Number of edges of any spanning tree of a graph G coming from a subgraph of G Consider a simple undirected, unweigthed graph $G=(V,E)$ and its edge-induced subgraph $G^{*}=(V^{*},E^{*})$. Suppose that $G$ and $G^{*}$ have $c$ and $c^{*}$ connected components, respectively. Is it true that exactly   $$|V^{*}|-\min\{0,c-c^{*}\}$$  edges can be selected for any spanning tree of $G$ from $E^{*}$? Could somebody give a proof/counterexample? Thank you for the help!
Number of edges of any spanning tree of a graph $G$ coming from a subgraph of $G$ Consider a simple undirected, unweigthed graph $G=(V,E)$ and its edge-induced subgraph $G^{*}=(V^{*},E^{*})$. Suppose that $G$ and $G^{*}$ have $c$ and $c^{*}$ connected components, respectively. Is it true that any spanning tree of $G$ has exactly  $$|V^{*}|-\min\{c^{*},c\}$$  edges from $E^{*}$? Could somebody give a proof/counterexample? Thank you for the help!
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Frequentism and priors Robby McKilliam says in a comment to  post:     It should be pointed out that, from the frequentists point of view, there is no reason that you can't incorporate the prior knowledge into the model. In this sense, the frequentist view is simpler, you only have a model and some data. There is no need to separate the prior information from the model   Also, , @jbowman says that frequentists use regularization by a cost/penalty function, while bayesians can make this a prior:     Frequentists realized that regularization was good, and use it quite commonly these days - and Bayesian priors can be easily interpreted as regularization.   So, my question is, can frequentists in general incorporate into their models what Bayesians specify as priors? Taking the regularization as an example, is the cost/penalty function really integrated into the model, or is this a purely artificial means of adjusting the solution (as well as making it unique)? How can I stream live video on YouTube? Nowadays, I see a lot of users live streaming on YouTube and wanted to try it for myself. However, when I go to my channel I don't see an option for live streaming.  I tried researching how to do this on YouTube and Google, but without luck.  So I was wondering, how can I live stream on YouTube?  I don't have any pro account or any of that, but I've watched livestreams from users who I don't think have pro accounts either.  So livestreaming is open for everyone, right? What is the significance of allowing nilpotents in scheme theory? A book on quantum mechanics supported by the high-level mathematics I'm interested in quantum mechanics book that uses high level mathematics (not only the usual functional analysis and the theory of generalised functions but the theory of pseudodifferential operators etc, certainly the modern mathematics). If there isn't something similar please give me a reference to the book that is strictly supported by mathematics (given a set of mathematically descripted axioms author develops the theory using mathematics as a main tool). Continuous and bounded imply uniform continuity? In a non-inertial reference frame, the centripetal force is balanced by a centrifugal force. Therefore, shouldn't the net acceleration be zero? Yet, you still feel an acceleration outwards? Why is that? Cardinality and infinite sets: naturals, integers, rationals, bijections Is it possible that your friends can join your Hamachi server when they don't have Hamachi? Does anyone know how to do that??!! In sketchup you can move an object based on two points. This makes it very easy to align a vertex in one object to vertex in another object. Here is an example:    I want to know how to do this in Blender. It seems that Translate can't align vertexs Triangle inequality for L2 norm I am posed with a problem as below.  Let $X$ be the collection of real valued functions that are measurable on $[0,1]$. For a given number $p\geq 1$ define the $L^p$ norm by $(\int_0^1|f(x)|^p)^\frac{1}{p}$.  I need to show for $p=1,2$ that this defines a semi norm on $X$. $p=1$ was simple as was all of $p=2$ except for the triangle inequality. I am having trouble proving the triangle inequality for this. Could anybody nudge me into the right direction or am I simply over thinking it? I think I can prove it for vectors but I'm not sure how to tackle it with integrals. Right now I can only use relatively basic results because this in for an introductory measure theory course.  Thanks! Analogue of closed graph theorem This is the analogue of closed graph theorem for compact space          Suppose that $X$ and $K$ are metric spaces, that $K$ is compact, and that the graph of $f: X \rightarrow K$ is a closed subset of $X \times K$. Prove that $f$ is continuous.      Because metric space is first countable, all we need is to prove that if $x = \lim_{n \rightarrow \infty} x_{n}$, then $f(x) = \lim_{n \rightarrow \infty} f(x_{n})$. Suppose $x = \lim_{n \rightarrow \infty} x_{n}$, then because $K$ is compact, sequence $\{f(x_{n})\}$ has a subsequence $\{f(x_{n_{i}})\}$ converge. Because the graph of $f$ is closed, we must have $\lim_{n \rightarrow \infty}\{f(x_{n_{i}})\} = f(x)$. All we need to prove now is $\{f(x_{n})\}$ converge to $f(x)$, but I can't prove that. Can any one help me. I really appreciate. Thanks Is it possible to dynamically set Views' pager settings? Say for instance, you have a view that displays when viewing a certain node type, but you want the pager settings to be different for each individual node. I was thinking of making a CCK field like "pager setting" and then give it an integer option of x-y. But I don't know if it's possible to somehow dynamically plug that field into the Views' settings. Or is there another way to do this maybe? How does the US federal income tax system work? General question here about US personal income federal tax rates.  Is it a cumulative effect where your income is first taxed at the lower tranches and move up, or is it one specific percent for your entire taxable income? pass 1 environment variable using sudo Note: This is an attempt to create a canonical question that covers all instances of &quot;low-graphics mode&quot; error that occurs to a user, including but not limited to installation of wrong drivers, incorrect or invalid lightdm greeters, low disk space, incorrect installation of graphics card like ATI and Nvidia, incorrect configuration of xorg.conf file while setting up multiple monitors among others. If you are experiencing the &quot;low-graphics mode&quot; error when trying to login but none of the following answers work for you, please do ask a new question and then update the answers of this canonical question as and when your new question gets answered.   When I try to boot into my computer, I am getting this error:  The system is running in low-graphics mode Your screen, graphics cards, and input device settings could not be detected correctly. You will need to configure these yourself.   How do I fix the failsafe X mode and login into my computer?  Answer index: I will try to be as explanatory as possible with my question. Please also note that I have done my share of googling and I am looking for simple language preferable with some example so that I can get some insight in this subject.  My question is what is so special about $c$? Why only $c$. Its like chicken and egg puzzle for me. Does Einstein reached to $c$ observing light or does he got to light using some number which turned out equal to $c$.   Why is $c$ not relative. If something has zero rest mass like a photon why they only travel at $c$ in vacuum and not with $c+1$ or $c-1$?