I'm trying to render out an animation, but I keep running into a lot of fireflies. I would like to try to avoid the denoise addon since Im rendering an animation, but at the very least I would like to get my samples down (I'm using 500). My scene is relatively simple, its two low poly hallways with lights and doorways, and the emission on the lights is cranked up the 60 (I use filmic blender color management so it doesn't blow out the image). The only thing in my scene that I think could cause all of these fireflies is the lights, but Im not sure why as Ive used bright lights with filmic blender before and it worked just fine. Also most materials in the scene are over .5 roughness, and I got rid of the metallic materials in my scene and the fireflies were still there. Any help would be appreciated, thanks!  Blend File:
I often use Cycles' built in Glass BSDF shader to achieve realistic refraction effects, but in many cases, it seems that Cycles' glass caustics creates an unfortunately large number of fireflies (pixels that are abnormally bright relative to their surroundings). In some models of mine, even when I render for 3000 or more samples/pixel, my final image is still covered in fireflies. While I can improve this in many cases by increasing the total amount of light available in the scene, this loses a lot of the potency of the refractive effects I'm looking for, especially in scenes with low light levels.  So my question is, how can I reduce the amount of fireflies while using Cycles' caustics without sacrificing the quality of the refractive effects?  Note: I am aware that Cycles' caustics will perform much better once Cycles has full support for bidirectional path-tracing, but it looks unlikely that that will become , unfortunately.
There are three random variables, $x,y,z$. The three correlations between the three variables are the same. That is,  $$\rho=\textrm{cor}(x,y)=\textrm{cor}(x,z)=\textrm{cor}(y,z)$$  What is the tightest bound you can give for $\rho$? I can't seem to find an answer to this simple question, which I need for some compliance documentation.    On a default install of CentOS 6.5 (OpenSSH 5.3p1-94.el6), after how long of being idle will a user's SSH session be terminated?  I believe the following can be set to increase the idle timeout, but they are commented out by default.  $ grep -i alive /etc/ssh/sshd_config #TCPKeepAlive yes #ClientAliveInterval 0 #ClientAliveCountMax 3   Also, is there a command to dump a list of the current sshd settings?  I don't see anything in man sshd. How does one change to engineering for graduate school after undergrad in a science field? I've always had an interest in electrical engineering (actually all engineering in general) and I've decided that I'd like to pursue a graduate degree.  I guess I'm wondering if I should really try to do this or just keep it as a fun hobby I spend all my free time doing.  So, the question is,  how does one change fields between undergrad and grad school?  A specific example follows, but please provide answers that address a more general case, so that your answers will be helpful to other people with the same general question.  I already have a BS and MS in chemistry with a minor in applied math.I published three papers during my MS, one first author in the top journal in my specialty and two as coauthor in an average and a so-so journal.  I hope those will count for something.  I finished my BS with around 3.1 and my MS with a 3.9 gpa.  I'm sure that my MS advisor would write me a great letter.  My GRE scores are too old to count so I'd have to retake it or try and get a pass on it since I already have a graduate degree.  Since finishing school I've been working for 4 years as a chemist.  I can get great recommendations from my supervisor who is a PhD chemist.      I've been working for a while to cover some of the things I would have learned in a BSEE to try and make up for my lack of one.  I worked through some texts on analog/digital design, lots and lots of programming, built a balancing robot that got a bit of internet attention, and I'm currently working on an autonomous robotics project that's looking really good so far.  After that I have more project ideas than I have time or money for.    I want to apply for PhD programs (specifically in controls/robotics) but I'm worried that my application will get tossed right away because I don't have a BSEE.  Maybe I should just apply to MS programs? I heard a lot about in-body image stabilization recently and I wonder how this technology works. I don't mean the mechanical part (I understand how the sensor moves along different axes to compensate), but the technique in principle:  When I take my 5D III and a lens like the 70-200 2.8 II IS USM with activated stabilization, the IS will constantly move one glas element within the lens in such a way that it directs the cone of light exactly onto the sensor, so that you can always use the full sensor area.  When the cone of light goes through an unstabilized lens and you need to move the sensor - how can it capture the same amount of light at all? The light enters the lens down a fixed axis right through the center of the lens. When the sensor currently moves down, for example, to compensate for a shake, I can't see how it then is still able to capture the full resolution? When a part of the sensor is below default/calm position, then there must be missing the same part on the opposite site?  There's only one way I could image IBIS to work without resolution reduction: if the sensor area was significantly bigger than the area exposed to light, so that there's still sensor available when it currently moves out of its initial position. But wouldn't this (partly) be the same as digital stabilization because the final image gets calculated from the full sensor area minus the not exposed area?  I really would like to understand this and it would be great if someone could "shed some light on this". Why does $\sum_{k=1}^{\infty}\frac{{\sin(k)}}{k}={\frac{\pi-1}{2}}$? Inspired by  (and far more straightforward, I am guessing), Mathematica tells us that $$\sum_{k=1}^{\infty}\dfrac{{\sin(k)}}{k}$$ converges to $\dfrac{\pi-1}{2}$.   Presumably, this can be derived from the similarity of the Leibniz expansion of $\pi$ $$4\sum_{k=1}^{\infty}\dfrac{(-1)^{k+1}}{2k-1}$$to the expansion of $\sin(x)$ as $$\sum_{k=0}^{\infty}\dfrac{(-1)^k}{(2k+1)!}x^{2k+1},$$ but I can't see how...  Could someone please explain how $\dfrac{\pi-1}{2}$ is arrived at? How do I install Ubuntu as the only OS on a MacBook?  That is, I'd like to completely remove Mac OS X and install Ubuntu. I've tried dual-booting, but I've had so much trouble that I prefer to have only Ubuntu installed. How do I set the default program? How do I set the default program that I use?  I have tried to open System Info → Default Program → change my program, but it won't work,  Any idea? Or should I use Terminal?   I want to replace the default movie player with VLC media player, because the current movie player is useless to me. How to get back to a site when viewing one of its questions When viewing a question on a site which I don't use (I never registered on it), I can't figure out how to get back to the list of questions of the site.  See this example  from the Stack Exchange Android app. Where should I click to see the list of questions? (I chose Mathematics Educators because it's the most recent site, it is very likely that most users don't have an account on this website)  Here is a random example:    When viewing a question, I don't know how to view the list of questions. This site can't be found in the sidebar since it's not in my Your Sites list. The top menu containing the site's title only opens the sidebar. Is there a limit to a total number of months a holder of UK multiple entry visitor visa can spend in UK per calendar year? A relative of mine has received a multiple-entry UK visa for 2 years. Is there a particular limit on how many months in total per year she can stay in UK? The way I understand the current rule is that such visitors can enter UK as many times as they want as long as they do not stay for longer than 6 months per single visit. However, does this mean that if she stayed with us for 5 months and went back home for 1-2 months, she could come and stay for another 5 months? I heard that it is advisable to spend only 180 day per calendar year in UK and not more, but I could not find that rule in any official UK Home Office documents. Thanks in advance for any info about this. When do I need to invoke \phantomsection? How do you address unknown peers in email? This is one bit of academic etiquette I'm not sure about. When I write to another researcher, a peer (we're both professors). I've never exchanged with before and never met him but I know of him (and he probably knows of me).  Erring on the side of formality, I write to him “Dear professor Mortimer”. I sign with my first name, because that's how I've always signed my emails (there's a formal signature block below, with full name and contact details) and I think it indicates a willingness to be called by my first name (which I prefer, but I don't want to force people one way or the other).  Is that going to be well received? Will people take the hint? Or should I just start using first names from the start? (it does sound weird to write “dear Philipp” to someone you don't know) How to run a complete GUI as root? I don't care if I mess up the system (which is likely if I do things like this), or the potential security issues that this could imply. Note, that I don't just want to "", but run my GUI from bottom-up as root. I don't want to type my password for anything but just for login and have total control over it.  Of course, I want any application running as root, like Firefox, LibreOffice, etc. without having to mess up with the terminal. So, how do I do this?     This question tries to address users that do not like the current permission system with root being separated from normal users. I do not endorse such behaviors, but if a user wants to, for whatever reason, this question should fit their corncerns.   Not advisable for new users $\sqrt{1-x^2}$ is not a rational function If $k$ is a field in which $1 + 1 \neq 0$, prove that $\sqrt{1-x^2}$ is not a rational function. Hint. Mimic the classical proof that $\sqrt{2}$ is irrational   It seems rather odd to discuss the square root function in the context of an arbitrary field, but here goes nothing!  Proof:   Suppose that $\sqrt{1-x^2}$ is a rational function in $k(x)$. Then there exist polynomials $p(x)$ and $q(x) \neq 0$ that are relatively prime with $\sqrt{1-x^2} = \frac{p(x)}{q(x)}$; clearly we may take $p(x) \neq 0$. Then $1-x^2 = \frac{p(x)^2}{q(x)^2}$ or $q(x)^2(1-x^2) = p(x)^2$, which says $q^2 | p^2$, and since $q | q^2$, we can infer $q | p^2$. Moreover, since $p|p^2$ and $(p,q)=1$, then $pq|p^2$ or $q|p$, which contradicts the fact that the polynomials are relatively prime. Hence $\sqrt{1-x^2}$ cannot be a rational function.    How does this sound? Aside from the problem of discussing $\sqrt{~~}$ in the context of an arbitrary field, I am worried about not using the fact that $1+1 \neq 0$, at least not explicitly. Where exactly is this assumed used, if at all?  EDIT:  Suppose that $q(x) =1$, and let $f(x) = a_n x^n + ... a_1 x + a_0$. Then $1-x^2 = f(x)^2$. If $x=0$, $1 = a^2_0$ and therefore $a_0$ must be a unit. Letting $x=1$ we get $0 = (a_n + ... + a_1 + a_0)^2$; and letting $x=-1$ we get $0 = (-a_n - ... - a_1  + a_0)^2$. Since we are working in a field, there can be no nonzero nilpotent which means that $a_n + ... + a_1 + a_0$ and $-a_n - ... - a_1  + a_0$ are both zero. Adding the two equations together yields $2a_0 = 0$, and since $1+1 \neq 0$, $a_0 = 0$ which contradicts the fact that $a_0$ is a unit.   How does this sound?  Another attemtpt:  If $f(x)^2 = 1-x^2$, then $2 = \deg(f^2) = 2 \deg (f)$ implies $\deg(f) = 1$. Yet if $x=-1$, we get $f(-1)^2 = 1 - 1 = 0$ and therefore $f(-1)=0$ since there are no nonzero nilpotent elements in a field; similarly, $f(1)=0$. Since $1+1 \neq 0$, $f$ has two distinct roots in $k$ yet is only a $1$-st degree polynomial, a contradiction. What is this "forbidden" icon in my systray? The following is a systray applet icon that appeared in my Unity session after I upgraded to Ubuntu 16.10. Its menu is tall, narrow and empty. What is it? Error in pip: unsupported operand type(s) for -=: 'Retry' and 'int' I'm trying install the requests, but I can't do that. I'm using Ubuntu 16.04  I tried to install using  $ sudo pip install requests   and I get the following error   Collecting requests Exception: Traceback (most recent call last):   File "/usr/lib/python2.7/dist-packages/pip/basecommand.py", line 209, in main     status = self.run(options, args)   File "/usr/lib/python2.7/dist-packages/pip/commands/install.py", line 317, in run     requirement_set.prepare_files(finder)   File "/usr/lib/python2.7/dist-packages/pip/req/req_set.py", line 360, in prepare_files     ignore_dependencies=self.ignore_dependencies))   File "/usr/lib/python2.7/dist-packages/pip/req/req_set.py", line 512, in _prepare_file     finder, self.upgrade, require_hashes)   File "/usr/lib/python2.7/dist-packages/pip/req/req_install.py", line 273, in populate_link     self.link = finder.find_requirement(self, upgrade)   File "/usr/lib/python2.7/dist-packages/pip/index.py", line 442, in find_requirement     all_candidates = self.find_all_candidates(req.name)   File "/usr/lib/python2.7/dist-packages/pip/index.py", line 400, in find_all_candidates     for page in self._get_pages(url_locations, project_name):   File "/usr/lib/python2.7/dist-packages/pip/index.py", line 545, in _get_pages     page = self._get_page(location)   File "/usr/lib/python2.7/dist-packages/pip/index.py", line 648, in _get_page     return HTMLPage.get_page(link, session=self.session)   File "/usr/lib/python2.7/dist-packages/pip/index.py", line 757, in get_page     "Cache-Control": "max-age=600",   File "/usr/share/python-wheels/requests-2.9.1-py2.py3-none-any.whl/requests/sessions.py", line 480, in get     return self.request('GET', url, **kwargs)   File "/usr/lib/python2.7/dist-packages/pip/download.py", line 378, in request     return super(PipSession, self).request(method, url, *args, **kwargs)   File "/usr/share/python-wheels/requests-2.9.1-py2.py3-none-any.whl/requests/sessions.py", line 468, in request     resp = self.send(prep, **send_kwargs)   File "/usr/share/python-wheels/requests-2.9.1-py2.py3-none-any.whl/requests/sessions.py", line 597, in send     history = [resp for resp in gen] if allow_redirects else []   File "/usr/share/python-wheels/requests-2.9.1-py2.py3-none-any.whl/requests/sessions.py", line 195, in resolve_redirects     **adapter_kwargs   File "/usr/share/python-wheels/requests-2.9.1-py2.py3-none-any.whl/requests/sessions.py", line 576, in send     r = adapter.send(request, **kwargs)   File "/usr/share/python-wheels/CacheControl-0.11.5-py2.py3-none-any.whl/cachecontrol/adapter.py", line 46, in send     resp = super(CacheControlAdapter, self).send(request, **kw)   File "/usr/share/python-wheels/requests-2.9.1-py2.py3-none-any.whl/requests/adapters.py", line 376, in send     timeout=timeout   File "/usr/share/python-wheels/urllib3-1.13.1-py2.py3-none-any.whl/urllib3/connectionpool.py", line 610, in urlopen     _stacktrace=sys.exc_info()[2])   File "/usr/share/python-wheels/urllib3-1.13.1-py2.py3-none-any.whl/urllib3/util/retry.py", line 228, in increment     total -= 1 TypeError: unsupported operand type(s) for -=: 'Retry' and 'int'   My python version is the 2.7 and the pip is 8.1.1 Connectivity, Path Connectivity and Differentiability I have two questions which pertain to differentiability, connectivity and path connectivity. Ocasionally, I will encounter an author who defines connectivity in the following way:   An open subset $U$ of $\mathbb{R}^n$ is said to be connected if and only if given two points $a$ and $b$ of $U$ there exists a differentiable mapping $\phi: \mathbb{R} \rightarrow U$ such that $\phi(0) = a$ and $\phi(1) = b$.   This particular example is from Edward's Advanced Calculus of Several Variables p 84. Now, this is obviously not the standard definition we learn from topology which has nothing to do with differentiability but rather whether there exists two nonempty open subsets that comprise a separation. It also seems to me that what the author is really defining what it means for a space to be "smoothly path connected", which of course implies connectivity and, it seems to me, considerably more.  My first question is: Is "smoothly path connected", as defined above actually equivalent to "connected", in the topological sense, in $\mathbb{R}^n$?  Next, in Vector Calculus by Baxandall and Liebeck on p 150 the authors state the existence of a continuous path $\alpha$ from the closed interval $[0,1]$ to an open subset $D$ of $\mathbb{R}^n$ where $\alpha (0) = a \in D$ and $\alpha(1) = b \in D$ guarantees the existence of a differentiable path with the same properties. This claim is stated without proof.   My second question is: Can someone provide a reference to a proof of the above claim or explain why it is so? Solving cubic equations with another method. I want to solve cubic equations with some pattern or formula. My maths teacher told me that if we want to solve cubic equations we have to find any one value for the equation which satisfy it and then divide the main equation by it. But I find that method little useless . Is there any other simplier or good method to solve it. You can explain it by an example. Cast Double to Integer in Java Galois group of irreducible cubic equation The polynomial $f(x) = x^3 -3x + 1$  is irreducible, and I'm trying to find the splitting field of the polynomial.  We've been given the hint by our lecturer to allow an arbitrary $\alpha$ to be a root and then to check that $\alpha$ is a root, then $2-\alpha^2$ is also root. I've done this with no problem.  My issue is since this is a polynomial of degree $3$, there must be another root that is somehow related to $\alpha$ so that I can create the splitting field and in turn the Galois group of my polynomial but I have no idea what this last root could be. Proof of when is $A=X^TX$ invertible?