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1
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1answer
170 views

Advice on career- interested in Mathemetics

I am sorry to post this question related to career over here. But I knew this is the place where my question can be viewed by the best people who can provide best advice. To introduce myself, I have ...
5
votes
0answers
268 views

Logical consequence of Euclid's theorem

Are there any far reaching non-trivial consequences of Euclid's infinitude of primes where theorems make use of it? Wikipedia does not have the list of applications of this theorem, rather modern ...
4
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4answers
135 views

Resources to help an 8yo struggling with math

Friends of mine asked me for suggestion for one of their children (age 8) who had bad scores at the local Star test (the family is based in California). Both parents work, so they have also limited ...
15
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8answers
930 views

Should a high school introductory calculus class teach $\varepsilon$-$\delta$ proofs?

It seems to me that most high school students are comfortable with the intuitive notion of a limit ("as $x$ gets arbitrarily close to $c$, $f(x)$ gets arbitrarily close to $L$") and gain little ...
28
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4answers
7k views

Are We Teaching Pre-Calc Wrong?

It took some 1,250 years to move from the integral of a quadratic to that of a fourth degree polynomial. When we jump too fast to the magical algorithm, when we fail to acknowledge the effort that ...
10
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3answers
304 views

categorical generalizations of familiar objects

A couple of days ago I've learned that you can define trace in a very abstract setting. Namely, suppose $F\colon A\to B$ is a functor between two categories. Suppose $E,G\colon B\to A$ are two ...
5
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4answers
808 views

Self-teaching myself math from pre-calc and beyond.

Going to be starting grade 12 (pre-calculus) shortly and looking to get ahead. I would like to try some more rigorous stuff on my own and have a couple questions. Ideally I would like to be prepared ...
8
votes
2answers
141 views

How much mathematicians rely on other results?

When mathematicians do research, do they think every detail so that they know exactly every step what it takes to achieve the result from axioms? Or is it possible to work as a researcher without ...
31
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7answers
1k views

Why are topological spaces interesting to study?

In introductory real analysis, I dealt only with $\mathbb{R}^n$. Then I saw that limits can be defined in more abstract spaces than $\mathbb{R}^n$, namely the metric spaces. This abstraction seemed ...
2
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4answers
467 views

Advanced undergraduate(?) Real Analysis book which is concise and lots of interesting problems

I have gone through the other book recommendations on Real Analysis, but I think my requirements and background is slightly different. I am a Physics undergrad teaching myself Pure math. My journey is ...
1
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0answers
71 views

Mathematical fields and countries

How can I know which mathematical field each country or university is the best in these days? For example, I've already heard that in my country there isn't so much research in algebra and my ...
1
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1answer
817 views

Applications of number theory and functional analysis in the real world/ job market

what do you think are the application of pure mathematics in the real world except research. would like to pursue a masters in it because i love it but is it leading me any where in the job market? ...
1
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1answer
152 views

Learning roadmap for additive combinatorics

I have read Calculus by Michael Spivak. Now I want to learn additive combinatorics though I have no experience with combinatorics or probability theory. To my understanding, there is a book on the ...
1
vote
1answer
230 views

What are the fertile areas of research in Analytic Number Theory?

My professor once told me that Analytic Number Theory was "dead," which at the time was something of a disappointment, and which I struggled to agree with. Surely any subject may appear inferrtile in ...
55
votes
12answers
7k views

Can the golden ratio accurately be expressed in terms of e and $\pi$

I was playing around with numbers when I noticed that $\sqrt e$ was very somewhat close to $\phi$ And so, I took it upon myself to try to find a way to express the golden ratio in terms of the ...
1
vote
0answers
69 views

Learning roadmap for additive combinatorics

I have read Calculus by Michael Spivak. Now I want to learn additive combinatorics though I have no experience with combinatorics or probability theory. To my understanding, there is a book on the ...
1
vote
1answer
56 views

Recursive application of a function : Symbol of [duplicate]

I need to apply a function $f(x)$ recursively/repeatedly for n times; how do I express it (mathematically) ? Is their a mathematical symbol which denotes $f(x)$ applied n times ie $g(x,n)$ ...
3
votes
1answer
812 views

Solving questions which appear to be pure hit-and-trial

Consider the following problem: There is an integer in each square of a $8 \times 8$ chessboard. In each move you can add $1$ to each of the integers in a smaller $4\times 4$ or $3 \times 3$ ...
3
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5answers
691 views

Have people tried to find the area under a curve by means other than integration?

It seems pretty frustrating that some definite integrals can only be evaluated numerically, so have people tried to find another method of finding the area under a curve that isn't numerical? I'm ...
1
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0answers
47 views

getting “pairs” from $3$ digit numbers

I'm in quite the weird position where I have to get "pairs" from three digit numbers For example, the number $123$ gets the "pairs" : $12$ $13$ $21$ $23$ $31$ $32$ - what is this called?
5
votes
7answers
322 views

Is there a name for this type of logical fallacy?

Consider a statement of the form: $A$ implies $B$, where $A$ and $B$ are true, but $B$ is not implied by $A$. Example: As $3$ is odd, $3$ is prime. In this case, it is true that $3$ is odd, and ...
6
votes
1answer
123 views

Are there treelike representations of axioms, theorems, lemmas and corollaries?

I was watching Bill Shillito Lectures on Higher Mathematics, in the second episode he says the basic stuff about axioms and theorems - that axioms are unproved statements in which we build theorems ...
5
votes
3answers
332 views

How to be good at angles and trigonometry

I am Computer Science Engineer and loved algebra side of Mathematics. But when it comes to trigonometry and angles and triangles, I do not understand anything since college time. And till now also ...
11
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3answers
526 views

What is the motivation for differential forms?

I am that point in my mathematical career where I am learning differential forms. I am reading from M.Spivak's Calculus on Manifolds. So far I have gone over the tensor and wedge products and their ...
2
votes
1answer
116 views

The largest number to break a conjecture [duplicate]

There are several conjectures in Mathematics that seem to be true but have not been proved. Of course, as computing power increased, folks have expanded their search for counterexamples ever and ever ...
48
votes
12answers
3k views

Do groups, rings and fields have practical applications in CS? If so, what are some?

This is ONE thing about my undergraduate studies in computer science that I haven't been able to 'link' in my real life (academic and professional). Almost everything I studied I've observed be ...
3
votes
1answer
911 views

Recommended maths book for beginner to study in computer science

I am going to study computer science next year. I am afraid I can't handle the mathematics in the university because I only know some basic mathematics, such as set theory, simple probability, simple ...
5
votes
1answer
144 views

Mathematical game-formalism and set theory

Are mathematical game-formalism and set theoretical foundations of mathematics (ZFC, for example) compatible? If not, where to start with a mathematical game-formalism foundations on their own? Are ...
18
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5answers
597 views

Is $3 \ge 1$ or is it just $3 > 1$?

Well, probably this might seem a really simple question (and it might be so too!), but off late me and my friends have been debating quite hard over this question. Is $3 \ge 1$ or is it just $3 ...
3
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4answers
145 views

Applications of logic

What are some applications of symbolic logic? I tried using Google and Bing but just got a bunch of book recommendations, and links to articles I did not understand.
7
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1answer
207 views

How to Remain Competetive at Mediocre University? [closed]

I am attending a university with a mediocre mathematics department. The courses at my school cover fewer topics and are less challenging than the same courses at top universities. How can I avoid ...
16
votes
7answers
1k views

Mathematical writing: why should we not use the phrase “we have that”?

In Knuth's Mathematical Writing, he writes on page 2, at number 8: Don't omit "that" when it helps the reader to parse the sentence. Bad: Assume $A$ is a group. Good: Assume ...
0
votes
1answer
86 views

Homogeneity Versus Heterogeneity in Student Groups

There is an overwhelming amount of research regarding homogeneous and heterogeneous grouping in education. The former refers to the practice of grouping "like" students together (regarding age, ...
2
votes
1answer
266 views

How impact the work of a pure mathematician in the society? [duplicate]

Firstable i explain my situation: On my University most of the careers are doing videos to explain what we do and try to atract more people to our careers. Im in a really bad position, because the ...
10
votes
1answer
309 views

Toddler introductions to higher mathematics

I bought my youngest (now 10 months old) "Introductory Calculus For Infants" by Omi Inouye a while back. It's actually an ABC book about the letter x and how no one ever plays with him until he meets ...
5
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3answers
2k views

Baby Rudin vs. Abbott

I am considering Stephen Abbott's Understanding Analysis and Walter Rudin's Principles of Mathematical Analysis. I am looking for a comparison between the two that addresses both of the following ...
4
votes
1answer
526 views

Self-studying through an undergraduate math course. Need Tao-like textbooks!

I'm a physics undergraduate student who always enjoyed math, and briefly studied it at a university but for various reasons (laziness, youth) gave up and changed 'majors'. But I always wanted to go ...
7
votes
1answer
268 views

The perception of mathematics

In my work I wrote the following sentence. "...there is a negative perception of mathematics and mathematicians, both within and outside of academia." Err. Right. So, I believe that this is ...
3
votes
4answers
217 views

Textbook Recommendation: Topological Dynamics

I need to take credits satisfying a topology requirement, and can structure it myself. My field of study is dynamical systems, can someone recommend a textbook that handles differential ...
9
votes
1answer
214 views

Journals or Magazines on Study Skills or How to Study Math

I am trying to find only journals or trustworthy magazines which can help math students to study math more efficiently and productively. I am not asking about books in this thread. In particular, I am ...
1
vote
1answer
107 views

numerical linear algebra 101

since I'm a programmer and I need linear algebra, I'm starting considering how to teach myself a little of numerical linear algebra, not really optimize things right from the start, but I would like ...
4
votes
1answer
130 views

Why are square functions important in analysis?

I have been reading through chapter 1 of E.M. Stein's textbook Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals. In chapter 1, Stein discusses the relationship ...
2
votes
3answers
306 views

What's the motivation to add inner product and wedge product together in geometric product

I am reading some geometric algebra notes. They all started from some axioms. But I am still confused on the motivation to add inner product and wedge product together by defining $$ ab = a\cdot b + ...
0
votes
1answer
196 views

Manifold learning/nonlinear dimensionality reduction for beginners

I'm a computer science graduate student. I recently discovered manifold learning. I think I understand the very basic, high-level concept of nonlinear dimensionality reduction, but I'd like a ...
4
votes
1answer
100 views

Trustworthiness of foundational systems

Naively, we might think that if a foundations of mathematics is consistent, then its fair game. Then we learn a bit more, and we realize that even if a foundations of mathematics is consistent, it may ...
7
votes
2answers
125 views

Popular textbooks and current research.

I am sure that I will be finding out first hand as I am entering a PhD program, but I will ask my question anyway. Say, for instance, a student has worked through the majority of a textbook like ...
0
votes
1answer
32 views

What is the difference between “model” and “method”

I am not sure which forum to ask this question since the answer may change depending on the scientific area. I am analysing some time series using linear regression. I predict data using the linear ...
1
vote
3answers
86 views

The ubiquitous “helper function” $\frac{f(z) - f(a)}{z - a}$

I've been looking at basic complex analysis recently, and have noticed (am imagining?) something which I've never really paid attention to before: The "helper function" $$g(z) = \frac{f(z) - f(a)}{z ...
4
votes
2answers
170 views

How to determine the big questions in a field of mathematics?

During my self-study (and soon to continue at a university) of mathematics, one thing I have been interested in is how to to effectively learn the material. An answer to a question provided by ...
2
votes
1answer
105 views

Topology and commutative algebra.

I don't know both of these subjects, but I was wondering if there was any topology in commutative algebra. I don't need any detailed answer (since I don't know any of them yet)...So would it be ...