Remove violet glowing effect
What is Chromatic Aberration?
Proof by Induction for inequality, $\sum_{k=1}^nk^{-2}\lt2-(1/n)$ Let $n$ be a positive natural number, $n\ge 2$. Then $\displaystyle\sum_{k=1}^n \frac{1}{k^2} \lt 2 - \frac{1}{n}.$   The basis step was easy but could someone give me a hint in the right direction as to how to do the induction step?  I tried this:  $\displaystyle\sum_{k=1}^k \frac{1}{k^2} + \frac{1}{(k + 1)^2} \lt 2 - \frac{1}{k + 1}$  But it's getting me nowhere or I am doing something wrong. I am no expert so a clear explanation would be appreciated. Thanks. How do I draw the graph of inequalities in the coordinate plane? I want to graph the following inequalities (done in desmos) in latex. I need the code. Is `if (condition = value)` the correct syntax for comparison? How long can you say "the late so and so"? When you refer to the deceased, you say "the late so and so." How long can you say that? Is JFK referred to as the late John F. Kennedy? How about Abraham Lincoln? Best books for mathematical background? I want to design a member login site in WP but I have a doubt that is WordPress can handle more than 40000 users on same database?  I am not sure about this so I am stooping my work here. So please help me if anyone know exactly about this to proceed my project with WP. Concatenate several columns into one in Google Sheets Complicated FOR loop nesting. Having trouble cleaning up legacy code Let $m$ be Lebesgue measure and $a \in R$. Suppose that $f : R \to R$ is integrable, and $\int_a^xf(y) \, dy = 0$ for all $x$. Then $f = 0$ a.e. This is a corollary to a proof in Bass, but I don't understand why it follows from the proof he gives. I follow everything up until the last statement. Why is it that proving that the integral is $0$ for all Borel measurable sets guarantees it is true for all Lebesgue measurable sets? Am I missing something vital about the relationship between Lebesgue measurable sets and Borel measurable sets? Statement: Let $m$ be Lebesgue measure and $a \in\mathbb R$. Suppose that $f : \mathbb R \to\mathbb R$ is integrable, and $\int_a^xf(y) \, dy = 0$ for all $x$. Then $f = 0$ a.e. Proof: We prove the conclusion first for intervals, then for unions of intervals, then for open sets, then for Borel measurable sets. For any interval $(c,d)$, we have $$\int_c^df = \int_a^d f - \int^c_a f = 0.$$ By linearity, if $G$ is the finite union of disjoint open intervals, then $\int_G f = 0$. Now, if $G$ is open, then $G$ is the countable union of disjoint, open intervals, $G = \bigcup^n_{i=1}I_i$. We have $$\int_G f = \int \lim_{n\to\infty} f \cdot \chi_{\bigcup^n_{i=1} I_i}$$ and since $f_n$ is integrable for all $n$, and $|f_n| \le f$ for all $n$, by the dominated convergence theorem, $$\int_G f = \lim_{n\to\infty} \int_{\bigcup^n_{i=1}I_i} f =0.$$ Now, if $G_n$ is a decreasing sequence of open sets converging to $H$, since $|f \cdot \chi_{G_n}| \le f \cdot \chi_{G_1}$ for all $n$, we again have by the DCT that $\int_H f = 0 = \lim_{n\to\infty} \int_{G_n}f = 0$. Finally, if $G$ is a Borel measurable set, then we know that there exists a sequence $G_n$ of decreasing, open sets, that converge to some set $H$, where $H \setminus G$ is a null set. Thus, $$\int_G f = \int_H f = 0.$$ We have found that for every Borel measurable set $G$, $\int_G f =0$. Since $f$ is real valued and measurable, $f = 0$ a.e. I'm having an issue with the following piece of (G)AWK script:  do {     ... } while (system("sleep 10"))   My intention is to break the loop when the user presses ^C during the sleep, but it's not working.  I believe the problem is that Bash exits with 0 when interrupted with ^C, at least when it's executed by AWK's system():  $ awk 'BEGIN { print "\n" system("sleep 2") }' (let the sleep complete) 0      $ awk 'BEGIN { print "\n" system("sleep 2") }' ^C 0   Why is it so?   Is this a bug in Bash or in (G)AWK?  Is there a simple solution that doesn't involve complicated Bash-specific syntax like trap?  The best I could come up with is this:  do {     ... } while (42 == system("sleep 10 &amp;&amp; exit 42"))   which still looks like a kludge to me. present continous or present perfect continous? Should I write 'I have been reading a book now' or 'I am reading a book now'? I started to read one week ago and I haven't ended yet. Thanks for help How to append multiple lines of text to a file? Rational + irrational = always irrational? I had a little back and forth with my logic professor earlier today about proving a number is irrational. I proposed that 1 + an irrational number is always irrational, thus if I could prove that 1 + irrational number is irrational, then it stood to reason that was also proving that the number in question was irrational.  Eg. $\sqrt2 + 1$ can be expressed as a continuous fraction, and through looking at the fraction, it can be assumed $\sqrt2 + 1$ is irrational. I suggested that because of this, $\sqrt2$ is also irrational.  My professor said this is not always true, but I can't think of an example that suggests this.  If $x+1$ is irrational, is $x$ always irrational?   Actually, a better question is: if $x$ is irrational, is $x+n$ irrational, provided $n$ is a rational number? Statistical Entropy and Information theory Object rotate in a strange way today i run into another problem. i dont know how i have done it (probably i clicked sth what i should not click) - anyway - since that moment my objects are behaving totally different when i try to rotate them. the objects are now rotating around the pivot point but they keep their alignment - not sure if i am clear. eg. if u would rotate a plane which is lies flat on the ground it makes the rotation but the faces of the plane would still be up and dont which shouldn't be the case with the "normal" rotation that i know. so i want to understand what i have done (because this kind of rotation could be also usefull sometimes for me) and how can i return it in the case it happens again. When I'm watching a film in Mythtv the screen turns to black every 10 - 15 mins and I have to log back into Ubuntu. Very annoying!  How do I disable the black screen / screensaver / logout in Unity?  There no longer seems to be any options to turn the screen saver off as there were in Ubuntu prior to Unity. Saving project with data source path as relative in QGIS? Is there an option in QGIS to save the project with the data source being a relative path?  I don't see the option here. QGIS Wroclaw1.7.3 I'm fairly new to blender and not a graphics person by any stretch of the imagination. I've been having some success though modelling semi-reflective glass for a project I wish to build (physically) and I'm wondering if it is possible to model the effects of a lens. The lens I wish to model is a double-convex type. It has a diameter of 120mm and a focal-length of 301mm. Is this something that can be modelled in blender and if so, how? I'd like to be able to position the camera close to this lens and reproduce the magnifying effects of the lens on the scene behind the lens as well as depth-of-field effects.  Thanks a lot. Rasterizing a plot make axes ticks invisible Error while writing strcat() using pointers