I want to create a new matrix environment with different spacing for a matrix instead of manually typing arraystretch locally for each matrix. I am new to more advanced LaTeX coding, and have not quite figured this out. I thought that I could just do \newenvironment{dbmatrix}{     \renewcommand*{\arraystretch}{1.5}     \ensuremath{\begin{bmatrix}  \end{bmatrix}} }}  But then, after searching through Stack, I thought maybe I need to use \bgroup,\egroup or \makeatletter, \makeatother type commands, but have not figured out how these commands really work and failed using them. It would also be great to have maybe an additional environment that took an argument to adjust \arraystretch, like \begin{dbmatrix}{1.5} \end{dbmatrix}. Would love some help and suggestions.
I'm trying to create a pmatrix whose components are rather complicated fractions (Christoffel symbols), and the line spacing is too small making the whole thing a bit cramped and hard to read.  Is there any easy way to increase the spacing (about 1.5 the default would be perfect)?
Zsh get terminfo keys with modifiers Normality of dependent variable = normality of residuals? This issue seems to rear its ugly head all the time, and I'm trying to decapitate it for my own understanding of statistics (and sanity!).   The assumptions of general linear models (t-test, ANOVA, regression etc.) include the "assumption of normality", but I have found this is rarely described clearly.   I often come across statistics textbooks / manuals / etc. simply stating that the "assumption of normality" applies to each group (i.e., categorical X variables), and we should we examining departures from normality for each group.   Questions:     does the assumption refer to the values of Y or the residuals of Y?  for a particular group, is it possible to have a strongly non-normal distribution of Y values (e.g., skewed) BUT an approximately (or at least more normal) distribution of residuals of Y?   Other sources describe that the assumption pertains to the residuals of the model (in cases where there are groups, e.g. t-tests / ANOVA), and we should be examining departures of normality of these residuals (i.e., only one Q-Q plot/test to run).  does normality of residuals for the model imply normality of residuals for the groups? In other words, should we just examine the model residuals (contrary to instructions in many texts)?  To put this in a context, consider this hypothetical example:   I want to compare tree height (Y) between two populations (X).   In one population the distribution of Y is strongly right-skewed (i.e., most trees short, very few tall), while the other is virtually normal  Height is higher overall in the normally distributed population (suggesting there may be a 'real' difference). Transformation of the data does not substantially improve the distribution of the first    population.  Firstly, is it valid to compare the groups given the radically different height distributions? How do I approach the "assumption of normality" here? Recall height in one population is not normally distributed. Do I examine residuals for both populations separately OR residuals for the model (t-test)?     Please refer to questions by number in replies, experience has shown me people get lost or sidetracked easily (especially me!). Keep in mind I am not a statistician; though I have a reasonably conceptual (i.e., not technical!) understanding of statistics.  P.S., I have searched the archives and read the following threads which have not cemented my understanding: Sum of two squares modulo p I have heard somewhere that for all primes $p$, for all $k$, there exist $x, y$ s.t. $x^2 + y^2\equiv k \pmod{p}$? I recall that the proof is very elementary, but I can't remember such a proof. How would one prove this? One way is to use Cauchy-Davenport, but I don't think that this is the simplest solution. Can't open snap-store A Light is shone out the side window of a spaceship ( moving close to light speed ), will the light still travel at the speed of light in the B axis? I have a dataframe with many observations and many variables. Some of them are categorical (unordered) and the others are numerical.  I'm looking for associations between these variables. I've been able to compute correlation for numerical variables (Spearman's correlation) but :   I don't know how to measure correlation between unordered categorical variables. I don't know how to measure correlation between unordered categorical variables and numerical variables.   Does anyone know how this could be done? If so, are there R functions implementing these methods? Contraction mapping in the context of $f(x_n)=x_{n+1}$. Should Indicate which Positions I've Already Applied For If I push or hit an object in space will it rotate or move along a straight line? If I push or hit an object in space (vacuum and no gravitation) in direction what is not going trough its centroid, will it rotate or move along in straight line?   I expect that on earth it will depend on what is less difficult for the object (rotation or linear movement). So the object will do some kind of combination of both movements (rotating and also moving along the direction of impulse or force).  But how could an object "decide" what to do in space, where is not resistance? What is a buffer overflow and how do I cause one? I have heard about a buffer overflow and I would like to know how to cause one.  Can someone show me a small buffer overflow example? New(And what they are used for?) Can a wizard using an arcane focus quarterstaff and a shield still cast somatic component spells? If there is a wizard that, by some means, has acquired proficiency in the use of shields, can said wizard use his shield in one hand and his arcane focus quarterstaff in the other and still cast somatic component spells?  Since the quarterstaff has been crafted to also be an arcane focus, and the Sage Advice Compendium states that you can provide the somatic components of spells with the same hand wielding the arcane focus, in this case an arcane focus quarterstaff, could that wizard still be able to perform somatic component spells while using a shield in the other?  I understand War Caster is the feat that allows you to cast somatic component spells even while using a shield and weapon or two weapons in each hand, but is this a simple way around those rules seeing as this "weapon" is your arcane focus? I would like to believe it is not, but I'm not fully fluent in the ways of spellcasting. Ok, so I have an index.php file which has to process many different file types.  how do I guess the filetype based on the REQUEST_URI.  If I request http://site/image.jpg, and all requests redirect through index.php, which looks like this  &lt;?php    include('/www/site'.$_SERVER['REQUEST_URI']); ?&gt;   How would I make that work correctly?  Should I test based on the extension of the file requested, or is there a way to get the filetype? Let $f$ be a continuous function on a set $E$.  Is it always true that $f^{-1}(A)$ is always measurable if $A$ is measurable?  I say no.  We know that $\bar{\psi(x)}:=\frac{\phi(x)+x}{2}$ where $\phi$ is the Cantor (or Cantor-Lebesgue) function and $\bar{\psi}:[0,1] \rightarrow [0,1]$.  We know this function maps a measurable subset of $C$ (the Cantor set) into a non-measurable set $W$.  A few observations about $\bar{\psi(x)}$.  It is strictly increasing and continuous, so it has a continuous inverse $\bar{\psi(x)}^{-1}$.  Thus $\bar{\psi(x)}^{-1}(W)=c \subset C$; that is, $\bar{\psi(x)}^{-1}(W)$ is non-measurable, but c is.  Is this correct? Does %* in batch file mean all command line arguments? What is the proper way to explain the twin paradox? The paradox in the  is that the situation appears symmetrical so each twin should think the other has aged less, which is of course impossible.  There are a thousand explanations out there for why this doesn't happen, but they all end up saying something vague like it's because one twin is accelerating or you need general relativity to understand it.  Please will someone give a simple and definitive explanation for why both twins agree on which twin is younger when they meet for the second time? Why "?v=hash" is added to my URLs? I'm new to wordpress, I set up everything, but there's something that bothers me: on every single URL or link, there's a "?v=hash" appended everywhere (example.com/?v=d21feabed96b).  I tried to see inspect every plugin, I don't understand how this parameter is added.  It looks like it's added in js, because if I see the source, there's no trace of this hash, but I can see in firebug live source.  Plugins installed:   jetpack sumome wp-piwik wp super cache woocommerce   Theme: Storefront  I tried to disable everything, but still I get this hash, does anyone knows how and why it's added?  I'm also using cloudflare Numerical Approximation of Differential Equations with Midpoint Method A question about how to express a fraction as ${1\over q_1}+{1\over q_2}+ \cdots+{1\over q_N}$ Let $x$ be a positive rational number, strictly between $0$ and $1$. Prove that there is a finite strictly increasing list of positive integers $2 \leq q_1&lt;q_2&lt;\cdots&lt;q_N $ such that $$x={1\over q_1}+{1\over q_2}+ \cdots+{1\over q_N}.$$  I have tried many methods, such as mathematical induction. I know it is obvious that $\frac{1}{m}$ is right for assumption, but when I begin to prove $\frac{2}{m}$, I feel it is hard to find some patterns for $\frac{2}{m}$. Thus, maybe my thoughts are wrong.  Also, I still tried to change the fraction so that the numerator will be smaller and smaller. But I still cannot find a way to lower the numerator. Can someone help me solve the question? Or, can someone give me some hints. I will appreciate you very much! How to print "$" in here-document How can I echo "$" in a here-doc in bash?  For example, I want to have a file with the content on remote server like $ABC=home_dir.  $ ssh hostname sudo -s &lt;&lt;EOF echo "$ABC=home_dir" &gt; file EOF   But it would be treated as a variable. How can I print a literal $? Prove : $p$(n│even number of ODD parts)=$p$(n│distinct parts ,number of ODD parts is even )