I was watching TV series &quot;Prison Break&quot; and in the first episode I came across this kind of strange to me sentence.  HALE: We're all clear on the Burrows execution. KELLERMAN: Good. HALE: Except for one thing. Bishop McMorrow is not in the fold. HALE: He's got a lot of influence with the Governor. They went to prep school together apparently. HALE: Look, the closer it gets, the more I'm worried that the bottom is  gonna fall out of this whole thing. KELLERMAN: Well, maybe it's time we arranged a visit with the good bishop  then. KELLERMAN: Look, in one month it'll all be over.  I don't get why there has been used a definite article. I learned that articles always come before nouns. In this situation &quot;more&quot; doesn't seem to be a noun, I would say it's an adverb. Maybe I'm missing something, any suggestions ?
Why is the definite article used in expressions like the more and the less? For example,  The more you study, the more you know. The less you study, the less you know.
What is the limit of $\frac{\prod\mathrm{Odd}}{\prod\mathrm{Even}}?$ What is this limit   $$   \frac{1\times3\times5\times\cdots}{2\times4\times6\times8\times\cdots} = \lim_{n \rightarrow \infty}\prod_{i=1}^{n}\frac{(2i-1)}{2i} $$   I remember that it was something involving $\pi$.  How can I compute it?    In addition; how can I compute it's sum series limit? Suppose we have a set $S$ of real numbers. Show that   $$\sum_{s\in S}|s-x| $$   is minimal if $x$ is equal to the median.    This is a sample exam question of one of the exams that I need to take and I don't know how to proceed. Why does the Contains() operator degrade Entity Framework' Linq queries? The product of all the elements of a finite abelian group I'm trying to prove the following statements.  Let $G$ be a finite abelian group $G = \{a_{1}, a_{2}, ..., a_{n}\}$.   If there is no element $x \neq e$ in $G$ such that $x = x^{-1}$, then $a_{1}a_{2} \cdot \cdot \cdot a_{n} = e$.   Since the only element in $G$ that is an inverse of itself is the identity element $e$, for every other element $k$, it must have an inverse $a_{k}^{-1} = a_{j}$ where $k \neq j$. Thus $a_{1}a_{1}^{-1}a_{2}a_{2}^{-1} \cdot \cdot \cdot a_{n}a_{n}^{-1} = e$.   If there is exactly one $x \neq e$ in $G$ such that $x = x^{-1}$, then $a_{1}a_{2} \cdot \cdot \cdot a_{n} = x$.   This is stating that $x$ is not the identity element but is its own inverse. Then every other element $p$ must also have an inverse $a_{p}^{-1} = a_{w}$ where $p \neq w$. Similarly to the first question, a rearrangement can be done: $a_{1}a_{1}^{-1}a_{2}a_{2}^{-1} \cdot \cdot \cdot xx^{-1} \cdot \cdot \cdot  a_{n}a_{n}^{-1} = xx^{-1} = e$. And this is where I am stuck since I proved another statement.    Any comments would be appreciated for both problems. How does one change to engineering for graduate school after undergrad in a science field? I've always had an interest in electrical engineering (actually all engineering in general) and I've decided that I'd like to pursue a graduate degree.  I guess I'm wondering if I should really try to do this or just keep it as a fun hobby I spend all my free time doing.  So, the question is,  how does one change fields between undergrad and grad school?  A specific example follows, but please provide answers that address a more general case, so that your answers will be helpful to other people with the same general question.  I already have a BS and MS in chemistry with a minor in applied math.I published three papers during my MS, one first author in the top journal in my specialty and two as coauthor in an average and a so-so journal.  I hope those will count for something.  I finished my BS with around 3.1 and my MS with a 3.9 gpa.  I'm sure that my MS advisor would write me a great letter.  My GRE scores are too old to count so I'd have to retake it or try and get a pass on it since I already have a graduate degree.  Since finishing school I've been working for 4 years as a chemist.  I can get great recommendations from my supervisor who is a PhD chemist.      I've been working for a while to cover some of the things I would have learned in a BSEE to try and make up for my lack of one.  I worked through some texts on analog/digital design, lots and lots of programming, built a balancing robot that got a bit of internet attention, and I'm currently working on an autonomous robotics project that's looking really good so far.  After that I have more project ideas than I have time or money for.    I want to apply for PhD programs (specifically in controls/robotics) but I'm worried that my application will get tossed right away because I don't have a BSEE.  Maybe I should just apply to MS programs? Age of universe? Well relativity teaches us that time interval between two events is a frame dependent quantity, then how can we say that our universe is 13.8 billion years old? Should it not depend on who is asking this question? How do I select all but two types of entities in Minecraft with the type selector? 'sudo su -' vs 'sudo -i' vs 'sudo /bin/bash' - when does it matter which is used, or does it matter at all? When I'm doing something that requires root be typed in dozens of times in a row, I prefer to switch my session to a root session. In the various tutorials and instructions I have used on the Internet, I see sudo su, sudo su -, sudo -i and sudo /bin/bash being used to open a root session, but I'm not clear on the difference between these and when or if that difference matters.  Can someone clear this up for me? What does 'close invalidated' mean Recently I flagged  as a duplicate. A few days later, when I checked it again to see if it was closed, I realized it wasn't.    Searching in Meta brought me to , where a comment suggested checking the timeline. So I checked the  and saw that the close flag was set as invalidated.  Does this mean that it was disputed? When I click on it I see only "This item is no longer reviewable." Do I need more reputation (3000+ ?) to view more detailed information? For what $n$ is $U_n$ cyclic? I often read that in case of Deep Learning models the usual practice is to apply mini batches (generally a small one, 32/64) over several training epochs. I cannot really fathom the reason behind this.  Unless I'm mistaken, the batch size is the number of training instances let seen by the model during a training iteration; and epoch is a full turn when each of the training instances have been seen by the model. If so, I cannot see the advantage of iterate over an almost insignificant subset of the training instances several times in contrast with applying a "max batch" by expose all the available training instances in each turn to the model (assuming, of course, enough the memory). What is the advantage of this approach? Why does increasing the number of features reduce performance? I'm trying to gain an intuition as to why increasing the number of features could reduce performance.  I'm currently using an LDA classifier which performs better bivariately among certain features but worse when looking at more features.  My classification accuracy is performed using a stratified 10-fold xval.    Is there a simple case of when a classifier would work better univariately than bivaraiately to gain a somewhat physical or spatial intuition of what is happening in these higher dimensions? I have a problem with the use of "the" after "and" where you would basically be connecting words. For example, which of the following is better:     The table and the chairs?   The table and chairs?   Or are both okay? What is the rule, and are there exceptions to it? Or is the rule just contextual and may vary between the sentences and meanings? Why is time special? In Special Relativity, the spacetime interval between two events is $s^2 = -(c{\Delta}t)^2+({\Delta}x)^2+({\Delta}y)^2+({\Delta}z)^2$ giving the Minkowski metric $\eta_{\mu\nu}=\text{diag}(-1, 1, 1, 1)$. What is the justification for making time have a negative coefficient, and how closely is that related to the 2nd law of thermodynamics? Sure, by letting $\eta = \text{diag}(1, 1, 1, 1)$, we get a pretty boring spacetime, and the boosts in the Poincaré group become trig instead of hyperbolic functions, but what's the physical reasoning behind this? Which style is preferred? Is there a good reason for one vs. the other?  Thanks in advance!  1) cmds.each do |cmd|    end  2) cmds.each { |cmd|    }   Example code:  cmds = [ "create", "update", "list", "help" ]  # Block style one # cmds.each do |cmd|   puts "loop1, cmd: #{cmd}" end  # Block style two # cmds.each { |cmd|   puts "loop2, cmd: #{cmd}" } command not found and syntax error near unexpected token If $G$ is a group, show that $x^2ax=a^{-1}$ has a solution if and only if $a$ is a cube in $G$ I was checking my old set of homework problems that I found this one:     If $G$ is a group, show that $x^2ax=a^{-1}$ has a solution if and only   if $a$ is a cube in $G$.   One direction is easy. If $a$ is a cube in $G$ then there exists $y \in G$ such that $a=y^3$. Now with direct manipulations it is shown that $x=y^{-2}$ is a solution to $x^2ax=a^{-1}$.  The other direction is harder though. I first attempted to isolate $x$ and then somehow show that it's necessary for $a$ to be a cube. But since I had no idea how to isolate $x$ I gave up. Then I decided to prove the existence of $y \in G: a=y^3$ by showing that a particular map is bijective so I can come up with a nice variable change to conclude that $a$ must be a cube, but this attempt failed as well. I'm tired now, so I think I'm giving up. I'm looking for a hint or a complete solution if a hint is not possible at this stage. Elastic net penalty What are the dangers of accepting gifts from unknown accounts on Steam? A user named 'steamfairy1' gifted me a game on Steam. I don't know this user, their profile shows no games, no friends and zero hours played. The profile of 'steamfairy1' was created the day I received the gift.  It very much looks like a scam, but I can't figure out how the scam is supposed to work. What's the worst thing that could happen if I accept the gift? Any idea what the trick here is? It could be just a random act of kindness and I'm just paranoid…  I contacted the Steam support, but they were unhelpful as usual: responding in the wrong language and with a canned one-liner that didn't make a whole lot of sense or had much to do with my question.  Keywords (not really tags, but I want this to be searchable): steamfairy, steam fairy Often I see code blocks pasted into questions with no formatting, or terrible formatting.  It's sometimes quite a chore to fix these: you have to either manually space out each line, or paste the code into an IDE, format it there, and paste it back.  Could we have a button/function where you could highlight a code block, and it would turn this:  foreach (var i in stuff) { if (i==1) { Whatever(); } }   into this:  foreach (var i in stuff) {   if (i==1) {     Whatever();   } }   I realise the function would have to work differently based on the language in the code block. Perhaps  it could use the same rules as the prettify code.  Also, it wouldn't be a big deal if it gets the language wrong and breaks the code, since you can always just ctrl+z.