Ways to go from A to B(advanced)
Ways to go from A to B(proof)
I am trying to work through Hartshorne's book and while working through one of the exercises, I need to show the following:  Let $k$ be an algebraically closed field. Let $Y \subseteq A^3$ be the set $Y=\{(t,t^2,t^3)\;|\;t \in k\}$. Show that $Y$ is an affine variety.  The rest of the problem I have completed. I'm just not sure how to show that this set is closed and irreducible. I assume that they intend for $A^3$ to have the Zariski topology. Bound for the correlation of three random variables Write space with TCPDF into a PDF file For example, in this text:     Lorem ipsum dolor sit amet, consectetur adipiscing elit. Nunc eu tellus vel nunc pretium lacinia. Proin sed lorem. Cras sed ipsum. Nunc a libero quis risus sollicitudin imperdiet.   I want to match the word after 'ipsum'. The Quotation Mark character in TeXworks editor has vanished Now, my version of TeXworks is 0.4.5r.1280 and I can't have the " character on the keyboard.    I have ``  or '' instead. Why?  And It's the same with Alt+034. I'm very disappointed with that. I'm a fan of extension methods in C#, but haven't had any success adding an extension method to a static class, such as Console.  For example, if I want to add an extension to Console, called 'WriteBlueLine', so that I can go:  Console.WriteBlueLine("This text is blue");   I tried this by adding a local, public static method, with Console as a 'this' parameter... but no dice!      public static class Helpers {     public static void WriteBlueLine(this Console c, string text)     {         Console.ForegroundColor = ConsoleColor.Blue;         Console.WriteLine(text);         Console.ResetColor();     } }   This didn't add a 'WriteBlueLine' method to Console... am I doing it wrong? Or asking for the impossible? Question about Conditional Expectation Alignment of equals sign in multiple align environments What can I do to manage my phone's internal storage? If you're not used to seeing the 'low on internal storage' notification, what is it that you're doing to keep your phone's internal memory from filling up so quickly?   The 'SD Card and Phone Storage' menu in the settings doesn't really tell me much other than how much is free. And, I don't really have that many applications installed. The top 5 storage hogging apps on my phone are Gmail (18.6 MB), Twitter (17.5 MB), Facebook (12.55 MB), Maps (9.63 MB), and Swype (6.06 MB).  So, are there apps out there (besides your normal file manager ones, like Astro) that will help monitor internal phone storage? Is there something I should be doing to keep my phone storage use to a minimum? What is the rigor behind U subsitution? $ \int f(g(x)) dx = \int \frac {f(u)}{u'} du$  requires that  $ \int f(x) dx = \int f(x) \cdot dx$  but dx just represents the variable that F(x) +c is a function of. So why is it legal for dx to be treated algebriacally?  I tried investigating this property by using riemann summation:  $\lim\limits_{n \to \infty}(\sum_{k=1}^n f(\frac{kx}{n} )\frac{x}{n}) = \int_0^x f(t)dt= (F(x)+c)-(F(0)+c)$  and so you can define   $ \lim\limits_{n \to \infty} (\sum_{k=1}^n f(\frac{kx}{n} )\frac{x}{n})+F(0)+c)  = \int f(x) dx$  you can write $\frac{x}{n} = dx$ and $k\frac{x}{n}=kdx= x_k$  then you have  $ \lim\limits_{n \to \infty} (\sum_{k=1}^n f(x_k)dx)+F(0)+c) = \int f(x) dx$  since  dx is being written to be multiplied by the series, then you can define dx in different terms to aquire an integral in terms of other variables.   but that exists only in abstraction. I can't quite sufficiently complete the task of doing so. Algebra of LDA. Fisher discrimination power of a variable and Linear Discriminant Analysis Apparently,     the Fisher analysis aims at simultaneously maximising the   between-class separation, while minimising the within-class   dispersion. A useful measure of the discrimination power of a variable   is hence given by the diagonal quantity: $B_{ii}/W_{ii}$.     I understand that the size (p x p) of the Between (B) and Within-Class (W) matrices are given by the number of input variables, p. Given this, how can $B_{ii}/W_{ii}$ be a "useful measure of the discrimination power" of a single variable? At least two variables are required to construct the matrices B and W, so the respective traces would represent more than one variable.  Update: Am I right in thinking that $B_{ii}/W_{ii}$ is not a trace over a trace, where the sum is implied, but the matrix element $B_{ii}$ divided by $W_{ii}$? Currently that is the only way I can reconcile the expression with the concept. How can I specify the default save directory for gnome-screenshot? Are there any sort of configuration options for specifying the default save location for gnome-screenshot, or is this hard-coded into the source code? It used to be ~/Desktop, which seems to have changed to ~/Pictures (in 12.04). The only possible solution I've seen is about  (as it includes time stamp information now instead of simply 'Screenshot#'), but that solution doesn't really seem ideal to me. Also,  post suggested that the last save location is remembered the next time you take a screenshot, but in my experience, this doesn't seem to be the case. And in any case, following on from that, that entry in gconf-editor doesn't even seem to accurately reflect the last location, so more than likely an entry related to an older version of gnome-screenshot. Table of contents: What information does the trace of a matrix give? I was recently thinking what information do we get from a matrix. So if we say the columns (or rows) of a matrix define the basis of a system, say vectors of 3 dimensional space. Then the determinant will tell about the volume of the space enclosed by those vectors. Does trace give any information about the space? Why do I need 50 reputation to comment? What can I do instead? Users with less than 50 reputation points cannot write comments on questions and answers that they don't own.  Why does this limit exist? But I want to contribute now. What should I do instead? But I can't write a good answer without more information! Shouldn't I be allowed to post comments if I have reputation on a different site? But I saw a new user posting a comment! How did they do it? Why do I need 50 reputation to comment on my own post?   See also:    in the Help Center I have asked this question before  The first guy solved it in a way, but  i would like to know is there any efficient way to do so, because i did the same for N = 13 and N = 15  which took long time.  This problem is from a pen and paper test, ZIO-2018 Can i solve it with an efficient algorithm, well i am pretty beginner in algos, so if there is one kindly descibe it how can i do so?  Thank you Dual monitors not working with an Nvidia card Even my main monitor is shown as unknown. I have tried setting up dual monitors using Nvidia settings with no joy however it does recognize that I have 2 monitors.  This was not an issue in 11.04. My problem with the Nvidia settings appears to be that my 2 monitors require different resolutions but even when I set them to a resolution that they both can handle, it is then only appearing in cloned mode. I really hope someone can help or I will have to roll back to 11.04.  Why won't Ubuntu 11.10 recognize that I have 2 Monitors?   Here is the contents of my xorg.conf:  # nvidia-settings: X configuration file generated by nvidia-settings # nvidia-settings:  version 280.13  (buildd@rothera)  Thu Aug 11 17:28:49 UTC 2011  Section "ServerLayout"     Identifier     "Layout0"     Screen      0  "Screen0" 0 0     InputDevice    "Keyboard0" "CoreKeyboard"     InputDevice    "Mouse0" "CorePointer"     Option         "Xinerama" "0" EndSection  Section "Files" EndSection  Section "InputDevice"     # generated from default     Identifier     "Mouse0"     Driver         "mouse"     Option         "Protocol" "auto"     Option         "Device" "/dev/psaux"     Option         "Emulate3Buttons" "no"     Option         "ZAxisMapping" "4 5" EndSection  Section "InputDevice"     # generated from default     Identifier     "Keyboard0"     Driver         "kbd" EndSection  Section "Monitor"     # HorizSync source: xconfig, VertRefresh source: xconfig     Identifier     "Monitor0"     VendorName     "Unknown"     ModelName      "CRT-0"     HorizSync       30.0 - 83.0     VertRefresh     50.0 - 76.0     Option         "DPMS" EndSection  Section "Device"     Identifier     "Device0"     Driver         "nvidia"     VendorName     "NVIDIA Corporation"     BoardName      "GeForce 7025 / nForce 630a" EndSection  Section "Screen"     Identifier     "Screen0"     Device         "Device0"     Monitor        "Monitor0"     DefaultDepth    24     Option         "TwinView" "1"     Option         "TwinViewXineramaInfoOrder" "CRT-0"     Option         "metamodes" "CRT: 1280x1024 +1024+0, DFP: 1024x768 +0+0"     SubSection     "Display"         Depth       24     EndSubSection EndSection"   ok I have tried disper -e. Problem is that Nvidia settings appear to be treating both monitors as the same one. so changing the settings to one has no effect, but change the other and it affects both screens identically.  Results of disper -l as follows:      carter@carter-desktop:~$ disper -l display DFP-0: HP vs17  resolutions: 320x240, 400x300, 416x312, 512x384, 576x432, 680x384, 640x480, 720x450, 640x512, 800x512, 840x525, 800x600, 960x540, 832x624, 960x600, 896x672, 1024x768, 1152x864, 1280x960, 1280x1024 display CRT-0: CRT-0  resolutions: 320x240, 400x300, 416x312, 512x384, 576x432, 680x384, 640x480, 720x450, 640x512, 800x512, 840x525, 800x600, 960x540, 832x624, 960x600, 896x672, 1024x768, 1152x864, 1360x768, 1280x960, 1440x900, 1280x1024, 1400x1050, 1600x1024, 1680x1050, 1600x1200, 1920x1080, 1920x1200, 1792x1344 carter@carter-desktop:~$ Every function on $\mathbb{R}^n$ that is continuous in each variable separately is Borel measurable. I'm trying to solve exercise 2.11 from Folland's Real Analysis.    Firstly, how do we show that $f_n$ is Borel measurable on $\mathbb{R} \times \mathbb{R}^k$? I really have no idea how to show this when we have separately Borel measurable functions.  Next, I can't show that $f_n\to f$ pointwise.   For any $(x,y)\in \mathbb{R}\times \mathbb{R}^k$,   $f_n(x,y)=n[f(a_{i+1},y)(x-a_i)-f(a_i,y)(x-a_{i+1})]=n[x[f(a_{i+1},y)-f(a_i,y)]+\frac{1}{n}[f(a_i,y)-f(a_{i+1},y)]]$,   but because of the $n$ outside the bracket, I can't show that this converges to $f$. How can I show this? Also, how can I conclude from this fact that every function on $R^n$ that is continuous in each variable separately is Borel measurable?  I would greatly appreciate any suggestions, hints, or solutions. Alpha and Omega limit sets (dynamical systems) What are all the possible $α$- and $ω$-limit sets of points for the linear system of equations $x'=Ax$ given by the following matrices $A$:  I guess I don't really know how to get started, Could I have some guidance? So I was told to calculate the eigenvalues (fine) and eigenvectors (I kind of forgot how to do that...)  I know that the $ω$-limit sets of points are the set of points that the system of equation approach as time goes to infinity, and the $α$-limit sets of points are the points approached as t goes to negative infinity.   $    A=   \left[ {\begin{array}{cc}    -4 &amp; -2 \\    3 &amp; 11 \\   \end{array} } \right] $   $λ_1=-10$ $λ_2=-5$  Since both eigenvalue are negative, this is a stable node, and the $ω$-limit set=0.  for $λ_1=-10$:  eigenvector $    =   \left[ {\begin{array}{cc}   1 \\    3 \\   \end{array} } \right] $  for $λ_2=-5$:  eigenvector $    =   \left[ {\begin{array}{cc}   2 \\    1 \\   \end{array} } \right] $   $    A=   \left[ {\begin{array}{cc}    1 &amp; 1 \\    3 &amp; 1 \\   \end{array} } \right] $   $λ_1=4$ $λ_2=-2$  Since one eigenvalue is positive and one is negative, it is not stable, so I can't use the same reasoning that I used above to find the $ω$-limit set... so I am not sure how to continue  for $λ_1=4$:  eigenvector $    =   \left[ {\begin{array}{cc}   1 \\    1 \\   \end{array} } \right] $  for $λ_2=-2$:  eigenvector $    =   \left[ {\begin{array}{cc}   -1 \\    1 \\   \end{array} } \right] $   $    A=   \left[ {\begin{array}{cc}    4 &amp; 5 \\    -5 &amp; -2 \\   \end{array} } \right] $   $λ_1=1+4i$ $λ_2=1-4i$  for $λ_1=1+4i$:  eigenvector $    =   \left[ {\begin{array}{cc}   -3/5-(4i)/5 \\    1 \\   \end{array} } \right] $  for $λ_1=1-4i$:  eigenvector $    =   \left[ {\begin{array}{cc}   -3/5+(4i)/5 \\    1 \\   \end{array} } \right]$ Ideas to improve website loading speed? Are there any ways in which I can improve the loading speed of a website?  I know that Google is really pushing loading speeds and with the popularity of mobile websites it's now imperative that sites load quickly. How can I fit my table of contents into a single page? When I use \tableofcontents I get all my contents in a single page except for a line that goes on the other page, how can I make all in the same page?