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9
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0answers
100 views

How are contest problems designed? [duplicate]

How are competition questions designed? What techniques do designers employ to design math competition questions? How they know a problem can be solved by introductory methods?Some contest math ...
1
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1answer
147 views

Books on coordinate transformations

Which are the best books on linear algebra that explain in detail coordinate transformations (that math about matrices also used in computer graphics)?
1
vote
1answer
153 views

What are good programs for writing expressions?

By writing expressions, I mean intermediate expressions to "show your work," such as the ones in the below image. LaTex would seem like overkill for such problems.
1
vote
0answers
82 views

Chi-square or chi-squared?

The $\chi^2$ test/distribution is referred to as either "chi-square" (more frequently) or else "chi-squared" (less frequently). What is the history behind the name? Footnote 2 in this paper by Peter ...
3
votes
1answer
208 views

prerequisites for serre's FAC?

Is the knowledge of undergraduate's basic algebra and general topology enough to reading FAC? Do I need learn some algebraic topology and homological algebra, commutative algebra, or several complex ...
8
votes
1answer
308 views

Do schemes help us understand elliptic curves?

I'm reading Silverman and Tate's "Rational Points on Elliptic Curves" and I'm very much enjoying learning about these objects, and in particular doing a bit of number theory. It's different to what ...
2
votes
1answer
127 views

Is there a branch of mathematics which studies functions of functions?

I'm not a mathematician, sorry... My question is best explained with an example: I have lots of functions (programs in a functional programming language) in which the domain and codomain are lists ...
27
votes
5answers
2k views

Can mathematics get from other sciences what it got from physics?

Throughout history, physics has been an unparalleled source of '' inspiration'' for discovering/inventing mathematical ideas, which is due to its ability to describe the physical world. But can this ...
3
votes
1answer
122 views

Math or Physics

I'm a Master Math Student And I'm very interested in Some fields in Physics Like Cosmology. I even Considered Changing my field and Apply for physics(Cosmology) but wasn't really possible (my ...
1
vote
2answers
244 views

Elementary Applications of Cayley's Theorem in Group Theory

The Cayley's theorem says that every group $G$ is a subgroup of some symmetric group. More precisely, if $G$ is a group of order $n$, then $G$ is a subgroup of $S_n$. In the course on group theory, ...
7
votes
1answer
600 views

Innocent looking open problems in real analysis

Are there any apparently easy problems or conjectures in basic real analysis (that is, calculus) that are still open? By apparently easy, I mean: so much so, that, if it was for the statement alone, ...
3
votes
1answer
248 views

Meaning of a long exact sequence

Edit: The setting for the question is some abelian category. From this question I learned that one way to view a short exact sequence $$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$$ is as ...
2
votes
1answer
93 views

What is a regular homotopy?

The definition of regular homotopy from Wikipedia says that two immersion $f,g:M\to N$ are regularly homotopic if they represent points in the same path-component of $\text{Imm}(M,N)$. What does ...
2
votes
2answers
670 views

Preparation for STEP Mathematics Examinations [closed]

First I will give some background, to put my question in context. I am currently in my final year of school, and in order to meet my university offer to study Mathematics this Autumn, I am required ...
2
votes
0answers
96 views

Background & Advice for a self-learner of Descriptive Set Theory

A rather straight to the point soft-question: What kind of background should have somebody who wants to study properly descriptive set theory? More specifically, how much analysis should she/he ...
1
vote
4answers
180 views

Mathematical concepts that permeate algebra, geometry, and analysis? [closed]

Arguably, the concept of a linear space penetrates algebra, geometry, and analysis because we can find examples in these subfields. Except for the number field concept and the linear space concept, ...
2
votes
2answers
93 views

Ice cream issue in Lem's 'Extraordinary Hotel'

Could you clarify the ice cream issue mentioned at the end of the story The Extraordinary Hotel (pages 189-190 here)?
2
votes
2answers
252 views

Is mathematics invented or discovered? [closed]

In physics for example, and in science in general, facts are "discovered" in the sense that they arise from observing nature. A particle is discovered if we can measure its existence in nature. A law ...
2
votes
0answers
66 views

Question regarding pattern recognition in a table

Knowing this is a seemingly weird question, I do believe I should provide some additional context . One of our university professor's has a terribly peculiar habit . Namely, each year, the students ...
11
votes
2answers
563 views

How to introduce type theory to newcomer

I want to introduce (dependent) type theory to some friends having background in mathematical logic and set theory. To make this introduction easy I would like to give an informal presentation that ...
9
votes
1answer
349 views

“Visualizing” Mathematical Objects - Tips & Tricks

It has been a while since I am kind of stuck with my skills concerning the visualization of mathematical objects. Here there is the problem. First of all, let me point out that I am completely ...
2
votes
1answer
359 views

Can I Start Analysis? Seeking Your Advice on My Journey to Mathematics! [closed]

I am a college sophomore with double majors in mathematics and microbiology, and I have been doing independent research in the mathematical/computational biology, which really led me to love the ...
4
votes
1answer
86 views

I'm taking rings and fields this semester but don't remember group theory. Is it necessary to review everything?

I took group theory a year and a half ago so I don't remember anything. I'm taking rings and fields this semester and I'm worried I won't be prepared (group theory was already a struggle back then). ...
5
votes
2answers
1k views

Applications (“in everyday life”) of graph theory

EDIT another idea someone gave me was to consider flows in a network that would not only depend on the node at the beginning and at the end of a vertice but also about the vertice itself, like a ...
20
votes
1answer
911 views

List of generally believed conjectures which cannot all be true

There are some conjectures which most leading experts believe in albeit no one can prove it yet. For example: $\mathcal{P} \neq \mathcal{NP}$, the Riemann hypothesis or the Collatz conjecture. My ...
18
votes
8answers
293 views

Intuitively, why should the coefficient of the derivative of $x^n$ be $n$?

I am able to differentiate $x^n$ with respect to $x$ from first principles using the definition of differentiation. Also it seems natural that the gradient of a finite polynomial will be one order ...
0
votes
0answers
96 views

Proof using Archimedean property and Bernoulli's inequality

I am trying to prove the theorem below (using both the Archimedean property and Bernoulli's inequality). As usual, I would like to write a highly intelligible proof. Any constructive feedback is ...
57
votes
13answers
5k views

How to stop forgetting proofs - for a first course in Real Analysis?

I am taking my first course in analysis. I like the subject. I study it almost on a daily basis. I try to prove theorems on my own without even looking at the hints. If I really get stuck I just read ...
3
votes
1answer
167 views

Motivating the compact-open topology

It has been a while since I studied algebraic topology, and I wanted to revisit homotopy theory. Determined to take a more sustainable approach, I started by questioning and verifying every result in ...
6
votes
1answer
130 views

Better Notation for Partial Derivatives

I'm constantly seeing questions here where people are confused about the notation $\frac {\partial f}{\partial x}$ or $\frac {\partial f}{\partial x} (x,y)$ or $\frac {\partial f(x,y)}{\partial x}$. ...
9
votes
3answers
5k views

Learning Mathematics using Khan Academy

I am in my late 20s learning mathematics using Khan Academy. I have always passed my math with borderline grades. Since last year I have joined Khan Academy, I have learned voracious and re-learn ...
3
votes
3answers
86 views

Which quadrant is the “first quadrant”?

In the coordinate plane split into four quadrants by the $x$- and $y$-axes, I learned (educated in a public school in the U.S.) that the "first quadrant" was the one with both $x$ and $y$ positive, ...
3
votes
1answer
363 views

Euler Vs. Diderot

I'm reading The Music of the Primes by Marcus Du Sautoy and I came across a page with the following excerpt about Leonhard Euler: "The role of the court mathematician is perfectly illustrated by a ...
2
votes
1answer
186 views

Modern algebra and set theory: ZFC vs. NBG

This may be somewhat of a philosophical question and is probably nitpicking, but it is also one that has always bothered me a little: Is it not more natural consider NBG set theory as the foundation ...
0
votes
0answers
95 views

What is the most rigorous book for knot theory?

I have read the knot book by Colin Adams and it has been very helpful with getting to grips with the basics. I struggle with showing things and proving things. I need a book with a lot of rigorous ...
0
votes
0answers
74 views

How to derive this step in a book called Brownian motion calculus?

How to derive the step which result in the magnitude of slope of the path in section 1.8.2? I know it is not the definition itself.
6
votes
5answers
549 views

Is $1234567891011121314151617181920212223…$ an integer?

This question came from that one and from that talk where it's noted that "integers have a finite count of digits", so that the "number" in the title is not at all a number (not integer nor rational ...
4
votes
2answers
122 views

Multiplication Operation

I am a father of two young boys and I looks forward to exploring mathematics with them for as long as they will let me :-). I would really like for them to have a deeper understanding of mathematics ...
2
votes
0answers
104 views

Who First Considered This Generalization of the Fibonacci Numbers?

I am looking for the author who originally researched a generalization of the Fibonacci numbers, which Koshy, in Chapter 7 his book Fibonacci and Lucas Numbers with Application refers to as the ...
20
votes
1answer
277 views

How are long proofs “planned”?

I just graduated with my bachelors in mathematics last year, so I have little experience in writing huge, very involved proofs. The longest proof I've ever written was about 10 pages, but it wasn't ...
0
votes
1answer
65 views

Are there general conditions under which minimal generating sets can be expected to exist?

There exist algebraic structures $X$ with no minimal generating set. For example, $\mathbb{Q},$ viewed as an Abelian group. There also exist algebraic structures whose every generating set includes ...
3
votes
1answer
114 views

Moduli spaces, stacks and homotopy theory

For my final-year project (not this academic year but next) I'm hoping to write a relatively complete account of the basic theory of schemes used in modern algebraic geometry. My supervisor thinks ...
1
vote
3answers
61 views

judging probability by intuition is not always correct?

Suppose I have a question which goes like,"A pack of cards is counted with face downwards, and it is found that one card is missing. Two cards are drawn and are found to be spades. The odds in favor ...
-1
votes
2answers
92 views

The set of the roots of all polynomials in one variable with integer coefficients

Is the following set countable:: The set of the roots of all polynomials in one variable with integer coefficients. Please show the mapping between $N$ and the above set if the set is so I think ...
73
votes
10answers
5k views

Besides proving new theorems, how can a person contribute to mathematics?

There are at least a few things a person can do to contribute to the mathematics community without necessarily obtaining novel results, for example: Organizing known results into a coherent ...
3
votes
1answer
125 views

Is there a system of mathematics where everything is a function?

I was wondering if there is a system of mathematics where everything (except sets) is a function. For example, 3 would be the 3 function $x \mapsto 3$. There would be basic operators, such as $+$, ...
4
votes
1answer
430 views

Real analysis “theory book” similar to Andreescu's Problems in Real Analysis: Advanced Calculus on the Real Axis

I am going through Andreescu et al.,Problems in Real Analysis: Advanced Calculus on the Real Axis and I am very impressed: the style of the book seems really modern and the material covered includes ...
10
votes
2answers
226 views

Is function $f:\mathbb C-\{0\}\rightarrow\mathbb C$ prescribed by $z\rightarrow \large \frac{1}{z}$ by definition discontinuous at $0$?

Is function $f:\mathbb C-\{0\}\rightarrow\mathbb C$ prescribed by $z\rightarrow \large{\frac{1}{z}}$ by definition discontinuous at $0$? Personally I would say: "no". In my view a function can only ...
3
votes
1answer
81 views

Generalizations of de l'Hospital rule

Are there any useful generalizations of de l'Hospital rule? Could you point out some references? Edit: Something in the spirit of http://arxiv.org/abs/1403.3006, but not necessarily for functions of ...
1
vote
1answer
78 views

English or French translation of a paper in German

Is there an English or French translation of the following paper (or at least a work in these languages that summarize its main results)? Stolz, O. "Ueber die Grenzwerthe der Quotienten." Math. ...