Does reduced casting cost affect casting of cards with variable casting cost?
How do I handle casting a spell with a cost of X when the casting cost is reduced?
Proof of continuity $f(x+y) = f(x) + f(y)$ Adding a citation after paper is accepted I am a junior researcher in mathematics. I worked simultaneously on proving Theorem A and Theorem B. They both generalize Theorem X in two different directions. Theorem X was proved several years ago (not by me). The proofs of Theorem A and Theorem B use very different techniques, and so I decided ahead of time to write one paper for each.  I finished proving Theorem A before I finished proving Theorem B. Then, I submitted Paper A to a journal (and the arXiv) proving Theorem A. Paper A includes a survey of related problems in the introduction. After a short while, I managed to prove Theorem B, and two months after the submission of Paper A, I submitted Paper B (to a different journal, and also to the arXiv). Now, six months after the submission of Paper A, it is accepted with requests for minor corrections. There is no response yet regarding Paper B.     In this situation, is it a reasonable/common to add one/two sentences to the survey in introduction of Paper A, explaining and citing the arXiv prepreint of Paper B, in addition to the minor corrections the referees asked for? (and letting the editor and referees know that I did that in my answer to their reports which comes along with the corrected manuscript).   Note that I am not asking if it is ok to cite a preprint. This is something I do, and so do other people, at least in math. My question is about making a change, after acceptance, which was not asked by the referees and the editor. Discriminant analysis vs logistic regression I found some pros of discriminant analysis and I've got questions about them. So:     When the classes are well-separated, the parameter estimates for   logistic regression are surprisingly unstable. Coefficients may go to   infinity. LDA doesn't suffer from this problem.      If number of features is small and the distribution of the predictors   $X$ is approximately normal in each of the classes, the linear   discriminant model is again more stable than the logistic regression   model.    What is stability and why is it important? (If logistic regression provides a good fit that does its job, then why should I care about stability?)      LDA is popular when we have more than two response classes, because it   also provides low-dimensional views of the data.    I just don't understand that. How does LDA provide low-dimensional views? If you can name more pros or cons, that would be nice. How do I bake chocolate chip cookies like Subway? I want a chocolate chip cookie like the ones Subway has. Those cookies have a smooth top and chewy texture.  I have a recipe for 3/4 cup fat, 0.5 cup sugar 0,5 cup brown sugar, 3/4 cup flour, 1 egg, 2 tsp extract. I get a semi stiff dough, but also an airy cookie, not as dense like Subway. I use room temperature margarine.  I have tried to bake cookies with 2 eggs and with 1 egg, but they always come out wrinkly on top, not as stiff. Is it possible that using an egg and a yolk would work? (But wouldn't that waste a lot of whites if that's how they do it?)  I think as part of this I want a stiff dough, that'll leave the bowl clean, to minimize spread and keep the tops from getting wrinkly as stiffness will keep the the cookies from expanding and contracting. Is that true? If so, I'm afraid that if I add more flour to stiffen the dough will make the cookies cakey - is that the case? How would I avoid that? I've got a custom loop that I'm using to display some Real Estate listings that will be available within 60 days. I'm calling it with the following function:   &lt;?php  $sixtydays = date('Y/m/d', strtotime('+60 days')); $paged = (get_query_var('paged')) ? get_query_var('paged') : 1; $query = new PostsOrderedByMetaQuery(array(   'post_type' =&gt; array('post', 'real-estate'),   'meta_key' =&gt; 'Time Available',   'meta_compare' =&gt; '&lt;=',   'meta_value' =&gt; $sixtydays,   'paged' =&gt; $paged,   'orderby_meta_key' =&gt; 'Price',   'orderby_order'    =&gt; 'ASC' )); ?&gt; &lt;?php while ($query-&gt;have_posts()) : $query-&gt;the_post(); ?&gt;   While the loop works great, I can't get it to paginate. It shows the first 10 (my default) posts but doesn't show the pagination. The only way to display all posts is to show them on one page by adding 'posts_per_page' =&gt; -1, I have similar loops on other pages that have no problem paginating. The only difference with this one is that there are two meta keys that are filtering the posts.   I'm using WP Page Navi for this and the rest of my pages. I'm closing the loop and adding the pagination using the following code:   &lt;?php endwhile; // End the loop. Whew. ?&gt; &lt;?php wp_pagenavi(); ?&gt; &lt;?php wp_reset_query(); ?&gt;   How can I go about fixing this? I know in TNG that all computer interfaces are now touch-screen, and that got me thinking: when was the first reference to touch-screen computer interfaces in Sci-Fi made? In the final episode of season 6 we see that Lyanna Stark is whispering something to Ned. What is she saying? I have to admit, I didn't even know spoilers were supported until today, but really bugs me that spoilers just look like empty block quotes before you mouse over them.   Is this a spoiler or just a long empty block quote? I can't tell right away, so it would be really nice if there was some indication of which one it was so I didn't have to push my mouse all the way over to it. What can I say, I'm incredibly lazy.  vs.         I suppose the author should include something that says 'Spoilers below:', but it would be nice if there was some other indication of what it was. Closure of the set of degree $n$ polynomials with $n$ real roots A very beautiful (but maybe classic) exercise:   Consider the set $\Omega_n$ of polynomials of degree $n$ with real coefficients and $n$ distinct real roots.   What is the closure of $\Omega_n$ in $R_n[x]$?  ($R_n[x]$ denotes the set of polynomials with real coefficients of degree at most $n$)    The answer is the set of polynomials in $R_n[x]$ all of whose roots are real, plus the constant polynomials. Showing that this set is contained in the closure is not too hard, however the reverse implication is more difficult and I'm wondering what is the most elementary method to prove it (without talking about holomorphic functions for example). Prove that if $d$ divides $n$, then $2^d -1$ divides $2^n -1$ Prove that if $d$ divides $n$, then $2^d  -1$ divides $2^n  -1$.  Use the identity $x^k  -1 = (x-1)*(x^{k-1}  +  x^{k-2}    + \cdots + x +1)$ The function auto.arima in the forecast package of R is a powerful tool to identify the best ARIMA(p,d,q) model for a given data series. In the   of the function, they report the following:  auto.arima(y, d=NA, max.p=5, max.q=5, max.order=5, max.d=2, start.p=2, start.q=2, ...)   According to the above function it seems that, by default, auto.arima tries to find the best model among all the combination having:   p equal or lower than 5 q equal or lower than 5 d equal or lower than 2 order equal or lower to 5, i.e. p + q + d  ≤ 5.   However, why does the default code stops at these conditions? Is there any particular reason why p, q and d should be equal or lower than 5, 5 and 2 and their sum lower than 5? Why is my ethernet interface called enp0s10 instead of eth0? Getting video dimension / resolution / width x height from ffmpeg What are the current travel restrictions on individuals entering the United States from regions affected by Covid-19? What is the difference between public, protected, package-private and private in Java? In Java, are there clear rules on when to use each of access modifiers, namely the default (package private), public, protected and private, while making class and interface and dealing with inheritance? Why is polyploidy lethal for some organisms while for others is not? What is the relativistic calculation of travel time to Proxima Centauri? It has already been  how fast a probe would have to travel to reach Alpha Centauri within 60 years.  NASA has done  into a probe that would take 100 years to make the trip.  But I am interested in this question from a slightly different perspective.  I would like to know how long the trip from our solar system to Proxima Centauri would take for both a ship and for an observer on Earth while accounting for the effects of general relativity on a .  The following assumptions may be made:   The ship starts and ends at 0 km/s relative to the Earth observer. The ship experiences a constant force of acceleration of 20 g until the half-way point and then a constant force of deceleration of 20 g until the destination. The path traveled is such that accleration is in only one spatial direction. The distance traveled is 4.25 light years.   Calculate (accounting for general relativistic effects):   Time of trip for Earth-based observer. Time of trip for ship-based observer. Maximum speed.   You don't need to consider mass, or life-support systems, or method of propulsion; I am really only interested in the effects of relativity on the perceived time of the trip. Would this look natural:     StackExchange is a helluva site! I found out about this site at Facebook.    Or should we use the preposition on:     StackExchange is a helluva site! I found out about this site on Facebook.    Or are both okay?   (Inspired by a ) Direct product of abelian groups Let $G_1,\dots,G_n$ be groups. Prove that the direct product $G_1\times\cdots\times G_n$ is abelian if, and only if, each of $G_1,\dots,G_n$ is abelian. To prove that the direct product is abelian is straightforward but what I don't understand is the converse. Initializing a member array in constructor initializer