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4
votes
1answer
366 views

Getting better at math? [duplicate]

I really want to get better at math, but I'm not sure how. I have a book with competition problems from which I could learn, but I really don't enjoy sitting there for half an hour trying to find some ...
2
votes
0answers
70 views

Intuition behind criterion for an irreducible Markov chain to be transient

I have been looking over my notes for Markov chains, and I have come across the following: Theorem: An irreducible Markov chain is transient iff for some state $i$ there exists a nonzero vector $y$ ...
6
votes
3answers
2k views

Importance of Neatness / Organization / Speed in Math?

Pretty simple question here but it does relate to math. I ask this as my writing is quite messy, possibly a cause of silly mistakes. How important is neatness in math? Does having messy writing put ...
9
votes
3answers
767 views

Rationale behind truth values

I originaly asked a question on Programmers.SE to know why $0$ was consider $\text{false}$ and all the other [integral] values were considered $\text{true}$. That was a huge debate and many said it ...
9
votes
3answers
563 views

How to start learning knot theory?

Knot theory really sounds cool and I'm very interested in it. But I'm wondering what basic knowledge it is required and how I should start learning about it. Thanks
40
votes
3answers
783 views

Conjectural closed-form representations of sums, products or integrals

What are some examples of infinite sums, products or definite integrals that have conjectural closed form representations that were confirmed by numerical calculations up to whatever maximum precision ...
11
votes
7answers
3k views

Guides/tutorials to learn abstract algebra?

I recently read up a bit on symmetry groups and was interested by how they apply to even the Rubik's cube. I'm also intrigued by how group theory helps prove that "polynomials of degree $\gt4$ are not ...
1
vote
0answers
127 views

How should we study maths? [duplicate]

what is the best way to study maths ? also , should we memorize the proofs of the theorems ? or just read them and understand them but not memorize ? also , should we follow one text or more than ...
16
votes
2answers
903 views

Applications of information geometry to the natural sciences

I am contemplating undergraduate thesis topics, and am searching for a topic that combines my favorite areas of analysis, differential geometry, graph theory, and probability, and that also has ...
3
votes
1answer
2k views

Differential Geometry Video Lectures

I know there's a similar question here, however since what I found there wasn't what I was looking for I thought on creating a new question. I'm studying Differential Geometry through Spivak's book "A ...
1
vote
1answer
97 views

Special numbers in patterns and the reasons they are special

I know there are several big list questions out there (e.g. Patterns that break down at certain numbers) that touch on classifications of mathematical structures where certain numbers don't fit in, ...
3
votes
1answer
65 views

Intended Audience for N Is a Number?

Is the film N is a Number appropriate (mathematically) for a student entering high school?
13
votes
1answer
548 views

List videos of interesting courses at the doctoral level.

Many mathematics departments has provided video lessons their courses (usually one semester) that are offered in their doctoral programs in mathematics. Most often these courses total average of 26 ...
9
votes
4answers
363 views

Should $\mathbb{N}$ contain $0$? [closed]

This is a classical question, that has led to many a heated argument: Should the symbol $\mathbb{N}$ stand for $0,1,2,3,\dots$ or $1,2,3,\dots$? It is immediately obvious that the question is ...
3
votes
4answers
2k views

PDEs in biology

I am student who mostly heard lectures on partial differential equations and homogenization. But I really like the idea of working in biology or with biologists - but (with my lack of overview) it ...
5
votes
3answers
403 views

Statistics Workshop for High School Students

We are going to hold an introductory workshop about the statistics. The participants will be students who have just finished their 8th or 9th grade. The workshop consists of 10 two-hour sessions. The ...
2
votes
0answers
63 views

Request for translation from Russian: 'Bayesian Sufficiency' from a paper by Kolmogorov

The following seminal paper by the great Kolmogorov introduced the important statistical concept of Bayesian Sufficiency. Kolmogorov, A. (1942). Sur l’estimation statistique des paramètres de la loi ...
16
votes
5answers
2k views

Do we really need polynomials (In contrast to polynomial functions)?

In the following I'm going to call a polynomial expression an element of a suitable algebraic structure (for example a ring, since it has an addition and a multiplication) that has the form ...
22
votes
4answers
752 views

Why do we study prime ideals?

I hope this isn't an inappropriate question here! I'd like to ask the following (perhaps slightly ill-posed) question: why do we study prime ideals in general (commutative or non-commutative) rings? ...
3
votes
0answers
350 views

Imagining four or higher dimensions and the difference to imagining three dimensions

I’m very interested in how people envision four or higher dimensions. And I’m especially interested in how geometers and topologists who actually work in four dimensions do. Now I know of the video ...
3
votes
2answers
61 views

Repeating u-substitution

A question about the integration technique of u-substitution: Is it allowed to apply u-substitution over and over again, to reduce the integral to a more manageable form?
18
votes
3answers
764 views

Math blogs, pros and cons for writers?

I regularly read blogs by three mathematicians, and occasionally run into others. Definitely they help me a lot studying mathematics. But now I am more interested in the writers' perspective, and I ...
3
votes
1answer
217 views

getting started self study

I am 14 and currently in year 10 (uk). I am very interested in maths and intend to pursue it. I currently can do C1, C2, C3 and parts of C4 edexcel exam board a level mathematics. However, when i ...
1
vote
2answers
131 views

Why do we name equations more in applied math?

In pure math, equations aren't often named. For instance, we might define that $f$ is a (real) polynomial function iff there exists a finite sequence of real numbers $a_i$ such that $$f(x)=a_0 + a_1 x ...
13
votes
6answers
493 views

Should every group be a monoid, or should no group be a monoid?

Question: What is more convenient/useful? Writing mathematics as if every group is a monoid, or as if these two classes are disjoint? Additional discussion. Define a monoid as follows. Defn 1. A ...
3
votes
1answer
165 views

Can we use proper classes in this way, to define a new infinity larger than |Ord|?

I believe there is a way to do this that makes sense, and I explain it below. I would like to know if I did some obvious mistake, or if the idea doesn't make sense for some reason I didn't figure it ...
7
votes
4answers
1k views

Connections between number theory and abstract algebra.

I haven't taken abstract algebra yet, but I am curious about connections between number theory and abstract algebra. Do the proofs of things like Fermat's little theorem, the law of quadratic ...
51
votes
2answers
2k views

Unexpected approximations which have led to important mathematical discoveries

On a regular basis, one sees at MSE approximate numerology questions like Prove $\log_{{1}/{4}} \frac{8}{7}> \log_{{1}/{5}} \frac{5}{4}$, Prove $\left(\dfrac{2}{5}\right)^{{2}/{5}}<\ln{2}$, ...
24
votes
7answers
764 views

Examples of Diophantine equations with a large finite number of solutions

I wonder, if there are examples of Diophantine equations (or systems of such equations) with integer coefficients fitting on a few lines that have been proven to have a finite, but really huge number ...
45
votes
5answers
1k views
27
votes
4answers
1k views

Is $\mathbb{N}$ impossible to pin down?

I don't know if this is appropriate for math.stackexchange, or whether philosophy.stackexchange would have been a better bet, but I'll post it here because the content is somewhat technical. In ZFC, ...
2
votes
1answer
50 views

The best softwares to understand the intersections of the 3D objects in the Euclidean space

What is the best software (Easy to follow and clear graphics) to draw the intersections between two spheres, Two spheres and a pyramid, for example. The centre and the radius of the spheres are given ...
6
votes
2answers
875 views

Philosophy of a Mathematician.

Introduction I don't study Mathematics at university, and probably I don't have any chances to have a little understanding of what mathematics in all its aspects. But I love to find structures and ...
2
votes
3answers
542 views

Is Vector Calculus useful for pure math?

I have the option to take a vector calculus class at my uni but I have received conflicting opinions from various professors about this class's use in pure math (my major emphasis). I was wondering ...
189
votes
24answers
15k views

Is mathematics one big tautology?

Is mathematics one big tautology? Let me put the question in clearer terms: Mathematics is a deductive system: it works by starting with arbitrary axioms, and deriving therefrom "new" properties ...
2
votes
2answers
145 views

Is measure identical to magnitude?

In an introduction to Euclid's Elements , D.E Joyce writes: Treating angles as magnitudes should not be confused with measuring angles. The angles themselves are the magnitudes. The only ...
2
votes
3answers
606 views

Thermodynamics for math majors

I'm about to wrap a course in partial differential equations. We've discussed the heat/wave equations and introductory Fourier Analysis. I'd like to do some reading into the field of thermodynamics. ...
15
votes
3answers
6k views

What makes a good mathematician?

A soft question but I believe important to help get maturity in maths. Not until I got admitted to a graduate applied math program that I started to learn math. Before that, I was a life science ...
11
votes
2answers
555 views

Path Algebra for Categories

For a while I had been thinking that the path algebra of a quiver $Q$ over a commutative ring $R$ is the same as the "category ring" $R[P]$ (analogous to "group ring", "monoid ring", "semigroup ring", ...
18
votes
6answers
2k views

Why is the definition of “limit” difficult to understand at first?

Tomorrow I teach my students about limits of sequences. I have heard that the definition of limit is often difficult for students to understand, and I want to make it easier. But first I need to ...
5
votes
3answers
439 views

Does Differential Topology or Differential Geometry play a larger role in Chaos Theory?

I'm an undergraduate on somewhat of a time constraint in school. I have room in my remaining schedule for a semester of either Differential Geometry or Differential Topology. I understand the ...
16
votes
3answers
1k views

Are there interesting rings without unity?

There are several introductory textbooks which define a ring without any reference to a unity. However, nearly all of the rings one encounters in various branches of mathematics are endowed with a ...
6
votes
2answers
341 views

How can I get better at math? [closed]

I am a high school junior taking calculus AB. What should I do to become the best I can at math? Can you suggest any books I should read, a list of problems I can do, etc.? Thanks
4
votes
3answers
5k views

What are the real-world applications of real analysis?

I've read the wikipedia article on mathematical analysis and this, but I can't exactly find an answer. Is real analysis just some pure math, or does it really have something to with physical ...
5
votes
3answers
312 views

learning/teaching approach to rigorous math with the goal of improving

I will state this now: yes, this is a subjective question. But I feel the answers people give may benefit students. I want to get better at doing non trivial proofs. Real analysis is standard ...
3
votes
4answers
1k views

Sports that use Mathematics

What kind of sports and games use mathematics beyond simple arithmetic? How is math applied to build strategies for these games? Sailing could use mathematics in terms of astronavigation, tying ...
7
votes
2answers
5k views

Simplest examples of real world situations that can be elegantly represented with complex numbers

Mathematics could be defined as the study of formally defined abstractions. These abstractions may or may not be useful for describing real world phenomenon. Indeed, Physics could be defined as the ...
7
votes
2answers
1k 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 ...
5
votes
2answers
246 views

Are algebraic numbers analogous to group elements with finite order?

Would you say that the "elements with finite order" in group theory are analogous to "algebraic numbers" in field theory? I thought this is the case since requiring an algebraic number $\alpha$ to be ...
7
votes
3answers
5k views

Is there a correct order to learning maths properly?

I am a high school student but I would like to self-learn higher level maths so is there a correct order to do that? I have learnt pre-calculus, calculus, algebra, series and sequences, combinatorics, ...