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0
votes
2answers
35 views

Continuous trapdoor functions?

Every trapdoor function I've seen has been a discrete function. Do there exist continuous trapdoor functions? If so, what's an example of a continuous trapdoor function? And if not, why not?
3
votes
1answer
75 views

Taking a Putnam (General Questions) [duplicate]

I've just discovered an undergrad math competition (William Lowell Putnam Competition) and that my school offers it. The competition looks extraordinarily difficult, but I thought I'd give it a go. ...
3
votes
2answers
129 views

Is “mixed math” a useful way to learn math?

I was reading a book about how mathematics was taught in Cambridge in the 19th century, and it struck me how much physics was included in the syllabus, and it wasn't optional but everyone had to learn ...
1
vote
2answers
116 views

How to upgrade Category Theory skills for Algebraic Geometry?

I am doing a second advanced graduate course in Algebraic Geometry, with Hartshorne as a textbook. The skillset I am least satisfied with is the application of the Category Theory to Algebraic ...
3
votes
1answer
49 views

Good confusion-avoiding notation for iterated commutators?

I am doing some complicated and tedious calculation on iterated commutators. A typical term in my calculation looks like $$[x_a,[[[x_b,x_c]-x_d,x_e],[x_f,x_g]]]\text{.}$$ (I am considering ...
1
vote
1answer
30 views

Applications of Singular Functions

For our purposes here, a singular function is a continuous function such that the part which is absolutely continuous with respect to Lebesgue measure is zero. For example, the Cantor function or ...
27
votes
7answers
5k views

Genius mathematicians who never published anything

Amongst philosophers, Socrates is an example of a genius with a great influence on human history who never wrote anything. Almost all facts which are known about his revolutionary ideas are written by ...
1
vote
1answer
77 views

What topics have complex analysis among their prerequisites?

I have one spot left in my bachelor's curriculum and am trying to decide between complex and functional analysis. What the latter is good for, is more or less clear to me: e.g. for advanced ...
5
votes
2answers
193 views

Good source on current 'views and thoughts' on mathematics

Recently I've become interested in the history of the way people think about mathematics. What I mean by this is for example how Godels proof basically put an end to the whole school of thought ...
2
votes
3answers
43 views

Restrictive definition of diagonalizable matrix

There is a theorem that says that every matrix of rank $r$ can be transformed by means of a finite number of elementary row and column operations into the matrix $$D=\begin{pmatrix} I_r & O_1 \\ ...
7
votes
1answer
153 views

How important is Differential Geometry for Number Theory?

The title pretty much says it. To elaborate slightly, I am, of course, aware of the huge role played by Algebraic Geometry in Number Theory but I'm not so sure about Differential Geometry. I would be ...
1
vote
2answers
75 views

What are the suggested prerequisites for topology?

I am interested in topology but I don't know if I can learn it without learning something else first. I've done: Algebra 1 and 2 Euclidean Geometry Calculus Is that enough if not please tell me what ...
1
vote
1answer
45 views

A book with heuristics or general techniques used in real analysis?

I have been looking for a book with some good heuristics for real analysis and point set topology. Any ideas?
1
vote
4answers
56 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
79 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 ...
33
votes
20answers
3k 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
108 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
84 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
52 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
61 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
158 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
32 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
56 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
52 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
132 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
647 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
230 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
457 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
89 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
83 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
38 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
172 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
53 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
35 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
3answers
111 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
3answers
79 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
125 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
61 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
88 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
23 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
30 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
166 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
77 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 ...
1
vote
1answer
29 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
85 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
31 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
55 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
63 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
152 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 ...