When I have data which has one dependent and one independent variable, I can clearly explain it in cartesian coordinate system  here is a regression line when my $y$ approximately equals to $2x + 3$ but what about a second variable ? what if my model was $y = 2x + z +3 $ ? would visualizing it be only possible with a three-dimensional coordinate system ? if it is, what about four and more ? My question might be a bit pointless, thanks in advance.
I am currently writing a paper with several multiple regression analyses. While visualizing univariate linear regression is easy via scatter plots, I was wondering whether there is any good way to visualize multiple linear regressions?   I am currently just plotting scatter plots like dependent variable vs. 1st independent variable, then vs. 2nd independent variable, etc. I would really appreciate any suggestions.
Why is my StackExchange OpenID so hard to find? Find the number of solutions of $x^2=x\mod m$ for any integer $m$.  I was thinking about bringing the $x$ over to the other side and somehow apply chinese remainder thm. Do you get Hex Warrior's Cha bonus with a two-handed Pact Weapon? Hex Warrior says:     you can touch one weapon with which you are proficient and that lacks the two-handed quality. When you attack with that weapon you can use your Charisma for attack and damage rolls.   later the ability says:     This benefit extends to every weapon conjured with the Pact of Blades feature, no matter the type.   (emphasis mine)  Does this allow me to use my charisma with 2 handed weapons(provided it is my Pact Weapon and my Hex Warrior Blade)?  This question is not the same as . That one is asking how many weapons, I'm asking if type means 2 handed or something else(Slashing or Ranged for examples). At the start of the movie it says:      "A long time ago in a galaxy far, far away...."   Does this mean that the entire Star Wars saga takes place in a single galaxy? To my understanding a galaxy isn't very big compared to the universe itself. Star Wars has many different sentient races, and the way they are presented suggests that most planets are host to only one of them (some exceptions like naboo). This strongly leads me to believe there should actually be more galaxies in the Star Wars universe (hehe).   Another thing, I think it's safe to assume that even if the entire saga takes place in one single galaxy, their researchers can observe other galaxies (I think we started doing that in the 1600s, and they are a lot more technologically advanced than we were back then). When Obi-wan is looking for Kamino (the rain planet with the cloners) the librarian says something like "If it's not in our records, it doesn't exist", so she must be convinced that the entire universe has been explored.    If the entire universe has been documented, why wouldn't they go to those other galaxies? The technology seems to be advanced enough to hop to another galaxy. Is there any canon material that suggest that traveling between galaxies is done? If not, is there a canon explanation why they don't? How can I get the Cartesian product (every possible combination of values) from a group of lists?  Input:   somelists = [    [1, 2, 3],    ['a', 'b'],    [4, 5] ]   Desired output:  [(1, 'a', 4), (1, 'a', 5), (1, 'b', 4), (1, 'b', 5), (2, 'a', 4), (2, 'a', 5) ...] Why does the review system disable the "First Posts" and "Late Answers" if I reached daily upvote limit? I was reviewing on , and I faced a weird thing. I can review "Close Votes", "Low Quality Posts", and "Suggested Edits" if I reached my daily vote limit, but I can't for "First Posts" and "Late Answers". Why?    If they just deserve an upvote/downvote then they should not include in the review system. But if we can improve them through leaving comments and by editing the posts then why can't we access them? Display the Number of Comment Upvotes In Recent Activity Pages It would be helpful if we could see the number of upvotes cast for our comments, or other people's comments in response to us directly in the user profile in either the activity tab or the response tab respectively. It should be quite easy to implement (e.g. sharing a single partial view for displaying comments).  It could be useful, since we can evaluate the legitimacy of the comment and address horrible mistakes sooner. As we know, any proposition can translate to Godel numbers, such as  $$( \exists x)\space (x=sy)$$Its Godel numbers is $2^8 × 3 ^4 × 5^ {13} × 7^ 9 × 11^8 × 13^{13} × 17^ 5 × 19^ 7 × 23^ {17} × 29^ 9=a \space big \space number $  We also could get the character sequence "$( \exists x)\space (x=sy)$" through prime factorization of this big number.  Such as$$243,000,000=2^6×3^5×5^6 $$ $$6\space\space\space\space\space 5\space\space\space\space\space 6 $$ $$\downarrow\space\space\downarrow\space\space\downarrow$$ $$0\space\space\space=\space\space\space 0 $$ So $243000000$ is the Godel numbers of "$0=0$"   There are four axioms in Godel's PM system, here are two of them: $$(p\lor p)\subset p$$ $$p\subset (p\lor q)$$ through these two axioms, we can prove "$(p\lor p)\subset(p\lor q)$" is a true theorem.   And these two axioms and a theorem all have a corresponding Godel numbers, can we prove this theorem is true through Godel numbers? If it works, how? I'm trying to understand at a deeper level the ubiquity of log-likelihood (and perhaps more generally log-probability) in statistics and probability theory.  Log-probabilities show up all over the place: we usually work with the log-likelihood for analysis (e.g. for maximization), the Fisher information is defined in terms of the second derivative of the log-likelihood, entropy is an expected log-probability, Kullback-Liebler divergence involves log-probabilities, the expected diviance is an expected log-likelihood, etc.    Now I appreciate the many practical and convenient reasons.  Many common and useful pdfs are from exponential families, which leads to elegantly simplified terms when log-transformed.  Sums are easier to work with than products (esp for differentiating).  Log-probs have a great floating point advantage over straight probs.  Log-transforming a pdf often converts a non-concave function into a concave function.  But what is the theoretical reason/justification/motivation for log-probs?  As an example of my perplexity, consider the Fisher information (FI).  The usual explanation for intuiting the FI is that the second derivative of the log-likelihood tells us how "peaked" the log-likehood is:  a highly peaked log-likelihood means the MLE is well-specified and we are relatively sure of its value, while a nearly flat log-likehood (low curvature) means many different parameter values are nearly as good (in terms of the log-likelihood) as the MLE, so our MLE is more uncertain.   This is all well-and-good, but isn't it more natural to just find the curvature of the likelihood function itself (NOT log-transformed)?  At first glance the emphasis on the log-transform seems arbitrary and wrong.  Surely we are more interested in the curvature of the actual likelihood function.  What was Fisher's motivation for working with the score function and the Hessian of the log-likelihood instead?  Is the answer simply that, in the end, we have nice results from the log-likelihood asymptotically? E.g., Cramer-Rao and normality of the MLE/posterior.  Or is there a deeper reason? Can I change the color of the last selected face? Can I change the color of the last selected face, so that it is the same as the other selected faces?  Here, I've selected the left face then the right face. Note the right face is some sort of checkered or dithered pattern, and just the outline has the nice orange 'selected' color.    Here I've selected the top face, note the right face has now the nice color.    And here I've deselected the top face. Now, both of the remaining selected faces have the nice color. This is the result I'd rather see instead of image 1. Inverting direction of mouse scroll wheel Is there a way to make a mouse scroll wheel invert its direction? I'd like to scroll upwards and have that action scroll downwards and vice-versa. There is no setting on the Mouse control panel that makes this possible.   Any pointers to a hack or a particular mouse model that has such a setting would be appreciated. I am using Windows 7. Why is there no absolute maximum temperature? If temperature makes particles vibrate faster, and movement is limited by the speed of light, then I would assume that temperature must be limited as well. Why is there no limit? Proving a function is Lipschitz continuous Show that the following function is Lipschitz continuous and find a Lipschitz constant  $$y\mapsto f(x,y)\\ f(x,y)=\frac{y}{x}\ln(\frac{y}{x})\text{  , } |x-1|\leq\frac{1}{2}\text{  , } |y-e|\leq\frac{e}{2} $$  I have no clue on where to begin to prove this.   My first question is how I should interpret this function, is $x$ a sort of constant? How do I make a table row on a web page look draggable? I have a table on a web page where I need to let the user reorder the rows. I was thinking of letting the user drag the rows to reorder them. But how would I let the user know that he could do that? Should I make an empty cell as the first column with dots or vertical lines like gmail has for each email in the inbox? Where can I get an image like that? Do I use a "grabbing hand" icon for the cursor or a "move" cursor? Do I show the cursor while hovering over the table or just the "grab area"? Do I let the user grab any part of the row to drag? Many anime do not follow the manga history and are made shorter. However there are a few that do follow it or try to finish the history with the manga like the big three (Naruto, Bleach, One Piece), Death Note, and FMA Brotherhood. But others finish the anime without the real ending like FMA, Ouran High School, Pandora Hearts, Tsubasa Reservoir Chronicle. Why is this so? Is it because of the high cost? Or the lack of popularity? I'll preface this by saying I'm very new to Blender - I'm trying to put a bevel on an edge, looks something like this:    As you can see, the vertical side is much longer than the horizontal side.  When I apply to bevel to the edge there, the bevel stretches proportionally - so there's 'more' bevel on the longer side.  I want to bevel without any 'ratio' like that - just a 45-degree bevel, if that makes any sense.  Do I have to put in a loop cut to make the side lengths equal?  I tried using a sub-surf modifier, but it applies on a bunch of nearby edges, when I just want to round this one corner. How to make a file executable? How can I make my file so that I can double click on it, and it runs. It is a .sh script, but I also have files that say:  executable (application/x-executable)  in the description of what they are.  I can't run any of these from terminal, or by double clicking.  If possible, I would like a way using either the GUI or a Terminal, but not a combination of the two.  Here is a screenshot of what I see when I right click then go on properties. The file first:    And the shell script here:    NB: I accept that this is a duplicate (I was looking for it, and couldn't find it, so asked + answered it, hoping that I would find it) however, I don't think the question about .desktop files is a duplicate. Question: Are there infinitely many primes of the form $4n+3$ and $4n+1$?  My attempt: Suppose the contrary that there exist finitely many primes of the form $4n+3$, say $k+1$ of them: $3,p_1,p_2,....,p_k$  Consider $N = 4p_1p_2p_3...p_k+3$, $N$ cannot be a prime of this form. So suppose that $N$=$q_1...q_r$, where $q_i∈P$  Claim: At least one of the $q_i$'s is of the form $4n+3$:  Proof for my claim: $N$ is odd $\Rightarrow q_1,...,q_r$ are odd $\Rightarrow q_i \equiv 1\ (\text{mod }4)$ or $q_i ≡ 3\ (\text{mod }4)$  If all $q_1,...q_r$ are of the form $4n+1$, then $(4n+1)(4m+1)=16nm+4n+4m+1 = 4(\cdots) +1$  Therefore, $N=q_1...q_r = 4m+1$. But $N=4p_1..p_k+3$, i.e. $N≡3\ (\text{mod }4)$, $N$ is congruent to $1\ \text{mod }4$ which is a contradiction.  Therefore, at least one of $q_i \equiv 3\ (\text{mod }4)$. Suppose $q_j\equiv 3\ (\text{mod }4)$  $\Rightarrow$ $q_j=p_i$ for some $1\leq i \leq k$ or $q_j =3$  If $q_j=p_i≠3$ then $q_j$ | $N = 4p_1...p_k + 3 \Rightarrow q_j=3$ Contradiction!  If $q_j=3$ ($\neq p_i$, $1\leq i \leq k$) then $q_j | N = 4 p_1...p_k + 3 \Rightarrow q_j=p_t$ for some $1 \leq i \leq k$ Contradiction!  In fact, there must be also infinitely many primes of the form $4n+1$ (according to my search), but the above method does not work for its proof. I could not understand why it does not work. Could you please show me?  Regards I am creating a model of a glass and I would like to put a drawing on it (a png picture).  I know how to do it using Blender Render, but not with Cycles Render. Help :) Device that makes metal malleable