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2
votes
0answers
39 views

Polynomial decomposition

I've just recently learned about the neat algorithm that, given a polynomial $f$ finds (non linear) polynomials $h,g$ such that $$f = g \circ h \quad (1),$$ or decides that there are no such ...
0
votes
1answer
30 views

Choosing final year project [closed]

I'm studying pure math now yet I would like to develop a career in actuarial science. For my final year project, I can choose to either the topic of probability analysis or statistic. While I know ...
2
votes
3answers
232 views

Existence of numbers such as $\pi^{-1}$

For my non-mathematics students (this particular class are computing), I would define $\displaystyle \frac{1}{n}$ for $n\in\mathbb{N}$ as the solution of the equation $$nx=1,$$ and then ...
4
votes
0answers
53 views

Can we consider a hypergeometric function as a closed-form?

Let's say a calculus problem like an integral or a series has a solution that inevitably involving a hypergeometric function. It turns out that hypergeometric function cannot be expressed in term of ...
6
votes
4answers
119 views

Combinatorial group theory books

I would please like some recommendations for an introductory level book on combinatorial group theory, by which I mean a group theory book which places emphasis on generators and relations and free ...
-1
votes
1answer
57 views

Most general mathematical framework

One can think of the same mathematical object in many different ways. For example take $\mathbb{R}$. One can think of this as (assume necessary hypotheses and so on) As a group. As a one ...
1
vote
1answer
71 views

Are the quaternions obsolete in pure mathematics?

I remember I read an article saying that "The quaternions $\Bbb{H}$ are obsolete in pure mathematics since the theory of vectors has been developed enough, however it is useful in computer science". ...
0
votes
1answer
24 views

2 x 2 matrix game and expected value

I have found the optimal strategy for the row player and column player. How do I find the expected value of the game for the row player and determine whether the game is favourable to the row player ...
2
votes
3answers
84 views

Prove that $S= \{ (x,y) : x^2 - y^2 <1 \}$ is open in $\mathbb{R}^2$

Prove that $S= \{ (x,y) : x^2 - y^2 < 1 \}$ is open in $\mathbb{R}^2.$ The question itself is rather easy and trivial by observing the $S$ in $\mathbb{R}^2$ geometrically. But if we are ...
0
votes
0answers
39 views

What are the most useful inequalities?

From a general point of view, when attacking problems where it could be useful, which are some of the most useful or handy inequalities that a mathematician can use as tools, based on your own ...
0
votes
1answer
22 views

Validity of the term 'cost' in 'cost function' [on hold]

I am studying machine learning and came across the term "cost function." Although I understand the basic idea, I am wondering how the term applies to specific problems (i.e., finding the cost of a ...
0
votes
1answer
26 views

Discuss the following graphs(Differential Equations)

So I have a differential equations midterm coming up soon, and in my last exam I messed the graphing question up. It was very similar to the one I am posting. All the questions said was "Discuss the ...
5
votes
4answers
107 views
+50

Software, techniques and tricks of experimental mathematics to conjecture possible closed forms

It often happens that people conjecture possible closed forms of integrals, series, and so on starting from a numerical value calculated to very high precision. What are the techniques, tricks, ...
0
votes
0answers
38 views

Exist some kind of irreversible transfomations on maths?

I know that this kind of transformation by itself without control can lead to contradiction because it value change depending the state of the function where you do the transformation. Anyway I want ...
4
votes
0answers
49 views

From algebraic master degree to algebraic geomery Phd

I am a foreign master student in algebra at the final year. I'm familar with categorical algebra and have interesting in algebraic geomery and number theory. I have learned some knowledge about scheme ...
28
votes
3answers
2k views

“Stick it to the man!” Mathematical discoveries that resulted in persecution.

As the old story goes, Pythagoras and his followers were adamant that all numbers were rational, until Hippasus came along and proved that $\sqrt{2}$ (the length of the diagonal of the unit square) is ...
3
votes
1answer
42 views

Recommendation on setting the reference axis for mathematical objects

(I don't know what the title should be for this post, please change it if you have a better title. Also tags) In many situations, there arises cases that one mathematical structure embeds into ...
2
votes
4answers
156 views

Theorems in number theory whose first proofs were long and difficult

What are the examples of important theorems of number theory that has been shown to have surprisingly simple proofs though their first demonstration wasn't at all simple enough. Now simple proof is an ...
5
votes
0answers
128 views

Can I use any theorem I know at an IMO? [closed]

What if I happen to know a (fairly well-known) theorem that trivializes a given problem set at a math contest? Could my answer be rejected (unless I provide proof)? For example, see this question on ...
2
votes
0answers
36 views

Typical course of study chart [closed]

I am an undergraduate student in math and am very interested in the field. I have bought a few books and self-studied on my own as i have free time. The next book i found on amazon was riemannian ...
2
votes
0answers
85 views

Algebraic approach to analysis

Can topics and foundations of real analysis be interpreted and profitably explained in terms of abstract algebraic structures? If so, what papers or books (accessible to undergraduate students) ...
2
votes
1answer
49 views

How we can drow a Blaschke $3$ ellipse?

Today I read the article Ellipses and Finite Blaschke Products (www.jstor.org/stable/3072367 Blaschke ellipses) by Ulrich Daepp, Pamela Gorkin, and Raymond Mortini. In there they have proved very nice ...
2
votes
3answers
166 views

Big list of books on counterexamples and other clever observations in different topics

This question is related to Counterexample Math Books, but I'm looking for books in areas which aren't covered there (for example, number theory). In addition, books that focus on clever ...
1
vote
1answer
34 views

Fixed-point theorem restriction in numerical analysis

The Banach fixed-point theorem states that if $f:[a,b]\to [a,b]$ is $\lambda$-Lipschitz where $\lambda\in[0,1)$ is such that satisfies $|f(x)-f(y)|\leq \lambda |x-y|$ for every $x,y\in [a,b]$ (I'm ...
14
votes
6answers
2k views

Why do some mathematical ideas seem counter-intuitive?

Suppose you play the following game: There's a certain buy-in, and at every turn you flip a coin. If anytime you flip a tail, you lose the game and leave with your winnings. If you flip a head on the ...
2
votes
3answers
105 views

Undergrad level presentation on homological algebra and some related topics

I'm a TA of an introductory course about modules, category theory and homological algebra and the students have to do a 2 hour long presentation as a final exam. There's one student who really likes ...
3
votes
0answers
75 views

Do mathematicians suffer from higher rates of madness?

After seeing this post I watched the BBC documentary Dangerous Knowledge. Overall I felt it worthwhile, but couldn't help bristle at the explicit connections made between mathematical abstraction and ...
3
votes
1answer
46 views

Intuition behind an algebraic identity

This problem Compute $$\sqrt{\sqrt{44\cdot 45 \cdot 46 \cdot 47+1}-44}$$ has a nice solution that relies on the identity $$n(n+1)(n+2)(n+3) +1 = \left(n^2 + 3n + 1\right)^2$$ a word form of ...
1
vote
1answer
42 views

What best explains the 'perspective' effect in this image?

I found this image and I would like to replicate the effect algorithmically. How would one describe this distortion effect mathematically? The rate of change of scale seems very familiar, and the ...
1
vote
2answers
65 views

Is it possible to not have irrational numbers?

(Math noob question): Is there a base that can be used like binary that produces no irrational numbers or numbers with an infinite amount of one number after the decimal (don't know the name)? I feel ...
2
votes
1answer
74 views

Intuitive understanding of logarithms

I know logarithms are supposed to be the inverse of exponential functions, and while this makes sense, it seems to me that a more intuitive and significant property is $$\log (ab) = \log(a)+\log(b)$$ ...
2
votes
0answers
93 views

Learning Roadmap to Mathematical Physics

Currently, I am a graduate student specializing in algebraic geometry. On the other hand, I have also become extremely interested in the mathematical physics. However, I am not sure what steps I ...
1
vote
1answer
48 views

Unit vector symbols/names

I am currently studying vectors and matrices in 3 dimensions, my book calls the unit vectors i j and k, however I have seen them being called in other ways, such as: x-hat, y-hat and z-hat; or simply ...
0
votes
0answers
18 views

Probability and Measure, Gravity and Curvature, and What Else?

Apart from the great discoveries of mathematical linkage such as Kolmogorov's axiomatic probability, which links probability to measure, and Einstein's general theory of relativity, which links ...
0
votes
2answers
86 views

Quicker way to compare numbers without calculator

Question: Find the order of $(1/2)^{1/2}$, $(1/e)^{1/e}$, $(1/3)^{1/4}$ without using calculator. Extra constraint: You only have about 150 seconds to do it, failing to do so will eh... make you run ...
0
votes
0answers
44 views

Prime Gap Conjectures

There are many conjectures which can be proven by proving a tight enough upper bound for the prime gap above $n$. Some of these conjectures are stronger than other conjectures and imply the weaker ...
8
votes
2answers
547 views

Should I be afraid of using known identities that I can't prove?

Recently I've noticed a pattern in all of my "researches" (if you can call them that), and that is I will not allow myself to use known identities if I can't prove them (or at least understand a given ...
2
votes
0answers
27 views

Why study physical differntial equations in$\mathbb{R}^n$

Studying equations such as the heat equation and the wave equation in n$\le$3 dimensions makes sense to me as these are physical processes. I can also justify studying PDEs in $\mathbb{R}^n$ because ...
1
vote
0answers
56 views

Big list of fun mathematical book to “play” with classmates

I am searching for some fun maths books "have fun" (mathematically) with my classmates. To give you a better idea of what I'm looking for, I'll mention some books that I find suitable: Roger B. ...
0
votes
0answers
67 views

Consequences of irrationality of e

We know that $e$, $\pi$ are irrational... But WHY do we know it? I am going to give a lecture about irrationality of e, and I'm looking for a reason why it is an interesting subject. I know the proof, ...
1
vote
2answers
70 views

How to teach Critical Thinking

I am currently tutoring a few students in an entry level physics course and had some trouble recently when it comes to helping them with problem solving. The students I am helping don't have many ...
2
votes
1answer
52 views

Career counselling undergraduation underconfidence

I don't know whether this is the right place to ask this, but I expect you to help me. I am a third year undergraduate student in the Indian Statistical Institute, Bangalore. After completing two ...
-2
votes
2answers
92 views

Best Mathematical Logic Books the Style of Which is Like a Mathematics Publication rather than a Logic Publication?

I found many good mathematical logic books are written like a publication in the field of Logic. For instance, in such books I would see such as "For every $x$, if $x$ is a real number then $x^{2} ...
1
vote
0answers
18 views

Geometric Realization of Finite Dimensional Abstract Simplicial Complex

I am learning the theory of complex. And there are two theorems presented by our teacher: Every abstract complex $K$ has its geometric realization. Every $n$-dimensional abstract complex $K$ has its ...
51
votes
11answers
9k views

Am I just not smart enough? [closed]

When I was doing math, let us say for example, introductory number theory, it seems to take me a lot of time to fully understand a theorem. By understanding, I mean, both intuitively and also ...
5
votes
1answer
99 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 ...
1
vote
1answer
55 views

Useful techniques of experimental mathematics (reference request)

I am searching for papers or books that explain thoroughly useful interesting techniques of experimental mathematics that can be understood and profitably applied by an undergraduate student.
4
votes
0answers
42 views

Numbers Made From Concatenating Prime Factorizations

I came across the following curious problem while playing around with my calculator. Take any positive integer $n$; for this example we'll use $216$. Create a sequence as follows: Factor $n$ into ...
7
votes
1answer
118 views

Reference request: books that describe application of physical reasoning to mathematical problems

I am searching for more books like Uspenski's Some applications of mechanics to mathematics and Levi's The Mathematical Mechanic. In other words, I am looking for books that show interesting and ...
0
votes
1answer
58 views

Physical significance of the fact that the cardinality of the real number line is the same as a finite interval of the real number line

It is known that the cardinality of the real number line is the same as a finite interval of the real number line. Is there a physical meaning of this apparently conter-intuitive statement?