For $x \geq 0$, $y \geq 0$, prove that $$ |x^p-y^p| \leq p|x-y|(x^{p-1}+y^{p-1}). $$  I thought it would be simple but I messed everything up. Here are my attempts. Fixing $x &gt; y \geq 0$, I thought about the function $$ h(p)=x^p-y^p-p(x-y)(x^{p-1}+y^{p-1}) $$ Then $h(1)=0$ and I wanted to prove that $h(p) \leq 0$ when $p \geq 1$ by discussing its derivative but the derivative is messy offering no way out. Also I thought about after fixing $y \geq 0$, consider the function $$ \varphi(x)=x^p-y^p-p(x-y)(x^{p-1}+y^{p-1}) $$ where $\varphi(y)=0$ and I want to prove that $\varphi(x) \leq 0$ when $x \geq y$ yet again it's a messy way. I also thought about restricting $p$ to $\mathbb{N}$ or $\mathbb{Q}$ using decomposition like $$ (x^p-y^p)=(x-y)(x^{p-1}+x^{p-1}y+x^{p-2}y^2+\cdots+y^{p-1}) $$ but the number of terms does not match.  But I do believe this should be a easy question with some special background. I must have missed something critical. Any hint/solution appreciated!
I got stuck on this inequality for a day. If $p$ is positive integer, then the problem becomes too easy, but I can't find how we deal with the general case when p can be any positive number. Can someone give me a hint? I really appreciate     Prove that $|x^p - y^p| \le p|x-y|(x^{p-1} + y^{p-1})$ provided that $1 \le p \lt \infty$ and $x, y \ge 0$
When do I use a dot, arrow, or double colon to refer to members of a class in C++? Boundary condition with spatial derivative is ignored by NDSolve How can we read and write values in a multidimensional array in verilog, i had read in  regarding different operations that can be done on multidimensional array. like if there is a part of code like this  input [15:0] me;  reg [15:0] p_array [7:0]; reg abc_pqr [2:0];  abc_pqr[0] &lt;= me[0]; abc_pqr[1] &lt;= me[1]; abc_pqr[2] &lt;= me[1]; p_array[abc_pqr[0]] &lt;= me[0]; p_array[abc_pqr[1]] &lt;= me[1]; p_array[abc_pqr[2]] &lt;= me[2];   so is this code will work and will not produce any error?  What is the difference between these two while assigning value  "&lt;=" and "=", i had seen "&lt;=" in most of the places during passing values. What is the Killing form of $\mathfrak{gl}_{m}$? Consider the Killing form of the Lie algebra $\mathfrak{gl}_{m}$. Then $\{e_{ij}\}$ is a basis for this Lie algebra where $e_{ij}$ is a matrix with 1 in the $i$th row, $j$th column and 0 everywhere else. Then every $A = (a_{ij}) \in \mathfrak{gl}_{m}$ can be written as $A = \sum_{i, j = 1}^{m}a_{ij}e_{ij}$. Since $[e_{ij}, e_{kl}] = \delta_{jk}e_{il}-\delta_{li}e_{kj}$, we have $ad(a)e_{ij} = \sum_{kl}(a_{ki}\delta_{lj} -a_{jl}\delta_{ki})e_{kl}$. From this we have $$ad(a)ad(b)e_{ij} = \sum_{k, r, s}(a_{rk}b_{ki}\delta_{sj} + a_{ks}b_{jk}\delta_{ri} - a_{ri}b_{jk}\delta_{sk} - a_{js}b_{ki}\delta_{rk})e_{rs}.$$ From this how can I see that the Killing form is $2m tr(ab) - 2tr(a)tr(b)$. I know that the Killing form is just the trace of the above centered equation, but I seem to be getting confused with all the indices. I have  experience where empty spaces cause unwanted effects. It is not easy to trace the cause of these unwanted effects. In order to eliminate any doubt, I often overuse % as follows.   \usepackage% [% left=3cm, right=3cm% ]% {geometry}   or  \newcommand{\mycommand}% {% This is my command.% }   I got an extreme example that will break what Leo Liu said     In fact, only spaces which would be   output have to be removed by comment.   The following code does not produce output, but we cannot remove the % and leave a blank line between two elements of a list below.  \newpsstyle{gridstyle} { gridwidth=0.4pt, % griddots=0 }   Shortly speaking, where are the necessary places to be appended with % to remove unwanted spaces? I would like to know how to make this effect of merged circle selection. Here are images to illustrate:     Basically I'm looking for this effect:    How the merge effect of the circles can be achieved ? I didn't found any explanation concerning this effect. I know that to project those texture I can develop a decal system but I don't know how to create the merging effect.  If possible, I'm looking for purely shaders solution. Good books for a high schooler self-studying Abstract Algebra? Does anyone have any suggestions for abstract algebra books particularly suited to self-study?     Here is some background and motivation, if it's helpful.  I'm currently a junior in high school, but I have some familiarity with groups, rings, polynomials and fields, as I went through Fraleigh's book. I liked its writing style, but many of the problems were not helpful. I would like to get better at algebra with some other book, particularly one that's aimed at someone who already has a little familiarity with the objects, but is by no means completely comfortable.  One of my longer-term goals is to learn a bit of algebraic number theory, as I've only been studying elementary number theory. That's why I feel it  good to learn more algebra first. (I'd also be interested in suggestions of other good subjects to learn prior to tackling algebraic number theory.)  I was considering using one of Lang's books, since they seem pretty universal, but I've heard rumors that they are very terse and not good for self-learning. Butterknife appears twice in the OSS attribution list of the Android app (v1.0.85). Based on the links, the second one should be JMustache. Write access without read access Is it possible for a user to have a write access to a file and not be able to read it? How is it possible?   I tried the following commands:  debianbox@debian:~/posix/io$ touch filetest debianbox@debian:~/posix/io$ ls -l filetest -rw-r--r-- 1 debianbox debianbox 0 14 oct.  03:10 filetest  debianbox@debian:~/posix/io$ echo "Hello World" &gt; filetest debianbox@debian:~/posix/io$ cat filetest Hello World  debianbox@debian:~/posix/io$ chmod u-r filetest debianbox@debian:~/posix/io$ cat filetest cat: filetest: Permission forbidden  debianbox@debian:~/posix/io$   As you can see here, I have write but not read access on this file. How can this be possible? Is this considered as a bug? If not, in what situation would this be useful? How to return multiple objects from a Java method? Double decimal formatting in Java Prevent windows from going into sleep when my program is running? Mass and Newton's Second Law Summary table with a field that shows components of each summarized record To show for following sequence $\lim_{n \to \infty} a_n = 0$ where $a_n$ = $1.3.5 ... (2n-1)\over 2.4.6...(2n)$ How can I show   $\lim_{n \to \infty} a_n = 0$  $a_n = {1.3.5 ... (2n-1)\over 2.4.6...(2n)}$  I have shown that $a_n$ is monotonically decreasing. I thought to shown sequence is bounded from below then it automatically would converge and hence my question will be solve. But I'm unable to show its boundedness... Or there maybe another method to prove this. Thanks How can I shorten my command line (bash) prompt? Hiding some slides in student handouts in Beamer I am preparing lecture slides for a class. I would like to hide some of the slides from the students. As an example, I would like to be able to compile students.pdf which has only student slides 1 and 2 (i.e., "teacher slide" is hidden); but teacher.pdf which has all three slides.  \documentclass{beamer}  \begin{document}  \begin{frame}{Student slide 1} blah blah blah \end{frame}  \begin{frame}{Teacher slide} secret stuff not to be shared with students \end{frame}  \begin{frame}{Student slide 2} blah blah blah \end{frame}  \end{document}   How would I do this in beamer? In a regular document, e.g., article, I would use the comment environment.  I feel that the approach in  is overkill for my needs. If you have an array of Java objects which have a primitive type (for example Byte, Integer, Char, etc).  Is there a neat way I can convert it into an array of the primitive type?  In particular can this be done without having to create a new array and loop through the contents.  So for example, given  Integer[] array   what is the neatest way to convert this into  int[] intArray   Unfortunately, this is something we have to do quite frequently when interfacing between Hibernate and some third party libraries over which we have no control.  It seems this would be a quite common operation so I would be surprised if there's no shortcut.  Thanks for your help! What's the past participle form of the word exit? Is it exit (irregular, like set)? exited? exitted? On one page I found exited but if that's the case why isn't it exitted (double t) like with the word emit - emitted? Is there a rule when the consonant at the end is doubled and when not? Which fruits and vegetables should be kept in a fridge, and which outside?