Ubuntu for pentium dual core
What are the system requirements for each flavour of Ubuntu Desktop?
Equivalence of the definition of the Subbasis of a Topology I came to know about 2 definitions of a subbasis of a topology. 1. A collection of subsets of X viz. S is said to be a subbasis of a topology T on X if S covers X and the finite intersections of elements of the subbasis form the basis elements (H.L.Royden). 2. If we consider the all the topologies on X containing S, then their intersection is also a topology on X and it is the smallest topology containing S. Then S is said to be a subbasis of the smallest topology, and the topology is said to be generated by S.  Now I want to prove their equivalency, i.e. S is a subbasis of a topology T iff T is generated by S. I have tried in the way: First let T is generated by S. Then by 2nd definition T is a topology containing S. Let B be the set of all finite intersections of elements of S.Then S is a subset of B and B is a subset of T (as T is a topology). LET T1 be the collection of possible unions of elements of B, we will show that T = T1. After this I am not able to proceed. Also I am unable to sketch the converse part. If something is wrong, please let me know, and also I hope to get help. Thank You. What's the difference between "I love singing" and "I love to sing"? I would like to know what the difference is between "I love doing something" and "I love to do something".  For example:        “I love singing.”      and,        “I love to sing.”      When I was young, I was told that "love doing something" is right, but one day, I found Jessie J wrote "I love to sing" on her Instagram. Is cross-posting a question on multiple Stack Exchange sites permitted if the question is on-topic for each site? It is possible to migrate a question from one Stack Exchange site to another by closing, but if I have a question that I think is on-topic for multiple Stack Exchange sites, is it OK to post it on both (multipost)?   For example, I have a question that's earned me the tumbleweed badge on SO and I'm not sure what the best thing to do with it is. It's about a web server so it might be answerable on Server Fault but it's really more of a programming thing, hence the posting the question on Stack Overflow.    Is there any way to make the question visible on multiple sites (crosspost) and then accept the answer wherever it came from? Show every nonempty compact Hausdorff space is not the countable union of nowhere dense sets I know this proof is somewhat similar, or related to the Baire's Category Theorem but I can't seem to figure out how to do it.  The Baire Category theorem asserts that if X is a complete metric space or a locally compact Hausdorff space (my case), then the complement of a countable union of nowhere dense sets is always nonempty.  How can I use this on the proof? Update  This has been confirmed resolved in a later version 376.33 and possibly the update prior.     Okay, I recently did a computer update. I used to run Blender 2.77 on a Dell desktop (5 GB RAM, 2.7 GHZ, Windows 8, build in graphics card). Since that computer wasn't fast enough for my preference, I saved up and got a new gaming laptop - the Asus ROG GL552 (Windows 10, 16 GB DDR4 RAM, Nvidia GeForce GTX 960M with 4 GB DDR5 RAM, 2.6-3.5 Ghz, and quad-core intel processor). That laptop has been known to have outdated drivers, so when I got it, I updated it to up to date Windows 10 drivers.  As soon as I started Blender, I immediately noticed its speed. Even doing simple things, like clicking buttons, or rotating the view with the middle mouse button, took too long to register in the program. I tried changing some User Preference System settings, but that didn't really help much. And the hardware seems to be fine - the render is lightning fast (renders the startup scene in under .1 second; it's just laggy when loading the uv screen), so I don't know why it's like this, or what to do. Also, I ran Blender 2.77 on this computer, and it was also slow (although it was fine on my previous computer).  It could be an issue with drivers perhaps (or maybe OpenGL?), but I ran Unreal Engine 4, and had none of the same issues - the speed was pretty decent on that; so I'm not sure why Blender would be so slow. Any help is appreciated - thanks guys! Low-RAM Linux Distro Specifically for serving static sites I'm looking for a near-zero-configuration linux distro that is designed specifically for serving static web sites and nothing else on a low-RAM machine (I'm planning on 128MB, but the option of less would be nice).  Any ideas of a linux distro specifically tuned for serving static sites in a small footprint? What graphics packages are there for creating graphics in LaTeX documents? What packages do you use and recommend for creating graphics in your LaTeX documents?     As this is a community wiki post, please add your package to the accepted answer (or add a comment, and someone with >100 rep will add it to the CW answer), and include a brief description of what differentiates it from others and how it can be used (GUI drawing tool which generates code, type in raw text, or generates image for inclusion in document).   We'll eventually sort these answers under headings. Linear Discriminant Analysis for $p=1$ I'm studying 'Introduction to Statistical Learning' by James, Witten, Hastie, Tibshirani.   In page 139, of their book, they began by introducing Bayes' Theorem $p_k(X)=P(Y=k|X=x) = \dfrac{\pi_kf_k(x)}{\sum_{l=1}^k \pi_l f_l(x)}$. $\pi$ is not mathematical constant, but denotes the prior probability. Nothing is strange in this equation.   The book claims that it wants to obtain an estimate for $f_k(x)$ that it can plug into the equation given above. To estimate $f_k(x)$, it assumes that is normal. In the one dimensional setting, $f_k(x)=\dfrac{1}{\sqrt{2\pi}\sigma}\exp(-\dfrac{1}{2\sigma^2}(x-\mu_k)^2)$, where $\mu_k$ and $\sigma^2_k$ are the mean and variance for the $k$th class. It is assumed that $\sigma^2_1 = \sigma^2_2 = \cdots = \sigma^2_K$. (I began to get confused from the last statement.)   Plugging $f_k$ into $p_x$, you have this quite messy equation (1) :   $$p_x(k)=\dfrac{\pi_k \frac{1}{\sqrt{2\pi}\sigma}\exp(-\frac{1}{2\sigma^2}(x-\mu_k)^2)}{\sum_{l=1}^K \pi_l \frac{1}{\sqrt{2\pi}\sigma}\exp(-\frac{1}{2\sigma^2}(x-\mu_l)^2)}. $$  Again, no surprises here since it is just substitution.     The Bayes' Classifier involves assigning an observation to the class for which equation (1) is the largest. Taking the log of Equation (1) and rearranging terms, it is not hard to show that this is equivalent to assigning the observation to the class for which the following is the largest:  $$\delta_k(x)=x \cdot \dfrac{\mu_k}{\sigma^2} - \dfrac{\mu_k^2}{2\sigma^2} + \log(\pi_k) $$  Question: I do not understand where this came from, and what it means. I tried doing the log of equation and it doesn't become this. Are we taking the derivative somewhere here, since this is the largest observation? I have an independent variable called "quality"; this variable has 3 modalities of response (bad quality; medium quality; high quality). I want to introduce this independent variable into my multiple linear regression. When I have a binary independent variable (dummy variable, I can code 0 / 1) it is easy to introduce it into a multiple linear regression model.  But with 3 modalities of response, I have tried to code this variable like this :  Bad quality      Medium quality      High quality       0                1                  0      1                0                  0      0                0                  1      0                1                  0   But there is a problem when I try to do my multiple linear regression: the modality Medium quality gives me NA:    Coefficients: (1 not defined because of singularities)    How can I code this variable "quality" with 3 modalities? Do I have to create a variable as a factor (factor in R) but then can I introduce this factor in a multiple linear regression? 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. Homology of $S^1 \times (S^1 \vee S^1)$ I'm trying to solve question 2.2.9(b) in Hatcher's Algebraic Topology.   Question: Calculate the homology groups of $X=S^1 \times (S^1 \vee S^1)$.  My attempt: I try to use the Mayer-Vietoris sequence.   Let $A=S^1 \times (S^1 \vee \text{small bit}) =S^1\times S^1$ and $B=S^1 \times ( \text{small bit} \vee S^1)=S^1 \times S^1$. (Unsure how to express this, but hopefully this is clear enough.)   Then $A\cap B=S^1\times \{\text{point}\}=S^1$ and $A\cup B=X$.  The Mayer-Vietoris sequence then gives us an exact sequence in reduced homology as follows:  $$0 \to \mathbb{Z}^2 \to \tilde{H}_2(X) \to \mathbb{Z} \to \mathbb{Z}^2 \to \tilde{H}_1(X) \to 0.$$  Is it possible from this exact sequence to determine the homology groups, in some algebraic fashion? Or do I need to determine what the maps are? How do I determine the maps? My system is not dual-boot, I run a standard Ubuntu desktop system "on the metal" (I think running it in a VM is the same).  Pressing c while booting does not cause the GRUB menu to appear (which I believe it does for a dual-boot system).  How can I get the GRUB-menu to present itself on a single-boot system? How can I make Firefox open magnet-links in Transmission? What OpenType math fonts are there available for usage in TeX/LaTeX, e.g. with the unicode-math package?  Here is a MWE which you can use to show off the font.  \documentclass{article} \pagestyle{empty}  \usepackage{amsmath} \DeclareMathOperator\Res{Res} \newcommand*\diff{\mathop{}\!\mathup{d}}  \usepackage{amsthm} \newtheorem{theorem}{Theorem}  \usepackage{unicode-math}  %%% % Set up you text and math fonts %%%  \begin{document}  \begin{theorem}[Residue theorem]   Let $f$ be analytic in the region $G$ except for the isolated   singularities $a_1,a_2,\dots,a_m$. If $\gamma$ is a closed   rectifiable curve in $G$ which does not pass through any of the   points $a_k$ and if $\gamma\approx 0$ in $G$, then   \[     \frac{1}{2\pi i} \int\limits_\gamma f\Bigl(x^{\mathbf{N}\in\mathbb{C}^{N\times 10}}\Bigr)     = \sum_{k=1}^m n(\gamma;a_k)\Res(f;a_k)\,.   \] \end{theorem}  \begin{theorem}[Maximum modulus]   Let $G$ be a bounded open set in $\BbbC$ and suppose that $f$ is a   continuous function on $G^-$ which is analytic in $G$. Then   \[     \max\{\, |f(z)|:z\in G^- \,\} = \max\{\, |f(z)|:z\in \partial G \,\}\,.   \] \end{theorem}  First some large operators both in text: $\iiint\limits_{Q}f(x,y,z) \diff x \diff y \diff z$ and $\prod_{\gamma\in\Gamma_{\bar{C}}}\partial(\tilde{X}_\gamma)$;\ and also on display \[   \iiiint\limits_{Q}f(w,x,y,z) \diff w \diff x \diff y \diff z   \leq   \oint_{\partial Q} f'\Biggl(\max\Biggl\{   \frac{\Vert w\Vert}{\vert w^2+x^2\vert};   \frac{\Vert z\Vert}{\vert y^2+z^2\vert};   \frac{\Vert w\oplus z\Vert}{\vert x\oplus y\vert}   \Biggr\}\Biggr)\,. \]  \end{document}   Here is a similar MWE for ConTeXt.  The images in the answer will only be shown for LaTeX output but the ConTeXt output looks very similar.  %%% % Set up you text and math fonts %%%  \setupmathlabeltext[Res=Res] \definemathcommand[Res][nolop]{\mfunctionlabeltext{Res}} \definemathcommand[diff]{\mathop{}\!\mfunctionlabeltext{diff}}  \defineenumeration   [theorem]   [text={Theorem},    style=italic,    title=yes,    titlestyle=normal,    distance=0pt,    titleleft={(},    titleright={).~},    alternative=serried,    width=fit]  \starttext    \starttheorem[title={Residue theorem}]     Let $f$ be analytic in the region $G$ except for the isolated     singularities $a_1,a_2,\dots,a_m$. If $\gamma$ is a closed     rectifiable curve in $G$ which does not pass through any of the     points $a_k$ and if $\gamma\approx 0$ in $G$, then     \startformula       \frac{1}{2\pi i} \int\limits_\gamma f\Bigl(x^{{\mathbf N}\in\mathbb{C}^{N\times 10}}\Bigr)       = \sum_{k=1}^m n(\gamma;a_k)\Res(f;a_k)\,.     \stopformula   \stoptheorem    \starttheorem[title={Maximum modulus}]     Let $G$ be a bounded open set in $\mathbb{C}$ and suppose that $f$ is a     continuous function on $G^-$ which is analytic in $G$. Then     \startformula       \max\{\, |f(z)|:z\in G^- \,\} = \max\{\, |f(z)|:z\in \partial G \,\}\,.     \stopformula   \stoptheorem    First some large operators both in text:   $\iiint\limits_{Q}f(x,y,z) \diff x \diff y \diff z$   and   $\prod_{\gamma\in\Gamma_{\bar{C}}}\partial(\tilde{X}_\gamma)$;\   and also on display   \startformula     \iiiint\limits_{Q}f(w,x,y,z) \diff w \diff x \diff y \diff z     \leq     \oint_{\partial Q} f'\Biggl(\max\Biggl\{     \frac{\Vert w\Vert}{\vert w^2+x^2\vert};     \frac{\Vert z\Vert}{\vert y^2+z^2\vert};     \frac{\Vert w\oplus z\Vert}{\vert x\oplus y\vert}     \Biggr\}\Biggr)\,.   \stopformula  \stoptext This question was prompted by questions about ls' -1 option and the recurring tendency of people to ask question and answers that includes processing the output of ls.   This reuse of the output ls seems understandable, e.g.: if you know how to sort a list of files with ls you might want to use the output in that way as input for something else.  If those Q&amp;A don't include a reference to the file name list produced consisting of nicely behaving file names (no special characters like spaces and newlines), they are often commented upon by someone pointing out the danger of the command sequence not working when there are files with newlines, spaces etc.  find, sort and other utilities solve the problem of communicating "difficult" file names to e.g. xargs by using an option to separate the file names with the NUL character/byte which is not a valid character in file name (the only one in addition to /?) on Unix/Linux filesystems.  I looked to the man page for ls and the output for ls --help (which has more options listed) and could not find that ls (from coreutils) has an option to specify NUL separated output. It does have a -1 option which can be interpreted as "output file names separated by newline")  Q: Is there are technical or philosophical reason why ls does not have a --zero or -0 option that would "output file names separated by NUL"?  If you do something that only outputs the file names (and not use e.g. -l) that could make sense:  ls -rt -0 | xargs -r0 …   I could be missing something why this would not work, or is there an alternative for this example that I overlooked and that is not much more complicated and/or obscure.    Addendum:  Doing ls -lrt -0 probably does not make much sense, but in the same way that find . -ls -print0 does not, so that is not a reason to not provide a -0/-z/--zero option. Proving the sum of two independent Cauchy Random Variables is Cauchy Reference help for "fuzzy mathematics" I want to study Fuzzy Sets and Fuzzy logic self. For that what would be better book to start from basic? What are the ways to position things absolutely on the page? I know that there is tikz, but also eso-pic and textpos.  I'd like to see a list of possible ways (not necessarily packages) for absolute positioning of things (which I believe amounts to "boxes") on the page, together with their advantages and disadvantages (for instance: tikz's current page node is very easy to use, but usually requires two LaTeX runs, and overhead from loading tikz if you don't need it for any other things is considerable).  Both LaTeX and ConTeXt solutions are welcome RF Energy Harvesting Why can energy invested in RF radiation not be harvested fully? What are the main factors that affect this? Is there any significant research nowadays that fully describes this process? I stand on the conclusion: "Energy moves from one form to another.", but in this case why can it not be harvested in that quantity to use it for powering low-power devices or store it in significant quantities in some batteries for later use? Why is it not profitable nowadays and why is it not used today like energy from water, wind and other energy sources? Say if I have a dropdown in a form and I have another nested class inside of this class . Now what's the best way to access this dropdown from the nested class?