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5
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2answers
543 views

Advice: How can one prepare for a maths entrance exam? How can one develop mathematical thinking? [closed]

I am Computer science student so pardon me if I am asking this in a wrong place! My inspiration of learning maths is mainly due to algorithms and its feels to me that without mathematics I am not ...
4
votes
3answers
319 views

Mental Math Techniques [closed]

What are some interesting mental math techniques that you know? Here's one that I got from my Grandmother who got it from a book: To square a two-digit number (from $26$ to $49$), take the number ...
4
votes
3answers
545 views

Most inspirational mathematical books [closed]

I would like to know which books on mathematics (from university texts to divulgative pop-math books) inspired you the most. My choice is Spivak's Calculus, which is, IMHO one of the most ...
4
votes
2answers
433 views

Homological algebra in PDE

I have been fascinated by the power and wide applicability of homological methods in algebra and topology. Because I am also interested in PDE, there arises a natural question for me. What is ...
4
votes
3answers
1k views

Help in self-studying mathematics.

Is this a reasonable list for who seek to learn mathematics by "self learning" program? and is it a well sorted list to follow? http://www.math.niu.edu/~rusin/known-math/index/index.html
3
votes
1answer
449 views

Tips for finding the Galois Group of a given polynomial

I am currently in an introductory Galois Theory course, and I thought it would be nice to compile a list of standard tricks for finding the Galois Groups of certain polynomials. I am studying from ...
3
votes
6answers
698 views

Are there broad or powerful theorems of rings that do not involve the familiar numerical operations (+) and (*) in some fundamental way?

I am of, and I would like to retain, a mindset that mathematics does not have to have numbers as the central object of interest. With that in mind, I have done a fair amount of self-study on topics in ...
2
votes
1answer
347 views

Analogy of ideals with Normal subgroups in groups.

I've started with Ideals in ring theory but still not comfortable with the analogy it has with normal subgroups in group theory.Like we can visualize normal subgroups as Is there some good intutive ...
2
votes
1answer
147 views

Understand a weird method of calculus

I see this method of calculus on youtube and my question: is this method valid? How we can understand it? Thanks.
17
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5answers
5k views

Is 'no solution' the same as 'undefined'?

Today in class my teacher wrote something along the lines of: $6^x = 0$ And proceed to heed a response from the class. A few people shouted undefined. So the teacher then writes: no solution ...
16
votes
5answers
741 views

Reflections on math education

Why is there such a big difference in math education between The Americas and (Europe and Asia) ? except for a few privileged who have the opportunity to access to math much earlier than the ordinary ...
16
votes
6answers
971 views

What are some examples of induction where the base case is difficult but the inductive step is trivial?

According to Wikipedia: ...proofs by mathematical induction have two parts: the "base case" that shows that the theorem is true for a particular initial value such as n = 0 or n = 1 and ...
14
votes
3answers
803 views

Transcendental number

While reading on Wikipedia about transcendental numbers, i asked myself: Why is it so hard and difficult to prove that $e +\pi, \pi - e, \pi e, \frac{\pi}{e}$ etc. are transcendental numbers? ...
13
votes
2answers
865 views

Studying mathematics efficiently

I am particularly angry at myself for the last few days. I noticed how inefficiently I work. Here is the general scenario: I decided to study abstract algebra and analysis some days back. I tried ...
13
votes
2answers
834 views

Who is a Math Historian?

In the context of classes, it is very often that discussion on the history of mathematics arises, whether it'd be on who should a lemma be attributed to or a certain event that occurred during the ...
12
votes
2answers
363 views

What was the largest ratio (result size)/(integrand size) you have seen?

Sometimes a definite or indefinite integral of a simple-looking one-liner integrand can give astonishingly huge result. What was the largest ratio of the size of shortest known closed-form result to ...
12
votes
7answers
686 views

Is it possible to alternate the law of mathematics?

I am freelance writer. Recently I have been planning a science fiction - just planning, nothing solid yet - and I was wondering would it be possible for some other universes that have different set of ...
12
votes
3answers
523 views

Elementary Geometry Nomenclature: why so bad?

A long-ish wall of text, and I apologize. Some background: when I was a first-year university student, my chemistry professor was lecturing and was trying to find the word to describe a shape. A ...
11
votes
2answers
190 views

What's the idea of an action of a group?

I know the formal definition of an action over a set. I'm not asking this. What I'm asking is: what's the intuition of it? It is a way to define an algebra over a set? Since an action can exist in ...
10
votes
0answers
295 views

Big geometry grad schools - for an average applicant [closed]

What are some schools that have a lot of geometry going on, but that might accept some middle-of-the-range applicants? Let me add some context... I left grad school (UC Davis) with an MS in 2012 ...
10
votes
5answers
343 views

A question of H.G. Wells' mathematics

H.G Wells' short story The Plattner Story is about a man who somehow ends up being "inverted" from left to right. So his heart has moved from left to right, his brain, and any other asymmetries ...
10
votes
1answer
207 views

Independence results that cannot be established by forcing.

I read the Wikipedia article on Absoluteness recently and found mention of Shoenfield’s Absoluteness Theorem, which states that if $ \phi $ is any $ \Sigma^{1}_{2} $- or $ \Pi^{1}_{2} $-sentence of ...
10
votes
2answers
2k views

*Recursive* vs. *inductive* definition

I once had an argument with a professor of mine, if the following definition was a recursive or inductive definition: Suppose you have sequence of real numbers. Define $a_0:=2$ and ...
10
votes
1answer
587 views

Undergrad Student Trying to Figure Out What to Study

this is my first time on stack exchange and I am seeking advice for my future studies. Some background first; I am a undergraduate student pursuing a degree in mathematics and I hope to pursue ...
10
votes
2answers
721 views

Recalling Proofs

When I am able to follow a proof presented in class or in a textbook, I usually can prove the same corollary or theorem a couple days later using the same arguments. But after a week of seeing the ...
10
votes
2answers
6k views

Collatz finally solved? [closed]

http://www.newscientist.com/blogs/shortsharpscience/2011/06/simple-number-puzzle-possibly.html What does mathematicians say of this proof, right or wrong?
9
votes
1answer
338 views

Why is the permanent of interest for complexity theorists?

Studying a bit about the determinant and the permanent, I'm told that although both concepts have very similar formulas, the permanent was of not much interest historically - it was until later that ...
9
votes
4answers
279 views

Math games for car journeys

On long car journeys with kids we are all familiar with "I spy" or "Twenty questions". What math related games can one play on a car journey instead that are fun and offer similar variety?
9
votes
1answer
188 views

Could we reasonably classify finite groups?

So I have been reading some work on $p$-groups, and I noticed a particularly disturbing sentence: "There is no hope of finding a finite set of invariants that will define every p-group up to ...
8
votes
2answers
120 views

How can using a different definition for the integral be useful?

It's often said that the Lebesgue integral is superior to the Riemann integral because it satisfies nicer properties, for instance things like $$\lim_{n\to\infty} \int f_n = \int \lim_{n\to\infty} ...
8
votes
1answer
226 views

Construction(s) of new integral domains from “old ones”

Given an integral domain $D$, there are several ways how to construct a new integral domain related to D. For example, one can consider a ring of polynomials/formal power series/formal Laurent series ...
8
votes
2answers
486 views

Suggestions for further topics in Commutative Algebra

I am currently taking a semester long course in Commutative Algebra. We have covered a lot of dimension theory, and today finished proving Zariski's Main Theorem, which was the professor's original ...
8
votes
3answers
758 views

Are all numbers real numbers?

If I go into the woods and pick up two sticks and measure the ratio of their lengths, it is conceivable that I could only get a rational number, namely if the universe was composed of tiny lego ...
8
votes
2answers
1k views

When can one expect a classical solution of a PDE?

When solving a PDE, there may be a classical solution or a weak solution (or distribution solution). But I am wondering that when people talk about "finding a solution" to some PDE, what do they refer ...
7
votes
1answer
164 views

How to properly use technology for back-of-the-envelope calculations?

I'm usually quite eager on using technology wherever sensibly applicable, however whenever I make some calculations I still end up using a pen and paper, by now resulting in an entire pile of sheets ...
7
votes
2answers
2k views

Algebraic geometry project ideas for high school students

I am teaching a "senior seminar" course for strong students at our local high school. For 6 weeks the students learned about basic/classical algebraic geometry. In a few weeks they will start projects ...
7
votes
1answer
616 views

I feel that (physics) notation for tensor calculus is awful. Are there any alternative notations worth looking into?

I am reading through Fung and Tong's "Classical and Computational Solid Mechanics", and feel that the Einstein summation convention saves a few symbols, at the expense of a lot of clarity. Along with ...
7
votes
1answer
243 views

Is it possible to avoid redundancy in a foundational work?

Imagine we're developing all of mathematics from scratch. We settle on using a set-theoretic foundation. Early on, we assert that an ordered pair $(x,y)$ can be abbreviated $xy$ whenever there is no ...
7
votes
2answers
324 views

A Question on RH relating to Prime Number theorem

Well, in a previous post regarding the explanation of Riemann Hypothesis Matt answered that: The prime number theorem states that the number of primes less than or equal to $x$ is approximately ...
7
votes
2answers
548 views

A question on FLT and Taniyama Shimura

Sometime back i watched the documentary of Andrew Wiles proving the Fermat's Last theorem. A truly inspiring video and i still watch it whenever i am in a depressed mood. There are certain ...
6
votes
2answers
614 views

Advice: Modern vs. Classics

First of all, my apologies if (well, I know I am but I don't know where to put it) I am posting this in the wrong place. So please feel free to move it to someplace else or to tag it differently if ...
6
votes
3answers
662 views

Why is Kunen inconsistency at the top of Cantor's upper attic?

Motivation: I have reproduced part of page 396 and 397 from Handbook of Mathematical Logic below: So if we start with a concept of number and play the game of naming the largest one, does Kunen ...
5
votes
1answer
275 views

Guide to mathematical physics?

I am currently a math phd student specializing in algebraic geometry aspiring to work at the boundaries of the the fields of mathematics and physics and so, was looking into the field of mathematical ...
5
votes
4answers
355 views

Are problems about finding the next term of a sequence mathematical? [closed]

I have run through many questions like "given the few first terms of a sequence, find the next one". My question is "are such problems mathematical?". This is something like given $f(1), f(2), f(3)$, ...
5
votes
1answer
84 views

Have arrows in a category with this property a special name?

Studying posets I encountered the notation $a\prec b$. It means that $a<b$ and no $c$ exists with $a<c<b$. If $a\prec b$ then in words $a$ is covered by $b$. Looking at a poset $P$ as a ...
5
votes
4answers
371 views

Would nonmath students be able to understand this?

For a course, I am required to do a presentation. The topic could either be something mundane, like a career strategy report, or something more interesting, such as a controversial topic, or an ...
5
votes
0answers
214 views

Convex Hulls vs Shrink Wrap

I was recently explaining to a friend what the convex hull of a set of points is using the analogy of an elastic band around a set of nails hammered into a board. I was about to say that we can ...
5
votes
3answers
322 views

Is there a branch of mathematics that requires the existence of sets that contain themselves?

I notice that Russell's paradox, Burali-Forti's paradox, and even Cantor's paradox, all depend on our tolerance of sets that contain themselves (at one level of depth or another). Thus, I was thinking ...
5
votes
1answer
1k views

Can I go through Hartshorne without knowing much analysis?

I know intro abstract algebra and some real analysis. Is this enough to study algebraic geometry from the book of Hartshorne?
5
votes
2answers
381 views

Map of Mathematical Logic

My undergraduate University does not offer advanced courses on logic, I know truth tables, Boolean algebra, propositional calculus. However I want to pursue Mathematical Logic on the long term as a ...