Before you read my text down below. I would want you to know that I am down below talking about education on all levels, but mostly on the levels that is below university level. But on university level, at least not in all parts of the world, non-standard analysis is not very widespread.  Non-standard analysis gives a more in-depth explanation and rigorous ground for understanding limits. While limits most often is taught implicitly and gives an implicit understanding - which makes it a hard subject to understand, for many new to the concepts of limits - non-standard analysis is a lot better method to use instead of limits, as it gives an explicit understanding. I would really believe that at least some areas of non-standard analysis should be given more place in regular education; the standard part function is a perfect example. Non-standard analysis are also a good and firm ground to have if you want to learn limits.   There exists an elementary book written on the subject, which unforunately did not make it all the way into our schools: .  I think non-standard analysis to be a good thing to educate students in, at an early level, if you want to give them understanding about certain parts of math. But traditional methods are closely held to.  Why are traditional methods so hard held to? Why not integrate non-standard analysis into current school systems?
121395
Most students are first taught standard analysis, and the might learn about NSA later on. However, what has kept NSA from becoming the standard? At least from what I've been told, it is much more intuitive than the standard. So why has it not been adopted as the mainstream analysis, especially for lower-level students?
89315
Since the  makes a statement about domains defined in terms of real numbers, it would appear to invalidate statements about nature that we derived by applying real analysis. My reasoning is this:  If you can "duplicate" an abstract 3-dimensional ball defined, in the usual way, using the domain of real numbers, then clearly the domain of real numbers must be unsuited to describing physical objects (for instance The Earth), because duplicating them would double their mass (and thereby energy). But we use real numbers to reason about nature and derive new laws (or basic results) all the time in physics.  In the same way as we could integrate over the earth's volume to determine its mass from its density (for instance), we could apply the Banach-Tarski theorem to show that we could generate as many earths as we liked, right?  Does this mean that the mathematical foundation of physics is flawed? (I hope this question is not too provocative!)
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I'm wondering what some standard, modern references might be for continuum mechanics.  I imagine most references are probably more used by mechanical engineers than physicists but it's still a classical mechanics question.   This came up in a conversation with my father (who is a mechanical engineer).  I was curious to see the types of mathematics they use in stress analysis.  Complex analysis gets used to study 2-dimensional isotropic material here:     N.I. Muskhelishvili. Some basic problems of the mathematical theory of elasticity.    3rd edition, Moscow-Leningrad. 1949. (Translated by J. R. M. Radok. Noordhoff. 1953.)    but that's quite an old reference and my library does not have it.  Any favourite more modern and easy-to-find references?   I'm fine with mathematical sophistication (I'm a mathematician) but I'm not particularly seeking it out.  I'm looking for the kind of references that would be valued by physicists and engineers.  References "for math types" are fine too but that's not really what I'm after.
32700
For a frame of reference, I'm an undergraduate in mathematics who has taken the introductory analysis series and the graduate level algebra sequence at my university.   In my analysis class, our book lists axioms describing the structure of the reals. This seemed unnatural to me, as we can often write definitions or lists of rules that no set actually satisfies. So the axiomatic approach doesn't reassure me that the thing we are discussing can actually exist.  Our teacher talked to us about dedekind cuts as a way of explicitly constructing reals, which seemed more useful to me.  In my modern algebra class we discussed completion of the rationals as a way to construct the real numbers.  But this leads me to wonder - How do we know that all of these approaches and constructions result in the same structure, namely $\mathbb{R}$? Also, are the real numbers  the only complete, totally ordered field? If so, why?
32049
Before I ask my question, let me give you a bit of background about what I know about statistics so that you have a better sense of the types of resources that I'm looking for.  I'm a graduate student in psychology, and as such, I use statistics almost every day.  By now I'm familiar with a pretty broad array of techniques, mostly as they are implemented in the general structural equation modeling framework.  However, my training has been in the use of these techniques and the interpretation of results -- I don't have much knowledge of the formal mathematical foundations of these techniques.  However, increasingly, I've had to read papers from statistics proper.  I've found that these papers often assume a working knowledge of mathematical concepts that I don't know much about, such as linear algebra.  I have therefore become convinced that if I wish to do more than blindly use the tools that I have been taught, it would be useful for me to learn some of the mathematical basis of statistics.  So, I have two related questions:   What mathematical techniques would be useful for me to know if I want to brush up on the mathematical foundation of statistics?  I've encountered linear algebra pretty often, and I'm sure that learning about probability theory would be useful, but are there any other areas of math that would be useful for me to learn about? What resources (online or in book form) can you recommend to me as someone who wants to know more about the mathematical foundations of statistics?
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When learning mathematics I tend to try to reduce all the concepts I come across to some matter of interaction between sets and functions (or if necessary the more general Relation) on them. Possibly with some extra axioms thrown in here and there if needed, but the fundamental idea is that of adding additional structure on sets and relations between them.  I've recently tried applying this view to calculus and have been running into some confusions. Most importantly I'm not sure how to interpret Limits. I've considered viewing them as a function that takes 3 arguments, a function, the function's domain and some value (the "approaches value") then outputs a single value.  However this "limit function" view requires defining the limit function over something other then the Reals or Complexes due to the notion of certain inputs and outputs being "infinity". This makes me uncomfortable and question whether my current approach to mathematics is really as elegant as I'd thought. Is this a reasonable approach to answering the question of what limits actually "are" in a general mathematical sense? How do mathematicians tend to categorize limits with the rest of mathematics?
112154
Many students - myself included - have a lot of problems in learning scheme theory. I don't think that the obstacle is the extreme abstraction of the subject, on the contrary, this is really the strong point of modern algebraic geometry. I'm reading many books such those written by Hartshorne, Gortz &amp; Wedhorn, Liu, Vakil (notes), Gathmann (notes), Shafarevich, Perrin and Milne (notes) and in my humble opinion the learning problems arise from the following considerations:   It is enlightening to read about "the aim" of modern algebraic geometry, so I'm referring to: motivations behind schemes, the correpondence between algebraic and geometric entities (so the duality between  the category of affine schemes and the category of rings), the importance of sheaves (so the concept of admissible functions) etc. But, despite this, when one goes into the actual construction of the new objects, all theorems, lemmas and propositions are missing details (that are left to the reader). For example the verification that  certain presheaves are sheaves, functorial properties of assignments between categories and details about limit/colimit constructions are often missing. Even if the student has a solid background in algebra and geometry, generally they don't have the time or the capacity to complete all the statements. Basically taking a course in algebraic geometry implies that one must take many statements as acts of faith. I realize that authors and professors may have the same difficulties (especially lack of time) in writing down all the boring details, and moreover that a book with all proofs may include thousands of pages, but in this way students are encouraged (read discouraged) to simply memorize the most important results without really understanding the constructions. Finally, a book or a course characterized by explanations and by motivating as complete proofs is much more instructive than a book or a course which covers many advanced arguments IMHO. In mathematics when two object are isomorphic, it is a common practise to "identify" them. Basically if $A\cong B$ but $A$ has a simple description we write $A$ instead of $B$, but formally we are thinking of $B$. This procedure is used very often in algebraic geometry, but in some cases without explaining the isomorphisms and in other cases the two objects in question are considered "really the same" even if this can provoke formal problems (look for example  ). This "abuse of identifications" often makes one lose sight of the essence of what one is studying and once again the "stupid student", exhausted, tends to simply memorize things. I point again that the problem is not the abstraction, but the fact that the excessive tendency to simplify notation, often leads to inconsistencies. Enough importance is not given to the following: the process of successive generalizations, put in place by the great mathematicians across history, which marked the birth of modern algebraic geometry. This process is fundamental in learning because it probably represents the most natural way whereby the human mind can deal with the subject.    In summary, because of the above issues (principally the first two), rigorous mathematical statements, incredibly seem to be informal dissertations at the eyes of the student that is eager for formalism.  In your opinion, what are the most common difficulties that a student encounters during their learning process of algebraic geometry? If my problems do arise precisely from the above considerations, can you give me some advice to solve them?
121554
I was on the tube and overheard a dad questioning his kids about maths.  The children were probably about 11 or 12 years old.   After several more mundane questions he asked his daughter what $1/0$ evaluated to. She stated that it had no answer. He asked who told her that and she said her teacher. He then stated that her teacher had "taught it wrong" and it was actually $∞$.  I thought the Dad's statement was a little irresponsible.  Does that seem like reasonable attitude? I suppose this question is partly about morality.
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Looking through old problems, it is not difficult to see that some users are beyond incredible at computing integrals. It only took a couple seconds to dig up an example like .  Especially in a world where most scientists compute their integrals numerically, I find it astounding that people exist that can solve these problems analytically. Sometimes it is a truly bizarre contour, some examples use auxiliary functions and differentiation under the integral sign in a way that feels more like integration by wizardry.  Some of these users seem to have mastered an incredibly large number of special functions to a level of fluency that is almost unimaginable to me. Manipulations involving the error function almost looks like the work of an infant compared to some of these seemingly casual manipulations involving the Airy function, Barnes G function, the Legendre chi function and many more. Every time I read a post by them, it turns into an exercise in reading about yet another special function I have never heard of.   My question is what sort of material do you have to study, or what kind of area do you have to work in to get so good? To me, it doesn't seem as obvious as saying something like, "oh, they know this material because they are a topologist".
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It's one of my real analysis professor's favourite sayings that "being obvious does not imply that it's true".  Now, I know a fair few examples of things that are obviously true and that can be proved to be true (like the Jordan curve theorem).  But what are some theorems (preferably short ones) which, when put into layman's terms, the average person would claim to be true, but, which, actually, are false (i.e. counter-intuitively-false theorems)?  The only ones that spring to my mind are the Monty Hall problem and the divergence of $\sum\limits_{n=1}^{\infty}\frac{1}{n}$ (counter-intuitive for me, at least, since $\frac{1}{n} \to 0$ ).  I suppose, also, that $$\lim\limits_{n \to \infty}\left(1+\frac{1}{n}\right)^n = e=\sum\limits_{n=0}^{\infty}\frac{1}{n!}$$ is not obvious, since one 'expects' that $\left(1+\frac{1}{n}\right)^n \to (1+0)^n=1$.  I'm looking just for theorems and not their (dis)proof -- I'm happy to research that myself.  Thanks!
95089
In watching The Fault in Our Stars I could not help but cringe at a line that flew in the face of mathematics and subsequently ruined the movie for me:     "There are infinite numbers between 0 and 1. There's .1 and .12 and .112 and an infinite collection of others. Of course, there is a bigger infinite set of numbers between 0 and 2, or between 0 and a million. Some infinities are bigger than other infinities." - John Green   While walking out of the theater I tried to explain to my friends why there were, in fact, exactly the same amount of numbers between 0 and 1 as 0 and 2, but Cantor and bijective functions are not great learning tools to English majors.  Does anybody have an eloquent or elegant way to enumerate this phenomenon using an example accessible to those not familiar with advanced mathematics?
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In defining convergence, one requires $d &lt; \epsilon$ for some relevant "distance" $d$ and arbitrary $\epsilon &gt; 0$. Why not $d \leq \epsilon$?  This still allows one to get arbitrarily close to (but not equal to) the limit. Given any tolerance, we can always get at least as close as that tolerance by taking (for example, in convergence of a sequence) $N$ large enough.   Using $d &lt; \epsilon$ does seem more intuitive.
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Dimensional analysis, and the notion that quantities with different units cannot be equal, is often used to justify very specific arguments, for example, you might use it to argue that a particular formula cannot possibly be the correct expression for a particular quantity. The usual approach to teaching this is to go "well kids, you can't add apples and oranges!" and then assume that the student will just find it obvious that you can't add meters and seconds.  I'm sorry, but... I don't. I'm not convinced. $5$ meters plus $10$ seconds is $15$! Screw your rules! What are the units? I don't know, I actually don't understand what that question means.  I'm specifically not convinced when this sort of thing is used to prove that certain formulae can't possibly be right. Maybe the speed of a comet is given by its period multiplied by its mass. Why not? It's a perfectly meaningful operation - just measure the quantities, multiply them, and I claim that the number you get will always equal the current speed of the comet. I don't see how "but it doesn't make sense to say mass times time is equal to distance divided by time" can be a valid counterargument, particularly because I don't really know what "mass times time" is, but that's a different issue.  If it's relevant, I'm a math student and know extremely little about physics.
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I'm in a numerical analysis course right now and it's pretty rigorous but I'm enjoying it a lot. I took a lower level course before that was more oriented towards implementation of numerical methods, so it's not my first time with the material.  Anyway, my professor wrote his own course notes and they're fantastic but I wanted some reference material since I'm currently outlining my own notes. Just wondering what the big ones out there are  edit: I would say I'm at an advanced undergraduate level
24345
I am curious about which areas of mathematics one should be comfortable with before learning QFT.  I am familiar with the "learn-it-as-you-go" approach often advocated in physics, but would like to know how to avoid that in learning QFT.  Naming of specific textbooks is appreciated.  (For the sake of this discussion, "learning QFT" can be taken to mean "learning at the level of/from Peskin and Schroeder's text.")  In particular, I would like to know what, in addition to the material in Rudin's Real and Complex Analysis, one should know.  Would this be sufficient?  How about Folland's Real Analysis?
1877
I am a math master student and have done fundamental math courses like probability theory, measure theory, linear algebra and know a little bit about functional analysis. What is good way for me to learn machine learning in depth?  I have read the classical text  last summer; my impression was that it was very ineffective to read the book chapter by chapter like a mathematical text. The book does not go deep enough for many algorithms and skip too many steps considered too technical by engineers.   Is there a machine learning book that maybe does not cover too many topics, but treat each one in depth and takes advantage of math when necessary? It will be great to be able connect fundamental mathematical objects with machine learning (I am thinking about Lp spaces, hilbert space etc).
81156
At work we were discussing this as my boss has never heard of normalization. In Linear Algebra, Normalization seems to refer to the dividing of a vector by its length. And in statistics, Standardization seems to refer to the subtraction of a mean then dividing by its SD. But they seem interchangeable with other possibilities as well.   When creating some kind of universal score, that makes up $2$ different metrics, which have different means and different SD's, would you Normalize, Standardize, or something else? One person told me it's just a matter of taking each metric and dividing them by their SD, individually. Then summing the two. And that will result in a universal score that can be used to judge both metrics.  For instance, say you had the number of people who take the subway to work (in NYC) and the number of people who drove to work (in NYC).   $$\text{Train} \longrightarrow x$$ $$\text{Car} \longrightarrow y$$  If you wanted to create a universal score to quickly report traffic fluctuations, you can't just add $\text{mean}(x)$ and $\text{mean}(y)$ because there will be a LOT more people who ride the train. There's 8 million people living in NYC, plus tourists. That's millions of people taking the train everyday verse hundreds of thousands of people in cars. So they need to be transformed to a similar scale in order to be compared.  If $\text{mean}(x) = 8,000,000$  and $\text{mean}(y) = 800,000$  Would you normalize $x$ &amp; $y$ then sum? Would you standardize $x$ &amp; $y$ then sum? Or would you divide each by their respective SD then sum?  In order to get to a number that when fluctuates, represents total traffic fluctuations.  Any article or chapters of books for reference would be much appreciated. THANKS!  Also here's another example of what I'm trying to do.   Imagine you're a college dean, and you're discussing admission requirements. You may want students with at least a certain GPA and a certain test score. It'd be nice if they were both on the same scale because then you could just add the two together and say, "anyone with at least a 7.0 can get admitted." That way, if a prospective student has a 4.0 GPA, they could get as low as a 3.0 test score and still get admitted.  Inversely, if someone had a 3.0 GPA, they could still get admitted with a 4.0 test score.  But it's not like that. The ACT is on a 36 point scale and most GPA's are on 4.0 (some are 4.3, yes annoying). Since I can't just add an ACT and GPA to get some kind of universal score, how can I transform them so they can be added, thus creating a universal admission score.  And then as a Dean, I could just automatically accept anyone with a score above a certain threshold.  Or even automatically accept everyone whose score is within the top 95%.... those sorts of things.  Would that be normalization? standardization? or just dividing each by their SD then summing?