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0
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0answers
49 views

Apostol's Calculus Vol II OR Hubburd's multivariable OR Shifrin's multivariable for self study

I'm trying to self study multivariable calculus which I took at university but mostly forgot about it! I'm looking for a textbook that also incorporates linear algebra and gives a coherent view of the ...
4
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0answers
46 views

Unexpected applications of row rank = column rank

The fact that the row rank of a matrix is the same as the column rank is quite surprising (to me atleast, hopefully to you too!). I am looking for unexpected applications of this fundamental fact, ...
0
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2answers
40 views

Problem Solving Practice

Are there any resources ( mainly books, but other things as well :)) which involve a lot of exercises/problems. These problems should be challenging but still solveable for a High School Student. The ...
0
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0answers
19 views

How do I prove that as 2 integers p, s tend to infinity, p/s tends to x?

Forgive me for asking such a broad question, but I really do have very little knowledge on how to do this and it came up in a problem that I have been working on for some time now, so any help would ...
4
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3answers
283 views

What things should one know in order to enjoy their undergraduate degree?

From looking at undergraduate mathematics programmes it's quite apparent that mathematics degrees are demanding, one could even say the work load is grueling. However I'm certain that there are ...
8
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4answers
567 views

What are some elementary results (number theory) using theorems that went long-unproven?

Firstly, I do not mind if there are examples from fields other than number theory! This was just the first field, and where I think the richest examples, may come from. Now to elaborate on the title, ...
3
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1answer
30 views

How can I improve my explanation that the ratio $\theta=\frac{s}{r}$ that defines the radian measure holds for all circles?

I'm trying to demonstrate why the ratio $\theta=\frac{s}{r}$ (where $s$ is an arc measuring some $s$-units in length and $r$ is the radius of the circle) which defines the radian measure holds for all ...
3
votes
1answer
73 views

Applications of computer science to mathematics

I have been introduced to algorithms, computability and computational complexity (as part of my minor in CS). What are some mathematical topics that I can tackle with the new perspectives I ...
1
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1answer
65 views

A more advanced book on real analysis

I have already studied "Advanced Calculus Second Edition by Patrick M. Fitzpatrick" on real analysis. It is the best book on real analysis I found that can be studied by self-learning with high-school ...
6
votes
7answers
630 views

A book for abstract algebra with high school level

Any book that I find on abstract algebra is somehow advanced and not OK for self-learning. I am high-school student with high-school math knowledge. Please someone tell me a book can be fine on ...
1
vote
2answers
155 views

How can I improve my explanation of why the ratio $\pi=\frac{C}{d}$ holds for all circles?

I'm trying to informally explain why $\pi$ holds for all circles. I would like to know if there is anything pertinent that I can add, or that is wrong with this explanation. It's an explanation, not ...
47
votes
10answers
5k views

Why are primes considered to be the “building blocks” of the integers?

I watched the video of Terence Tao on Stephen Colbert the other day (here), and he stated, like many mathematicians do, that the primes are the building blocks of the integers. I've always had ...
57
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23answers
8k views

An example of a problem which is difficult but is made easier when a diagram is drawn

I am writing a blog post related to problem solving and one of the main techniques used in problem solving is drawing a diagram. Essentially, I want to illustrate that some hard problems (for example, ...
2
votes
1answer
53 views

Insight into Abel's impossibility theorem

Let 's consider the polynomial of degree $7$:$$f(x)=x^7-28x^6+ 322x^5-1960x^4+6769x^3-13132x^2+13068x-5040 $$ I am trying to get some insights into Abel's impossibility theorem. Does this theorem ...
8
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0answers
90 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
38 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
115 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
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0answers
44 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 ...
2
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1answer
74 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 ...
7
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1answer
76 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
98 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 ...
24
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
62 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
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2answers
87 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, ...
2
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0answers
65 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
84 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 ...
0
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1answer
29 views

What does associativity really do for Moufang Loops?

I've been trying to tackle the question of what is the real importance of every group axiom by looking at all the structures "below" a group like a magma, semigroup, monoid, quasigroup, and loop. I ...
2
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1answer
45 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 ...
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2answers
102 views

Preparation for STEP Mathematics Examinations

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
69 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
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4answers
92 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
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2answers
51 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
128 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
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0answers
37 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 ...
8
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1answer
374 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 ...
6
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1answer
86 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 ...
0
votes
1answer
34 views

How does one learn to be creative? can one learn that?

Lately I've been wondering whether or not one can "learn" to be creative in math, i.e solve unusual questions and have unique ideas on how to approach proofs/study/etc.. This is the whole nature v.s. ...
1
vote
1answer
102 views

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

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
75 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). ...
0
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0answers
25 views

What other words are good replacements for 'heat' in the phrase “heat equation” (the famous PDE), apart from 'diffusion'?

Historically, the term 'heat' has value as it hearkens back to the context in which Fourier studied the heat equation. Pedagogically, the term 'diffusion' has value as it imparts a more general, ...
4
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2answers
89 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 ...
18
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1answer
856 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
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8answers
241 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
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0answers
37 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 ...
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0answers
133 views

Does anyone know the bulgarian mathematician Georgi Bonev and his sequence?

This is a topic of seminarian work and I need some information about bulgarian mathematician Georgi Bonev and his sequence. I don't know where else I should ask. Here is some infos: The sequence ...
48
votes
13answers
3k 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 ...
2
votes
1answer
54 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 ...
4
votes
1answer
75 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 $\frac {\partial f(x,y)}{\partial x}$. ...
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0answers
25 views

How can I make sure the problem I want to ask has not been presented before? [migrated]

e.g. I have a question on proving an inequality statement. If I google it Google will not recognize it and if I use Math StackExchange'search engine, it will not respond. Thank you for your help.
7
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3answers
398 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 ...