I am having trouble with this proof. I believe the way to do it is through induction. This is what I have so far.  Proof:  We begin by induction on n. For the case that n = 1, we have $a^1-b^1= (a-b) \geq (a-b)(1)(b^0) = (a-b) $.  $(a-b) \geq (a-b)$   Now we assume that this is true for some natural number k. $a^k-b^k \geq (a-b)kb^{k-1}$  Now we must show it is true for k + 1. So $a^{k+1} - b^{k+1} = a^ka-b^kb.$  I am not really sure how to proceed from this point. Where can I use the inductive hypothesis?
I was trying to prove this inequality using induction, but couldn't do.  Question: Suppose $a$ and $b$ are real numbers with $0 &lt; b &lt; a$. Prove that if $n$ is a positive integer, then:  $$a^n-b^n \leq na^{n-1}(a-b)$$
Confused about AMD/Intel When I go to the downloads page to get Ubuntu, it automatically downloads the AMD version, but my computer has no hardware with AMD on it. I only have Intel and Nvidia. Also, when I install this AMD version on my computer, it works (other than the constant random freezing of the computer when I use Ubuntu). I don't really understand why the amd version works on a computer with an intel processor. How to calculate Area Under the Curve (AUC), or the c-statistic, by hand I am interested in calculating area under the curve (AUC), or the c-statistic, by hand for a binary logistic regression model.  For example, in the validation dataset, I have the true value for the dependent variable, retention (1 = retained; 0 = not retained), as well as a predicted retention status for each observation generated by my regression analysis using a model that was built using the training set (this will range from 0 to 1).  My initial thoughts were to identify the "correct" number of model classifications and simply divide the number of "correct" observations by the number of total observations to calculate the c-statistic. By "correct", if the true retention status of an observation = 1 and the predicted retention status is > 0.5 then that is a "correct" classification. Additionally, if the true retention status of an observation = 0 and the predicted retention status is &lt; 0.5 then that is also a "correct" classification. I assume a "tie" would occur when the predicted value = 0.5, but that phenomenon does not occur in my validation dataset. On the other hand, "incorrect" classifications would be if the true retention status of an observation = 1 and the predicted retention status is &lt; 0.5 or if the true retention status for an outcome = 0 and the predicted retention status is > 0.5. I am aware of TP, FP, FN, TN, but not aware of how to calculate the c-statistic given this information. If $(\log_4 x)^2= (\log_2 x)(\log_a x)$ (the $4$ is the little number next to log by the way) , find the value of $a$. I have some troble understanding the Kernel density estimation.  If a consider the next example:  s&lt;-0 z&lt;-density(rnorm(25)) f&lt;-function(i)approx(z$x,z$y,xout=i)$y for(i in seq(min(z$x),max(z$x),length.out = 10000)) s&lt;-s+f(i) s   I expected getting that s is approximately equals to one since I am considering an approximation for the integral of the Kernel density function (that I suppose is also a density). However, I don't get the expected result. Why it doesn's sum one?  The idea behind this is that I wanted to use the Kernel as the density of my data points. I was interpolating to get my explicit function and use with other x values. Or do you know another way to get a density function estimated by fitting? Pluralization issue on new question statistics underneath title, with views It looks like the new question statistics underneath the question title introduced a pluralization issue with views:        To be clear here, this was not an issue with the old sidebar-based question statistics; that used to properly say "viewed 1 time". Well, luckily I got this error on my test environment. I'm so scared about this error that I disabled all logon triggers I was using on production environment.   I created the famous logon trigger, and I put a database in offline mode just to mess with something else. Then I discover that because the trigger can't find the table, I can't access the instance anymore. There are plenty of questions related to this, but even doing everything I could find, I still can't acces the instance, not even with SQLCMD.  I'm doing this:  sqlcmd -S server\instance -U sa -P &lt;pass&gt;  -A   If I don't use -A I receive the famous error:  logon failed for sa due trigger execution.   If I use -A:     Sqlcmd: Error: Microsoft ODBC Driver 11 for SQL Server : SQL Server   Network Interfaces: An error occurred while obtaining the dedicated   administrator connection (DAC) port. Make sure that SQL Browser is   running, or check the error log for the port number [xFFFFFFFF]. .   Sqlcmd: Error: Microsoft ODBC Driver 11 for SQL Server : Login timeout   expired. Sqlcmd: Error: Microsoft ODBC Driver 11 for SQL Server : A   network-related or instance-specific error has occurred while   establishing a connection to SQL Server. Server is not found or not   accessible. Check if instance name is correct and if SQL Server is   configured to allow remote connections. For more information see SQL   Server Books Online..   Why i'm not able to connect to the sql server    sql server express here.  I could enable SQL Browser, but I still can't access it.  I don't think this question is duplicated because is not about the trigger, is about Why I can't access the server with SQLCMD. I pretty much know how to fix this. I just don't know how can I acces sql server without triggering the trigger. Cross product: matrix transformation identity How can one prove the following identity of the cross product?  $$(M a)\times (M b)=\det(M) (M^{\rm T})^{-1}(a\times b)$$  $a$ and $b$ are 3-vectors, and $M$ is an invertible real $3 \times 3$ matrix. I want to write all zeros to my hard drive, so I can re-install windows 7 To find minimum distance between two curves Using StartCoroutine() and WaitForSeconds() I am trying to perform two operations. First I want to increase the size of a GameObject and then delay for 0.1s before returning it back to its original size. However, when I try using StartCoroutine(), it keeps giving me the error :  An object reference is required to access not-static members   even though I made my Delay() function static. Can somebody please tell me what I'm doing wrong?  public static void MoveSelected(int choice) {     GameObject rock1 = GameObject.Find("Rock1");     GameObject paper1 = GameObject.Find("Paper1");     GameObject scissors1 = GameObject.Find("Scissors1");      Vector3 scl;      if (choice == 1)     {         scl = rock1.transform.localScale;         scl.x *= 1.1f;         scl.y *= 1.1f;         rock1.transform.localScale = scl;          StartCoroutine(Delay());          scl.x /= 1.1f;         scl.y /= 1.1f;         rock1.transform.localScale = scl;     }     else if (choice == 2)     {         scl = paper1.transform.localScale;         scl.x *= 1.1f;         scl.y *= 1.1f;         paper1.transform.localScale = scl;          StartCoroutine(Delay());          scl.x /= 1.1f;         scl.y /= 1.1f;         rock1.transform.localScale = scl;     }     else if (choice == 3)     {         scl = scissors1.transform.localScale;         scl.x *= 1.1f;         scl.y *= 1.1f;         scissors1.transform.localScale = scl;          StartCoroutine(Delay());          scl.x /= 1.1f;         scl.y /= 1.1f;         rock1.transform.localScale = scl;     }  }  static IEnumerator Delay() {     yield return new WaitForSeconds(0.1f); } How do I permanently allow a website to access my location? In iOS, I only need to be asked once whether I want to allow a website to share my location data. Is it possible to permanently do this in OS X version of Safari as well? MCMC in a frequentist setting I have been trying to get a sense of the different problems in frequentist settings where MCMC is used. I am familiar that MCMC (or Monte Carlo) is used in fitting GLMMs and in maybe Monte Carlo EM algorithms. Are there more frequentist problems where MCMC is used? Modifying .sty files? Yesterday I did some changes in the layout of my TeX document. I couldn't accomplish the changes until I opened the .sty files and modified some of their code. I am sure this is not the best idea. For one reason, after the packages got an update, I will have to modify the file again (and again and again).  Another way would be to create my own version of that .sty file and call that instead of the original one, but this might not be the best idea either. (again: at least the updates speak against it)  How can I accomplish the following changes by adding code to a local file (e.g. my .tex preamble) instead of changing global files (like the package .sty files)?  I wanted to change the following lines in ledmac.sty:  \def\printlines#1|#2|#3|#4|#5|#6|#7|{\begingroup   \setprintlines{#1}{#2}{#3}{#4}{#5}{#6}%   \ifl@d@pnum #1\fullstop\fi   \ifledplinenum \linenumrep{#2}\else \symplinenum\fi     %% #2 into: \arabtxt{\n{#2}}   \ifl@d@ssub \fullstop \sublinenumrep{#3}\fi   \ifl@d@dash \endashchar\fi   \ifl@d@pnum #4\fullstop\fi   \ifl@d@elin \linenumrep{#5}\fi   \ifl@d@esl \ifl@d@elin \fullstop\fi \sublinenumrep{#6}\fi \endgroup}   The second thing is a line in ledpar.sty:  \newcommand*{\l@dlinenumR}{%   \numlabfont\linenumrepR{\line@numR}\Rlineflag%       %% numlabfont into: numlabfontR   \ifsublines@     \ifnum\subline@num&gt;\z@       \unskip\fullstop\sublinenumrepR{\subline@numR}%     \fi   \fi}   (In the latter case, just adding \renewcommand with the modified code did not work.) Is (or will be) there any sort of light linking in Cycles, or a method to get a light to affect only certain objects that doesn't involve compositing two or more different render layers? Sort of what 3Dstudio does with the Include/Exclude methods. Propose Newbie-Overflow site or section Show that the countable product of metric spaces is metrizable Show that $x^{n-1}+\cdots +x+1$ is irreducible over $\mathbb Z$ if and only if $n$ is a prime. Display Custom Fields in a SharePoint 2010 Content Query Web Part Using SharePoint 2010, Having Custom List With 4 Columns, Now I am Using Content Query Web-part To Display List Data ,But Currently It Display Only First Column, SO How to Display Custom Fields in a SharePoint 2010 Content By Query Web Part How do you prove the second law of thermodynamics from statistical mechanics? How do you prove the  from statistical mechanics? To prove entropy will only increase with time? How to prove? Please guide. Similar to my old question about  are there any limits to the animal forms an Animagus can be?     We know that the animal form can be much smaller than human (cat, rat, beetle) so I don't think size is a factor.   I'm not aware of any Animagus who transforms into a magical creature (e.g. unicorn or phoenix) so maybe those aren't possible.   Ultimately, what I'd like to know is whether there could be a T-Rex Animagus... (prompted by )