How to convert the wilcox.test W value output for two independent groups to U in R? I need the U values of Mann-Whitney test, but the wilcox.test in R outputs only W values. I do not think they are identical, so here is my question - how to convert W to U?
Is the W statistic output by wilcox.test() in R the same as the U statistic? I've recently been reading about the Mann-Whitney U test. It turns out that to carry out this test in R you actually need to run a Wilcoxon test!  My question: is the W statistic of wilcox.test in R identical to the U statistic?
I am looking for a short proof that $$\int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}.$$ What do you think?  It is kind of amazing that $$\int_0^\infty \frac{\sin x}{x} \mathrm dx$$ is also $\frac{\pi}{2}.$ Many proofs of this latter one are already in this . Combinatorial proof of $\sum_{i = 0}^{n} \binom{i}{r - 1} = \binom{n + 1}{r}$ and then use result to find a formula for $1^2 + 2^2 + \ldots + n^2$ Trigger to check for duplicate contacts on lead conversion I'd like to write a Lead trigger, to check for duplicate Contacts before inserting the lead, or converting it.   Contacts fields I'd like to check:   Firstname - should not be the same as the leads name Lastname - should not be the same as the leads name Email - should not be the same as the leads phone   After checking these fields &amp; failing to find a duplicate, only then should the lead be converted. How to "grep" for line length in a given range? Sending packets captured with Wireshark Why does extended piston become stuck if air above it is powered? To reproduce this problem make layout like this and then power the piston. Once the piston is extended, removing the power source will not retract it:    Since piston head block is transparent, I see no explanation for this. Also, even if it wasn't transparent, it still wouldn't power a piston. For example this doesn't power the piston:    But if I place redstone and THEN place the piston, it's powered:    It also remains powered no matter what, as long as there isn't a block update event:    I suppose this behavior can be used to make a block update detector, but in my case this sucks. I wanted such setup for automatic smooth stone generator (without any clocks):    But due to this bug, it won't work - the piston will be always extended. I can't (vote to) close a question I voted to close before but then retracted? For example...  I closed a vote for 'too broad'. But then I realized I wanted to close it as a specific duplicate of another question instead. So I retracted my close vote.  That worked. But I can't now vote to close as duplicate and the title says I already voted to close.  Maybe this is intentional but it doesn't feel that way. What techniques have been used to repeat geometry or other detail to the infinite horizon? When I composed my     I would have liked to have the camera pitch up enough to show the horizon, except I didn't have enough geometry to extend the cobblestone street and the sidewalk all the way to infinity (mostly because that would require infinite stuff).  Even in that shot it is clear that I need to widen the "sidewalk" to cover up the "sky"box.    Contrast with     where the ray tracer can use the procedurally replicated geometry.   is a failed attempt at faking out the horizon, even with low detail.  It fails because lighting's interaction with the geometry gives it away.  Some techniques that prevent you from having to deal with infinity include   put the scene in a box (usually a room) put the scene in a canyon (which is just a box without a top so you can see the relatively featureless sky) route the road into a tunnel, or around a corner and hide the horizon with other geometry (which is really just a variant on the canyon). have no detail on the ground at all (flat color or gradient)   Those are all dodging the question of interest to me:  What techniques have people used in blender to fake various kinds of detail repeating into the distance (instead of just obstructing the view of infinity)? In Bash, two integers can be compared using conditional expression     arg1 OP arg2      OP is one of -eq, -ne, -lt, -le, -gt, or -ge. These   arithmetic binary operators return true if arg1 is equal to, not equal   to, less than, less than or equal to, greater than, or greater than or   equal to arg2, respectively. Arg1 and arg2 may be positive or negative   integers.   or arithmetic expression:     &lt;= &gt;= &lt; &gt; comparison      == != equality and inequality   Why do we have two different ways for comparing two integers? When to use which?  For example, [[ 3 -lt 2 ]] uses conditional expression, and (( 3 &lt; 2 )) uses arithmetic expression. Both returns 0 when the comparison is true  When comparing two integers, can these two methods always be used interchangeably? If yes, why does Bash have two methods rather than one? Does going outside of range or line of sight after casting a spell have any effect? Application launchers in '/usr/share/applications' not working in Nautilus Terminating with Master's after several years in a PhD program. How to indicate on CV/Resume? I am not happy with my PhD program, and after several years I am graduating with a master's this month. I earned the M.S. along the way as a PhD student, but have never formally requested it before this semester. What is the best way to indicate this situation on a CV or resume? I don't want to put just M.S. because then it looks like it took me a really long time to get a master's and I can't put that I am a PhD student because I no longer am.  Right now, I am thinking something like  M.S. in &lt;field&gt;, &lt;University&gt;, May 2016 PhD student in &lt;field&gt;, &lt;University&gt;, Aug. 2011 - May 2016 B.S. in &lt;field&gt;, &lt;Undergrad University&gt;, Aug. 2007 - May 2011   Would that be clear what's going on? Alternatively, should I list PhD above M.S. even though I didn't complete it? Should I list both M.S. and PhD in same line since it is the same university and span of time? MySQL strip non-numeric characters to compare How can I group numerical data into naturally forming "brackets"? (e.g. income) The following describes what I'm trying to accomplish, but it's possible an alternative problem statement can describe my goal:  I want to    divide the following numbers into groups where the variances of the numbers within each group are not too large, and the differences between the averages of groups are not too small  compare the distribution obtained in the end with the "perfect" ones and see how "different" it is from being perfect.     Layman's explanation of goal  , and determine the "income brackets" each population is in.  The income bracket is supposed to be self-adjusting based on the input data.  My goal is to ultimately measure or calculate the difference between the income brackets.  I assume there will be many brackets, and want to see how far "apart" each tier is.  Here is a sample of hourly income for a sample set of a population of 20, and a total income of 3587:  Population= 10                   pop=2   population=5              population =3 10, 11,13,14,14,14,14,14,15,20,  40,50  ,90,91,92,93,94      999,999,900    How can I use mathematical concepts to group, sort and analyze data that acts like income distribution over a given population?  At the end of the calculation, I want to determine tiered income distribution, where a perfect distribution would look (something) like this   (each person makes $10 more per hour than the previous; total is 3587) 89, 99, 109, 119, 129, 139, 149, 159, 169, 179, 189, 199, 209, 219, 229, 239, 249, 259, 269, 279   or this:   (evenly distributed groups of people make the same per hour)  (gaps between income groups is consistent and not "too far") (income total is 3587) 99 99 99   129 129 129   159 159 159   199 199 199  229 229 229  269 269 269    Question  How should I analyze the population groups, and measure the gap in a way that will tell me how much is needed to make it more like the last two model sets listed above? A different approach in distributing $8$ distinct balls into $6$ distinct boxes Find the number of ways in distributing $8$ distinct balls into $6$ distinct boxes such that there is at least $1$ ball in each box.  We are well acquainted with the traditional Inclusion-Exclusion principle in solving this and it will come up as $191520$ ways.  I tried in this way:  When we distribute $8$ balls into $6$ boxes we have two possible cases  Case $1.$ There are exactly $2$ boxes with $2$ balls each  Case $2.$ There is exactly $1$ Box with $3$ balls in it.  For Case $1$, the distribution is done by choosing $2$ boxes from $6$ boxes and $2$ balls from $8$ balls and those two balls can be arranged in $2!$ ways. But I require another two balls from remaining $6$ balls to be placed in each of these two boxes which can be done in $\binom{6}{2} \times 2!$ ways. Finally remaining $4$ balls can be distributed in remaining $4$ boxes in $4!$ ways. So total number of ways for Case $1.$ is  $$\binom{6}{2} \times \binom{8}{2} \times 2! \times \binom{6}{2} \times 2! \times 4!=604800$$  For Case $2$, the number of ways is  $$\binom{6}{1} \times \binom{8}{3} \times 5!=40320$$  I am pretty sure the value obtained for Case $2.$ is correct. but I could not find where I went wrong in Case $1.$ What is $P(X > 0 \mid X + Y > 0)$ given that $X, Y$ are i.i.d standard normal? What are alternatives to ExtJS? So what I'm looking for is a javascript framework I can use that has several UI controls. I have taken a look at  but those controls are very basic compared to ExtJS. Are there any other competitive alternatives?     This question exists because it has historical significance, but it is not considered a good, on-topic question for this site, so please do not use it as evidence that you can ask similar questions here. More info: . EmployeeNumber = string.IsNullOrEmpty(employeeNumberTextBox.Text)     ? null     : Convert.ToInt32(employeeNumberTextBox.Text),  I often find myself wanting to do things like this (EmployeeNumber is a Nullable&lt;int&gt; as it's a property on a LINQ-to-SQL dbml object where the column allows NULL values).  Unfortunately, the compiler feels that  There is no implicit conversion between 'null' and 'int'  even though both types would be valid in an assignment operation to a nullable int on their own. Using the null coalescing operator is not an option as far as I can see because of the inline conversion that needs to happen on the .Text string if it's not null. As far as I know the only way to do this is to use an if statement and/or assign it in two steps.  In this particular case I find that very frustrating because I wanted to use the object initializer syntax and this assignment would be in the initialization block... Does anyone know a more elegant solution? Check if user is using IE Let $a,b,c,x,y,z&gt;0$ and $x+ y+ z= 1$. Show that $$ax+ by+ cz\geq  a^{x}.b^{y}.c^{z}$$ I have found the generalization of this inequality: For 2 sets of numbers $\left \{ a_{i} \right \}, \left \{ b_{i} \right \}&gt; 0$ and $\sum_{1}^{n}b_{i}= 1$, we have: $$\sum_{1}^{n}\left ( a_{i}b_{i} \right )\geq  \prod_{1}^{n}\left ( a_{i}^{b_{i}} \right )$$ First solution Simply compare the logs: as the natural logarithm is a concave function, one has $$\ln\bigl(a^x b^y c^z)=x\ln a+y\ln b+z\ln c\le \ln(ax+by+cz)$$ for all non-negative $x,y,z$ such that $x+y+z=1$. How about another solution? I really hope to see that. Thanks!