If $dF_p$ is nonsingular, then $F(p)\in$ Int$N$
Let $M$ be a smooth manifold and let $N$ be a manifold with boundary. If $dF_{p}$ is an isomorphism, then $F\left(p\right)\in\mbox{int}M $.
Windows 7 Automatically Connecting To Unsecured Wireless Networks On Startup Most of the questions on this topic related to folks connecting to somebody else's wireless network when their own was available and could remedy the situation by going to their connections and unchecking the "connect automatically" box.  See this: "  " as an example.  In my situation, I've noticed that Win 7 will automatically connect to any unsecured wifi network - even if I have never connected to it in the past.  If I am traveling and boot Win 7, it will start and connect to what appears to be the best signaled unsecured network without prompting me for confirmation (note: in the above link, "Naveen" seems to have same problem).  Obviously, that is a security concern to me.  Further, when I open "Network and Sharing" and "Manage wireless networks"  the network is not displayed (probably because I labelled it a public network).  Again, these are new, never connected with before, wireless networks.  I always promptly disconnect from them but don't want to have to be on constant guard for an auto connection to a malicious network.  This began about a month ago, as I recall, Win 7 did not behave like this in the past, I didn't monkey with wifi settings, and don't use a 3rd party connection manager.  I did have to download some internet security certificates for army website access but I don't think that should mess with network settings.  Any ideas how I can tell Win7 cease automatically connecting to networks or, at least, to prompt me for a confirmation before connecting? Proper analyses for 2x2 contingency tables So I have multiple questions about the different types of analyses that can be used on a 2x2 contingency table and when different analyses should be used.   To open, I'll note that this stems from my efforts to find the appropriate analyses to run for research using categorical data in a 2x2 contingency table. In my case I have two conditions: control and experimental and within the experimental group there are two items of interest (say, Item A and Item B). In all cases I'm investigating whether an outcome occurs or not: How many experimental participants show the outcome on Item A? How many experimental participants show the outcome on Item B? How many control participants show the outcome? So in addition to comparing Item A with the Control group (A vs. C) and Item B with the Control group (B vs. C), I also want to compare Item A with Item B (A vs. B).   My understanding is that for between-subjects comparisons (e.g., A vs. C   &amp;   B vs. C) a  is traditionally used and that for within-subjects comparisons (e.g., A vs. B) that  is a common choice.   Question 1: is this an accurate understanding of the common choices for statistical tests using 2x2 contingency tables?    My follow-up questions, and where most of my confusion lies, are related to what to do when multiple observed cell counts are low (less than 10). I've explored numerous threads here, the Wikipedia pages for different topics (Yates's correction, Fisher's exact, etc.), a few journal articles, and descriptions in R-packages, but I have found conflicting views.   Some sources indicate that you should use , others say it's out-dated because, in addition to being overly conservative, it was only preferred because the other options were computationally intense.   Others say that you should use , but I've seen additional information suggesting that Fisher's exact test is only appropriate if you know both the row and column totals and that other options (Boschloo, Barnard, etc.) are generally more powerful. However, which of those to use is also not clear.  Finally, even others suggest that the Pearson's $\chi^{2}$ test is appropriate as long as the expected cell counts are not low.   I've seen some indication that these arguments occur for McNemar's test as well (e.g., continuity correction &amp; McNemar's exact test).  Question 2: Is the assumption regarding cell counts about expected cell counts or observed cell counts? If the observed cell counts are low, does that matter to the type of test that should be used?  Question 3: What is an exact test and when should exact tests be used?    In addition to answers to these questions, I would also greatly appreciate any suggestions as to books or articles I can read to learn more about this. Part of my struggle with all of this has been just general difficult finding clear information about these topics and how they relate to one another. This  shows the dual cone of $K = \{(x,t) \mid \| \boldsymbol{x} \|_1 \le t \}$ is $K^{*} = \{(x,t) \mid  \| \boldsymbol{x} \|_{\infty} \le t\}$. Is it right? How is it proved? Initializing C# auto-properties barchart with pgfplot - missing bar I am using PGFplot to make a barchart. There are 4 entries in my csv file and unfortunately I have only 3 bars in my figure.   my LaTeX code:  \begin{figure}     \centering     \begin{tikzpicture}          \begin{axis}[             x tick label style={/pgf/number format/1000 sep=},                 xlabel=Sequence,             ylabel=Heigth,                 ymin=0.7,             enlargelimits=0.05,             legend style={at={(0.5,-0.15)},                 anchor=north,legend columns=-1},             ybar interval=0.7,             width=\textwidth,             ]             \addplot table [col sep=comma,trim cells=true,y=height] {numericaltests/data/test.csv};              \addplot[red,sharp plot,update limits=false]                 coordinates {(-1,1.00) (34,1.00)}                 node[above] at (axis cs:15,1.00) {};             \legend{Ca2Fe2As2}         \end{axis}     \end{tikzpicture}     \caption{Sequence}     \label{fig:sequenceHeight} \end{figure}   my test.csv file:  sequence,height 1,2.0 2,1.5 3,1.8 4,2.4   Barchart: Problems with fonts I'm an average user of LaTeX, but not proficiency. I'm trying to learn from book templates I've found on the web, but I get this error message:  Trying to make PK font ccicons at 1493 DPI... Running miktex-makemf.exe... miktex-makemf: The ccicons source file could not be found. Running ttf2pk.exe... miktex-makemf: PK font ccicons could not be created.  LaTeX Font Warning: Some font shapes were not available, defaults substituted.  !pdfTeX error: pdflatex.exe (file ccicons): Font ccicons at 1493 not found  ==&gt; Fatal error occurred, no output PDF file produced!   I'm using MiKTeX (over Windows 7), and according to the MiKTeX package manager the "ccicons" package is already installed.  Could you please help me solve this problem? Detect Chrome extension first run / update Show that there exists a unique $v_0 \in H^1(0,1)$ such that $u(0)=\int_0^1(u'v_0'+uv_0), \forall u \in H^1(0,1)$ Show that there exists a unique $v_0 \in H^1(0,1)$ such that $u(0)=\int_0^1(u'v_0'+uv_0), \forall u \in H^1(0,1)$. Further Show that $v_0$ is the solution of some differential equation with appropriate boundary conditions. Compute $v_0$ explicitly.   Let $f: H^1(0,1) \to \mathbb{R}$ be defined by $f(u)=u(0)$. Then I showed that $f$ is linear and continuous. Hence there exists $v_0 \in H^1(0,1)$ such that $u(0)=\int_{0}^1u'v_0'+\int_{0}^1uv_0$  The differential equation is given by $f=-u''+u$ with the conditions $u'(0)=u'(1)=0$.   How do I compute $v_0$ explicitly?? Angle notation for complex numbers in polar form I am looking for suggestions on how to typeset complex numbers in the modulo-argument form, sometimes called phasor notation. Have already checked  but nobody mentions this in particular. I am explicitly excluding exponential and sine-cosine notations. My MWE is as follows  \documentclass{article} \begin{document}     $z = r \angle \phi$ \end{document}   I think it looks ugly, specially if the angle has many digits, ie  $z = 1.19 \angle -78.2039^{\circ}$   I was thinking of an \angle replacement which maybe would extend the lower segment up to the last digit. What do you think?  EDIT: It looks like the steinmetz package doesn't do a good job with the vertical space if the \phasor{} argument has a \fraction{}{} with parenthesis, see:  \documentclass{article} \usepackage{steinmetz}  \begin{document}     Something.     \[I = I \phase{\left(\frac{A_0}{B_0}\right)}\]     This line is too close to the upper equation, don't you think?      \[I = I \phase{\left(\frac{A_0}{B_0}\right)}\]     Whereas this line is NOT so close to the upper equation. \end{document}   Strangely, commenting the first line (% Something.) fixes the problem. Is there any way to fix this? Why does "Object reference not set to an instance of an object" not refer the variable name or column? LaTeX package "titling", titles with hard line breaks, and \MakeUpperCase I have a document with a title that is spread over several lines using hard line breaks. To configure the title formatting, I use the titling package with code adapted from its documentation:  \documentclass{article}  \usepackage{titling} \pretitle{\begin{center}\LARGE\sffamily\bfseries} \posttitle{\par\end{center}\vskip 0.5em}  \title{Higgelty Pigglety Pop! \\ or \\ There Must Be More to Life}  \begin{document} \maketitle \end{document}   …and it works perfectly.  I now want the title additionally to be in all caps and therefore change the \pretitle command:  \pretitle{\begin{center}\LARGE\sffamily\bfseries\MakeUppercase}   …and I get the following error message:  ! Use of \@icentercr doesn't match its definition. &lt;argument&gt; \def  l.12 \maketitle   This only happens with a title that includes hard line breaks.  What does this error message mean? How can I fix it? Is my expansion of the state $| x \rangle$ correct? In my quantum mechanics textbook it says that the relation between the basis $|x\rangle$ and $|p\rangle$ is given by:  $\langle p | x \rangle = \Large \frac{e^{-ip x/ \hbar}}{\sqrt{2\pi \hbar}} \, .$  However, i'm not sure how to go about proving this relation. My idea was that the eigenstates of momentum can be written as below:  $\phi(x,p) = \Large \frac{1}{\sqrt{2\pi \hbar}} e^{-ipx/\hbar},$  which form an orthogonal basis.  But, we can expand any state, $|\text{state}\rangle$, in terms of the othorgonal basis right? So is my notation below correct for the expansion of the state $| x \rangle$ ?  $|x\rangle = \int_{-\infty}^{+\infty} \phi(p', x) |p'\rangle dp'$  If it were correct then the following would also be correct?  Using the fact that $\langle p | p' \rangle = \delta(p - p')$, $\langle p | x \rangle$ can be computed:  \begin{align} \langle p | x \rangle =&amp; \int_{-\infty}^{+\infty} \phi(p', x) \langle p | p' \rangle dp' \\ =&amp; \int_{-\infty}^{+\infty} \phi(p', x) \delta(p - p') dp' \\ =&amp; \phi(p, x) \\ =&amp; \Large \frac{e^{-ip x/ \hbar}}{\sqrt{2\pi \hbar}} \end{align} as required.  Does the logic work here? Given that   $x^x = y$; and  given some value for $y$   is there a way to expressly solve that equation for $x$? Suppress black box at end of line Under TeX, when I write a line too long, it prints a fat black box at the end of the line.  That is probably a feature, but I dislike it.  How to turn that OFF? Is there a command-line option or something else?   I usually use LaTeX, but I have a complicated file which happens to be plain TeX. What is the best way to convert an array to a hash in Ruby In Ruby, given an array in one of the following forms...  [apple, 1, banana, 2] [[apple, 1], [banana, 2]]   ...what is the best way to convert this into a hash in the form of...  {apple =&gt; 1, banana =&gt; 2} Windows update doesn't progress beyond 0 KB 0% I have Windows&nbsp;7 64-bit and the  isn't working properly. What can I do? Prove $2^n-1$ is divisible by $(2^m-1)^2$ iff... Prove that for any $m,n\in\mathbb{Z}^+$, the number $2^n-1$ is divisible by $(2^m-1)^2$ iff the number $n$ is divisible by $m(2^m-1)$.  My work so far:  It's an if and only if statement, so the common approach is to split the proof into two "if then" proofs ($A\rightarrow B \text{ and } B\rightarrow A$).  Starting with $A\rightarrow B$, for a direct proof, I would assume $A$ and prove $B$. I can use the definition of divisibility:  Assume $n=m(2^m-1)k$ for some integer $k$  Then I need to prove that $2^n-1=(2^m-1)^2h$ for some integer $h$  So I'm thinking I need to start with the expression $2^n-1$ and transform it to $(2^m-1)^2h$, or vice versa, and I need to use the assumption $n=m(2^m-1)k$ somehow.  I notice that the LS of the equality contains the variable $n$ while the RS contains $m$, and the assumption relates $n$ to $m$, so that could be a way forward. For example, simple substitution gives $2^{m(2^m-1)k}-1$, but I'm not sure what I could do from here. Perhaps there's a divisibility theorem that could help?  Other things I notice: $2^n-1$ and $(2^m-1)^2$ are always odd. Also, the divisors in both cases are multiples of $2^m-1$.  Any help/hints appreciated. Does the proof of Bolzano-Weierstrass theorem require axiom of choice? When selecting the terms of subsequence from each bisections, I thought axiom of choice might be required. But I'm not so sure whether or not, so please tell me. [edited] I'm sorry for the lack of explanation. I want to prove this statement: Let $a_1, a_2, \ldots \in \mathbf{R}$, and $(a_n)_{n\in\mathbf{N}}$ is bounded, then $(a_n)$ has some convergent subsequence.  The proof is as follows. Since $(a_n)$ is bounded, for all $n\in\mathbf{N}$,   $a_n \in I = [b, c]$.  Now, let $I_0 = I$ and if $I_n = [b_n, c_n]$,  we define $d_n = (b_n+c_n)/2$ and  if infinite terms of $(a_n)$ is included in $[b_n, d_n]$(resp. $[d_n, c_n]$), we will define  $I_{n+1} = [b_n, d_n]$(resp. $[d_n, c_n]$).If both intervals contain infinite terms, let $I_{n+1}$ be $[d_n, c_n]$. For all $n\in \mathbf{N}$, infinite numbers of $m \in \mathbf{N}$ exist such that $a_m \in I_n$ suffices. We take the sequence of natural numbers $(n(k))_{k\in\mathbf{N}}$which suffices $n(0) &lt; n(1) &lt; \cdots &lt; n(k) &lt; \cdots$ following this procedure: Now we have already selected $a_{n(1)}, \ldots, a_{n(k)}$, there are infinite numbers of $m\in \mathbf{N}$ which suffices $n(k)&lt;m, a_m \in I_{k+1}$, so let's take the minimum m out of it. Applying this process recursively, we obtain a infinite convergent subsequence(?). I think intuitively, by only repeating this process we can't obtain countable infinite terms of subsequence because we have to repeat infinite times. Evaluating $\int_0^\infty \frac{dx}{1+x^4}$. Can anyone give me a hint to evaluate this integral?  $$\int_0^\infty \frac{dx}{1+x^4}$$  I know it will involve the gamma function, but how? Can we build capacitors on a PCB board?