Original problem setup   \begin{equation}\tag{1} \begin{array}{c} \min_{\mathbf{b}} \hspace{4mm}  \mathbf{b}^{T}_{}\mathbf{A}^{}_{}\mathbf{b}^{}_{} \\ s.t \hspace{5mm} \mathbf{b} \in \mathbb{R}^{4} \\ \hspace{9mm}b_0=1\\ \hspace{15mm}b_3=b_1b_2  \end{array} \end{equation}  where $\mathbf{A}$ is a $4\times 4$ positive semi-definite matrix.  I Reformulated this to  \begin{equation}\tag{2} \begin{array}{c} \min_{\mathbf{b}} \hspace{4mm}  \mathbf{b}^{T}_{}\mathbf{A}^{}_{}\mathbf{b}^{}_{} + \lambda(\mathbf{b}^{T}_{}\mathbf{Z}^{}_{}\mathbf{b}^{}_{}-1) \end{array} \end{equation}  where  \begin{equation} \mathbf{Z} = \begin{bmatrix}     1 &amp; 0 &amp; 0 &amp; 0 \\     0 &amp; 0 &amp; -1 &amp; 0 \\     0 &amp; -1 &amp; 0 &amp; 0 \\     0 &amp; 0 &amp; 0 &amp; 2 \\ \end{bmatrix} \end{equation}  My Question:   Are (1) and (2) exactly same? Does (2) result in convexification of non-convex constraint $b_3=b_1b_2$ If (2) is correct, then do we have a closed-form solution for $\mathbf{b}$   (Can someone please edit to place $\mathbf{b}$ exactly below min)
Here is the problem setup  \begin{equation} \begin{array}{c} \text{min} \hspace{4mm}  \mathbf{b}^{T}_{}\mathbf{A}^{}_{}\mathbf{b}^{}_{} \\ s.t \hspace{5mm} \mathbf{b} \in \mathbb{R}^{N} \\ \hspace{9mm}b_0=1\\ \hspace{62mm}b_i=b_{i-2}b_{i-1} \hspace{5mm} \forall \hspace{5mm} 2&lt;i\leq N-1 \end{array} \end{equation}  where $\mathbf{A}$ is a $N\times N$ positive semi-definite matrix  To handle $b_0=1$ constrain, I know I can introduce Lagrange multiplier in objective function  \begin{equation} \begin{array}{c} \text{min} \hspace{4mm}  \mathbf{b}^{T}_{}\mathbf{A}^{}_{}\mathbf{b}^{}_{} + \lambda(\mathbf{b}^{T}_{}\mathbf{u}^{}_{}-1) \end{array} \end{equation}  where $\mathbf{u}_{}^{}$ is a $N\times 1$ column vector and $u_0=1$ and $u_i=0 \forall 1&lt;i\leq N-1$.  But I dont know how to handle second set of constraints?   EDIT: Example  Let $N=4$. Therefore I am looking for $\mathbf{b}=[b_0 \quad b_1 \quad b_2 \quad b_3]^T = [1 \quad b_1 \quad b_2 \quad b_1b_2]^T$. How do I include $b_3=b_1b_2$ constraint in objective function
Windows 8 Setting an image on the login screen In Windows 8.1 when you log in there's the prelogin screen that has an image you can change, but then there's also the login (password entry) screen that as far as I can see is just a solid color.  How could I replace that with an actual picture? (ie, if I wanted a company logo to be displayed in the large space underneath the password entry box. creating variable using variable value as part of new variable name I'm trying to create a new variable using the value of an existing variable as part of the variable name.  filemsg"$word1"=" "   I've also tried   filemsg$word1=" "  filemsg${word1}=" "   on all attempts I get the following when that line executes,  cicserrors.sh[45]: filemsgCICS= : not found [No such file or directory] Compiler warning: null reference exception I have the following code in Visual Studio 2005.      Dim OutFile As System.IO.StreamWriter     Try         OutFile = New System.IO.StreamWriter(Filename)        // Do stuff with OutFile     Catch Ex As Exception        // Handle Exception     Finally        If OutFile IsNot Nothing Then OutFile.Close()     End Try   But VS2005 brings up the warning for the line "If OutFile IsNot.." that      Variable 'OutFile' is used before it has been assigned a value. A null reference exception could result at runtime.   Is there some way of removing this warning by subtly altering the code or is there just a better way of doing what I'm trying to do?  Thanks  Rob Can “does” be contracted? Just wondering, I'm about to buy it and it sounds awesome, but how many characters can I have in Skyrim on the PS3? Am I limited to just 1, or can I make more than one character at once? Find the ratio of $\frac{\int_{0}^{1} \left(1-x^{50}\right)^{100} dx}{\int_{0}^{1} \left(1-x^{50}\right)^{101} dx}$ What is numeric promotion? 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: I have heard that there are some Pokémon that are region specific.  Could someone please clarify which Pokémon are region specific and in which regions that they can be found in? Show that $x^2+x+4$ is irreducible over $\mathbb Z_{11}$.    The problem can be solved by putting every element from $\mathbb{Z}_{11}$ and checking if there is any root.  $0^2 + 0 + 4 = 4 \\ 1^2 + 1 + 4 = 6 \\ 2^2 + 2 + 4 = 10 \\ 3^2 + 3 + 4 = 5 \\ 4^2 + 4 + 4 = 2 \\ 5^2 + 5 + 4 = 1 \\ 6^2 + 6 + 4 = 2 \\ 7^2 + 7 + 4 = 5 \\ 8^2 + 8 + 4 = 10 \\ 9^2 + 9 + 4 = 6 \\ 10^2 + 10 + 4 = 4 $   But this process is very long one. I am seeking a short method by which I can solve such problem.   Now suppose that it is reducible, then the factors should be linear. So how can we solve the problem based on this kind of observation? Any help will be great. Thanks. I'm trying to create a command, which calls on an environment, which defines a custom column definition with one argument using \newcolumntype, and defines a second command, containing parameters for a table column.  Then, still within the first command, I begin the tabular environment and call the custom column, with the second command as its argument. I always get an undefined control sequence error, though. Any thoughts on why that might be?  I'm sure that was clear as mud, but I've cut it down to a minimal example, which follows:  \documentclass{article} \usepackage{array} \usepackage[table,dvipsnames]{xcolor} \usepackage{colortbl}  \newenvironment{tablecolor}{     \newcommand{\tlight}{\columncolor{SkyBlue!30}\color{black}}     \newcolumntype{C}[1]{&gt;{##1}c} }{}  \newcommand{\tablegoals}{ \begin{tablecolor} \begin{tabular}{l|C{\tlight}} Goal 7  &amp; \\ \end{tabular} \end{tablecolor} }  \begin{document}     \tablegoals \end{document}     Here's my second attempt. Different approach, same undefined control sequence error with \columncolor.  \documentclass{article} \usepackage{array} \usepackage[table,dvipsnames]{xcolor} \usepackage{colortbl} \usepackage{ifthen}  \newenvironment{tablecolor}{     \newcolumntype{C}[1]{&gt;{         \ifthenelse{\equal{dark}{##1}}{             \columncolor{Cyan}\color{white}             }{             \columncolor{SkyBlue!30}\color{black}         }     }c} }{ }  \newcommand{\tablegoals}[1]{     \begin{tablecolor}         \begin{tabular}{l|C{dark}}             Goal 7  &amp;   #1         \end{tabular}     \end{tablecolor} }  \begin{document}     \tablegoals{Check mark} \end{document} Since my previous questions were not answered I will try to ask again more precisely.  I previously had the Nvidia driver installed and working correctly.  Then the ubuntu software update did something which broke it.  In fact, it completely de-configured it so that the standard display driver is used when the computer reboots.  I would like to re-enable the existing Nvidia  driver without re-installing the Nvidia driver.  How do I do that?  This machine is no longer connected to the internet.  NOTE: I am not asking how to disable automatic updates in this question.  I asked that elsewhere (and it wasn't answered).  NOTE: I am not asking how to stop the Ubuntu software updater from breaking the Nvidia driver the next time it updates the system.  That would be nice to know, but I asked that already and did not receive the correct answer either.  NOTE: Please don't make this as duplicate simply because someone else asked a question about Ubuntu and the software updater.  Try to read the actual question and see how it's different and try to realize how annoying it is when users have to spend hours of their own time to correct the problems that other people causing by incorrectly designing software updaters. Maintain the aspect ratio of a div with CSS Inline image alignment issue for app and mobile web get the Eyes of the Dark feature (XGtE, p. 51):     When you reach 3rd level in this class you learn the darkness spell [...] If you cast it with sorcery points, you can see through the darkness created by the spell.   Does this mean that if a Shadow Sorcerer casts darkness using sorcery points and then moves into a different area of magical darkness (not the one s/he created), the Sorcerer would not be able to see?  Or does the ability mean that the Sorcerer would possess the ability to see through the magical darkness created by the darkness spell in general (provided they were maintaining their own casting of the spell)? Why is there a peeking duck in my profile pic? In , you may see something odd: a peeking duck.    I've just recently uploaded this new profile picture of mine. I checked it after the upload and everything's fine. After a few hours, I opened my profile and there it is, the duck has somehow become afraid of the people viewing it.  Is this a bug, a feature, or is it just the duck?  I am about to re-upload my profile pic, but I've decided to drop first this topic to meta.  And if you're curious,  is the original picture.    ADDITIONAL: I've used the non-gravatar way to upload the picture.  ADDITIONAL 2: The problem only shows in the profile view. No problem in chat, less info profile view, and in the questions and answers.  UPDATE: Re-uploaded picture. Status normal (for now, at least). "Insufficient storage" when trying to install new app Every time I go to install a new app, I get the message "Insufficient storage", and an unable to install anything!  I am using Samsung GT19060 my internal memory is 8GB. I have not downloaded anything but anytime I try to download something it will tell me low memory space. I am sketching out my 2012 programming project, and Im planning to make a classic fantasy RPG game.  The world design, plots, characters and all that I can make, but being a avid tabletop role-player, I would love to use a stable and tested rpg system like Pathfinder, since its balanced and interesting.  Would there be a problem if I make a game (indie dev, but still planning on selling it) using the pathfinder license ? Do I have to purchase a license? And would I able to say that it's based on the Pathfinder ruleset, or should i keep that under wraps.  All big official D&amp;D games boast with that they are official, so doesn't this mean that unofficial games can exist too ?  It's not because I wan't to jump the lowest fence, and all storylines and the game setting will be by my own hand. But I love the Pathfinder system and actually think it could use some publicity as well.  If not, I guess I could make a Pathfinder-like system, with a few measely differences... But I would rather not, since it would feel a bit of a rip... Let $\,f : \mathbb{R} \rightarrow \mathbb{R}$ be a infinitely differentiable function that is  increasing and bounded. Then is it true that $\lim_{x\to \infty}f'(x)=0$? $I_m - AB$ is invertible if and only if $I_n - BA$ is invertible. The problem is from  Algebra by Artin, Chapter 1, Miscellaneous Problems, Exercise 8. I have been trying for a long time now to solve it but I am unsuccessful.      Let $A,B$ be $m \times n$ and $n \times m$ matrices. Prove that $I_m - AB$ is invertible if and only if $I_n - BA$ is invertible.    Please provide with only a hint.