How we show that $Z := \{(x, y, z) ∈ R^3 | x^2 + y^2 − z^2 = 0\}$ is not a manifold (even it is not a topology manifold)
How do we prove that this set is not manifold?
Resizing Controls in MFC I am writing a program which has two panes (via CSplitter), however I am having problems figuring out out to resize the controls in each frame. For simplicity, can someone tell me how I would do it for a basic frame with a single CEdit control?   I'm fairly sure it is to do with the CEdit::OnSize() function... But I'm not really getting anywhere...  Thanks! :) I don't want to watch the prisoner cart ride to the execution: is it possible to skip this?   If so, how? Can I replace the intro movie file by some empty video? If $\sum_{1}^{\infty}(a_n)^3$ diverges, does $\sum_{1}^{\infty}(a_n)$? How to determine the encoding of text? I received some text that is encoded, but I don't know what charset was used. Is there a way to determine the encoding of a text file using Python?  deals with C#. strcmp on a line read with fgets Quick question about curly braces not showing up Ii am writing a short code, so I doubt anything else in the code is causing the problem. This is the problematic line of code:  \[ M_R := \max {|f(z)|: |z - z_0| = R }\le M\]   It doesn't show the curly braces. I've tried using \left{ and \right} but to no avail.  What should I do?  Thank you How can I use an array of function pointers? How should I use array of function pointers in C?  How can I initialize them? RegEx for matching UK Postcodes Why a backslash displays as "W"+strikethrough in Netbeans? I'm facing a strange problem with Netbeans. In the editor, when I type a backslash(), I see a W with a strikethrough (as shown in the attached image).  So the c:\\abc is displaying as:    But, when I copy and paste that string into notepad, it gets pasted normally (i.e. c:\\abc).  Has anyone faced this problem before? How to fix it? Passive voice phrasing of "The bottle contains two liters" I want to write a function in Python that returns different fixed values based on the value of an input index.    In other languages I would use a switch or case statement, but Python does not appear to have a switch statement.  What are the recommended Python solutions in this scenario? How to prove that for all natural numbers, $4^n > n^3$? This is a problem set I have, it's not a homework but it's very important practice...  Send me some hints please, I don't want an answer I need to get it by myself but I'm failing miserably...   The problem is:     Prove that for all natural numbers $n$,  $4^n &gt; n^3$ unbalaced data set in classification tasks Render water-paint in iOS Comparing mixed-effects and fixed-effects models (testing significance of random effects) Given three variables, y and x, which are positive continuous, and z, which is categorical, I have two candidate models given by:  fit.me &lt;- lmer( y ~ 1 + x + ( 1 + x | factor(z) ) )   and  fit.fe &lt;- lm( y ~ 1 + x )   I hope to compare these models to determine which model is more appropriate. It seems to me that in some sense fit.fe is nested within fit.me. Typically, when this general scenario holds, a chi-squared test can be performed. In R, we can perform this test with the following command,  anova(fit.fe,fit.me)   When both models contain random-effects (generated by lmer from the lme4 package), the anova() command works fine. Owing to boundary parameters, it is normally advisable to test the resulting Chi-Square statistic via simulation, nonetheless, we can still use the statistic in the simulation procedure.  When both models contain only fixed-effects, this approach---and, the associated anova() command---work fine.  However, when one model contains random effects and the reduced model contains only fixed-effects, as in the above scenario, the anova() command doesn't work.  More specifically, I get the following error:   &gt; anova(fit.fe, fit.me)  Error: $ operator not defined for this S4 class   Is there anything wrong with using the Chi-Square approach from above (with simulation)? Or is this simply a problem of anova() not knowing how to deal with linear models generated by different functions?  In other words, would it be appropriate to manually generate the Chi-Square statistic derived from the models? If so, what are the appropriate degrees of freedom for comparing these models? By my reckoning:  $$ F = \frac{\left((SSE_{reduced}-SSE_{full})/(p-k)\right)}{\left((SSE_{full})/(n-p-1)\right)} \sim F_{p-k,n-p-1} $$  We are estimating two parameters in the fixed effects model (slope and intercept) and two more parameters (variance parameters for the random slope and random intercept) in the mixed-effects model. Typically, the intercept parameter isn't counted in the degrees of freedom computation, so that implies that $k=1$ and $p=k+2=3$; having said that I'm not sure if the variance parameters for the random-effects parameters should be included in the degrees of freedom computation; the variance estimates for fixed-effect parameters are not considered, but I believe that to be because the parameter estimates for fixed effects are assumed to be unknown constants whilst they are considered to be unknowable random variables for mixed effects. I would appreciate some assistance on this issue.  Finally, does anybody have a more appropriate (R-based) solution to comparing these models? How do I turn on workspaces? (Why do I only have one workspace?) What is a word that could define someone who likes to cause conflict? What's the difference between $\mathbb{R}^2$ and the complex plane? I haven't taken any complex analysis course yet, but now I have this question that relates to it.  Let's have a look at a very simple example. Suppose $x,y$ and $z$ are the Cartesian coordinates and we have a function $z=f(x,y)=\cos(x)+\sin(y)$. However, now I change the $\mathbb{R}^2$ plane $x,y$ to complex plane and make a new function, $z=\cos(t)+i\sin(t)$.  So, can anyone tell me some famous and fundamental differences between complex plane and $\mathbb{R}^2$ by this example, like some features $\mathbb{R}^2 $ has but complex plane doesn't or the other way around. (Actually I am trying to understand why electrical engineers always want to put signal into the complex numbers rather than $\mathbb{R}^2$, if a signal is affected by 2 components)   Thanks for help me out! A great Python IDE for Ubuntu Possible Duplicate:          I want to develop Python apps in Ubuntu, and I'm tired of working with gedit one time and another.  So I'm trying to find a great Python IDE (a true IDE, not just a syntax-highlighting editor) on the Ubuntu repositories. Which one do you recommend me? Let $A,B \subseteq \mathbb{R}^d$ be non-empty sets. Define their distance to be $$ d(A,B) = \inf \{ \|x-y\| : x \in A, \; \; y \in B \} $$ For any $A,B$, I want to prove that $d(A,B) = d( \overline{A}, \overline{B} )$. My Attempt Put $\alpha = d(A,B)$. Therefore, $\alpha = \|x_0 - y_0 \|$ for some $x_0 \in A$ and some $y_0 \in B$. But, notice $x_0 \in \overline{A}$ and $y_0 \in \overline{B}$ by definition. hence $$ \| x_0 - y_0\| \geq \inf\{ \|x' - y'\| : x' \in \overline{A}, \; \; y' \in \overline{B} \} =  d( \overline{A}, \overline{B} )$$ $$\therefore d(A,B) \geq d( \overline{A}, \overline{B} )$$ I am stuck trying to show the other direction: $d(A,B) \leq d( \overline{A}, \overline{B} )$ Can someone help me? thanks a lot