Learning about father's WWII US Army military experience
Finding specific records of US WW2 veterans?
integral of $\int_{0}^{1}\frac{\ln(x^{2}+1)}{x+1}dx$ using contour integration? I have an interest in contour integration. I am not that good at it, but I enjoy learning what I can about it.  Here is a version of a rational log integral rarely encountered.  $\displaystyle \int_{0}^{1}\frac{\ln(x^{2}+1)}{x+1}dx=\frac{3}{4}\ln^{2}(2)-\frac{{\pi}^{2}}{48}$  I can do this using real methods (via double integral and substitution). I can post my workings if anyone would be interested.  My question is, can this be evaluated using contour integration due to the limits being $[0,1]$ instead of $[0,\infty)$?.  Contours may not be the most efficient way to go about it, but what is the course of action when the limits are 0 to 1 instead of 0 to infinity?. Why does a RegExp with global flag give wrong results? Setting PATH variable in /etc/environment vs .profile Where is the preferred place to set the PATH envvar?  ~/.profile or /etc/environment?  What is the case when PATH is set in both places? Is the end result a concatenation of both values set in those two places? How can I find the right lens cap for my lens? I own a Nikon D3000 Camera with the AF-S 18-55mm VR Kit Lens. Last week I lost the lens cap and now I want to buy a new one.  But when I went to , I got confused with the different ranges of lens names —  LC-62, LC-78, etc.  My concern is what if I order one lens cap and it won't fit correctly on my lens?  If you guys have an D3000, can you please let me know how to find the correct lens cap names? Prove that $\int_0^\infty \frac{x\,\log x}{(1+x^2)^2} = 0$ I have been asked to prove that $$\int_0^\infty \frac{x\,\log x}{(1+x^2)^2}\,dx = 0$$  I tried proceeding as follows: \begin{align} \int_0^\infty \frac{x\,\log x}{(1+x^2)^2}\,dx &amp;= \int_0^1 \frac{x\,\log x}{(1+x^2)^2}\,dx + \int_1^\infty \frac{x\,\log x}{(1+x^2)^2}\,dx \\ &amp; = \int_0^1 \frac{x\,\log x}{(1+x^2)^2}\,dx + \int_{u=1}^0 \frac{u\,\log u}{(1+u^2)^2}\,du\quad (\text{where }u=1/x) \\ &amp;= \int_0^1 \frac{x\,\log x}{(1+x^2)^2}\,dx - \int_0^1 \frac{u\,\log u}{(1+u^2)^2}\,du \\ &amp;=^? \int_0^1 0 \,dx \\ &amp;= 0 \end{align}  But I think that I should first show that the improper integral $\int_0^1 {x\,\log x}/{(1+x^2)^2}\,dx$ is convergent.  Do I need to show it? If yes, how to do so?  (I have been taught basic methods such as comparison, limit comparison and Dirichlet test.) Derivation of survival copula $$ \begin{align}F(x,y) &amp;= P(X\leq x, Y\leq y) \\  &amp;= C(F(x), F(y)) \end{align}$$ The copula of a bivariate distribution is equal to its CDF, $C(u,v) = F(x,y)$. The density of the data's CDF therefore can be expressed as $$f(x,y) = c(u,v) \times f_X(x) f_Y(y)$$ where $c(u,v)$ is the copula density and $f_X(x)$ is a marginal. If $\hat{C}(u,v)$ is the survival copula for determining $$ \begin{align}\bar{F}(x,y) &amp;= P(X&gt; x, Y &gt; y) \\  &amp;= \hat{C}(1-F(x), 1-F(y)) \end{align}$$ how is the first equation of the ordinary copula, $C(u,v)$, linked to the survival copula as $$\hat{C}(u,v) = u + v - 1 + C(1-u,1-v)$$ are $u$ and $v$ merely vectors containing the (ranked) observations of the uniform marginals? A derivation in integrals and probabilities (rather than $F$s) would be good. Repeatedly rolling a six sided die four times and summing the highest three results gives you a distribution with what mean and standard deviation? Repeatedly rolling a six sided die four times and summing the highest three results gives you a distribution with what mean and standard deviation?  I've only taken AP statistics, but I would like to learn how to do this. Dual-boot boot menu does not show up after installing Ubuntu 15.10 alongside Windows 10 Need intuition about independence of events Suppose that we have the situation as depicted in the figure: a random experiment which has 4 outcomes $x_1, ..., x_4$ and two events $A$ and $B$. Also assume that $P(x_i)=0.25$.  Now, since $P(A \cap B) = P(A)P(B) = 0.25$, by definition the events $A$ and $B$ are independent. This does not make sense to me. Why are they independent? Is there an intuitive explanation?  Another thing is, suppose that there is a 5th outcome, $x_5$, outside of $A \cup B$, in that case the events $A$ and $B$ are no longer independent since  $P(A \cap B) \ne P(A)P(B)$. This result also does not make sense to me. Why do C# and Java bother with the "new" operator? Why does the  exist in modern languages such as C# and Java? Is it purely a self documenting code feature, or does it serve any actual purpose?  For instance the following example:  Class1 obj = new Class1();  Class1 foo() {     return new Class1(); }   Is as easy to read as the more Pythonesque way of writing it:  Class1 obj = Class1();  Class1 foo() {     return Class1(); }   EDIT:  hit the nail on the head with the clarification of the question: Why did they choose this syntax? I am trying to install a Ralink wifi adapter in Ubuntu Desktop 14.04.  The system is not detecting the wifi.   lsusb lists the device as:  Bus 002 Device 004: ID 148f:7601 Ralink Technology, Corp.  I have the rt2800 USB driver listed in "/etc/modules"; Following some instructions on the web, I tried this command:  echo 148F 7601 | sudo tee /sys/bus/usb/drivers/rt2800usb/new_i  lsmod | grep rt2800 returns the following:  rt2800usb              26581  0  rt2x00usb              20041  1 rt2800usb rt2800lib              83150  1 rt2800usb rt2x00lib              48886  3 rt2x00usb,rt2800lib,rt2800usb mac80211              545990  3 rt2x00lib,rt2x00usb,rt2800lib crc_ccitt              12627  1 rt2800lib I am trying to install reaver 1.4, but when I run ./configure ; make I get this message at the end  int ret_val = EXIT_FAILURE, r = 0;   I have searched a fair bit and cannot find an answer, any help would be appreciated. I apologize if this has been answered and I've just failed to find it. Plotting complex numbers as an Argand Diagram Why doesn't Mac OS X source ~/.bashrc? Checking to see if a user has a licence for a managed package So I'm aware that there is a method on UserInfo called isCurrentUserLicensed which lets you check if the user running the code has a licence for a namespace that you specify, i.e. for a particular managed package.  What I want to know is whether there is a way I can test to see if some other user (assume I have their ID) has a licence for a particular package. Has anybody found a way to do this? How to get video duration, dimension and size in PHP? Should I use "who" or "which" when referring to a specific animal species? I'd like to know what option to use, specifically when referring to a species by its scientific name. Necessary condition for square integrable functions? I'm studying Quantum Mechanics and I came across this which I don't quite understand: For a vector space of functions $f(x)$ to be square integrable (i.e $\int{|f(x)|^2dx &lt; \infty)}$, the necessary condition is $\lim_{|x| \to \infty}f(x) \to 0$. Can someone help me understand why that must be the case? I sort of understand that it must not blow up at the infinities but can a (continuous or piece-wise continuous) function have a non-zero finite value at the infinities and still be square integrable? Is it possible/sensible to add a suspension fork to an old mountain bike? I have a ~1993 Giant mountain bike.  It's not great, but I am a very new and casual rider.    The bike has a solid fork and it makes for the predictable rough trail ride.  The trails are generally fairly modest (I live in North Texas, a 100' change in altitude makes our ears pop here) and the surface is generally dirt (very few rocky areas).  So, I am thinking that it would be very nice to have a suspension bike.  And seeing that a modest fork can be purchased for a modest price, I am wondering if it makes sense to add a fork to this bike.   asked about going the other direction, and there were some cautionary words about changing steering geometry.  So, on balance, is this something that I would be best off to avoid?  I suspect that if I have to replace a bunch of steering parts, the cost will quickly approach that of a new low end bike. Consider the curve $\frac{1}{x}$  where $x \geq 1$. Rotate this curve around the x-axis.   One Dimension - Clearly this structure is infinitely long.  Two Dimensions - Surface Area = $2\pi\int_∞^1\frac{1}{x}dx = 2\pi(\ln ∞ - \ln 1) = ∞$   Three Dimensions -  Volume = $\pi\int_∞^1{x}^{-2}dx = \pi(-\frac{1}{∞} + \frac{1}{1}) = \pi$   So this structure has infinite length and infinite surface area. However it has finite volume, which just does not make sense.  Even more interesting, the "walls" of this structure are infinitely thin. Since the volume is finite, we could fill this structure with a finite amount of paint. To fill the structure the paint would need to cover the complete surface area of the inside of this structure. Since the "walls" are infinitely thin, why would a finite amount of paint not be able to cover the outside of the "walls" too?  Please help me make sense of this whole thing.