let $\varepsilon &gt;0$. show that $\forall \alpha,\beta \in \mathbb{R}$  $\alpha\beta \leq \frac{1}{2\varepsilon}\alpha^2+\frac{\varepsilon}{2}\beta^2.$  Its looks like holder inequality with $p=2$ and $q=2$.   some hints would be appreciated.
let a,b be real  numbers and $\varepsilon&gt;0$. prove the above inequality. I dont know how to even start. please, I need your help.
reflecting direct light on glossy surface in Cycles Working in Cycles.  Say I have a "disco ball" with a glossy shader. I'm pointing a spot light to it, and I'm trying to have the reflections of that light illuminate the diffuse material on the walls with the patterns from the ball.    I get this:    but I'd like to get something like:    For the illustration I placed a point light inside the ball an removed faces. But what I'd like to do is move the light around the ball so that the light reflected from it changes like it would do in real life.   Is that possible at this point?  EDIT:  Just for the record here's the image rendered in LuxRender (yet another program to learn...) I'm getting pretty close to the reputation where I can start to edit others' posts, and I'm wondering what the etiquette for when (and when not) to do so. I figure that I should edit in the following circumstances:  Grammatical or spelling errors. Clarification where the meaning is not changed. For instance if the title does not describe the question very well. Where the user has made a very minor mistake that doesn't justify a full post to clarify. To add related resources or links that will help someone answer the post, or provide context  Before the commenting feature was added, I noticed that some moderators edit posts in order to insert a reply or extension to the post. I feel this isn't good etiquette (but I could be wrong) because I don't know what to do with their changes when I come to re-edit or respond. I also feel that I should be very careful not to change the meaning of any question or answer, even if that meaning is misguided. What other rules would be best practice?  Signatures and fluff are discouraged and are fair game to be removed.  See, for example, this post by  and this post by  and also see .  Also see Jeff Atwood's response in the comments to How do malloc() and free() work? Number theory proof mod p For a given prime $p$ and integer $k$,  $k∈&lt;1, p-1&gt;$ $S(k)$ is defined as a sum of remainders of $k, k^2, k^3, ..., k^{p-1}$ when divided by $p$. Prove that $S(k)=\frac{1}{2}p(p-1)$ is true for an odd number of different k. I managed to prove that $S(p-1)=\frac{1}{2}p(p-1)$ but can't see how to do it further. Is it possible to use the laptop's audio via headphone jack and Bluetooth speakers at the same time? Lolo Joe wants earphone jack to plug and use big speakers but at the same time he wants to partner the music coming out the earphone jack and his Bluetooth system and speakers. Is it even possible to use the earplug connection and the Bluetooth system and devices at the same time? Suppose I have a line segment of length $L$.  I now select two points at random along the segment.  What is the expected value of the distance between the two points, and why? Why "was" not "were" in "Nearly £20 was taken from my bank account" I've always said "$100 were taken" not "$100 was taken" because I thought $100 is plural. Could you explain why "was" not "were"? Any other helpful notes about the issue would be appreciated.     Nearly £20 was taken from my bank account Copy files excluding x,y,z causing error in shell script Parsing input with scanf in C I've been having a lot of problems trying to figure out how to use scanf().  It seems to work fine with integers, being fairly straight forward scanf("%d", &amp;i).  Where I am running into issues is using scanf() in loops trying to read input.  For example:  do {   printf("counter: %d: ", counter);   scanf("%c %c%d", &amp;command, &amp;prefix, &amp;input); } while (command != 'q');    When I enter in a validly structured input like c P101, it seems to loop again before prompting me.  This seems to happen even with a single:  scanf("%c", &amp;c)    in a while loop.  It'll do the loop twice before prompting me again.  What is making it loop twice, and how do I stop it? When I enter in less amount of input that programmatically wouldn't have another character or number such as q, pressing enter seems to prompt me to enter more.  How do I get scanf() to process both single and double character entries? Prove that $R$ is a commutative ring if $x^3=x$ Let $R$ be a ring satisfying: $\forall x\in R, \; x^3=x$. Prove that $R$ is a commutative ring. How are the measurements used by Nielsen evolving to meet new methods of content delivery and viewing? I was interested to find out that in the 1950s, Nielsen used devices in the homes of the families that were surveyed to record their viewing preferences, and only later introduced their system of using a diary to record habits:  ...in 1950...This information was collected on a device that was attached to a television that recorded what was being watched.  In 1953, the company began sending out diaries to a smaller sample of homes (“Nielsen families”) within the survey to have them record what they had watched from   So, presumably as multiple different viewers in the home were considered in the sample, the means through which the ratings were recorded changed. Given that so many people are able to record shows digitally and watch when convenient and that so many view their favorites on outlets such as iTunes, how have the Nielsen ratings systems adapted (or how does the company plan to adapt them) to capture our modern viewing preferences? While a DVR-recorded show still technically makes its original &quot;timeslot&quot; relevant to the equation, will Nielsen consider timeslot-specific data a hindrance to comparisons between those shows viewed on cable and the same show viewed on iTunes? Show that if $a$ and $b$ are positive integers with $(a,b)=1$ then $(a^n, b^n) = 1$ for all positive integers n Recently my audio input, from USB headset and USB guitar cable have stopped working. Going into Windows 'Sounds' and enabling 'Listen to this device' outputs the audio as expected; the problem is not with the devices. However, as soon as I try to use them with any other program, discord, for example, no audio input is detected by discord.  I have tried everything I can think of and some suggestions from Googling around;   Updating everything - windows, drivers, all related software Rolling windows back to the last build Trying different USB ports Reinstalling drivers and software Running Windows troubleshooters Playing with device properties settings   But nothing seems to work, any suggestions? I'm having some trouble understanding the purpose of a salt to a password.  It's my understanding that the primary use is to hamper a rainbow table attack.  However, the methods I've seen to implement this don't seem to really make the problem harder.  I've seen many tutorials suggesting that the salt be used as the following:  $hash =  md5($salt.$password)   The reasoning being that the hash now maps not to the original password, but a combination of the password and the salt.  But say $salt=foo and $password=bar and $hash=3858f62230ac3c915f300c664312c63f.  Now somebody with a rainbow table could reverse the hash and come up with the input "foobar".  They could then try all combinations of passwords (f, fo, foo, ... oobar, obar, bar, ar, ar).  It might take a few more milliseconds to get the password, but not much else.  The other use I've seen is on my linux system.  In the /etc/shadow the hashed passwords are actually stored with the salt.  For example, a salt of "foo" and password of "bar" would hash to this: $1$foo$te5SBM.7C25fFDu6bIRbX1.  If a hacker somehow were able to get his hands on this file, I don't see what purpose the salt serves, since the reverse hash of te5SBM.7C25fFDu6bIRbX is known to contain "foo".  Thanks for any light anybody can shed on this.  EDIT:  Thanks for the help.  To summarize what I understand, the salt makes the hashed password more complex, thus making it much less likely to exist in a precomputed rainbow table.  What I misunderstood before was that I was assuming a rainbow table existed for ALL hashes. How to stop constantly blinking Wifi LED? koma-script/scrlayer-scrpage: How to get rid of the rulers at the page edge when enabling the draft class option? Is it possible to get rid of these rulers when enabling draft class option?  \RequirePackage{luatex85} \documentclass[draft]{scrreprt}  \usepackage{scrlayer-scrpage}  \usepackage[tuenc]{fontspec} \usepackage{graphicx,mwe}  \begin{document}  \begin{figure}[h]     \centering     \includegraphics[width=\textwidth]{example-image} \end{figure}  \end{document} After I saw that post i wanted to solve the first one which is $a\mid b^2,b^2\mid a^3,a^3\mid b^4,b^4\mid a^5\cdots$ Prove that $a=b$  Now i started by proving that $a$ and $b$ have same prime divisors,proving that is trivial,after doing so I checked do those factors have to have the same power.I tried with $$a=p^2,b=p^3$$ I noticed it exactly goes for 3 terms,or that $a^3\mid b^4$ doing $$a=p^{n-1},b=p^n\\p^{n-1}\mid p^{2n}\\p^{2n}\mid p^{3n-3}\\p^{3n-3}\mid p^{4n}\\p^{4n}\mid p^{5n-5}\\\cdots\\p^{n(n-1)}\mid p^{n(n-1)}\\p^{n(n-1)}\mid p^{n(n+1)}\\p^{n(n+1)}\not\mid p^{(n-1)(n+1)}$$ Now yeah I guess that would be a proof,but wouldn't setting $n=n+1$ infinitely many times make every term dividable? Does SQLDeveloper support executing scripts? There was a foolish (in my opinion) question asked on Scifi.SE       James bond keeps returning with a different face, albeit usually a similar one, and often operates with high-tech gadgetry that seem to be on par with the Doctor's sonic screw driver.   In my opinion each new actor represents a reboot of the franchise using the new face, not a continuation of the story.   Is there any continuity between versions of James Bond which change actors? I'm not talking about characters like M or Q, I'm talking about events or past advisories being referenced in a different Bond movie where the main actor was someone else. Reading shapefile using C#? I am new to ArcGIS and trying a few things..   I'm trying to read shapefile (which is supplied as an input) using C# and I want to generate a PDF file which contains underlying metadata information (similar to the Identify tool) ?