How do I get the position, rather than the value, of the highest one bit in integer? I know I can get the value like this:  // Prints 8; in 1101, the highest bit is the one denoting 8 int f = 13; System.out.println("Highest: " + Integer.highestOneBit(f));   However, I'd like to get position of the highest one bit - in this case, it would be 4 (since 1000 in binary == 8, the highest one bit is the fourth one).
This is probably pretty basic, but to save me an hour or so of grief can anyone tell me how you can work out the number of bits required to represent a given positive integer in Java?  e.g. I get a decimal 11, (1011). I need to get the answer, 4.  I figured if I could work out how to set all the bits other than the most significant bit to 0, and then >>> it, I'd get my answer. But... I can't.
Where should the comma be placed in the salutation of a letter? Sometimes I see a comma after the proper name:     Hello Mr. Black,      In order to give you....   But my native language is not English and I think that the comma in this phrase should be placed before the proper name:      Hello, Mr. Black.      In order to give you....   What is the correct punctuation in English? Integer solutions for $x^2-y^3 = 23$. As the title stated, I am wondering the integers $x,y$ that satisfy the equation $x^2-y^3 = 23$. When you create a script via the Unity editor, it generates a script with some pre-formatted code.  using System.Collections; using System.Collections.Generic; using UnityEngine;  public class GenericClass : MonoBehaviour {      // Use this for initialization     void Start () {      }      // Update is called once per frame     void Update () {      } }   When I create a script, I am generally guaranteed to use additional code, such as a namespace or a custom editor. Furthermore, I almost always delete content from the automatically-generated script. Is there a way to change the automatic code generated by Unity? reCaptcha v1 is shutting down on March 31, but SEDE still uses it According to this , I thought the new recaptcha API was used network-wide. Though today, when going to  (which for whatever reasons never recognized me automatically), I got this message instead of a usable captcha image.   V1 Unsupported Please direct siteowner to   (Doing a second try doesn't show this message anymore, and the recaptcha v1 works as intended.)  According to , the reCAPTCHA v1 API will stop working on March 31, 2018:  What happens to reCAPTCHA v1? Any calls to the v1 API will not work after March 31, 2018. Starting in November 2017, a percentage of reCAPTCHA v1 traffic will begin to show a notice informing users that the old API will soon be retired. Most websites around the world have already switched to reCAPTCHA v2, where humans pass effortlessly. With our advanced risk analysis engine, reCAPTCHA v2 can effectively separate humans from bots and always stay ahead of the attackers. If your site still uses reCAPTCHA v1, please upgrade to reCAPTCHA v2 immediately. We no longer support reCAPTCHA v1. Any calls to the v1 API will not work after March 31, 2018. Please register a new key and upgrade to v2.  I'm really hoping this will not take  to fix, because we've only got three and a half weeks left! I have 3 old computers lying around right now running a P2 at 800 MHz(?), Intel Mobile 1.6 GHz, AMD Athlon XP 2000+ at 1.66 GHz, and (might not use this) P4 at 2.7 GHz, all with 512 MB Ram, and am considering clustering them together for fun/knowledge. They would be running an undecided version of linux, preferably ubuntu based.   The issue is what I want to use it for: general computing and occasional video encoding. By general computing I mean day to day tasks. However I'm not sure if every program started by a single X session is going to exist on the same machine, defeating the purpose of such a system. Will programs be split up or exist on one machine?  Second, assuming this is running 100baseT ethernet (not sure if the PCI slot itself could handle Gigabit), would the speed of having a program exist over the network be an issue? It seems that the constant asking of various things in RAM would be quite slow.  And before you say "buy another computer!", that's not the point of this question. I'm asking would it be usable, not necessarily practical. And yes I know, this is going to be extreamly power consuming. Let $ D = 2^\mathbb{N} $, i.e., D is the set of all sets of natural numbers.   What's the meaning of this definition? Intuitively, I would suggest that $ D = \{1,2,4,...\} $ but the explanation "set of all sets" leads me to the guess that this is wrong. How to select a clustering method? How to validate a cluster solution (to warrant the method choice)? One of the biggest issue with cluster analysis is that we may happen to have to derive different conclusion when base on different clustering methods used  (including different linkage methods in hierarchical clustering).  I would like to know your opinion on this - which method will you select, and how. One might say "the best method of clustering is which gives you the right answer"; but I may question in response that cluster analysis is supposed to be an unsupervised technique - so how do I know which method or linkage is the right answer?  In general: is a clustering alone robust enough to rely on? Or we need a second method and get a shared result to be based on both?  My question is not only about possible ways to validate / evaluate clustering performance, but is broader - on what basis do we select/prefer one clustering method/algorithm over another one. Also, is there common warnings that we should look around when we are selecting a method to cluster our data?  I know that it is very general question and very difficult to answer. I only would like to know if you have any comment or any advise or any suggestion for me to learn more about this. Windows 7 / 8 stop modal dialogue flash on parent focus Is there some setting in Windows where I can change the default behavior that causes the border and title of a modal dialogue box to flash like 6 or 7 times when I click on the parent window?  Simply focusing it would be plenty to let me know what I did was wrong.   I don't feel the need for all the sirens and bull horns and strobe lights to let me know I clicked in the wrong place. A positive polynomial is the sum of two squares in $\mathbb{R}[X]$ Let $P\in\mathbb{R}[X]$ be a positive polynomial.  I want to show that there exists $A,B\in\mathbb{R}[X]$ so that $P=A^2+B^2$    $\displaystyle P=a\prod_{k=0}^q(x-a_k)\prod_{k=0}^{2p}(x-b_k)=a\prod_{k=0}^q(x-a_k)\prod_{k=0}^p(x^2-2\Re(b_k)+|b_k|^2)$ where $a$ is the leading coefficient, $a_k$ are the real rooks, $b_k$ are the complex roots.  What can i do now ? I know that the identity $(a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc)^2$ should come in handy, but I can't figure out :  1) How to show that the product real roots is a square  2) How to use the above identity for the complex roots.  If I can solve those two points, then I can solve the problem. How can I diagnose and repair missing drive space? Difference of the \dots* I have a question about calculating the degree of a finite field extension over $\mathbb{Q}$.  This is problem 18 in chapter 1 of Patrick Morandi's Field and Galois Theory. The problem asked to show that $$ \left[ \mathbb{Q}{ \left( \sqrt[4]{2},\sqrt{3} \right):\mathbb{Q} }\right] =8 $$ Here's my work : First, we have $$  \left[ \mathbb{Q} ( \sqrt[4]{2},\sqrt{3}):\mathbb{Q}\right] = \left[ \mathbb{Q}(\sqrt[4]{2},\sqrt{3}) : \mathbb{Q}({\sqrt[4]{2}}) \right] . \left[ \mathbb{Q}{(\sqrt[4]{2})}: \mathbb{Q} \right]$$ It's easy to show that $\left[ \mathbb{Q}{(\sqrt[4]{2})}: \mathbb{Q} \right] =4 $ because the minimal polynomial of $\sqrt[4]{2}$ over $\mathbb{Q}$ is $x^4 -2$. Next, I'm going to show that  $$ \left[ \mathbb{Q}(\sqrt[4]{2},\sqrt{3}) : \mathbb{Q}({\sqrt[4]{2}}) \right] =2$$ Consider the polynomial $f(x) = x^2 -3$. It's monic and $\sqrt{3}$ is a root of $f$, but I'm stuck in proving $\sqrt{3} \notin \mathbb{Q}(\sqrt[4]{2})$ (hence $f$ is irreducible over $\mathbb{Q}(\sqrt[4]{2}))$.  My question is: Is there any proof to show that $\sqrt{3} \notin \mathbb{Q}(\sqrt[4]{2})$ or another way to caculate the degree of this field extension?  Thank you for your help. Function is Continuous When Restricted to Closed Sets Which Cover the Space Suppose that $\{A_i\}$ is a finite collection of closed sets in $X$ such that $X = \bigcup A_i$, and $f : X \to Y$ restricted to each $A_i$ is continuous. Then $f$ is continuous.   Okay. The base case ($n=2$) is just the pasting lemma. I am having a little trouble with the inductive case, specifically with what the hypothesis is and how to deal with continuity with respect to subspaces. I believe the inductive hypothesis is to assume that any function over a topological space that is the union of $n-1$ closed sets such that the function restricted to each of these $n-1$ closed set is continuous must be continuous over the whole space. Then I assume I have a topological space $X = \bigcup_{i=1}^n A_i$, where each $A_i$ is closed, and a function $f : X \to Y$ such that $f|_{A_i}$ is continuous for each $i$. Thus $X = \bigcup_{i=1}^{n-1} A_i \cup A_n$, where $C = \bigcup_{i=1}^{n-1} A_i$ is closed since it is a finite union of closed sets. If I can show $f|_C$ is continuous, then I simply apply the pasting lemma to finish the problem. Give $C$ the subspace topology, and notice that each $A_i$ is closed in $C$. Thus we have a topological space $C$ that is the union of $n-1$ closed. But do I still have continuity $f$ restricted to $A_i$ since I am no longer considering them subspaces of $X$ but of $C$? When one says $f|_{A_i}$ is continuous, is one saying that $f$ is continuous when restricted to $A_i$ and $A_i$ is given the subspace topology?  EDIT:  I think I figured out the lemma I need. Claim: Let $A$ and $C$ be subspaces of $X$ such that $A \subseteq C$. Topologize $C$ with subspace topology with respect to $X$. Then the subspace topology on $A$ as a subspace of $X$ (call it $\tau_X$) is the same as the subspace topology of $A$ as a subspace of $C$ (call it $\tau_C$).   Proof: Let $U$ be open in $X$. Then $A \cap U$ is some open set in $A$ as a subspace of $X$. Since $C$ is a subspace of $X$, $C \cap U$ is open in $C$ as a subspace of $X$. Then $A \cap (C \cap U) = (A \cap C) \cap U = A \cap U$ is open in $A$ as a subspace of $C$, thereby proving $\tau_X \subseteq \tau_C$. Now let $O$ be some open set in $C$ as a subspace of $X$. Then $A \cap O$ is some open subset of $A$ as a subspace of $C$. But $O = C \cap U$ for some open set in $X$. Therefore $A \cap O = A \cap (C \cap U)= A \cap U$, which is also open in $A$ as a subspace of $X$. Hence $\tau_C \subseteq \tau_X$  Kind of obvious. But it was hard to spot the need for this lemma when I was considering subspace of subspaces and continuous functions, etc. Am I right in thinking this is the lemma I need? Reset password in Drupal 6? I'd like to show that the class of all limit ordinals, LIM, is cofinal in the class of all ordinals ON.  I thought that I could do this by contradiction and assume that it wasn't cofinal in ON. Then there is an $\alpha \in$ ON such that for all $\beta \in $ LIM: $\beta &lt; \alpha$, i.e. $\beta \in \alpha$ and hence LIM $\subset \alpha$.  This means that LIM is a set so we can consider $\bigcup$ LIM. Then we have $\beta &lt; \bigcup$ LIM for all $\beta \in$ LIM by definition of $\bigcup$ LIM. So LIM $\subseteq \bigcup$ LIM.   $\bigcup$ LIM is either a limit or a successor ordinal.  If it is a limit ordinal then $\bigcup$ LIM $\in$ LIM and by transitivity of ON, $\bigcup$ LIM $\subseteq$ LIM, so $LIM = \bigcup$ LIM in this case which would mean that $\bigcup$ LIM $\in$ LIM $= \bigcup$ LIM which would be a contradiction because sets cannot contain themselves. Hence $\bigcup$ LIM cannot be a limit ordinal.  If $\bigcup$ LIM is a successor ordinal then $\bigcup$ LIM = $\beta \cup \{&#160;\beta \}$ for some $\beta \in $ ON.  Now I'm not sure how to proceed from here. I'm also not sure about whether I'm on the right track with this proof because I think there should be a shorter way  to prove this.  Thanks for your help. When writing answers on here, it's often necessary to create a new script and chmod +x it to make it executable. I won't go into whether this is a good thing or not, but explaining to people that they need to fire in three separate command (and teaching them to use nano!) is a bit of a waffle.  Is there anything easier? Why is my USB drive showing corrupted data when plugged as an internal SATA drive? Surjectivity of Composition of Surjective Functions Suppose we have two functions, $f:X\rightarrow Y$ and $g:Y\rightarrow Z$. If both of these functions are onto, how can we show that $g\circ f:X\rightarrow Z$ is also onto? Clipping Rasters with a mask layer error I've got a NDVI layer that I'm trying to clip with a vector file but when I try and use GDAL Tools clipper I get an error.  That shows up like this:  What am I doing wrong?  This is before I even pick the mask layer!  Traceback (most recent call last):   File "C:/PROGRA~1/QUANTU~1/apps/qgis/./python/plugins\GdalTools\tools\widgetPluginBase.py", line 28, in someValueChanged     self.emit(SIGNAL("valuesChanged(const QStringList &amp;)"), self.getArguments())   File "C:/PROGRA~1/QUANTU~1/apps/qgis/./python/plugins\GdalTools\tools\doClipper.py", line 105, in getArguments     return self.getArgsModeMask()   File "C:/PROGRA~1/QUANTU~1/apps/qgis/./python/plugins\GdalTools\tools\doClipper.py", line 143, in getArgsModeMask     if self.alphaBandCheck.isChecked(): TypeError: 'sip.methoddescriptor' object is not callable  Python version: 2.5.2 (r252:60911, Feb 21 2008, 13:11:45) [MSC v.1310 32 bit (Intel)]  QGIS version: 1.7.3-Wroclaw Wroclaw, 00624b3  Python path: ['C:\Users\dean.walker\.qgis\python\plugins\GeoCoding', 'C:\Users\dean.walker\.qgis\python\plugins\GeoCoding', 'C:/PROGRA~1/QUANTU~1/apps/qgis/./python', 'C:/Users/dean.walker/.qgis/python', 'C:/Users/dean.walker/.qgis/python/plugins', 'C:/PROGRA~1/QUANTU~1/apps/qgis/./python/plugins', 'C:\PROGRA~1\QUANTU~1\bin\python25.zip', 'C:\PROGRA~1\QUANTU~1\apps\Python25\DLLs', 'C:\PROGRA~1\QUANTU~1\apps\Python25\lib', 'C:\PROGRA~1\QUANTU~1\apps\Python25\lib\plat-win', 'C:\PROGRA~1\QUANTU~1\apps\Python25\lib\lib-tk', 'C:\PROGRA~1\QUANTU~1\apps\qgis\bin', 'C:\PROGRA~1\QUANTU~1\apps\Python25', 'C:\PROGRA~1\QUANTU~1\apps\Python25\lib\site-packages', 'C:\PROGRA~1\QUANTU~1\apps\Python25\lib\site-packages\win32', 'C:\PROGRA~1\QUANTU~1\apps\Python25\lib\site-packages\win32\lib', 'C:\PROGRA~1\QUANTU~1\apps\Python25\lib\site-packages\Pythonwin', 'C:\PROGRA~1\QUANTU~1\apps\Python25\lib\site-packages\wx-2.8-msw-unicode', '/usr/share/qgis/python', 'C:\PROGRA~1\QUANTU~1\apps\qgis\python\plugins\fTools\tools'] Integrate JavaScript in JSF composite component, the clean way