The course descriptions is : A theoretical course in calculus; emphasizing proofs and techniques. Trigonometric identities. Limits and continuity; least upper bounds, intermediate and extreme value theorems. Derivatives, mean value and inverse function theorems. Integrals; fundamental theorem; elementary transcendental functions. Techniques of integration. Taylor's theorem; sequences and series; uniform convergence and power series.  This course is using Spivak's calculus, but I'm also looking for another proof based book accompany with Spivak, thx
This question is a  of the common question asking for calculus references. It is here to  away the repetition, and give a canonical resource for calculus references.      I'm looking for a resources to learn single-variable calculus.
Can the 5E abjurer wizard recover temp hp on his arcane ward any time it is below max or only when it reaches 0? On p115 of the PHB under Arcane Ward there are a few facts that I think are clear:   The ward is created by the first abjuration spell you cast in a day It has a max hp of level &times; 2 + int modifier. The ward does not absorb damage once it reaches 0 hp, but it remains until you take a long rest The ward can be recharged by casting abjuration spells, at a rate of 2 x spell level worth of hp 3.   However, the way they wrote the paragraph about recharging seems unclear to me. Can you recharge the ward whenever it is below max hp or only when it is at 0? My understanding of temporary hp would say it is the latter, but the writing on this ability infers the former.  An example.  If I were a level 10 abjurer with 20 int and cast Mage Armor, it would give the ward 25 hp. If I took 30 damage from an enemy, the ward would absorb 25 damage, leaving it at 0, and I would take the remaining 5 damage. If I cast Stoneskin (level 4 abjuration) the ward would be recharged with 8 hp. If I then cast Planar Binding (level 5 abjuration) which would be correct?   The ward remain at 8 hp The ward would recharge an additional 10 hp to a total of 18 hp Proof of Neumann Lemma Prove that if $\|A\| &lt; 1$, then $I-A$ is invertible. Here, $\|\cdot\|$ is a matrix norm induced by a vector norm.  This lemma is referred to as Neumann Lemma.  Any ideas on how to go ahead with this?  Thanks. Surprisingly many network interfaces on macOS: are these legitimate? ifconfig reveals a surprisingly long list of network interfaces on my MacBook, which is running macOS High Sierra (10.13.6): lo0, gif0, stf0, XHC20, en0, p2p0, awdl0, en1, en2, bridge0, utun0.  There is an F5 VPN client installed, but I've rebooted the MacBook since the last VPN connection.  Are all of these network interfaces legitimate or is this something I should worry about? Everywhere I get to read statements like EM radiations and EM fields carry "energy as well as momentum " . I wonder if energy and momentum are inalienable . Is it possible for a field to carry only energy without having any momentum term ? I mean isn't it sufficient to say that fields or EM waves carry energy or Doesn't the ability to transfer energy, of the fields, automatically imply that they can change the momentum of another particle while transferring energy to them? How can I better optimize my tables and a LEFT JOIN query to find items that don't yet exist in the right table? I have two tables that represent a list of urls and their related word indexes. Here are the table definitions for reference.  desc urllist; +-------+---------------------+------+-----+---------+----------------+ | Field | Type                | Null | Key | Default | Extra          | +-------+---------------------+------+-----+---------+----------------+ | id    | bigint(20) unsigned | NO   | PRI | NULL    | auto_increment | | url   | text                | NO   |     | NULL    |                | +-------+---------------------+------+-----+---------+----------------+   and  desc wordlocation; +----------+---------------------+------+-----+---------+-------+ | Field    | Type                | Null | Key | Default | Extra | +----------+---------------------+------+-----+---------+-------+ | urlid    | bigint(20) unsigned | NO   |     | NULL    |       | | wordid   | bigint(20) unsigned | NO   |     | NULL    |       | | location | int(10) unsigned    | NO   |     | NULL    |       | +----------+---------------------+------+-----+---------+-------+   The software app is a web spider. It crawls a list of urls, extracts those urls, and inserts these into the urllist table. Then, an indexer checks to see what urls have not yet been indexed, and then proceeds to index said urls.  Here is the query I am using to find items in the left table (urllist) that have not yet been indexed in the right table (wordlocation). This query is as suggested on the :  select * from urllist ul  left join wordlocation wl on ul.id = wl.urlid  where wl.urlid IS NULL;   As of this writing my test database has only 600 indexed urls, and the wordlocation table has 1.3 million rows.  However, my CPU is at 100%, and the longest I've waited to see if the query would complete is a half hour (which, it never did by the way).  To be thorough, here is the explanation of the query:  explain select * from urllist ul left join wordlocation wl on ul.id = wl.urlid where wl.urlid IS NULL; +----+-------------+-------+------+---------------+------+---------+------+---------+-------------------------+ | id | select_type | table | type | possible_keys | key  | key_len | ref  | rows    | Extra                   | +----+-------------+-------+------+---------------+------+---------+------+---------+-------------------------+ |  1 | SIMPLE      | ul    | ALL  | NULL          | NULL | NULL    | NULL |   50364 |                         | |  1 | SIMPLE      | wl    | ALL  | NULL          | NULL | NULL    | NULL | 1351371 | Using where; Not exists | +----+-------------+-------+------+---------------+------+---------+------+---------+-------------------------+   I need this query to finish in seconds, not minutes. Also, I'm worried about scalability. I have 40,000 unique urls that are waiting to be added to the index, so how do I take that into consideration with my table and query design? 400,000 urls?  Just a couple notes about my decisions on the current table structure.  I have no intention of stopping at 400,000 urls, but perhaps bigint(20) is a bit overzealous?   Url as text is for more practical reasons. I index a lot of Asian and other foreign language domains which do not appear as their equivalent Kanji or other characters in the database and frequently take more than 255 characters.  I'm using MySQL. I'm definitely open to suggestions as to better table and query design. Please let me know if I can provide more information. Why does Windows only show about 3.5 GB of my 4 GB of RAM? I recently upgraded my computer's RAM to 4&nbsp;GB. My 32-bit Windows installation shows only 3574&nbsp;MB of the memory. How can I make Windows use the full amount of RAM? Is conservation of momentum and energy valid for non-inertial frames? I have two complete sentences; the second is in parentheses, essentially a comment on the first.  Where do I place the periods?  I would use this:     They played opera. (I covered my ears.)   Is this punctuation correct? I was asked to prove the following:  If $\{ E_1,E_2,\ldots,E_n\}$ is a fully independent set of events, prove that:  $$ P\left(\bigcup_{i = 1}^n E_i \right) = 1 - \prod_{i = 1}^n \left[ 1 - P(E_i) \right]  $$  (Hint: use complements)  I got the following:  Let $\alpha = E_1\cup E_2 \cup \cdots \cup E_n = \{E^C_1 \cap E^C_2 \cap \cdots\cap E_n^C\}^C$. Then, $\alpha^C = \{ E^C_1 \cap E^C_2 \cap \cdots \cap E^C\} $.  Assuming that the set of complements of each ${E_i}$ is also fully independent, we have:  \begin{align} &amp; P(\alpha^C) = \prod_{i = 1}^n P(E_i^C)  = \prod_{i = 1}^n \left[ 1 - P(E_i) \right] \\ &amp; P(\alpha) = 1 - P(\alpha^C) = 1 - \prod_{i = 1}^n \left[ 1 - P(E_i) \right] \\ &amp; P(\bigcup_{i = 1}^n E_i) = 1 - \prod_{i = 1}^n \left[ 1 - P(E_i) \right] \end{align}  However, even though I know (by Googling it) that my assumption is true, I'm unable to prove it so I could validly use it as a lemma. Is there another way to prove this equality? Say I have 12 and have to factorize it in 2 factors. Now possible factors are 1, 12 2, 6 3, 4   Now 3,4 is such that value of maximum factor i.e. 4 is minimized among all pairs.  How to find the solution in generic case where a number N is to be divided in K factors minimizing maximum value. We've had a number of instances over on Stack Overflow involving users who get to the reputation level where they can comment, then spam comments across the site. Due to the lack of community review around comments, these spammed comments can remain on the site for months before someone notices and flags one. Moderators then have to go and manually remove each of these comments. In the last few months, I've personally had to deal with two users who spammed their website across 200+ questions each.  This isn't just limited to people spamming comments with a URL in them. We've had a number of users spray the same nonsense comment or insult across tons of posts and have no one seem to notice for months.  What I'd like is some kind of system-generated notification when a user has posted a barrage of copied-and-pasted comments. This could possibly be triggered when a user has posted a set number of identical comments in a row (4 of them, maybe) or maybe even if they've posted the same comment multiple times. We receive similar system-generated flags when users post duplicate answers, edit many of their posts in a row, or delete many of their own posts at a time.  This system-generated flag should ignore comments generated by review (which have the same text in them). I'm thinking that it shouldn't ignore comments generated by custom off-topic reasons, because someone using the exact same close reason over and over again might be something we want to look into.  The flag itself won't do anything to a user, and would only appear to moderators. I imagine there will be a number of false positives, but it should be easy for us to check these and identify the well-intentioned stock comments from the problematic ones.   We currently have no means of detecting comment spam, so even if this didn't have a great hit rate it would still be beneficial on the balance. Running sample queries for this seems to indicate that it wouldn't have that much noise.  This would allow us to catch comment spam early, and prevent it from living on the site or developing into a more serious problem. Determinant of circulant matrix Find the determinant of the following matrix in the terms of $a_1,a_2,\cdots,a_n$ explicitly, $$ \begin{bmatrix} a_1 &amp; a_2 &amp; a_3 &amp; \cdots &amp; a_n\\ a_2 &amp; a_3 &amp; a_4 &amp; \cdots &amp; a_1\\ a_3 &amp; a_4 &amp; a_5 &amp; \cdots &amp; a_2\\ \vdots &amp; \vdots &amp; \vdots &amp; \ddots &amp; \vdots\\ a_n &amp; a_1 &amp; a_2 &amp; \cdots &amp; a_{n-1}\\ \end{bmatrix} $$ When the determinant is zero? Is $n^{th}$ root of $2$ an irrational number? Possible Duplicate:          Will every $n^{th}$ root of $2$ be an irrational number? If yes, how can I prove that? Is it possible that your friends can join your Hamachi server when they don't have Hamachi? Does anyone know how to do that??!! Analysing Bounce Rate and Time on Page Is it possible to assign a base class object to a derived class reference with an explicit typecast in C#?.  I have tried it and it creates a run-time error. I'm developing a messenger application with encrypted chats.   In the first version of the app I've used PBKDF2 (10000 iterations, SHA1, random salt) to extend a short user password and generate keys to encrypt (AES256) and sign message (HMAC).  In the current version of the app I generate a random 130-bit key and use it instead of a user password. I think that using PBKDF2 with such a key has some overhead.   I see 2 possible ways to improve the current approach:   Keep using PBKDF2 and decrease iterations to 1 or 10. Don't use PBKDF2 and use HKDF instead.   What are the advantages and disadvantages of each approach? Are there any other options? Seems like a stupid question, but I'm after a nice solid answer if possible.  I joined SO a while ago to ask the odd question, but recently I'm starting to use it more often.  I've been reviewing the 40 max items (20 first posts/20 late answers) per day, and editing anything I see wrong with them (lots of +2's)... flagging them... etc.  I've also been  per day.  But other than commenting where I can and answering the odd question, I'm only  (1k at the moment).  I might just be overeager, but I was hoping some high-rep users might be able to write a little something on how they earned their stripes?  I've , and to be honest I was hoping I'd get a hit, because it sounds like a stupid question... anyways, here you go!  Also, I'm not referring to how to gain rep on Meta, since it's a bit of a fluke I've earned the rep I have on here. I'm asking about main SO. How to successfully put a bibliography in a report on TexShop? I'm trying to put a bibliography in my LaTeX document, but I get various errors when I run it in BibTeX (saying no \citation found etc.), and no bibliography is printed in the PDF when I typeset it in LaTeX. My code reads as follows:  \documentclass[12pt]{report} ... \usepackage{biblatex} \addbibresource{SUSYBib.bib} ... \begin{document} ...\cite{1}... ...\cite{2}... \nocite{*} \printbibliography \end{document}   And my .bib file reads as  @book{1, title={The Schrödinger Equation}, author={F. A. Berezin, M. A. Shubin}, year={1983}, publisher={Moscow State University, Moscow}, pages={57-59} } @book{2, title={An Introduction to Measure Theory}, author={Terence Tao}, year={2011}, publisher={American Mathematical Society}, pages={91} } I know that I can make certain blocks only placeable on a specified block, but what is the command to give an item that can only BREAK certain blocks?