Can the newly discovered mass of the neutrino explain in part, the missing matter
How do we know Dark Matter isn't simply Neutrinos?
How do you find the distance from a point to a plane? I am having trouble with this:      Find the distance from the point $(1,1,1)$ to the plane $2x+2y+z=0$.   Any ideas? Thanks. How prove this $\prod_{1\le i<j\le n}\frac{a_{j}-a_{i}}{j-i}$ is integer Chirp Transform and Convolution I was reading about the discrete fourier transform from the CLR algorithms book and I came upon an exercise whose hint confuses me. The exercise reads as follows:  The chirp transform of a vector $a=(a_0,a_1,\ldots,a_n)$ is a vector $y=(y_0,y_1,\ldots,y_n)$, where $y_k=\sum_{j=0}^{n-1}a_j (z^{k})^j$ and $z$ is any complex number. The DFT is therefore a special case of the chirp transform, obtained by taking $z=\omega_n$. Prove that the chirp transform can be evaluated in time $O(n\lg n)$ for any complex number $z$. (Hint: use the equation $$y_k = z^{k^2/2}\sum_{j=0}^{n-1}(a_j z^{j^2/2})(z^{-(k-j)^2/2})$$ to view the chirp transform as a convolution.  My question is about what is a convolution. I am familiar with taking two vectors $a=(a_0,a_1,\ldots,a_n)$ and $b=(b_0,b_1,\ldots,b_n)$ and forming the convolution $c=(c_0,c_1,\ldots,c_n)$ where $c_k = \sum_{j=0}^k a_j b_{k-j}$. Here the index of $j$ runs from $0$ to $k$ unlike in the hint given above where the index runs from $0$ to $n-1$. In what sense is the hint above a convolution? How do you configure an OpenFileDialog to select folders? In VS .NET, when you are selecting a folder for a project, a dialog that looks like an OpenFileDialog or SaveFileDialog is displayed, but is set up to accept only folders.  Ever since I've seen this I've wanted to know how it's done.  I am aware of the FolderBrowserDialog, but I've never really liked that dialog.  It starts too small and doesn't let me take advantage of being able to type a path.  I'm almost certain by now there's not a way to do this from .NET, but I'm just as curious how you do it from unmanaged code as well.  Short of completely reimplementing the dialog from scratch, how do you modify the dialog to have this behavior?  I'd also like to restate that I am aware of the FolderBrowserDialog but sometimes I don't like to use it, in addition to being genuinely curious how to configure a dialog in this manner.  Telling me to just use the FolderBrowserDialog helps me maintain a consistent UI experience but doesn't satisfy my curiosity so it won't count as an answer.  It's not a Vista-specific thing either; I've been seeing this dialog since VS .NET 2003, so it is doable in Win2k and WinXP.  This is less of a "I want to know the proper way to do this" question and more of a "I have been curious about this since I first wanted to do it in VS 2003" question.  I understand that Vista's file dialog has an option to do this, but it's been working in XP so I know they did something to get it to work.  Vista-specific answers are not answers, because Vista doesn't exist in the question context.  Update: I'm accepting Scott Wisniewski's answer because it comes with a working sample, but I think Serge deserves credit for pointing to the dialog customization (which is admittedly nasty from .NET but it does work) and Mark Ransom for figuring out that MS probably rolled a custom dialog for this task. What is the difference between image size and image quality? How to make two people play at the same time? My brother owns Skyrim on steam. We use Steam Family Share, but we were wondering how both of us could play it at the same time? Rename All Files with a Certain Name I'm trying to find certain files with the name "stringx" and replace the name (but not the extension) with "stringy". So basically for stringx.txt and stingx.cs, I'd want stringy.txt and stringy.cs. I've attempted to test my rename command regex using following, but it returns no results:  rename -n 'stringx./s/\*$stringy./\/' stringx.*   I'm running csh under Centos. I can see that my regex is probably not correct, but I can't see any results to verify this. I can see that there are files that are named stringx. with the following:  find ./ -name 'stringx.*' $a$ and $b$ are natural numbers and $p$ is a prime number. Prove that if $a^{p}-b^{p}$ is divisible by $p$, then it is also divisible by $p^{2}$.  My attempt: Based on Fermat's theorem $(a^{p}-a)$ and $(b^{p}-b)$ are divisible by p, therefore their difference is divisible by $p$. I.e. $(a^{p}-a)-(b^{p}-b)=(a^{p}-b^{p})-(a-b)$ is divisible by $p$. Since the left hand side of this equation is divisible by $p$, the right hand side also should be divisible by $p$. Based on the problem's assumption $(a^{p}-b^{p})$ is divisible by $p$ therefore we conclude $(a-b)$ must be divisible by $p$. Now We can factorize $(a^{p}-b^{p})$ $$(a^{p}-b^{p})=(a-b)(a^{p-1}+a^{p-2}b+....+b^{p-1})$$ we have to prove $(a^{p}-b^{p})$ is multiple of $p^2$ which means we have to show the right hand side should be multiple of $p^{2}$. But $(a-b)$ is multiple of p therefore we have to show  $(a^{p-1}+a^{p-2}b+\cdots+b^{p-1})$ is multiple of $p$ and I am stuck here. Any help would be appreciated. biblatex: \parencite in note field prints spurious ; with authoryear-icomp style Slow Network Printing over OpenVPN / Windows Server 2008 Our setup is as follows:  Two offices (A and B) connected with OpenVPN Access Server.  One Windows Server 2008 R2 Standard    Housed in remote datacenter. Terminals at both offices connect through RDP in order to use a particular piece of software concurrently. Network Connected printers (Deployed with GPO) that are connected to separate terminals at each office (and shared) are used with this software to print things throughout the day.   We are experiencing extremely slow printing times.  Opening printer properties can take 30-60 seconds and printing times range from 30-180 seconds with no apparent reason.  File sizes remain &lt; 1mb.  This happens with each of our printers.  All different models.  The server massively underutilized with a 6-core AMD, 16GB ram, 4x 120GB SSDs, and a 100 megabit port.  Our VPN does not seem to be a bottleneck either.  I've been attempting to find ways to increase print speed.  So far no success.  What can you advise? Proving Eigenvalue squared is Eigenvalue of $A^2$ The question is: Prove that if $\lambda$ is an eigenvalue of a matrix A with corresponding eigenvector x, then $\lambda^2$ is an eigenvalue of $A^2$ with corresponding eigenvector x.  I assume I need to start with the equation $Ax=\lambda x$ and end up with $A^2 x=\lambda^2 x$ but between those I am kind of lost. I have manipulated the equations several different ways and just can't seem to end up where I need to be. Help would be greatly appreciated as I believe this will be on a test tomorrow. I submitted a manuscript last year to an IEEE journal, and the paper was accepted for publication. I recently submitted a new manuscript to the same journal, but, this time, the paper was rejected, and I was told that I should submit to another journal since my paper was not within the journal's scope. The two papers both target similar applications. Furthermore, the second paper uses some work in the first paper, making substantial improvements at solving a much more difficult problem. Based on the reviewer's comments (and my own personal assessment), I do not believe that the work presented in the second paper was "incremental" in nature. Far from it. Two of the three reviewers gave positive reviews. The third reviewer, however, mentioned the "outside of journal's scope" issue and the associate editor and editor-in-chief both sided with the 3rd reviewer, and the paper was stopped dead in its tracks.  I'm scratching my head trying to come to terms with how in the world my first paper could be accepted by this journal, but the second paper focused on the same application space and achieving substantial improvements on a much more difficult problem (backed up with measured results) could be rejected. Any ideas? One thing that seems to be missing in the Top&nbsp;Bar is a link to the Stack&nbsp;Exchange portal. It only displays under the "Current&nbsp;Community" section when you are in the community, but it disappears as soon as you go to another one.  Would it be possible to add a link that would always be there no matter which community you are currently on? How to jump to the region header from the endregion tag in c# visual studio 2012? What is a simple example of an unprovable statement? I am considering to use iCloud Photo Library to store a growing number of family photo albums, shared to web or set of AppleIDs.  I dont need to keep original photos on iDevices for 99.9% of photos.  How much space will be used on the photo-originating set of devices (iOS, OS X, Windows)? I.e. what (roughly) is the function from the original size and number of photos, device storage space available, any other variables? I would like to know how can I customize the Ubuntu installer NOT customize Ubuntu, I just want to modify the installer.  We have a large environment here with lots of developers and since we don't use Windows, I want to customize the installer to automatically set the timezone to NYC, the keyboard layout to English USA, put the company logo in the installer etc... If a product of relatively prime integers is an $n$th power, then each is an $n$th power Show that if $n$, $a$, $b$, and $c$ are positive integers with $\gcd(a, b) = 1$ and $ab = c^n$, then there are positive integers $d$, and $e$ such that $a = d^n$ and $b = e^n$.   I know that (by Bezout) $\gcd\left(a,b\right) = 1$ implies $ax + by = 1$ for some integers $x$ and $y$, and also that $\gcd\left(a^n,b^n\right) = 1$, but this does not help me. Why do we need sigma-algebras to define probability spaces? We have a random experiment with different outcomes forming the sample space $\Omega,$ on which we look with interest at certain patterns, called events $\mathscr{F}.$ Sigma-algebras (or sigma-fields) are made up of events to which a probability measure $\mathbb{P}$ can be assigned. Certain properties are fulfilled, including the inclusion of the null set $\varnothing$ and the entire sample space, and an algebra that describes unions and intersections with Venn diagrams.  Probability is defined as a function between the $\sigma$-algebra and the interval $[0,1]$.  Altogether, the triple $(\Omega, \mathscr{F}, \mathbb{P})$ forms a probability space.  Could someone explain in plain English why the probability edifice would collapse if we didn't have a $\sigma$-algebra? They are just wedged in the middle with that impossibly calligraphic "F". I trust they are necessary; I see that an event is different from an outcome, but what would go awry without a $\sigma$-algebras?     The question is: In what type of probability problems the definition of a probability space including a $\sigma$-algebra becomes a necessity?       provides a plain English accessible explanation. The idea is a spinning pointer rotating counterclockwise on a circle of unit perimeter:       We begin by constructing a spinner, which consists of a circle of unit   circumference and a pointer as shown in [the] Figure. We pick a point   on the circle and label it $0$, and then label every other point on the   circle with the distance, say $x$, from $0$ to that point, measured   counterclockwise. The  experiment  consists  of spinning the pointer and    recording the label of the point at the tip of the pointer. We let the random   variable $X$ denote the value of this outcome.  The sample space is   clearly the interval $[0,1)$.  We would like to construct a   probability model in which each outcome is equally likely to occur. If   we proceed as we did [...] for experiments with a finite number of   possible outcomes,  then we must assign the probability $0$ to each   outcome,  since otherwise,  the  sum  of  the  probabilities,  over    all  of  the  possible  outcomes,  would not equal 1.  (In fact,   summing an uncountable number of real numbers is a tricky business;    in  particular,  in  order  for  such  a  sum  to  have  any  meaning,   at  most countably many of the summands can be different than $0$.)    However, if all of the assigned probabilities are $0$, then the sum is   $0$, not $1$, as it should be.   So if we assigned to each point any probability, and given that there is an (uncountably) infinity number of points, their sum would add up to $&gt; 1$. How do I read from /proc/$pid/mem under Linux? The  tells me that /proc/$pid/mem “can be used to access the pages of a process's memory”. But a straightforward attempt to use it only gives me  $ cat /proc/$$/mem /proc/self/mem cat: /proc/3065/mem: No such process cat: /proc/self/mem: Input/output error   Why isn't cat able to print its own memory (/proc/self/mem)? And what is this strange “no such process” error when I try to print the shell's memory (/proc/$$/mem, obviously the process exists)? How can I read from /proc/$pid/mem, then?