What are the prerequisites for studying modern algebraic geometry? I have studied differential geometry , thought of starting to study algebraic geometry , but it seems completely different from differential geometry . It uses lots of abstract algebraic tools . I wonder how much abstract algebra do I need to learn in order to learn algebraic geometry . Possibly , some reference of algebraic  geometry books will much helpful .  Another thing I see a lot in lecture notes on introductory algebraic geometry is the usage of angle brackets , i suppose they something completely different . What do they mean , anyway ?
Prerequisites for Algebraic Geometry Is it possible to get into algebraic geometry by just knowing calculus and linear algebra or is this too far of a stretch?  If not could anyone give me a list of book/lecture notes recommendations in chronological order to dive into algebraic geometry with my state of knowledge?  Thanks
How should the "and/or" text of a card be interpreted? How should the "and/or" text of a card be interpreted? For example on a card like  the text reads:     Nahiri's Wrath deals damage equal to the total converted mana cost of the discarded cards to each of up to X target creatures and/or planeswalkers.   For the remainder of this question, let's fix X at 2 to keep things simple.  Should the "and/or" be interpreted such that the effect is applied to a combination of targets the player chooses. That is to say for Nahiri's Wrath the player could perform one of the following options:   Nahiri's Wrath deals damage to each of two target creatures. Nahiri's Wrath deals damage to each of two target planeswalkers. Nahiri's Wrath deals damage to each of one target creature and one target planeswalker.   Or should the card's effect be expanded and applied to both sides of the "and/or". With this interpretation the player would have to the following two options when playing Nahiri's Wrath:   Nahiri's Wrath deals damage to each of X target creatures and Nahiri's Wrath deals damage to each of X target planeswalkers Nahiri's Wrath deals damage to each of X target creatures or Nahiri's Wrath deals damage to each of X target planeswalkers   The confusion stems from  (which was formerly formerly worded using the same phrasing) where some commenters have said things like:     So this lets you do a total of 12 damage? 3 each to 2 target creatures AND 3 each to 2 target players, one of which could be a planeswalker? That's how the "and/or" works, right? You can choose. Otherwise it would just be "or"... How to draw tangent vectors and component vectors on a curve I am searching for a way to have a figure drawn using LaTeX. I am aware of some packages TikZ and PSTricks that could accomplish this, and have been tinkering around with them recently but do not have enough skill set at the moment to accomplish this drawing. Is there a way someone can accomplish the drawing, preferably with TikZ but could be any package your most comfortable with in a reduced amount of coding (non-advanced) to where I could understand what is happening from the code to the picture. If this is not possible, then I will have to study the package more and work with it until I understand the method (code) to create the graphic. The drawing is below.  Note: This is a rough sketch. The optimal output would have smooth quadratic curves and all quantities are vectors. Prove $2^k+1$ divisible by 3 for odd K For example, "dude," "man," "buddy," "pal," etc, when used to stand in for someone's name.  "Hey, pal, how's it going?"  Is there a word for terms like these?  Or is "colloquialism" as close as we can get? What is the maximum convergent $x$ in the power tower $x^{x^{x^{x\cdots}}}$? In the power tower $x^{x^{x^{x\cdots}}}$ where there is an infinite stack of $x$'s, what is the maximum convergent number? I know the answer by playing with the form $x^y=y$ and using Mathematica, but I don't know how to solve this by hand. What's the origin of "I'm down with it"? I understand it's an expression of agreement. What exactly does it mean and where did it originate from? What are the responsibilities of each Pseudo-Terminal (PTY) component (software, master side, slave side)? I am trying to figure out how a tty works1 (the work-flow and responsibilities of each elements). I have read several interesting articles about it, but there are still some blurry areas.  This is what I understand so far:   The emulated terminal makes different system calls to /dev/ptmx, the master part of the pseudo-terminal. The master part of the pseudo terminal allocates a file in /dev/pts/[0-N], corresponding to the obsolete serial port, and "attaches" a slave pseudo terminal to it. The slave pseudo terminal keeps information such as session Id, foreground job, screen size.   Here are my questions:   Has ptmx any purpose besides allocating the slave part? Does it provide some kind of "intelligence", or the emulated terminal (xterm for instance) has all the intelligence of behaving like a terminal? Why does xterm has to interact with the master part, as it only forwards the stdout and stdin of the slave part? Why can't it directly write and read from the pts file? Is a session ID always attached to one pts file and vice versa? Could I type a ps command and found 2 sessionId for the same /dev/pts/X? What other information does the pts store? Does Xterm update all fields by himself, or does the ptm add some "intelligence" on it?     1. I base my understanding on the , and the  posts, as on several other Why are the lower 3 bits of curve25519/ed25519 secret keys cleared during creation? I am currently experimenting with ed25519 and I noticed that on secret key creation, bit 254 is always set and the lower 3 bits are always cleared. I found that bit 254 is always set to protect against timing attacks in this question:   But why are the lower 3 bits cleared. Obviously it has to do with the formula in the curve25519 paper:  The set of secret keys is defined to be $\{\underline{n} : n \in 2^{254} + 8\{0, 1, 2, 3,\ldots, 2^{251}-1\}\}$  It's because of the 8 in there, but why is that 8 there? I suspect it has something to do with theorem 2.1 in the curve25519 paper, but I am not sure, because I do not understand fully what is being proven there.   I am experimenting with ed25519 primitives in some cryptographic routines which need me to add scalars to the secret key (add secret keys). Even if I add two well formed ed25519 secret keys, the result will not always have bit 254 set and the lower 3 bits cleared. Is this a security problem? I understand the risk for bit 254 but not the lower 3. User's suspension period has expired, but they're still suspended? I came across a user with a notification that states:     This account is temporarily suspended network-wide. The suspension period ends on Dec 2 '18 at 2:35.   Screencap for posterity:    Unless this user is  (in which case the full year should be shown, or at least a tooltip), it seems like they should have been unbanned 4 months ago. Peppers evolved chemicals like capsaicin to discourage animals from eating them, and that chemical itself can cause internal damage if consumed in large quantities, though this is usually only a concern for the hottest of peppers (which people do eat).   Still, why is it that humans seem to be the only animal that often "like" spicy food? This came to my mind after I learned the following from :  where T : struct   We, C# developers, all know the basics of C#. I mean declarations, conditionals, loops, operators, etc.  Some of us even mastered the stuff like , , , , ...  But what are the most hidden features or tricks of C# that even C# fans, addicts, experts barely know?  Here are the revealed features so far:    Keywords    by  var by  using() statement by  readonly by  as by  as / is by  as / is (improved) by  default by  global:: by  using() blocks by  volatile by  extern alias by    Attributes    by   by   by   and  by   by   by   by    Syntax    (coalesce nulls) operator by  Number flaggings by  where T:new by  Implicit generics by  One-parameter lambdas by  Auto properties by  Namespace aliases by  Verbatim string literals with @ by  enum values by  @variablenames by  event operators by  Format string brackets by  Property accessor accessibility modifiers by  Conditional (ternary) operator (?:) by  checked and unchecked operators by  implicit and explicit operators by    Language Features   Nullable types by  Anonymous types by  __makeref __reftype __refvalue by  Object initializers by  Format strings by  Extension Methods by  partial methods by  Preprocessor directives by  DEBUG pre-processor directive by  Operator overloading by  Type inferrence by  Boolean operators  by  Pass value-type variable as interface without boxing by  Programmatically determine declared variable type by  Static Constructors by  Easier-on-the-eyes / condensed ORM-mapping using LINQ by  __arglist by    Visual Studio Features   Select block of text in editor by  Snippets by      Framework   TransactionScope by  DependantTransaction by  Nullable&lt;T&gt; by  Mutex by  System.IO.Path by  WeakReference by    Methods and Properties   String.IsNullOrEmpty() method by  List.ForEach() method by  BeginInvoke(), EndInvoke() methods by  Nullable&lt;T&gt;.HasValue and Nullable&lt;T&gt;.Value properties by  GetValueOrDefault method by    Tips &amp; Tricks   Nice method for event handlers by  Uppercase comparisons by  Access anonymous types without reflection by  A quick way to lazily instantiate collection properties by  JavaScript-like anonymous inline-functions by    Other   netmodules by     by     by Pronunciation differences between "finite" and "infinite" In my experience, "finite" is pronounced (IPA) ˈfaɪnaɪt while "infinite" is ˈɪnfɪnɪt.  In general, the prefix "in" negates an adjective, but does not change the pronunciation. Based on this, I would expect ˈɪnfaɪnaɪt.  Is there a reason for this "deviation from the norm"? Are there dialects/regions that actually say ˈɪnfaɪnaɪt (or ˈfɪnɪt, for that matter)? I've got some photos with quite random names, but the dates (not exif, but when they were last edited) are correct. I'd like to rename them in that order, preferably so that the first photo is named 1.jpg and so on, but anything goes. How can I do that? Define a relation and find its equivalence classes. Is normality testing 'essentially useless'? Is it possible to extend an Indian business visa from India? I'm an Egyptian girl from Cairo. I'm planning a six-month trip to India in December for training purposes, but getting a six-month visa is so hard from my country.  If I can only get 3 months instead of 6 months, is it possible to extend the business visa in India? Overfull box and broken pagination with page that contains only section headings Wrong view count for questions? I have a question with 1508 views, but in the personal page list it gets reported as 2k.  Bug or aggressive approximation? What happened to the voting buttons? Starting from yesterday, the up-/downvote buttons seem to have moved from the left of a post towards above (in the middle) of the bodies of all questions and answers. Here is a screenprint of a sample (containing such buttons for both a question and an answer to it):    This happens on an iOS device (running 9.3.5), using various browsers and this in any SE site. No idea so far how it looks like in the most recent iOS version. Same for the (green) checkbox below those voting buttons (to mark an answer as accepted). Is this intended to be like this (IMO it looks ugly ... and a huge waste of available space), or is it rather some sort of CSS related issue that needs to be fixed? PS: Looking at the same page around the same time on a Win PC (using any of the common browsers) seems to work fine. Should the mean be used when data are skewed? Often introductory applied statistics texts distinguish the mean from the median (often in the the context of descriptive statistics and motivating the summarization of central tendency using the mean, median and mode) by explaining that the mean is sensitive to outliers in sample data and/or to skewed population distributions, and this is used as a justification for an assertion that the median is to be preferred when the data are not symmetrical.  For example:     The best measure of central tendency for a given set of data often depends on the way in which the values are distributed.... When data are not symmetric, the median is often the best measure of central tendency. Because the mean is sensitive to extreme observations, it is pulled in the direction of the outlying data values, and as a result might end up excessively inflated or excessively deflated."   —Pagano and Gauvreau, (2000) Principles of Biostatistics, 2nd ed. (P&amp;G were at hand, BTW, not singling them out per se.)   The authors define "central tendency" thus: "The most commonly investigated characteristic of a set of data is its center, or the point about which observations tend to cluster."  This strikes me as a less-than forthright way of saying only use the median, period, because only using the mean when the data/distributions are symmetrical is the same thing as saying only use the mean when it equals the median. Edit: whuber rightly points out that I am conflating robust measures of central tendency with the median. So it is important to keep in mind that I am discussing the specific framing of the arithmetic mean versus the median in introductory applied statistics (where, mode aside, other measures of central tendency are not motivated).  Rather than judging the utility of the mean by how much it departs from the behavior of the median, ought we not simply understand these as two different measures of centrality? In other words being sensitive to skewness is a feature of the mean. One could just as validly argue "well the median is no good because it is largely insensitive to skewness, so only use it when it equals the mean."  (The mode is quite sensibly not getting involved with this question.)