Square matrix A is singular $\iff \operatorname{rank}(A)<n$?
A square matrix $n \times n$ is an invertible matrix iff the rank of the matrix is $n$.
I'm fitting a multiple linear regression model between 4 categorical variables (with 4 levels each) and a numerical output. My dataset has 43 observations.  Regression gives me the following $p$-values from the $t$-test for every slope coefficient: $.15, .67, .27, .02$. Thus, the coefficient for the 4th predictor is significant at $\alpha = .05$ confidence level.  On the other hand, the regression gives me a $p$-value from an overall $F$-test of the null hypothesis that all my slope coefficients are equal to zero. For my dataset, this $p$-value is $.11$.  My question: how should I interpret these results? Which $p$-value should I use and why? Is the coefficient for the 4th variable significantly different from $0$ at the $\alpha = .05$ confidence level?  I've seen a related question, , but there was an opposite situation: high $t$-test $p$-values and low $F$-test $p$-value. Honestly, I don't quite understand why we would need an $F$-test in addition to a $t$-test to see if linear regression coefficients are significantly different from zero. Is it possible to create 3D red/cyan anaglyphs in a hand drawing? There are plenty tutorials online on how to create 3D red/cyan anaglyphs from existing photos in Photoshop or similar software. However, I'm wondering if it is possible to draw a 3D image by hand on for example a whiteboard using markers.  My plan is to tape together a red and blue marker and draw with it using the same rotation/inclination of the marker across the drawing. Will this give a 3D effect when viewed through red/cyan glasses? Or is the 3D anaglyph method more complex than just offsetting red and cyan lines? How can I distribute a native executable for a Python program? Lately all of my work has been in Python (and Ruby), and I'm in love with those two languages. There are many libraries for game development on Python but the problem I see is that many of those frameworks don't cover the "distribution" aspect of game creation. I'd like to code my game on Python (and any other pure Python extra libraries I need) and be able to distribute an executable for the main PC platforms (Windows, Mac and Linux).  Any options out there or what are the best combinations for this? (maybe pyglet + cx_freeze) To show that $\sum a_nb_n$ is absolutely convergent Assume that $\sum a_n$ is convergent and $\sum b_n$ is absolutely convergent.To show that $\sum a_nb_n$ is  absolutely convergent  My try::Consider the sequence of partial sums of $\sum |a_nb_n|$  $S_n=|a_1b_1|+|a_2b_2|+......+|a_nb_n|$  Now $(a_1b_1+a_2b_2+......+a_nb_n)\leq ( (a_1^2+a_2^2+..+a_n^2))^\frac{1}{2}((b_1^2+b_2^2+..+b_n^2)^\frac{1}{2})$  How to proceed from here? Another problem from quora.  What can be said about $v =\prod_{s=2}^{\infty} \zeta(s) $?  Wolfy says that $v \approx 2.294856591673313794183 $.  The Inverse Symbolic Calculator () finds nothing.  Here are the close values:  $2294856391585950 = (0064) sum((7/6*n^3-3*n^2+65/6*n-3)/(Fibo(n)+1),n=1..inf)\\ 2294856397653578 = (0001) GAM(5/6)*BesI(1,2)^{GAM(7/12)}\\ 2294856469349552 = (0006) 1/4357571\\ 2294856473388934 = (0324) 1/10*(10+14^{1/2}*10^{1/4})^{1/2}*10^{3/4}\\ 2294856488490537 = (0001) BesJ(0,1)^{GAM(1/3)/GAM(5/12)}\\ 2294856524215021 = (0314) sin(Pi*8/45)-sin(Pi*14/51)\\ 2294856591673313 \text{ would be here}\\ 2294856740350177 = (0011) sum((2*n^3-5*n^2+22*n-14)/(n!+2),n=1..inf)\\ 2294856991319600 = (0324) 23^{1/2}/(12^{3/4}-3^{2/3})^{1/2}\\ 2294857017919943 = (0001) Feig2*Li4(1/2)/GAM(5/6)\\ 2294857089879260 = (0248) F(11/26;27/50;1) \\ 2294857493303450 = (0404) Psi(1/21)+Psi(19/21)+Psi(13/14)\\ $  I tried using Euler's zeta product and partition sum, but this didn't seem to help. Here is what resulted:  $\zeta(s) =\dfrac1{\prod_p (1-p^{-s})} $.  $\begin{array}\\ v &amp;=\prod_{s=2}^{\infty} \zeta(s)\\ &amp;=\prod_{s=2}^{\infty} \dfrac1{\prod_p (1-p^{-s})}\\ &amp;=\prod_{s=2}^{\infty} \prod_p\dfrac1{ (1-p^{-s})}\\ &amp;= \prod_p\prod_{s=2}^{\infty}\dfrac1{ (1-p^{-s})}\\ \end{array} $  Euler's product identity states that $\prod_{n=1}^{\infty}\dfrac1{1-zx^n} =\sum_{n=1}^{\infty} \dfrac{x^nz^n}{\prod_{k=1}^n(1-x^k)} $. Putting $x=z=\dfrac1{p} $, $\prod_{n=2}^{\infty}\dfrac1{1-p^{-n}} =\sum_{n=1}^{\infty} \dfrac{p^{-2n}}{\prod_{k=1}^n(1-p^{-k})} $.  That's as far as I can go. What characters are valid in a Java class name? What other rules govern Java class names (for instance, Java class names cannot begin with a number)? How to negate a specific symbol? I want to negate the symbol whose code is \models (this would be specific to Formal Logic). How can I do that? I tried a number of combination, but didn't find how this could be done. I tried \neqmodels, \nemodels, \notmodels, \nmodels  but all of them are wrong.  Is there a general way to negate symbols? If $F: \mathbb R^2 \to \mathbb R$ and $F_x$ (partial derivative of $F$ wrt $x$) and $F_y$ exist at $(x_0,y_0)$ then the function is continuous at that point. Is this true? If not what could be a counter-example? Can someone please explain how I can install  in the following ways?   By using the package manager? By downloading and installing a pre-built package? Why are we in love "with" someone? I have some auto-generated T-SQL, which is probably valid, but I don't really understand.  ALTER TABLE [dbo].[MyTable] WITH CHECK CHECK CONSTRAINT [My_FORIEGN_KEY];   I know what a foreign key constraint is, but what's the CHECK CHECK? Semigroup analogue to the classification of finite simple groups? Is there a semigroup analogue to the classification of finite simple groups?  If so what are some of the major results?  --Edit, reference links--- I know roughly and informally what a confidence interval is.  However, I can't seem to wrap my head around one rather important detail:  According to Wikipedia:     A confidence interval does not predict that the true value of the parameter has a particular probability of being in the confidence interval given the data actually obtained.     I've also seen similar points made in several places on this site.  A more correct definition, also from Wikipedia, is:     if confidence intervals are constructed across many separate data analyses of repeated (and possibly different) experiments, the proportion of such intervals that contain the true value of the parameter will approximately match the confidence level   Again, I've seen similar points made in several places on this site.  I don't get it.  If, under repeated experiments, the fraction of computed confidence intervals that contain the true parameter $\theta$ is $(1 - \alpha)$, then how can the probability that $\theta$ is in the confidence interval computed for the actual experiment be anything other than $(1 - \alpha)$?  I'm looking for the following in an answer:   Clarification of the distinction between the incorrect and correct definitions above. A formal, precise definition of a confidence interval that clearly shows why the first definition is wrong. A concrete example of a case where the first definition is spectacularly wrong, even if the underlying model is correct. Group cohomology intuition Group cohomology is one thing I don't seem to get my head around.  I understand $H^0$ as being the "fixed points" but when it comes to anything higher I have no idea what the notions are meant to be capturing...if anything.  For example I can see that 1-cocycles are homomorphisms with a specific property but that property doesn't seem to mean anything to me...and I think it should!  Is there an intuition or example that explains the meaning or is it just something that "stuck" and seems to work in lots of places? Is there a name based virtual host SSH reverse proxy? I've grown quite fond of HTTP reverse proxies in our development environment and found the DNS based virtual host reverse proxy quite useful. Having only one port (and the standard one) open on the firewall makes it much easier for management.  I'd like to find something similar to do SSH connections but haven't had much luck. I'd prefer not to simply use SSH tunneling since that requires opening port ranges other than the standard. Is there anything out there that can do this?  Could HAProxy do this? Is there console mode in Mac OS X? Which computer science / programming Stack Exchange sites do I post on? Problem with solving systems of linear congruence of two variables The problem I am given is finding a solution to the following pair of equations:  $3x + 4y$ $\equiv$ $5$ $mod$ $13$  $2x + 5y$ $\equiv$ $7$ $mod$ $13$  By reading the methodology here:   Since the modulos in my problem are also the same, I am able to use substitution, but this leaves fractions in my expressions.  I then used matrix multiplication as follows:  $\begin{bmatrix} 3 &amp; 4\\ 2 &amp; 5\end{bmatrix}$ $\begin{bmatrix}\ x\\ y\end{bmatrix}$ = $\begin{bmatrix}\ 5\\ 7\end{bmatrix}$   And solving this also gives me a matrix with fractions, so I am a bit lost on how to solve this problem.  Any help is appreciated. Thank you very much. Differences between the introduction of a sandwich thesis and a paper I would like to know what are the main differences between the introductory part of a paper and of a thesis. My thesis would be a sandwich type, so I would have to wrap everything there.  Some of the differences are:   Longer in thesis. Deeper in thesis. Extended definition of key terms, concepts and methodology in thesis. Clear definition of the whole goal in thesis, whereas paper smaller goal.   To narrow the possible answers, I do research on the molecular aspects of the infection steps of a pathogenic swine bacteria. Also develop strategies to block infection by using vaccines. How to bond/connect wires to neodymium magnets? After reading , I learned it is not advisable to solder a wire to neodymium magnets.  After more reading I found  post describing the use of conductive epoxy to bond electrical wires to neodymium magnets, a very messy process which takes several hours of waiting.  I want to ask this community if they have or suggest better methods to bond/connect a wire to a neodymium magnet to act as a conductive connection?