I am having trouble proving this inequality: $2ab\leq a^2+b^2$  I can transpose the equation and change around signs. But I am not sure If I need to use k+1 here or just prove the inequality. In discrete math what would my professor be looking for as a proper answer?
I am having trouble with this problem from my latest homework.  Prove the arithmetic-geometric mean inequality. That is, for two positive real numbers $x,y$, we have $$ \sqrt{xy}≤ \frac{x+y}{2} .$$ Furthermore, equality occurs if and only if $x = y$.  Any and all help would be appreciated.
Is it possible to configure a TP_Link ADSL router to work with a cable connection? I want to show that for $S,T\in B(H)$ bounded operators on Hilbertspace with $S\geq 0,T\geq 0$ and $ST=TS$,  we have $S+T\geq 0$, and $ST\geq 0$.  $T\geq 0$ means $(Tx,x)\geq 0$.    To me it seems that $((S+T)x,x) = (Sx,x)+(Tx,x)\geq 0$. But for $ST$ i like some help.   There is a theorem in Rudin (12.32) that says that for every $T\in B(H)$ we have an equivalence:   $T$ is positive $\Leftrightarrow$ $T$ is self adjoint, i.e. $T=T^*$ and $\sigma(T)\subset [0,\infty)$.   Since $S, T$ are self adjoint we know that $ST=TS = S^*T^* = T^*S^*$ ($ST$ is selfadjoint since $S,T$ commute). Also I read somewhere that if $S,T $ commute then $\sigma(ST)\subset \sigma(S)\sigma(T)$, but I am not sure if this is true. Clearly I cant prove it, but if it were true by this ''Theorem'' it would follow that $ST \geq 0$.   Is there a more straightforward way? Keyword for the outer class from an anonymous inner class In the following snippet:  public class a {     public void otherMethod(){}     public void doStuff(String str, InnerClass b){}     public void method(a){         doStuff("asd",             new InnerClass(){                 public void innerMethod(){                     otherMethod();                 }             }         );     } }   Is there a keyword to refer to the outer class from the inner class?  Basically what I want to do is outer.otherMethod(), or something of the like, but can't seem to find anything. $I_m - AB$ is invertible if and only if $I_n - BA$ is invertible. The problem is from  Algebra by Artin, Chapter 1, Miscellaneous Problems, Exercise 8. I have been trying for a long time now to solve it but I am unsuccessful.      Let $A,B$ be $m \times n$ and $n \times m$ matrices. Prove that $I_m - AB$ is invertible if and only if $I_n - BA$ is invertible.    Please provide with only a hint. Can a new save withdraw from the pokemon bank? I've been considering restarting my save and replaying the story with access to the entire National Pokedex (Minus a few Event-only Pokemon I'm missing) from the start. But I would first like to confirm: Can I withdraw my stored Pokemon from the bank if my game is reset? It seems that the only contributing factor is that I end up signed in with the same Nintendo Network ID, so as long as my network ID remains the same, I can withdraw stored Pokemon regardless of what save-game or cartridge is in my system, correct?  While I'm reasonably confident it works the way I think it does, past experience (My Pokemon Ranch being tied to specific trainer IDs, and something about the dream world being tied to the game's wireless settings) has made me just cautious enough to ask before attempting. Lost passport with Schengen visa How to trick windows 7 to use only a part of the screen Possible Duplicate:          I have a broken 20" LCD monitor which displays random pixels in the top 200 lines of pixels. I want to limit the display in such a way that when I maximize a window it will not be in the bad area and to limit the cursor position so it won't be in the bad area. What's the most efficient way of determine individual card values for my collection? Potential difference across a zero resistance wire I have some pages on my site that I want to keep search engines away from, so I disallowed them in my robots.txt file like this: User-Agent: * Disallow: /email  Yet I recently noticed that Google still sometimes returns links to those pages in their search results.  Why does this happen, and how can I stop it? Background: Several years ago, I made a simple web site for a club a relative of mine was involved in.  They wanted to have e-mail links on their pages, so, to try and keep those e-mail addresses from ending up on too many spam lists, instead of using direct mailto: links I made those links point to a simple redirector /  script running on my own site.  This script would return either a 301 redirect to the actual mailto: URL, or, if it detected a suspicious access pattern, a page containing lots of random fake e-mail addresses and links to more such pages.  To keep legitimate search bots away from the trap, I set up the robots.txt rule shown above, disallowing the entire space of both legit redirector links and trap pages. Just recently, however, one of the people in the club searched Google for their own name and was quite surprised when one of the results on the first page was a link to the redirector script, with a title consisting of their e-mail address followed by my name.  Of course, they immediately e-mailed me and wanted to know how to get their address out of Google's index.  I was quite surprised too, since I had no idea that Google would index such URLs at all, seemingly in violation of my robots.txt rule. I did manage to submit a removal request to Google, and it seems to have worked, but I'd like to know why and how Google is circumventing my robots.txt like that and how to make sure that none of the disallowed pages will show up in their search results. Ps. I actually found out a possible explanation and solution, which I'll post below, while preparing this question, but  in case someone else might have the same problem.  Please do feel free to post your own answers.  I'd also be interested in knowing if other search engines do this too, and whether the same solutions work for them also. What is projection of WGS84 in GIS software? WGS84 is a reference ellipsoid that determines coordinates of every point on Earth using latitude, longitude and height about the its surface.  This ellipsoid is often used as a base to make maps. Though, to make a map on a flat paper or a flat digital screen you need a projection too. An example of a popular projection would be UTM.   Mainstream GIS software give plethora of projection alternatives to display your data layers. Among them, they also let you display your data using WGS84 only.  I think they are using a certain projection but what projection is that? convergence to infinity of$ (2^n + 3^n)^{\frac{1}{n}}$ (as $n \to \infty$) Just checking. $2^n$ ($n \to \infty$) tends to $\infty$. + $3^n$ also ($n \to \infty$) tends to $\infty$  so the sum gets me $\infty$.  Now $(\infty)^{1/\infty}$ : $(\infty)^0 = 1$  I see no other way.  Theorem: $n^{\frac{1}{n}} = 1$ as $n \to \infty$.  Analogous:  $(2^n - 3^n)^\frac{1}{n}$ = $(\infty  - \infty)^0$ = 1 ???  Is it ok to suppose infinity on any integer $k^n$ as $n$ goes to infinity ? Is it possible to train a neural network without backpropagation? Many neural network books and tutorials spend a lot of time on the backpropagation algorithm, which is essentially a tool to compute the gradient.  Let's assume we are building a model with ~10K parameters / weights. Is it possible to run the optimization using some gradient free optimization algorithms?   I think computing the numerical gradient would be too slow, but how about other methods such as Nelder-Mead, Simulated Annealing or a Genetic Algorithm?  All the algorithms would suffer from local minima, why obsessed with gradient? How can I disable the 2nd hard drive in my 27" iMac? how to stop firing oncomplete if save action has errors My requirement is to stop oncomplete call if save action has errors. I know I am doing some silly mistake.   Here I am unable to stop calling oncomplete function.  Requirement is, from VFP save method, data will get saved and then perform callout from oncomplete function. If there are any errors during save then stop making callouts.  Visualforce  &lt;apex:page id="hasErrorTestPage" controller="ErrorMessageFinder"  showHeader="false"&gt;     &lt;apex:pageMessages id="msgId"/&gt;     &lt;script&gt;         function performCallOutMethod()         {             var msg = document.getElementById('{!$Component.MSEForm.errorMsgInd}').value;             alert(msg);              if(msg == 'false' || typeof(msg) == 'undefined')             {                 alert('entering callout')                 performCallOut();                 return true;             }             else             {                 return false;             }         }     &lt;/script&gt;     &lt;apex:form id="MSEForm"&gt;         &lt;apex:actionFunction name="performCallOut" action="{!submit}" reRender="panelId,msgId"/&gt;          &lt;apex:outputPanel id="panelId"&gt;             &lt;apex:outputText value="{!hasError}" id="errorMsgInd"/&gt;         &lt;/apex:outputPanel&gt;         &lt;apex:inputText id="customerNm" value="{!CustomerName}"/&gt;&lt;br/&gt;         &lt;apex:commandButton value="Save" action="{!save}"                       oncomplete="return performCallOutMethod();" reRender="panelId,msgId"/&gt;     &lt;/apex:form&gt; &lt;/apex:page&gt;   Controller  public class ErrorMessageFinder {     public String CustomerName {get;set;}      public Boolean hasError {          get {              return ApexPages.hasMessages();          }            }      public PageReference save()     {            if(String.isBlank(CustomerName))         {             ApexPages.Message msg = new ApexPages.Message(ApexPages.Severity.Error, 'Name cannot be null');             ApexPages.addMessage(msg);             return null;         }         return null;      }       public PageReference submit()     {         try         {   //navigate to view page             PageReference pg = new PageReference('/'+ '0019000001UwnVS');             pg.setRedirect(true);             return pg;         }         catch (Exception ex)         {             System.debug('Error in submit ' + ex);             ApexPages.Message msg = new ApexPages.Message(ApexPages.Severity.Error, ex.getMessage());             ApexPages.addMessage(msg);              return null;         }     } }   Based on this code, if I hit save button without entering CustomerName then it is not showing error after save. It is showing error during oncomplete call and then navigating to view page, which I do not want.  Any help is appreciated. I have put a compiled code so that anyone can try with this code. Stop asking me if Community auto-flags were helpful Grub not showing on startup for Windows 8.1 Ubuntu 13.10 Dual boot In The Witness, I was walking along the coast, and I came upon this:    Now, since I've learned      that mazes can be embedded in the environment,   I see the puzzle here, but I just can't get the pieces to line up well enough.  Standing on the couch and walking all around it doesn't seem to get me a       functional maze.   Where do I need to stand to activate the puzzle successfully? Convergence of $\int\limits_1^{\infty}x^\alpha\sin(x^\beta)dx$ I need to show that $$ \int\limits_1^{\infty}x^\alpha\sin(x^\beta)dx $$ converges if and only if $\alpha + 1 &lt; \beta$. I used substitution and integration to get $$ \lim\limits_{t\to\infty}\int\limits_1^tx^\alpha\sin(x^\beta)dx \overset{u = x^\beta}{\underset{du = \beta x^{\beta-1}}{=} }\lim\limits_{t\to\infty}\left(\frac{1}{\beta}\int\limits_1^{t^\beta}u^{\frac{\alpha + 1 - \beta}{\beta}}\sin{u}du\right) $$ If $\alpha + 1 -\beta &lt; 0$, then we have a product if a function with bounded anti-derivative and a differentiable decreasing function which tend to 0, and from Dirichlet's criterion, the integral converges.   For the converse, I used integration by parts $$  = \frac{1}{\beta}\lim\limits_{t\to\infty}\left(\left.-u^{\frac{\alpha + 1-\beta}{\beta}}\cos{u}\right|_1^{t^\beta} + \frac{\alpha+1-\beta}{\beta}\int\limits_1^{t^\beta}u^{\frac{\alpha + 1-2\beta}{\beta}}\cos{u}du\right) $$ For $\alpha + 1 -\beta\geq 0$, I get that the first term indeed diverges, but this is not enough to show total divergence. How can one show it? Can you clone a larger drive onto a smaller drive (used space fits on the smaller) I have a 500GB drive which is about 50GB full. I got a 80GB SSD, which has 74.5GB of available space. I want to clone the system drive onto the SSD.   DriveImage XML will not let me copy a larger drive to a smaller one.