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0
votes
2answers
151 views

Why do Matrices work the way they do?

You get taught about matrices and how they work but nobody ever tells you WHY they work in the way that they do. What was the idea that sparked the creation of matrices?
4
votes
1answer
75 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 ...
2
votes
1answer
105 views

How to improve my grades.

I am 20 and in my 3rd year of undergraduate study. Today I wrote an exam (Measure and integration theory) and well, it was not easy for me, although I was prepared well. My motivation for this ...
2
votes
4answers
328 views

The fundamental group of Cayley graph

Today I read a math book and find interested in Theorem. Every group has its graph representation. And we call it cayley graph. Now we sort out the question Firstly, if we have a group, ...
4
votes
0answers
104 views

How can math students (not instructors) improve studying by Learning through Testing?

I hit upon Alex K. Chen's answer on What learning strategies do people who are "quick learners" follow? which links to this New York Times article To Really Learn, Quit Studying and Take a Test . ...
4
votes
0answers
178 views

For maximum productivity, how many math subjects or subfields should be studied daily? [closed]

I quote Ben Baert on Quora at http://qr.ae/hkuCd: Study every subject only for short periods and take regular breaks: during my 5 or 6 hours - preferably spread out over the course of the day, I try ...
16
votes
1answer
277 views

How often should math students take day breaks and longer breaks or vacations? Any research?

John Edensor Littlewood wrote in page 197 of Littlewood's Miscellany "For a week without teaching duties - and here I think I am preaching to the converted - I believe in on afternoon and the ...
0
votes
2answers
51 views

Mathematical structure of a dictionary.

Please correct me if I am wrong. I understand that during the last couple of centuries we have put a lot of effort in trying to understand structures, and for most of them we have been able to reduce ...
2
votes
1answer
76 views

Knot Theory: Calculating the Alexander Polynomial

I am going to be giving a talk about knot thoery in a few weeks and I will be discussing different knot invariants-one of which being the alexander polynomial I am having a problem understanding how ...
53
votes
19answers
18k views

How do I convince my students that the choice of variable of integration is irrelevant?

I will be TA this semester for the second course on Calculus, which contains the definite integral. I have thought this since the time I took this course, so how do I convince my students that for a ...
1
vote
2answers
228 views

A problem for math lovers to count the digits

Today a classmate of mine asked a question which is based on counting. Question. Find a positive integer which when multiplied up to $6$ times will give numbers having the same digits but rearranged ...
10
votes
2answers
218 views

When the mathematical community consider the inclusion of a new axiom?.

At first I was thinking about the axiom of choice, but let's keep it general. What motivates the inclusion of new axioms (or change the ones we already have in an already defined axiomatic theory?. It ...
17
votes
3answers
307 views

Results in graph theory proved using other areas of math, and vice versa

I'm curious about learning graph theory, as it seems to pop up in some unexpected places. In order to get a partial feel for the subject, I was wondering if anyone could point me to some survey ...
1
vote
0answers
70 views

Are forward references in a book OK?

In a math book I wrote, there is one forward reference (I'm sure not a proof loop). Should I work hard to eliminate forward references, or one such ref is OK?
2
votes
1answer
122 views

When have you 'read' a math book?

I have been pondering the following for some time: What does it take to say ''I have read that math book''? I realize that I want to $\textit{read}$ some math books, but I'm not quite sure when I can ...
6
votes
5answers
987 views

Teaching irrational numbers?

I'm interested in teaching the irrational numbers to high-school students, and I need your ideas on how to do this in an 'optimal' and innovative way. And my question is: What should the teacher know ...
4
votes
2answers
749 views

Definition: Theorem, Lemma, Proposition, Conjecture and Principle etc.

Definition: Theorem, Lemma, Proposition, Corollary, Postulate, Statement, Fact, Observation, Expression, Fact, Property, Conjecture and Principle Most of the time a mathematical statement is ...
2
votes
0answers
103 views

Exercises or courses to improve logical rigor and reasoning skills

There is plenty of math that is beautiful without needing much explanation of theory, such as fractals, geometric patterns and the Game of Life, that may interest beginners in mathematics. However, if ...
1
vote
1answer
42 views

Introduction to stochastic control

I'm looking for an introductory text on stochastic control. Any suggestions?
2
votes
1answer
117 views

Are there mathematical blogs/websites which publish “pop-math”?

Are there mathematical blogs/websites which publish "pop-math" (that is, simple and nice articles on interesting topics aimed at ...
4
votes
0answers
848 views

Is there any English version of Récoltes et Semailles?

I felt like my question isn't appropriate for MO, so I though maybe I should post it here. I want to read Alexander Grothendieck's "Récoltes et Semailles", but I don't know any French. I can easily ...
6
votes
6answers
182 views

Any ideas how I can rewire my brain such that $\varphi \leq \psi$ “obviously” means that $\varphi$ implies $\psi$?

The Boolean domain $B=\{\mathrm{False},\mathrm{True}\},$ can be viewed as a partially ordered set in two different ways. In the best approach, $\mathrm{False}$ is the least element and $\mathrm{True}$ ...
1
vote
0answers
35 views

Schoenfeld's limits & almost primes

If I am correct in my understanding, the Tao-Green theorem employed almost primes as density normalisers. If $N_k(x)$ is the counting function of almost primes, is the study of almost primes 'useful' ...
2
votes
3answers
276 views

Resources for properly developing a modern understanding tensors

I am currently learning about tensors as they come up in the mathematics behind continuum mechanics. I was fairly disappointed with my initial foray into tensors, as presented in the book Classical ...
1
vote
1answer
45 views

Complement Theorem and Steinitz Theorem, are the same thing?

There is a theorem about basis of a vector space that says: Let $u_{1}, u_{2},...,u_{p}$ be linear independent vectors of some vector space, $E$. Suppose that each $u_{i}$, with $i=\{1,2,3,...p\}$ ...
2
votes
1answer
42 views

What does “Good properties” mean?

in my mathmatics skript the professor wrote: "Warning: an inverse function does not automatically inherit all of the good properties of a function!" I don't really know what he means with "good". ...
7
votes
1answer
595 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 ...
2
votes
1answer
70 views

If $X$ and $Y$ are objects of $\mathrm{Set}$, is there any reason not to regard $\mathrm{Hom}(X,Y)$ as an object, too?

Suppose $X$ and $Y$ denote objects of $\mathrm{Set}$. Is there any reason not to say: therefore, $\mathrm{Hom}(X,Y)$ is itself an object of $\mathrm{Set}$? Furthermore, is there any reason not to ...
12
votes
2answers
204 views

How can you describe topology to a non-mathematician without using continuous deformations?

One of the most frequently used ways to describe topology to non-mathematicians is that it studies the properties of objects that are preserved under deformations where ripping or tearing is not ...
2
votes
2answers
94 views

Publishing mathematics content in a blog [closed]

I would like just to publish few math-related blog entries on blog.com. For drawing and animation, I can use pictures (for animation, they would be animated gifs). However, what to do with equation, ...
0
votes
1answer
181 views

Convergence of a sequence in trigonometric functions

Is the sequence $(a_n)$ defined as $$a_n :=\dfrac {\sin\Big( \dfrac {\pi}4+\dfrac n2 \Big)\sin\Big(\dfrac{n+1}2 \Big)}{\sin\Big(2n+2\Big)\sin\Big( \dfrac {\pi}4+\dfrac {n-1}2 \Big)\sin\Big(\dfrac{n}2 ...
2
votes
1answer
54 views

Comparison of notation for sets

Different authors use different notation, no question here...but doesn't this make the study of maths a little more difficult, always chasing different definitions of how a set is represented? I ask ...
3
votes
4answers
299 views

Why do logarithms produce such difficult problems

This was inspired by Fun logarithm question, because it made me remember a question I accidentally asked on a quiz some time ago. It was suppose to have both log bases the same, 3 or 5. ...
-1
votes
1answer
126 views

Insistence on classical geometry [closed]

This post may surprise and even irritate some people here but I am sure that if any sort of discussion springs out of it, we will get to read some interesting opinions. For the rest of the post, if I ...
0
votes
1answer
76 views

Path or One-Dimensional Manifold

The terminology of curve, path and one dimensional manifold is always in the textbooks about topology, differential manifold and riemannian geometry. The definition of path and one dimensional ...
10
votes
0answers
272 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 ...
45
votes
8answers
2k views

How to maintain enthusiasm and joy in teaching when the material grows stale

I recently finished my third semester of teaching calculus to freshman college students. This means I was drawing the same pictures, solving the same example problems, and discussing the same ...
8
votes
1answer
390 views

“All math is useful eventually”

We have all heard the argument : a lot of mathematics that was thought to be useless, abstract constructions with no links to the real world ended up being of use, like some arithmetic is useful in ...
6
votes
8answers
567 views

How t0 stop being obsessive about small, trivial things [closed]

I'm a student of math, my problem is that I tend to overformalize and overanalize some things and definitions and so I keep wasting time and effort in pointless things. For example, one often finds ...
12
votes
5answers
1k views

Self studying math, how can I learn the most?

I am currently studying Pre-Calculus on my own. I have a few texts I am working with but feel like I could learning a lot more than I am. When people typically ask these kind of questions the common ...
2
votes
0answers
45 views

Managing research ideas

I apologize if my question is not appropriate for this site, but I am interested in the opinion of mathematicians... in fact I would like to learn as many different opinions as possible. The question ...
4
votes
1answer
51 views

Is $2^\alpha=2^\beta\Rightarrow \alpha=\beta$ a $\sf ZFC$-independence result?

In a lecture recently one of my lecturers was proving something to do size of basis or something (I can't remember exactly) and somewhere near the end of the proof we had the following: ...
14
votes
6answers
695 views

When to use the contrapositive to prove a statment

My question tries to address the intuition or situations when using the contrapositive to prove a mathematical statement is an adequate attempt. Whenever we have a mathematical statement of the form ...
5
votes
5answers
597 views

Why are second-order 'things' studied so much in mathematics?

In many areas of math, I've been surprised at how much research, past and present, focuses on second order 'things'. Examples: Number theory: quadratic reciprocity, quadratic number fields Analysis: ...
2
votes
1answer
2k views

How to stop making stupid mistakes

I know that this question has been asked before by others however I just can't get around this problem. I am in my final year of high school and I need to find a solution before my final exams. For ...
2
votes
5answers
119 views

I am going to learn these Mathematics Topics. I need advice and suggestions please .

I am really horrible when it comes to maths since I never had any maths background in my High school. I am fairly good at programming ( C++ and Java) but without mathematics I cant advance in any ...
1
vote
0answers
128 views

Soft question - having difficulty with rigorous proofs

What exactly is in your opinion the best way to understand rigorous proofs?I seem to be having a number of problems when it comes to learning mathematics. .While I easily get the intuitive part, the ...
7
votes
1answer
163 views

General Topology: “Follow your Nose Approach”

So, this is definitely a soft question and I apologize. I've been in point set topology for about a week and I have two questions, Everything spews from definition, so should I dismiss my geometric ...
21
votes
9answers
3k views

Is there an example that a theorem in number theory is useful in another field in mathematics?

I know there are two advanced approaches to number theory. That is, algebraic number theory and analytic number theory. I have heard that algebraic geometry, which generally seems completely different ...
4
votes
1answer
202 views

How to Generalize Pick Theorem

I am interested in Pick Theorem. Pick Theorem Assume $P$ is a convex lattice point polygon. If $B$ is the number of vertexes of $P$ and $I$ is the number of lattice points which in the interior of ...