How do get rid of (1 not defined because of singularities) in R?
Qualitative variable coding in regression leads to "singularities"
Axiom of Choice: Where does my argument for proving the axiom of choice fail? Help me understand why this is an axiom, and not a theorem. In terms of purely set theory, the axiom of choice says that for any set $A$, its power set (with empty set removed) has a choice function, i.e. there exists a function $f\colon \mathcal{P}^*(A)\rightarrow A$ such that for any subset $S$ of $A$, $f(S)\in S.$ Is this correct?  My question then is about proving this fact, so that we do not need to put it as an axiom. Now as per the research done on this single object- Axiom of Choice, I believe here that there should be some falsity in my argument. I do not find the mistake.  For any $S\in \mathcal{P}^*(A)$, since $S\neq \emptyset$, $\exists s\in S$. Define $f(S)=s$. Then $f$ is a choice function.  This was showing me that the axiom of choice is proved, but then why it had been put as an axiom? For example, in  book, the author asserts that      It is a metatheorem of mathematical logic that it is impossible to specify the function that assigns to each non-empty subset of $\mathbb{R}$, an element of itself.     There are several notes and books on axiom of choice, but here I am trying to understand through doing some argument for some problem, where problem actually arises. How exactly to compute the ridge regression penalty parameter given the constraint? The accepted answer in  does a great job of showing that there is a one-to-one correspondence between $c$ and $\lambda$ in the two formulations of the ridge regression: $$ \underset{\beta}{min}(y-X\beta)^T(y-X\beta) + \lambda\beta^T\beta $$ and  $$ \underset{\beta}{min}(y-X\beta)^T(y-X\beta) \text{ s.t. }\beta^T\beta\leq{c} $$  The linked answer shows this in the orthogonal case.    In a general (non-orthogonal case), how can I compute $\lambda$ from $c$?  Update Here is an answer for going from $\lambda$ to $c$:   Assuming that the coefs are constrained by the penalty, $$ \beta^T\beta = c $$ and  $$ \beta = (X^TX + \lambda I)^{-1}X^Ty\\ \beta^T\beta = c = \beta^T(X^TX + \lambda I)^{-1}X^Ty $$  Still working on going the other way Orbiting a specific point - Orbiting Camera How would I create a camera that with the mouse, it allows you to orbit around a specifc position, with the camera always facing that point? I would also like it to be able to always be at a constant distance from the point, and be able to orbit around it in every direction.   Any ideas? Should we write “allow” or ”allows” in “require that the class allow[s]”? Can you explain to me whether I should or should not use ‑s at the end of the  verb allow in this sentence?     This function requires that the container class allow[s] random access   The sentence in the book I read wrote allow without ‑s. However, the container class is a third-person singular, so the ‑s must be required. Is that right? How can I tell if an ETF is expensive? For stocks, I can easily look at the price/earnings ratio, and, according to The Intelligent Investor, stand clear of those with a ratio higher than 20.  How can I tell if the current price of an ETF is a good price to buy it? Are there such simple metrics? Prove that any conjugate of $a$ has the same order as $a$. I am trying to prove that any conjugate of $a$ has the same order as $a$.  Let $G$ be a group and let $a \in G$.  An element $b \in G$ is called a conjugate of $a$ if $b=xax^{-1}$.  My professor gave us a hint.  Prove $a^k=e$ iff $b^k=e$ for all $k \in \mathbb{Z}$ implies $o(a) = o(b)$.  My first move is to $b^k = (xax^{-1})^k = (x^{-1})^ka^kx^k = (x^{-1})^kex^k = (x^{-1})^kx^k=x^{-k+k}=x^0$  I am not sure that I am doing this correctly?  I am trying to get $b^k=e$.  However when I raise $(xax^{-1})^k$, I am not sure if I what I have done is correct?  I am kind of confused here. What should I do when I see a "question ban warning"? When I  question, It says following:     Wait! Some of your past questions have not been well-received, and   you're in danger of being blocked from asking any more.      For help formulating a clear, useful question, see: How do I ask a   good question?      Also, edit your previous questions to improve formatting and clarity.   Here is screen-shot:   So, What should I do? (How long to wait?) timestamp, modification time, and created time of a file Reset lightning:input textbox Field Value to Null I am trying to create a button which will update a field value to null (RESET it). However, it's not working and not even giving any error. So when I click on the button, nothing happens.  lc.cmp:  &lt;aura:attribute name="myObj" type="Account" default="{'SObject':'Account'}"/&gt; &lt;lightning:input type="text" name="accountPhn" label="Enter Account Phone" value="{!v.myObj.Name}"/&gt; &lt;lightning:button label="Reset" onclick="{!c.resetMe}"/&gt;   lc.js:  resetMe : function(component, event, helper) {         //alert("Called");         component.set("v.accountPhn","");     }   Appreciate help around this. Mock vs MagicMock Union of subgroups is a subgroup if and only if one subgroup is a subset of the other Let $H$ and $K$ denote two subgroups of a group $G$.  Prove that the union $H \cup K$ is a subgroup of $G$ if and only if $H \subset K$ or $K \subset H$. How to randomize (shuffle) a JavaScript array? Algebraic Proof that $\sum\limits_{i=0}^n \binom{n}{i}=2^n$ Why doesn't Ubuntu 16.04 in Windows have ubuntu.exe? Can I color text by using an image, so that the color isn't flat? I'm using tikz to attempt to make a book cover, and I have a "wallpaper" image that looks like leather (not that I've handled that many leather-bound books). I'm using the node command to place text over the top of it, but none of the colors I've experimented with look decent.  Can I use an image of gold-leaf as the fill/stroke for the text? Or for that matter, for the path lines as well?  I've tried googling for this, but I can't figure out a set of keywords non-generic enough to pull back useful results. How do I boot into single-user mode from GRUB? Maybe this question is too obvious to everyone.      In Noetherian ring $R$, ideal $I\subset R$, $P\in\mathrm{Ass}(I)$, then $P=I:a$ for some $a\in R$ where $\mathrm{Ass}(I)$ denotes associated primes of $I$.   This is an exercise 8.29 in Gathmann's Commutative Algebra notes. I am supposed to find this statement obvious. However, up to now, I still did not find this statement obvious. Can someone explain intuitively why noetherianness here allows me removing radical $\sqrt{I:a}$ without going into details?  If $Q$ is a primary ideal of $I$, then certainly $\sqrt{Q:a}=P$ for $a\not\in P$. It seems taking $(p_1,\dots, p_n)=P$, $p_i^{n_i}a\in Q$. Since $Q$ primary, I must have $p_i\in Q$ and $p_i^{n_i}\in Q$. (This argument seems very wrong as this deduces the same conclusion without noetherian property.) Windows 7 shows red "X" on network icon, but I'm connected How do I draw the graph of inequalities in the coordinate plane? I want to graph the following inequalities (done in desmos) in latex. I need the code. Why does Google not consider dot in usernames of Gmail addresses? E.g.: If your Gmail ID is abc.xyz@gmail.com, it considers this the same as abcxyz@gmail.com.  Why is this so?