This is an excercise from Spivak's Calculus. Show that if $x^n=y^n$ and $n$ is odd, then $x=y$. Hint: first explain why it suffices to consider only the case x and y greater than 0, then show that x smaller than y or greater y are both impossible.  I try to prove this by using induction. The base case $n=1$ is correct. Assume that $n=2k+1$ is true, prove that $n=2k+3$ is also true. But now I don't know how to proceed.
Here, we suppose that $x,y\in\mathbb{R}$ and that $x^n=y^n$, where $n$ is odd. I want to prove that $x=y$.  Maybe we can use that $x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+...+xy^{n-2}+y^{n-1})$  So, it suffices to show that if $x^n=y^n$ then $x^{n-1}+x^{n-2}y+...+xy^{n-2}+y^{n-1}\neq 0$  Any hint?
Replace and Association I have seen some very weird for loops when reading other people's code. I have been trying to search for a full syntax explanation for the for loop in C but it is very hard because the word "for" appears in unrelated sentences making the search almost impossible to Google effectively.  This question came to my mind after reading  which made me curious again.  The for here:   for(p=0;p+=(a&amp;1)*b,a!=1;a&gt;&gt;=1,b&lt;&lt;=1);   In the middle condition there is a comma separating the two pieces of code, what does this comma do? The comma on the right side I understand as it makes both a&gt;&gt;=1 and b&lt;&lt;=1.   But within a loop exit condition, what happens? Does it exit when p==0, when a==1 or when both happen?  It would be great if anyone could help me understand this and maybe point me in the direction of a full for loop syntax description. Closed form solution for slope coefficients in bivariate regression In a univariate regression, $Y=a+bX+e$, the solution for slope b is given by $COV(X,Y)/VAR(X)$.  Is there a similar expression for a bivariate regression $Y=a+bX+cZ+e$. What is the closed form solution for b and c in this case? I'm studying the Darboux definition of integrability, which I completely explained . There's an exercise that asks me to prove that the Darboux Integrability is equivalent to Riemann Integrability, but this Riemann integral is defined as the following:  It first defines a 'pointed' partition (I don't know how to say it in ensligh) by the following: a 'pointed' partition $[a,b]$ is a pair $P^*=(E,ξ)$, where $P=\{t_0, t_1, \cdots, t_n\}$ is a partition of $[a,b]$ and $ξ = (ξ_1, \cdots, ξ_n)$ is a list of $n$ chosen numbers such that $t_{i-1}\le ξ_i\le t_i$ for each $i=1,\cdots ,n$.  Now, the Riemann Integral is defined as:  $$\sum(f,P^*) = \sum_{i=1}^n f(ξ_i)(t_i-t_{i-1})$$  (I didn't understand the notation for the left hand side of the equation, by the way)  Finally, I'm asked to prove the following: given $f,g:[a,b]\to \mathbb{R}$ integrable functions, for the entire partition $P=\{t_0, \cdots, t_n\}$ of $[a,b]$ let $P^* = (P,ξ)$ and $P^{\#} = (P, η)$ be pointed partitions of $P$, then:  $$\lim_{|P|\to 0}\sum f(ξ_i)g(η_i)(t_i-t_{i-1}) = \int_a^b f(x)g(x) \ dx$$  I guess here I need to prove that the Riemann Integral of the product of two functions if the darboux integral of $f(x)g(x)$, but it seems too obvious, I just need to verify that $f,g$ are integrable, then their product is too, isn't it? I'm pretty sure this should be a hard question. Is there another interpretation that I'm missing? Chance of selecting the last k pages in correct order from a set of n pages I have a set of documents numbered 1 to n. If I arrange the documents randomly, what is the chance of selecting the last k pages in the correct order if I select them at random from the documents?   Because order is important, I use the following formula:  $\displaystyle\frac{n!}{(n-k)!}$  From this I calculate how many ways there are to select k pages from n pages in the correct order. I need to have the last k pages. Which I can do in only one way. So is the chance   1/$\displaystyle\frac{n!}{(n-k)!}$?   It seems unlikely. It seems to me spontaneously that the chance should be smaller. Mobile web refresher, part 2 After a sizeable amount of work we are releasing a new version of the mobile web site, this time focusing on the question and answer page. We have also fixed or implemented a minimal large part of the issues and requests mentioned in the . Changes Question and answer page  SVG icons for upvote, downvote, favorite, comment upvote white background color for legibility general styling issues with notices (they actually exist now) bounty link is now in question menu (next to flag)   Ask question page  general styling and sizing of element use of placeholders where appropriate instead of page text styling of error messages   Pop-ups  Popups are now inlined in the page instead of being sometimes floating and draggable Many overdue styling issues Close button is SVG Flag indicators are readable Buttons   Misc pages  404 page did not exist. Now it does. Login, logout are formatted Pre-ask and pre-answer warnings are now formatted  Bugfixes  made search magnifying glass bigger fixed language in user timeline so it's consistent with the desktop site added login link on top bar made topbar icons ever so slightly smaller  Restating the obvious: send your &lt;3 to  and   for the most excellent design! Feedback , bugs and suggestions while we move to the next iteration? Please mark bugs with On found exact duplicates, should these be voted to close directly against a preferred, or ultimate, canonical question?   Or is a chain of possible duplicates (where one question is closed as duplicate of another, which itself is a duplicate of another, and so on and so on) allowable or OK?  One example of this situation can be found . LAN not working on 17.04 Actually yesterday I've upgraded from Ubuntu 16.04 to 17.04 and I got a problem that, if connect LAN cable it does not showing anything and  remains disconnected. But i can connect with WiFi. I don't know what's the problem. Actually its showing "device not managed" under the "Ethernet Network" and both were disabled. I've searched for the solution, but did not found yet. Is there a way of creating a filesystem object akin to this:  mknod files p cat file1 file2 ... fileN &gt; files   but such that it can be seeked in, as if it were a regular file? How to see download count of an Android app on Google Play store? I cannot see the download counts of one app on Google Play.   How can we know that number of one app? Generating random numbers without repetitions STM32 & ST-LINK - Cannot connect to MCU after successful programming I have built my own board with STM32F7-45VGT6. I have successfully programmed it with ST-LINK v2 (not the original one though) and now I cannot even connect with MCU.  I use ST-Link Utility from ST and SWD interface. It can be the case that I use SWD pins as output and in my code I set them as GPIO output. Can it be the case?  Nevertheless, I connect my reset pin to GND and set "Connect under reset" option in ST-Link Utility but it doesn't work... What can I do?  On the Internet, I have found something about using BOOT0 Pin, but I don't know exactly... Print LinkedList Recursively using C++ Itemize: Identation is different between first and second line How to remove GNOME Shell from Ubuntu 20.04 LTS to install other desktop environment from scratch? I use Chrome across 4 different workstations and I have my extensions synced across all of them.  The problem I have is that one of the extensions () works great if you've got an actual mouse, but on a trackpad on Ubuntu, it just gets in the way.  If I disable it on the Ubuntu machine, it gets disabled on all Chrome installations due to its .  Q: Can I selectively disable an extension on just a single machine? My book says  \begin{equation} (a,b)=\{\{a\},\{a,b\}\} \end{equation}  I have been staring at this for a bit and it doesn't make sense to me. I have read several others posts on this, but none made any sense to me.  For example,   Based on how my ignorant brain is viewing this, I don't see why the definition could not be.  \begin{equation} (a,b)=\{\{a\},\{b\},\{a,b\}\} \end{equation}  aka the power set. What is the significance of the {a} in that definition? Please keep things simple if possible. Normally definitions have a valid and clear reason for being defined that way.    Clarification  First, I understand what an ordered pair is. I just don't see how the set notation says that.   Second,  \begin{equation} (a,b) = \{\{a\},\{a,b\}\}=\{\{a,b\},\{a\}\} \end{equation} Sets don't preserve order, but ordered pairs do. How does the third part of the equality apply to the definition?  Third, another issue with the notation that I have starts with the Product Property of Sets  \begin{equation} \text{Let $X$ and $Y$ be sets} :\ X=\{a,b,c\}\text{ and }Y=\{a,d,e\}. \end{equation} \begin{equation} \text{Then }X \times Y = \{(a,a),(a,d),(a,e),(b,a),(b,d),(b,e),\dots,(c,e)\} \end{equation} If we look at the first ordered pair and our given definition we have  \begin{equation} (a,a)=\{\{a\},\{a,a\}\} \end{equation}   How can this be so, you can't have duplicates in sets? I guess what I am looking for in an answer, is not a proof or a definition of ordered pairs, but rather something like, "This notation says what it says because...". Except for the second to last point I get the terminology, I just don't get the connection between the two different uses of notation. I'm working with a date in this format: yyyy-mm-dd.  How can I increment this date by one day? Automatically make various multiple stroke colour the same as various underlying fill colours. I have auto traced a photograph in illustrator - its got thousands of vectors once I expand it and illustrator has converted each of them to a filled colour - I want to now trace the edges of them with a grungy brush but using the underlying fill colour. Where in illustrator can I stroke or outline these fills using the underlying colour - surely this is something that the program automatically does?   I've found this script online but don't know how to implement it. I'm new to this so any help appreciated. Many thanks.   James   var idoc = app.activeDocument;   var sel = idoc.selection;    for (i=0; i&lt;sel.length; i++) {       var ipath = sel[i];       ipath.stroked = true;       ipath.strokeColor = ipath.fillColor;   } Can "what kind" be plural? Is it correct to say, "What kind of patents are being issued in these sophisticated times?" It seems like it should be What kind of patents is being issued..., but that sounds wrong.  What kind of patent is being issued... is correct and sounds right, but loses a little bit of the original meaning.  Is there something magic about what kind that allows it to be plural, or do people just misuse it a lot?