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1
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4answers
49 views

Applying math knowledge [closed]

Currently I'm in the middle of my first year of college studying informatics engineering. I was never great at math, but if I put some effort, I understand it and constantly get good grades. However, ...
4
votes
2answers
73 views

Why do we focus so much in math on functions (as a subclass of relations)?

Why is it that math so focuses on the subclass of relations known as functions? I.e. why is it so useful for us in nearly all branches of mathematics to focus on relations which are left-total and ...
31
votes
20answers
2k views

Ways to write “50” [closed]

A really good friend of mine is an elementary school math teacher. He is turning 50, and we want to put a mathematical expression that equals 50 on his birthday cake but goes beyond the typical ...
6
votes
2answers
79 views

Which mathematical topics is knot theory related to?

I wonder if knot theory is related to any other topic in mathematics. I've not read much about it, but it seems to be living isolated. I also wonder if there any particular mathematical background ...
4
votes
3answers
80 views

Should I go back and start with a more “proof” based approach?

So I'm currently a calculus student, next semester I'll take calculus 2. I'm wondering if I should go to a book like the one by Spivak which is entirely different from the book used for my course, and ...
1
vote
1answer
48 views

What does “if and only if” mean in definitions?

Consider the following definition: A sequence $\{p_n\}$ is Cauchy if we have that for every $n, m \ge N$: $$|p_n - p_m| < \epsilon$$ Although if and only if is not used, we know that if a ...
3
votes
0answers
60 views

Topology of the space of “loops” [closed]

I have a question that I'm not even sure I can put into words, but please bear with me! I want to define some sort of "loop space" and I want to understand it's topology enough that I can compare it ...
4
votes
1answer
141 views

Is there any mathematician who felt guilty for one of his math discoveries ever?

Quoted from Wikipedia: In 1888 Alfred Nobel's brother Ludvig died while visiting Cannes and a French newspaper erroneously published Alfred's obituary. It condemned him for his invention of ...
2
votes
0answers
22 views

The idea behind the Sobolev embedding

Sobolev embedding and compact embedding are the most popular theorems in Sobolev space we actually used in research. But after I use them so many times, I am still wondering, why, philosophically, ...
1
vote
1answer
37 views

Studying mathematics in France (universities and grandes ecoles)

I am currently a senior in Albania and would like to study mathematics in France. However, I'm not quite sure if the universities or the so-called "grandes ecoles" provide the best quality of ...
5
votes
0answers
49 views

Why should the open mapping theorem be expected?

Soft question alert. I want to know why to expect the open mapping theorem to be true. My thoughts: I know that one nice consequence of the OMT could be thought of as the universal property of ...
26
votes
11answers
3k views

Great contributions to mathematics by older mathematicians [closed]

It is often said that mathematicians hit their prime in their twenties, and some even say that no great mathematics is created after that age, or that older mathematicians have their best days behind ...
9
votes
1answer
129 views

What would it mean if there was a link between e and $\pi$?

It is not even know if $\pi+e$ is rational, and the same is true for other similar expressions involving $\pi$ and $e$, but does this have an impact? If it were, for example, proven that $\pi=ae$ or ...
8
votes
4answers
635 views

Generalized graph theory

This question may be kind of 'out there' but it got me thinking. In graph theory we have a set of vertices $V$ and a set of edges $E$ which is made up of 2-element subsets of $V$ (either unordered or ...
5
votes
5answers
205 views

Should i study Mathematics? [closed]

I would like to ask you guys for some advice. I'm currently studying Bsc. engineering(2nd year). I really like mathematics and for example I really liked the calculus courses from 1st year and am ...
14
votes
9answers
402 views

Definite integrals with interesting results [closed]

I just stumbled across the fact that $\int_{-\infty}^{+\infty}{e^{-x^2}dx}=\sqrt{\pi}$. This intrigued my already-existing interest in integrals. It made me wonder, are there other integrals with ...
1
vote
1answer
81 views

Pure Mathematics vs Mathematical Statistics

I see that these majors are usually offered separately at universities. 1) Does Pure Math not cover ALL math including statistics? 2) If you choose Pure Math - will there be things you will NOT ...
2
votes
1answer
61 views

Getting stuck on difficult problems.

First, a little background: I hope to go to graduate school in mathematics, but for financial reasons I will be unable to go back to school any sooner than the fall of 2016. However, since I feel ...
1
vote
1answer
36 views

Catching own conceptual mistakes during tests

I am looking for strategies for catching mistakes in graduate exams eg. qualifying. The more people suggest, the better because what is obvious to you, might not be to others. Most of the advice in ...
3
votes
2answers
148 views

More than one pair of “nice” adjoint functors between different concrete categories

Though adjoint functors provide a universal description for many concrete mathematical constructions, these constructions usually revolve around finding a single "canonical" way to transform one type ...
2
votes
1answer
47 views

Can Spivak's 5-volume series on differential geometry be effective without exercises?

I was scouring the internet for information about these books and I learned that the latter 4 volumes have no exercises. Would I be able to attain mastery with no exercises?
1
vote
2answers
32 views

Is there a finite vector subspace over the reals?

I cannot think of a finite vector space over the reals, because we must sum these two elements and get a new element also in the vector space. And over the reals, I can't think of a sum that, at some ...
8
votes
2answers
78 views

How to google search mathematical notions and expressions?

It is usually not difficult to google search mathematical notions; for example, one can search (with quotation marks) the term "brunnian braid" and find the definition and other related materials. ...
0
votes
0answers
29 views

Is there a way on MSE to search for previously asked questions using LaTeX code? [migrated]

Before I post a question, I always try to see if it has been previously posted. But for some questions with mostly "math symbols" and few words, I find it difficult to search effectively. For example, ...
0
votes
3answers
63 views

What are some textbooks on the same level as Ross's “A First Course in Probability”?

Ideally, I would like to have at least six standard probability texts so that I can compare them to each other. Thank You.
5
votes
2answers
99 views

Still forget even if theorem-proof “self-discovered”; Importance of intuition/proficiency of concepts in research work…

It is widely said if we go through concepts/theorems/proof on our own by actively doing instead of passively reading, the idea will be ingrained in mind. I agree with that, it really often helps. ...
2
votes
1answer
52 views

What does $S^z$ mean for each $z\in\mathbb{C}$?

Let $S$ be a set. What does $S^z$ mean for each $z\in\mathbb{C}$? In Set Theory numbers are sets and for any two sets $A$ and $B$, we define $B^A$ as the set of maps from $A$ to $B$. Well okay, ...
2
votes
1answer
80 views

Why is polynomial long division being taught in schools instead of Horner's method? [closed]

The Horner´s method is by a long shot easier than the Polynomial long division and serves the same purpose. Why isnt it being taught in school (in germany at least)?
1
vote
0answers
22 views

Generators of Intersection of two Subgroups

Let $G$ be a group and let $A$ be the subgroup of $G$ generated by $\{a_i\}_{i\in I}$; let $B$ be the subgroup of $G$ generated by $\{b_j\}_{j\in J}$, where $I$ and $J$ are index sets. Is there a way ...
0
votes
1answer
19 views

Prove that the function is of exponential order and proving in mathematics

I'm currently learning about the Laplace transform and in my textbook in college and I have this definiton: Function $f$ is of exponential order if there exist constants $M>0$ and $a$ such that ...
1
vote
3answers
152 views

How much of Mathematics is limited by our writing? [closed]

I'm sorry if this question is too vague or otherwise a stupid question. Suppose the mathematicians in some alien civilisation similar to ours sculpted their Mathematics in three dimensions (or ...
0
votes
3answers
64 views

Is there an elementary “proof” that the first degree equation has only one solution?

I was asked from a student why the first degree equation has only one solution (if it has a solution) Let's consider the equation $2x+5-3x=-4x+14$ for example. How can I explain to a 13 year old ...
4
votes
0answers
64 views

Proof strategies repeatedly used by Erdős and Hilbert [closed]

I just read in an article that most of the proofs by the mathematician Paul Erdős had a few tricks that he used over and over again. Hilbert's paper too had some tricks used repeatedly. Since I am ...
1
vote
1answer
27 views

Best algo for finding no. of steps required to convert a sequence to a palindromic sequence

[My first question of Math SE, so, HI!] I'm not sure of what the rules are around the place, but I have a straightforward question as follows... The sequences 23, 45, 23 and 23, 45, 56, 23, 23, ...
4
votes
1answer
78 views

Is cellular automata something that is studied in mathematics departments?

I am interested in studying cellular automata but am unsure if I should be looking at CS or mathematics graduate departments. Symbolic dynamics seems to have some tie to cellular automata but I ...
1
vote
0answers
24 views

Advice needed in Cryptography

I'm currently in my undergrad studies (3rd Year in 2015) , majoring in pure mathematics and statistics. I'm thinking of pursuing cryptography for my Honours project, as its the closest thing that ...
0
votes
2answers
50 views

Pitfalls/subtleties of $O$ notation

What are some examples of $O$ subtleties? I'm not only thinking of the asymmetry of the $O$ relation, but of the ways in which $O$ constants can depend on nearby parameters, and the fact that the ...
2
votes
1answer
53 views

What kind of programming does a mathematician/mathematical engineer need to know?

I was thinking about which program to choose for university studies, and I will probably study an engineering program kind of like mathematical engineering. It is kind of hard to specify a ...
7
votes
5answers
146 views

Composition of Inverse Functions

$f$ and $g$ are inverses of each other when $f(g(x)) = x = g(f(x))$. However, can there be 2 functions where $f(g(x)) = x$ but $g(f(x))$ does not equal to $x$? I feel like there are but I cannot find ...
2
votes
1answer
64 views

resources to get a taste of advanced (graduate) math?

There are a lot of ideas which I'd like to learn more about but that would take years to reach if I follow a traditional path (where "traditional path" means a kind of education where things are ...
1
vote
0answers
17 views

Introductory text on numerical analysis [duplicate]

I was wondering if anyone has a good suggestion for a textbook on numerical analysis. I am an undergraduate with little prior knowledge about topics in numerical analysis since I have never taken a ...
1
vote
3answers
27 views

How likely are extreme observations in a probability distribution?

Given a measurement that follows a probability distribution (for the sake of argument, Gaussian) how likely is it that repeated observations on the distribution are an extreme of low or high? I ...
4
votes
0answers
45 views

The “muscle” behind the fact that ergodic measures are mutually singular

This is really motivated by the soft question at the end, but let me begin with something more circumscribed: Let $(X,\mathcal{B})$ be a measurable space and let $T:X\circlearrowleft$ be a self-map ...
1
vote
1answer
26 views

Exterior derivative cohomology

Let $\Omega^k (U)$ denote the set of differential $k$-forms on an open subset $U\subseteq \mathbb{R}^n$. For each $k\in \mathbb{N}$ the exterior derivative $d_k=d : \Omega^{k-1} (U) \rightarrow ...
3
votes
3answers
56 views

If $1$ was a prime, could it be possible for the prime factorization of any number to go on forever?

If $1$ was a prime, could it be possible for the prime factorization of any number to go on forever? I think this would happen because if you multiply anything by $1$, you get the first factor ...
3
votes
1answer
77 views

Studying for analysis- advice

I find that studying for analysis is unlike other math classes that I've taken. I dedicate a lot of time to studying for it, but it seems like no matter how much time I put into it I am not getting ...
9
votes
2answers
1k views

The Big Book of Proof

Some time ago, I came across an anecdotal story about the "Big Book of Proofs" that God always keeps up in the heaven, which records valid proofs of all theorems in the world. A noted mathematician ...
7
votes
2answers
88 views

How can using a different definition for the integral be useful?

It's often said that the Lebesgue integral is superior to the Riemann integral because it satisfies nicer properties, for instance things like $$\lim_{n\to\infty} \int f_n = \int \lim_{n\to\infty} ...
5
votes
1answer
55 views

Is a thorough study of algorithms useful for a mathematician?

In my university, there is a core course called "basics computer science for mathematicians". The topics covered range from algorithms, to the bases of programming, to theory of computability. The ...
15
votes
9answers
2k views

“Honest” introductory real analysis book

I was asked if I could suggest an "honest" introductory real analysis book, where "honest" means: with every single theorem proved (that is, no "left to the reader" or "you can easily see"); with ...