Solve for x in $a = x^b$ (mod n)
Modular Arithmetic - Find the Square Root
Relation between rank and number of distinct eigenvalues of a matrix Let $T : V\to V$ be a linear transformation such that $\dim\operatorname{Range}(T)=k\leq n$, where $n=\dim V$. Show that $T$ can have at most $k+1$ distinct eigenvalues.   I can realize that the rank  will correspond to the number of non-zero eigenvalues (counted up to multiplicity) and the nullity will correspond to the 0 eigenvalue (counted up to multiplicity), but I cannot design an analytical proof of this.  Thanks for any help . C program to convert Fahrenheit to Celsius always prints zero Why do some Debian packages have a "+deb7u2" suffix? I figured the "7u2" part stood for "Debian 7.2" (aka "wheezy"). But I'm wondering if anybody knows the official answer as to why some packages have it and others do not.  Example:   The only time "+deb7u2" is appended to the version is when the package gets put in the "stable" release. Does this mean that's a "tag" to indicate a checkin to stable from another repo? Are there any banned subjects on Area 51? I know of the following rules for creating a proposal on Area 51:  The scope must be reasonably well-defined, not too unclear (e.g. &quot;cool stuff&quot;), too opinion-based (e.g. &quot;car reviews&quot;), too broad (e.g. &quot;life on Earth&quot;), or too narrow (e.g. &quot;The 1995 academic misconduct scandal at East Oak Mews High School&quot;). One  come to Area 51 with an already existing community rather than hope to build one on Area 51. Proposals that are subsets of existing sites are not acceptable (e.g.  — deleted). These get closed as &quot;This subject is already well-covered by a live Stack Exchange site. We generally do not split off subjects simply to give them their own space.&quot; Proposals for non-English sites .  Other than the criteria above, are there any per se restrictions on site content for an Area 51 proposal? If so, what topics are currently banned? I would expect a proposal for a &quot;Let's organize a lawsuit against Stack Exchange&quot; or &quot;[Specific SE staff member] sucks, amirite?&quot; would get closed for obvious reasons, but I'm interested if there are any non-obvious topics that are not currently allowed in proposals regardless of how well-defined, well-scoped, original, and well-supported by a community they are. Having a list of banned topics would be helpful to potential proposers as well as the communities that back them. For example, I wouldn't want to go to all the effort of organizing a community, bringing them to Area 51, etc., only to have my proposal closed with a curt &quot;Sorry, due to the political climate, we are not currently accepting proposals for sites about real-life weapons.&quot; custom closure. To be clear, I am asking solely about topic restrictions (e.g. &quot;no sites about fishing, hunting, or weapons&quot;). I'm not asking about clarity requirements, the requirement to have a pre-existing community, not duplicating an existing site, or the requirement to not use sock puppets to artificially inflate proposal stats. Those requirements are already well-defined elsewhere or else are common sense. Also to be clear, I don't have a specific proposal idea in mind right now or the community to make it happen. I'm asking in general. The question could be rephrased as &quot;What are those topics where I shouldn't even bother to try to define, gather a community, etc. because any attempt to propose a site will be summarily closed due to covering a banned topic?&quot; This is not a duplicate of  because that question is about the appropriateness of expanding a proposal to cover things that are acceptable proposal topics but arguably don't make sense in combination with the original proposal scope. My question is about topics that are completely banned from proposals regardless of whether they are included in the original proposal description or tacked on by the community later. Solving functional differential equation $f'(x)=2f(2x)-f(x)$ Show that there is at least a nonzero function $f$, differentiable on $[0,+\infty)$, satisfying $$f'(x)=2f(2x)-f(x) \qquad \forall x&gt;0 $$   $$M_n:=\int_{0}^{\infty}x^nf(x)dx&lt;\infty \qquad \forall n\in  \mathbb{N} $$   My best idea so far is to assume that the solution is a power series, i.e.  $$ f(x)=\sum_{n=0}^{\infty}a_nx^n\qquad \forall x&gt;0$$ Then the equation becomes $$ \sum_{n=0}^{\infty}na_nx^{n-1}=2\sum_{n=0}^{\infty}a_n2^nx^n-\sum_{n=0}^{\infty}a_nx^n$$ equating all the coefficients of the same degree I get $$na_n=(2^{n}-1)a_{n-1}\qquad \forall n\geq 1$$ So setting $a_0=1$, I get $$a_{n}=\frac{1}{n!}\prod_{k=1}^{n}(2^k-1) \qquad \forall n$$ But does the power series actually converge? Using Hadamard's formula, and that $2^{k}-1\geq 2^{k-1}$, $$ |a_n|^{1/n}\geq\frac{1}{(n!)^{1/n}}\left[2^{n(n-1)/2}\right]^{1/n}\sim\frac{e}{n(2\pi n)^{1/2n}}2^{(n-1)/2}\to \infty$$ so the radius of converge of the series is $0$, so it doesn't actually define a solution on $[0,+\infty)$. Notice these sentences:   Shops are open late in summer.   Summer is a traditionally viewed as a slow season for games.   At the summer solstice, the days are longest and the nights are shortest.   it's time to start thinking about what 2016's Song of the Summer will be.    In English class, they told us that "the" is used for the things that are known.  One example for this was a phrase like "The man who killed his wife".  So when we are speaking of that man, we know what he did. Somehow we know something more than nothing about him. He's not just every man in the world.     So Now I'm confused about why "the" is not used in the first and second sentences. Can anyone please explain it for me? Android EditText listener for cursor position change Issue with particles in blender donut tutorial I have been recently doing the blender donut tutorial and ran into a problem regarding the sprinkle particles. The particles (hair) keep appearing on the underside of the icing. As seen in the image below, most of the hair particles only come out from the bottom of the icing. I tried to readjust the normals but I realised the normals were already correct. (Blue colour donut in image) Someone suggested following through the weight painting part of the tutorial. However, the sprinkles still remain on the underside of the icing. What is the actual problem and how do I solve this?  Particle settings below  Edit: @moonboots Thanks the hair particle problem has been solved. How do I prevent the submersion of the sprinkle particles? Creating lots of random cuts on a surface Here's an image of a cube with lots of edges added. It was created using knife tool and lots of clicking:    What would be the best way to randomly generate such cuts with less effort...? How is consensus determined on Meta sites? On our main sites &quot;the best answer&quot; is  as the superposition of the highest voted answer and the accepted answer.  Truth be told, the system is rather insistent on allowing multiple &quot;right answers&quot; and that's one of its strengths: you can be the judge of who you believe. But that doesn't always translate well on the per-site metas, which serve as a sort of policy-creation body.  People who are active on meta help decide how policies are enforced on the main site.  Moderators who have ♦-powers tend to take mandates on meta very seriously.  , accepted answers to meta-questions are no different than any other answer.  Does that leave just votes to determine consensus? Closely related: Not everyone participates on meta-sites.  It can be hard to tell if enough people have looked and voted on questions for real consensus to be declared.  Size of the site determines how many people could be voting on questions.  What counts as a quorum? Finally, sometimes people change their minds about a meta-question, but, , it's not really possible to change one's mind without a pointless edit to someone else's answer.  When should meta-questions be revisited?  On graduated sites, ♦ moderators are elected, which makes them representatives of the active participants, but on beta sites, we are appointed.  It's doubly pressing for beta sites to determine &quot;the will of the people&quot; since part of the point of a beta is to get a working policy framework in place before electing permanent moderators. I'm interested in figuring out how sites already answer these questions and what might be done network-wide to make discovering consensus a little bit easier.  N.B.: Obviously this assumes that consensus matters.  It does to me when I'm making my decisions and I suspect it matters to others.  But that isn't to say that it always makes a difference.  I'm really not interested in answers that hope to use this question as a grindstone for all the axes stored up against certain people who have made unpopular decisions and have the power to make those decisions stick.  Feel free to ask your own question(s).  (Thank you.) Do sets, whose power sets have the same cardinality, have the same cardinality? Is it generally true that if $|P(A)|=|P(B)|$ then $|A|=|B|$?  Why? Thanks. How do I save terminal output to a file? How do I save the output of a command to a file?  Is there a way without using any software? I would like to know how. I read news on the BBC (on its website) this morning. When I was reading the news I noticed the definite article the was missing in a sentence. As per my opinion, the reporter missed the article the. But I also thought that the reporter also had some point as he did not use the article there. So I just wanted to confirm so landed here to get some help on it.     On the website:      It also proclaimed the group's leader, XYZ  (name), as caliph and   "leader for Muslims everywhere".      As per my opinion it should have been:      It also proclaimed the group's leader, XYZ  (name), as the caliph and   "leader for Muslims everywhere".   Source link: How to interpret this yum dependency error? I'm trying to install XRDP and I'm hitting the following:  [root@box ~]# yum -y install xrdp Loaded plugins: fastestmirror, langpacks Loading mirror speeds from cached hostfile  * base: centos.serverspace.co.uk  * epel: mirrors.coreix.net  * extras: mirror.sov.uk.goscomb.net  * updates: mirror.sov.uk.goscomb.net Resolving Dependencies --&gt; Running transaction check ---&gt; Package xrdp.x86_64 1:0.9.10-1.el7 will be installed --&gt; Processing Dependency: xrdp-selinux = 1:0.9.10-1.el7 for package: 1:xrdp-0.9.10-1.el7.x86_64 --&gt; Processing Dependency: xorgxrdp for package: 1:xrdp-0.9.10-1.el7.x86_64 --&gt; Running transaction check ---&gt; Package xorgxrdp.x86_64 0:0.2.10-4.el7 will be installed --&gt; Processing Dependency: xorg-x11-server-Xorg(x86-64) = 1.20.4 for package: xorgxrdp-0.2.10-4.el7.x86_64 ---&gt; Package xrdp-selinux.x86_64 1:0.9.10-1.el7 will be installed --&gt; Finished Dependency Resolution Error: Package: xorgxrdp-0.2.10-4.el7.x86_64 (epel)            Requires: xorg-x11-server-Xorg(x86-64) = 1.20.4            Installed: xorg-x11-server-Xorg-1.20.1-5.6.el7_6.x86_64 (@updates)                xorg-x11-server-Xorg(x86-64) = 1.20.1-5.6.el7_6            Available: xorg-x11-server-Xorg-1.20.1-3.el7.x86_64 (base)                xorg-x11-server-Xorg(x86-64) = 1.20.1-3.el7            Available: xorg-x11-server-Xorg-1.20.1-5.el7.x86_64 (updates)                xorg-x11-server-Xorg(x86-64) = 1.20.1-5.el7            Available: xorg-x11-server-Xorg-1.20.1-5.1.el7.x86_64 (updates)                xorg-x11-server-Xorg(x86-64) = 1.20.1-5.1.el7            Available: xorg-x11-server-Xorg-1.20.1-5.2.el7_6.x86_64 (updates)                xorg-x11-server-Xorg(x86-64) = 1.20.1-5.2.el7_6            Available: xorg-x11-server-Xorg-1.20.1-5.3.el7_6.x86_64 (updates)                xorg-x11-server-Xorg(x86-64) = 1.20.1-5.3.el7_6  You could try using --skip-broken to work around the problem  You could try running: rpm -Va --nofiles --nodigest   Now, I get that it's missing a dependency, however, it appears to be saying it requires xorg-x11-server-Xorg version 1.20.4.   It then continues to say version 1.20.1-5.6.el7_6 is installed.  From where I'm sitting, 4 is in the range 1-5 so I'm unclear what the issue is. They all seem to be for the right architecture.  Is 1-5 a version string in and of itself as opposed to a range, or -if not- what am I missing?  Solution: Thanks to , here's a workaround:     Having the same issue, I installed manually archived xorgxrdp   yum -y install https://archive.fedoraproject.org/pub/archive/epel/7/x86_64/Packages/x/xorgxrdp-0.2.9-1.el7.x86_64.rpm   then   yum -y install xrdp and it works If A and B play a game with a fair coin such that A wins by getting consecutive TT for the first time, while B wins by getting a consecutive TH.  Who has higher probability of winning the game?   I tried the following method but not sure:  For A to win,     TT, HTT, HHTT ... that is, 2-k for each game length k   For B to win,     TH, HTH, HHTH ... that is, 2-k for each game length k   So ultimately I am getting same probability of each to win. I want to create a simple figure where images are connected with arrows. Here is what I did:  \begin{tikzpicture} \node[](A) {\includegraphics[width=0.2\textwidth]{a}}; \node[right=1mm of A](B) {\includegraphics[width=0.2\textwidth]{b}}; \draw[-&gt;] (A) to (B); \end{tikzpicture}     But I want the arrow to start and end right at the images with no space between. How can I do that? Distribution of the maximum of two correlated normal variables Say I have two standard normal random variables $X_1$ and $X_2$ that are jointly normal with correlation coefficient $r$.   What is the distribution function of $\max(X_1, X_2)$? How to understand the Ubuntu file system layout? I've read the , and I don't get it. I am really interested, but I've read the explanation there a couple of times and I still don't get it how the Duff's device works.  What would a more detailed explanation be? How can I connect the top loop to the down loop with faces Hey I want to connect the bottom loop with the top loop so that it forms a solid.  I have tried selecting multiple vertices and pressing F but that just creates a large face which I do not want. I have tried using grid fill bu that doesn't work either. The solution that seems to work is by selecting four vertices and pressing F and then selecting the next 4 and pressing F and on and on. That just takes too long.