $(a * b)^5 = a^5 * b^5$ for all $a, b ∈ G$. Show that $(G, *)$ is abelian.   My attempt:  I need to show $ab = ba$.  $$(ab)^3 = a^3b^3 \implies a(ba)^2b = a^3b^3 \implies (ba)^2 = a^2b^2$$ $$(ab)^5 = a^5b^5 \implies a(ba)^4b = a^5b^5 \implies (ba)^4 = a^4b^4$$ $$(ba)^4 = ((ba)^2)^2=(a^2b^2)^2 = a^4b^4$$ $$a^2b^2a^2b^2=a^4b^4$$ $$b^2a^2=a^2b^2$$ But what next?
Let G be a group, where $(ab)^3=(a^3)(b^3)$ and $(ab)^5=(a^5)(b^5)$. Prove that $G$ is abelian group.  Thank you in advance. Any help is appreciated.
Use of pizza stone in gas oven I have a new gas kitchen stove.  The oven seems to burn so easily. I have a thermometer in there and it is registering the temp that I have the oven set on.  I have seen where people have used a pizza stone to even out the heat.  I want to know if you place a rectangle pizza stone directly on the bottom of the oven or should it be placed on a rack on the lowest setting?  I would appreciate replies from anyone who has used this and if it had helped the problem Let $R$ be a finite ring with unity. Prove that every nonzero element of $R$ is either a unit or a zero-divisor. Opposite of "straight talk" What is the opposite for the straight talk idiom? How do I best call the activity when someone makes a very long preamble before he says what he wants? Why do statisticians say a non-significant result means "you can't reject the null" as opposed to accepting the null hypothesis? Traditional statistical tests, like the two sample t-test, focus on trying to eliminate the hypothesis that there is no difference between a function of two independent samples. Then, we choose a confidence level and say that if the difference of means is beyond the 95% level, we can reject the null hypothesis. If not, we "can't reject the null hypothesis". This seems to imply that we can't accept it either. Does it mean we're not sure if the null hypothesis is true?   Now, I want to design a test where my hypothesis is that a function of two samples is the same (which is the opposite of traditional statistics tests where the hypothesis is that the two samples are different). So, my null hypothesis becomes that the two samples are different. How should I design such a test? Will it be as simple as saying that if the p-value is lesser than 5% we can accept the hypothesis that there is no significant difference? How do I go about evaluating the following by contour integration?  $$ \int^1_0 \frac{dx}{(x^{2} - x^{3})^{1/3}} $$  The question does not fit in the standard form of : $\int^{2\pi}_0$ or $\int^\infty_{-\infty}$  etc.  I tried substituting $ x = cos(\theta) $ but got stuck. I just installed ubuntu 11.10 from a usb stick.   After installation, none of the software which I had collected and run on 11.10 or 11.04 runs on it. Even my newly downloaded opera or chrome installations does not work.  I do not have a internet connection.  Whenever i try to install any .deb file by double clicking it,software center opens up but the install button is grayed out. Is "funnest" a word? We seem to be stuck at an impasse on this issue.  Is funnest a word or not?  If so, does it mean "most fun"? Prove $\mathbb{Z}_2 \times \mathbb{Z}_2$ is not isomorphic to $\mathbb{Z}_4$ modify specific column with sed or awk I have a CSV file that looks like this:  qw12er,foo,0 bn5mgh rt8yp,foo,10 gh78jk bn852mv,foo,852 78ghjkh tgbr,foo,10 ujmyhn wsx2d,foo,0000 ui52ohn tgbr,foo,7418529 ujmyhn ikl896o,foo,22 wsxdc52   I want to modify the 3rd column and remove all numbers and space from the beginning of the 3rd column.  Then output would be as follows:  qw12er,foo,bn5mgh rt8yp,foo,gh78jk bn852mv,foo,78ghjkh tgbr,foo,ujmyhn wsx2d,foo,ui52ohn tgbr,foo,ujmyhn ikl896o,foo,wsxdc52 How large would the steam explosion at Chernobyl have been? So the  of the HBO series began to cover the  that led to them sending three divers into the water below the reactor to drain the tanks.  This occurred after the initial explosion that destroyed the reactor, and after the fire in the core had been put out. But at this point the decay heat and remaining fission reaction kept the core at more than 1200°C, causing it to melt through the concrete floors below the reactor.  And below the reactor were water tanks which contained 7,000 cubic meters of water (according to the TV show. If anyone has a real figure, I'd love to hear). When the lava of the melted core hit it, it would cause an enormous .  Finally, my question: About how large would this explosion have been? The character in the show says "2-4 megatons" (of TNT equivalent, I assume). I'm pretty sure this is absurd and impossible. But real estimates are hard to come by. Other sources vary wildly, some repeating the "megatons" idea, and others saying it would've "". This still seems crazy.  tl;dr:  I know a lot of it hinges on unknowns and the dynamics of the structures and materials involved, so I can simplify it to a constrained physics question:  Assuming 7,000 cubic meters of water instantly flashes to steam, how much potential energy is momentarily stored in that volume of steam occupying the same volume as the water did?  I don't know what to assume the temperature of the steam is. There were hundreds of tons of core material at temperatures near 1200°C, so worst case scenario you could assume all the steam becomes that temperature as the materials mix. Best case scenario, I guess we could assume normal atmospheric boiling point (100°C)? Converting Polygon into single or all Points using QGIS? Error entering equation in DSolve Show that $(\mathbb{Q}^*,\cdot)$ and $(\mathbb{R}^*,\cdot)$ aren't cyclic I'm reading a book about abstract algebra, but I'm having trouble solving this excercise: "Show that $(\mathbb{Q}^*,\cdot)$ and $(\mathbb{R}^*,\cdot)$ aren't cyclic"  Where $(\mathbb{Q}^*,\cdot)$ is the group of nonzero rational numbers under multiplication and $(\mathbb{R}^*,\cdot)$ is the group of nonzero real numbers under multiplication.    Here is my attempt for the first.  Suppose $(\mathbb{Q}^*,\cdot)$ is cyclic, then $\mathbb{Q}^*=\langle\frac{p}{q}\rangle=\{(\frac{p}{q})^n,n\in\mathbb{Z}\}$, where $p$ and $q$ are coprime.  $\frac{2p}{q}$ is also in $\mathbb{Q}^*$ so it must be equal to $(\frac{p}{q})^n$ for some $n\in\mathbb{Z}$.  To solve $\frac{2p}{q}=(\frac{p}{q})^n$, I take a logarithm of both sides and end up with $1+\log_\frac{p}{q}(2)=n$, since $n$ is an integer $\log_\frac{p}{q}(2)$ must be an integer too, but it is possible only when $\frac{p}{q}=2^{\frac{1}{k}}, k\in\mathbb{N}$, (i.e. $\frac{p}{q}$ is a k-th root of $2$), but $k$ must be $1$ for $2^\frac{1}{k}$ to be rational so $\frac{p}{q}=2$ contradicting the hypothesis of $p$ and $q$ being coprime.    However I don't know whether this is a proper proof and the same reasoning cannot be applied to $\mathbb{R}^*$, I'd like you to just give me an hint towards a proof, without telling me the whole proof, if possible. sed fails to remove newline character Seems like this is the most powerful weapon of all: transport your enemy into a bulkhead, planet, etc. Transport your enemy's ship inside a planet, etc.  Is there some technical reason that it can not be used this way? new finding by Minev et al. seems to suggest that transitions between atomic states are not instantaneous, but continuous processes wherein a superposition smoothly adjusts from favoring one state to another (if I understand it correctly). The authors also claim to be able to catch a system "mid-jump" and reverse it. Popular articles are  and .  I am curious if this finding rules out any interpretations of QM. It seems to generally go against the Copenhagen attitude, which describes measurements as collapsing physical systems into a definite classical state. The popular articles indeed claim that the founders of QM would have been surprised by the new finding.  The link with the asterisk mentions that something called "quantum trajectories theory" predicts what was observed. Is this an interpretation, or a theory? And are they implying that other interpretations/theories don't work? One thing that made me curious while watching a football game on TV was the numbers that appear on the ground. For instance, there's a "1st &amp; 10" beside one team, but there was a "3rd &amp; 6" a few minutes prior (and I think it was beside the other team). What do these numbers mean? Question from HNQ widget on Android doesn't load, showing last viewed question instead I have found myself in an uncomfortable situation after clicking the Hot Network Question widget.   I click the widget, it opens Stack Exchange app and doesn't refresh. That causes me seeing the question I haven been reading a while ago. Fact is that I haven't killed the app since using it last time.   Is there any known workaround to prevent this bug occurring every time I want to read a question opened from a widget? Killing the app/deleting data every time is not a convenient option.   It happens on all Android devices, so device specs aren't relevant, I guess.   Edit: it isn't reproducible 100% of the time. Sometimes it works. Concatenating string with space I want to install 13.04 (or higher) using wubi, is that possible?