Integrate $\int^1_{0} \frac{\ln (x+1)}{x^2+1}dx$
Evaluate the integral: $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$
When reading a stack trace like:  [FormatException: Input string was not in a correct format.]    System.Number.StringToNumber(String str, NumberStyles options, NumberBuffer&amp; number, NumberFormatInfo info, Boolean parseDecimal) +2755599    System.Number.ParseInt32(String s, NumberStyles style, NumberFormatInfo info) +112    System.Convert.ToInt32(String value) +68   What do the +68, +112, etc. numbers mean. I'm guessing that they're offsets to something. If so, what? Some examples include:     We fear the damned.   He honored our fallen.   This is a given.   You are the chosen.   The lost were among us.   They obey the venerated.   My beloved kissed me. (TIL “beloved” is a verb thanks to RegDwight АΑA)   Is there a word to describe the past participle used as a noun? I’m hesitant to refer to it as a gerund, since it doesn’t end in -ing. Another possibility is adjectival noun, if you originally treat the word as a participle (e.g. “the damned man”) that has given up its modified noun.  Also, why does English tend to preface such nouns with the? Contacting prospective advisors before graduate school applications - courtesy or annoyance? Last year, rather late during the MSc application cycle (after one round of funding was allocated - tardiness due to an abrupt change in my life situation), I emailed two professors from two schools that I most wanted to go to. I had perused their webpages and decided these professors had the specific research interests closest to my own.  My reason for contacting them before finishing my applications for their respective schools was that the applications included a large section for specific research questions that I wanted to investigate. So, I emailed these professors asking a bit about their work, because their webpages of course featured surface-level stuff.  One professor was very receptive and invited me to Skype with her so she could better explain herself and I could ask her questions. With her support, I was able to write an application that met her standards quite quickly.  The other professor told me - paraphrasing, of course - that he was too far into the process of choosing a new student for his research group to spend time helping me finish the application.  Let me be clear that the latter professor didn't say any harsh words and I have nothing bad to say about him. That said, the purpose of my question is to decide whether it's better for me as a prospective MSc student to email professors ahead of time like this. Was the latter's rejection likely due to my tardiness rather my emailing him? The more receptive professor made the point that she always wanted to have prospective MSc students write their own research proposals and get her approval before submitting the application. Is this common? If so, then I guess reaching out beforehand is a good thing to do, but earlier in the application cycle?  Thank you. How to typeset asymmetrical confidence level limits / uncertainties? How to make unit selection circles merge? I would like to know how to make this effect of merged circle selection. Here are images to illustrate:     Basically I'm looking for this effect:    How the merge effect of the circles can be achieved ? I didn't found any explanation concerning this effect. I know that to project those texture I can develop a decal system but I don't know how to create the merging effect.  If possible, I'm looking for purely shaders solution. I was thinking of the noted 1980 paper by Sidney Coleman and Frank de Luccia--"Gravitational effects of and on vacuum decay"-- about metastable vacuum states that could tunnel to a lower energy "true vacuum" with catastrophic results. I suspect the answer is that if our vacuum is a false one, we wouldn't "see" the true one til it hit us. Read two column CSV as dict with 1st column as key Looking at the picture below, it's easy to see why the perimeter of a polygon inscribed in a circle is an underestimation of the circle's perimeter. This follows from the triangle inequality: Any side (say $AB$) of the polygon is shorter than the circular arc with the same endpoints ($\stackrel{\frown}{AB}$). Summing all these inequalities shows the perimeter of the inscribed polygon is indeed smaller than that of the circle.  I'm wondering if there is proof that the perimeter of a circumscribed polygon always overestimates the perimeter of the circle, which is as simple as that of the inscribed polygon case. Thanks! Can a hyphen be used without anything on the right side? There are 'n' candies and 't' boxes. Find the number of ways to place the candies in the boxes for each of the conditions (given in the problem). There are 'n' candies and 't' boxes. Find the number of ways to place the candies in the boxes for each of the conditions (all the candies must be spread out) : (a) candies and boxes are different; (b) candies of the same diiferent boxes should not be empty cartons: (c) candies equally new, boxes are different; Edit :(d) candies are different, the boxes are the same, there should be no empty boxes; (e) candies of different boxes alike.(Edit : candies are different, boxes are equal) Specify the display type that matches the placement, if possible. My Answers : (a) Each layout is encoded with a word of $n$ letters from the alphabet of 't' letters $\implies$ possible $n^t$ variants. (b) Write, $n$ candies in the form of balls ina line, we need to put $(t-1)$ partition in $(n-1)$ place, but we can't put two partitions on one place, so we get : $^{t-1}C_{n-1}$. (c) First we need to choose a candy in a box (in the first box $n$ ways, in the second box $(n-1)$, $\cdots$ in the $t-th$ : $(n-t+1)$ ways $\implies$ total $n!/(n-t)!$ methods, and then we distribute the remaining $(n-t)$ candies into $t$ boxes, this is encoded with owrds from the alphabets (for each candy $t$ variants) $\implies$ $t^{n-t}$. The Answer is : $\frac {n!}{(n-t)!}*t^(n-t)$ Are my answers (a), (b) &amp; (c) are correct? For (d) &amp; (e) I don't know how to proceed? Please help me. I have a file with the lines like below.  title1:A1 title2:A2 title3:A3 title4:A4 title5:A5  title1:B1 title2:B2 title3:B3 title4:B4 title5:B5  title1:C1 title2:C2 title3:C3 title4:C4 title5:C5  title1:D1 title2:D2 title3:D3 title4:D4 title5:D5  How can I achieve this?   title1    title2     title3    title4 A1         A2         A3         A4 B1         B2         B3         B4 C1         C2         C3         C4 D1         D2         D3         D4 Sigma algebra and algebra difference An algebra is a collection of subsets closed under finite unions and intersections. A sigma algebra is a collection closed under countable unions and intersections.   Whats the difference between finite and countable unions and intersections? Does "countable" mean it implies there can be infinitely many unions and intersections?    Secondly, I was reading a definition  For an algebra on a set: By De Morgan's law, $A \cap B = (A^c \cup B^c)^c$, thus an algebra is a collection of subsets closed under finite unions and intersections.  What law are they using here to get $A \cap B = (A^c \cup B^c)^c$? I thought de morgan's law was $(A\cap B)^c = A^c \cup B^c$?  Finally, what exactly do they mean by "closed under finite unions and intersections? Continuous weighted random distribution, biased toward one end I am currently contributing to a particle system for our game and developing some emitter shapes.  My uniform random distribution along a line or along a rectangular area works fine - no problem.  But now I would like to have something like a 1 dimensional gradient in this distribution. This would mean for example lower values are more common than higher values.  I don't know what would be appropriate mathematical terms for this problem, so my search skills are rather useless with this one. I need something that is computationally simple, as the particle system needs to be efficient. Should the first word after a colon be capitalized? Proof with 3D vectors Let ${a} = \begin{pmatrix}x_a\\y_a\\z_a\end{pmatrix}$, ${b} = \begin{pmatrix}x_b\\y_b\\z_b\end{pmatrix}$, and ${c} = \begin{pmatrix}x_c\\y_c\\z_c\end{pmatrix}$. Show that $(x_a,y_a,z_a)$, $(x_b,y_b,z_b)$, and $(x_c,y_c,z_c)$ are collinear if and only if ${a} \times {b} + {b} \times {c} + {c} \times {a} ={0}.$  Hello,  Is there any other way to do this problem without bashing?  I can't seem to find a nice and slick solution to this.  Thanks in advance! 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? Numpy installation? I used VIRTUALBOX to install Ubuntu 12.04 on my windows pc. I used apt-get to install Python 2.7 that shipped with 12.04. I followed the directives I saw on this site to install Numpy, Scipy, Matplot all failed. I download tar source file of these modules. How do I install them manually? Someone pls help, I'm getting frustrated. I am playing a bit around with Qgis 2.14.6 to understand how it deals with distance measurement.  I took a line-shapefile that I saved in 3 different coordinate systems (4326, 32642 and a national one that doesn't exist in Qgis yet).  Using 32642 and the national one, the length is quite similar (6442m vs. 6428m)  But using 4326 (unprojected WGS84), I get 8057m  In all cases, I just used the Field Calculator and introduced $length.  I was already positively surprised that the result using 4326 was in meters and not in degrees.  To calculate the distance on the dataset in 4326, I assume that Qgis does some kind of re-projection in the background. Does anyone know which one?   Shapefiles here if anyone wants to play around:   EDIT - adding more info after more tests  As you can see, now both my shapefile and the project are in EPSG:4326, so unprojected:    Here the rough measurement of my shapefile, with the Qgis measure tool:   When I use the field calculator, the preview gives me a value in degrees, while the result seems to be in meters:      This would mean that the function $length, like the measurement tool, always gives a result in meters, but using a very coarse approximation of the degrees => meters transformation:    Is this a bug? or should it really behave like this? I find it at best counter-intuitive (but maybe it's just me) and, at worse, completely misleading (it makes very little sense to use, everywhere on earth, the length of 1 degree at the Equator). Any input or opinion here? Should I open a bug report (at least for the inconsistency between the preview and the actual result)? For $V$ vector space of dimension $n$ under $\mathbb C$ and $T: V \to V $ linear transformation , show $V= \ker T^n \oplus $ Im $T^n$ UV map / model turned black on saving - how to recover? In the middle of touching up a new texture, I pressed save and suddenly my textures all turned black on the model and UV texture - just vanished.  I tried reopening the file in another instance of Blender, I tried to CtrlZ too and it just did this weird patchy thing on my texture.   How do I recover it and what did I do wrong?