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5
votes
5answers
329 views

Are there finitely many interesting theorems? [closed]

I'm not a logician, I read, a long time ago, about Gödel etc... A theorem is a provable proposition under a system of axiom. Let's take the usual system of ZFC. Of course, the notion interesting is ...
1
vote
4answers
116 views

What is the most awe-inspiring math equation you have come across [closed]

What is the favorite equation of your life? I know this might be a subjective question, and may be not-so-on-topic here, so if anyone decides to close this, could you link me somewhere I can ask this? ...
1
vote
1answer
38 views

Should $\sum_{i=0}^n a + b$ be interpreted as $(\sum_{i=0}^n a) + b$ or $\sum_{i=0}^n (a + b)$

Should $\sum_{i=0}^n a + b$ be interpreted as $(\sum_{i=0}^n a) + b$ or $\sum_{i=0}^n (a + b)$ I often see the expression $\sum_{i=0}^n a + b$ in books and wonder whether we take the sum ...
3
votes
1answer
303 views

How can I study efficiently from a textbook that doesn't have answers?

My lecturer has informed us that tutorial questions will be insufficient, and has asked us to practice liberally from the textbook. Unfortunately, none of the exercises have answers supplied and I ...
5
votes
1answer
125 views

More unknown / underappreciated results of Euler

What are some of the more unknown and/or underappreciated things that Euler discovered? The man has done so much that there's bound to be notable results that most people aren't aware of. This could ...
0
votes
1answer
25 views

How to come up with formula for this number growth

I have difficulties to come up with a formula to achieve this in my programming challenge : The input number ranges between 100 to 10.000 (all integers), then the output would be 0.1 - 10, so the ...
5
votes
4answers
164 views

Mathematicians who started late, similar to my situation?

I came here since I know this is the best place to ask a question. I'm a first year student who changed his major to applied mathematics. In middle school I was a garbage math student, but I ...
2
votes
2answers
110 views

When do we write “we are done”?

This may seem like a bit of a silly question, but I notice that in some proofs (a remarkable amount), the author writes: "We are done." after completing a proof. Is this the equivalent of writing one ...
74
votes
29answers
15k views

Examples of mathematical results discovered “late”

What are examples of mathematical results that were discovered surprisingly late in history? Maybe the result is a straightforward corollary of an established theorem, or maybe it's just so simple ...
5
votes
2answers
349 views

Is maths = set theory + logic? [closed]

It seems to me that most branches of mathematics are just an application of set theory and the rules of logic. You just definite particular types of sets and study their properties. For example, in ...
1
vote
2answers
41 views

How could I show that if $a$ is an integer, then $a^3 \equiv 1, 0, 6 \mod 7$?

I've literally tried everything including proofs by cases. If I were to try a an odd integer, then I would get $8k^3 + 4k^2 + 8k^2 + 4k + 1$, which obviously couldn't convert to a $\mod 7$, and I'm ...
0
votes
2answers
49 views

A question about notation

Earlier this week, a friend asked me what the most complicated equation I could think of was that was equal to $1$. The answer I gave was this: Let $G_n$ denote the n$th$ number in the grandi series, ...
2
votes
0answers
68 views

What are some cool projects that can be done in a high school math class?

I'm studying to be a secondary math teacher and will be starting student teaching next month (an algebra 2 class). It's difficult to find activities and projects that are actually informative and time ...
4
votes
1answer
125 views

The method of “Mathematical Induction” as explained in my book.

I am reading the book "A Treatise on Advanced Calculus" by Philip Franklin. I found this book in our city's central library and liked it at the first reading, am continuing this book as a reference. ...
10
votes
3answers
172 views

Natural uses for the co-product of sets?

I had come across countless uses of the (Cartesian) product of sets long before I first ever met the concept of a "co-product"1 of sets. In fact, anyone who has learned basic analytic geometry in ...
0
votes
0answers
47 views

The difference between Dynamic Optimization, Stochastic Programming, Optimal control and Markov Decision Processes

I've seen the following terms thrown around somewhat interchangeably, and I'm confused. What are the distinctions between them, and what are some representative problems that each deals with? ...
1
vote
3answers
55 views

Study of polynomials and root approximation algorithms

I am currently beginning a long term investigation (I will have 6 months of time). I wish to better understand fundamental properties of polynomials and eventually come up with a good root finding ...
2
votes
0answers
17 views

Motivation behind Binet form and its generalization to arbitrary coefficients

Binet form (http://www.proofwiki.org/wiki/Binet_Form) gives a closed-form solution to $n^{th}$ term in a series with recurrence relation as below $U_{n}=m\cdot U_{m-1}+U_{m-2}$ I have two questions ...
0
votes
0answers
34 views

Standardizing the terminology of graph theory

I took a course in graph theory many years ago and remember being frustrated that the terminology was not at all standardized (or didn't seem to be). Various sources used vastly different words or ...
3
votes
0answers
58 views

Was the Weierstrass function constructed or discovered?

Reading Halmos' I want to be a mathematician, he mentions a continuous function without a tangent. Naturally, I was curious to see how such a function could possibly exist, and I imagined it to be ...
1
vote
2answers
67 views

From Engineering-Style to Proper Mathematics

I currently have an engineering-style education in mathematics. We covered quite a lot of material (e.g. real and complex analysis, some probability theory and graph theory), but more often than not ...
2
votes
1answer
61 views

What does quotienting by a congruence mean?

I have come across quotient algebras in my different mathematics courses. I know of quotienting with normal groups, quotienting with ideals etc. While studying Boolean Algebra I encounter quotienting ...
1
vote
0answers
11 views

Is there a specific term for such collections of filters?

Let $U$ be a set. Concept = "a set $\mathscr{C}$ of proper filters on $U$ such that if $X\in\mathscr{C}$ and $Y$ is a proper filter on $U$ and $Y\supseteq X$, then $Y\in\mathscr{C}$." Is there a ...
1
vote
2answers
69 views

Evaluating $\lim_{n\to\infty}\left(\frac{1-i}{4}\right)^n$

It's been a while since I have taken Calculus II so my experiences on sequences and series has gone down the drain. I'm trying to find the limit of the sequence ...
0
votes
0answers
34 views

Notations in Riemannian Geometry

Let $f:M\rightarrow N$ be differential map. We denote tangent map $$f_*:TM\rightarrow TN$$ and cotangent map $$f^*:T^*N\rightarrow T^*M$$ Now let $M$, $N$ be Riemannian manifolds, and ...
3
votes
0answers
23 views

History of Moment Generating Functions

I am beginning to appreciate how important Moment Generating Functions (MGFs) are regarding various common probability distributions and the ways their expectations/variances are calculated. My ...
4
votes
2answers
164 views

Careers in Mathematics?

I am a college freshman, and I really like to have goals for my life, one of the big ones is my career of choice. Previously, I have always wanted to be a programmer, and I have written a lot of code. ...
0
votes
0answers
33 views

Events for high school students which can look good on a cv.

I'm a european high school student and I'll apply to university in July. Are there any interesting activities/events which can look good on the curriculum I'll submit with the application? Further ...
15
votes
3answers
553 views

Starting sentences with mathematical symbols.

I apologise if this is a duplicate in any way or is too opinion-based. To what extent is it best not to start a sentence with a mathematical symbol? I find that when trying to solve a problem or ...
2
votes
1answer
81 views

How to proceed doing number theory?

I'm an undergrad majoring in mathematics. Being in first year I'm still exploring new branches of mathematics and till now, It is analysis and Number theory that I've come to have a great interest ...
12
votes
9answers
579 views

Strangest Notation? [closed]

While this may be a fruitless pursuit of anecdotes, I still ask: what is the strangest (or most blatantly wrong (at least in the eyes of common notation)) mathematical notation you have ever seen?
6
votes
5answers
380 views

Why has $\int \sin (\sin x) dx$ not been solved yet?

I have Calculus 2 background, so please try to keep your answers around that level. I inly want a brief explanation. What is it about $\sin (\sin x)$ that makes it difficult to integrate? Also, what ...
6
votes
8answers
575 views

The line between axiom and theorem

Consider the intermediate value theorem. The theorem is very intuitive and may be described as obvious: if you go from $A$ to $B$ without teleporting, you have been everywhere between $A$ and $B$. ...
1
vote
0answers
30 views

Modular arithmetic - Suggestions to begin

I've always wanted to start studying modular arithmetic to try to solve problems like: $$\text{find } n \in \mathbb{N} : 4n^2 \equiv 1 ~(\text{mod }{10^4})$$ I've good basis of mathematical analysis ...
2
votes
2answers
67 views

Can things go wrong if we declare objects to be arrows?

I am familiar with categories and also with 'categories without objects'. In fact a category is completely determined by its set of arrows and the objects can be missed. Nevertheless the objects are ...
7
votes
2answers
633 views

Algebraic geometry project ideas for high school students

I am teaching a "senior seminar" course for strong students at our local high school. For 6 weeks the students learned about basic/classical algebraic geometry. In a few weeks they will start projects ...
12
votes
2answers
274 views

What problems are easier to solve in a higher dimension, i.e. 3D vs 2D?

I'd be interested in knowing if there are any problems that are easier to solve in a higher dimension, i.e. using solutions in a higher dimension that don't have an equally optimal counterpart in a ...
11
votes
4answers
188 views

Topics in Differential geometry $\cap$ Algebraic geometry

I find (both: differential and algebraic) geometry fascinating. I'm just beginning my graduate studies, but I'd like to know some topics/theorems in the intersection of these two (since I don't really ...
2
votes
3answers
65 views

Limit notation convention

I've seen in different sources that there is a prevalent notation convention regarding to limits. If $f: X \rightarrow \mathbb{R}$ is a function and $x_0$ is an adherent point of $X$. It's very ...
0
votes
0answers
18 views

The “dynamics” of foliations

My mathematical interests revolve mainly around dynamical systems, and (unfortunately) I don't have as much general culture in geometry as I would like (and I should...) I often see foliations come ...
1
vote
1answer
116 views

Have any definitions in mathematics been redefined

Based on certain intuitions and motivations we make certain definitions and then proceed to use these concepts in further developing our intuition. For example, we have an intuition that a line has ...
27
votes
10answers
2k views

How to be a successful math undergraduate student?

I am an undergratuate student in my first year of combined bachelor of electrical engineering and bachelor of mathematics. For my mathematics degree, this year I am supposed to take two math courses ...
8
votes
4answers
345 views

Topics on Number theory for undergraduate to do a project

Im an undergraduate in the mathematics field ..So i wanna be alittle more productive and wanted to do an essay or project mostly on number theory or Algebra(Rings or Groups) and i want to ask if you ...
0
votes
1answer
32 views

What is the “lowest” set of axioms that can be used in proofs?

What is the most basic set of axioms that one can use in proofs? As in, the axioms are irreducible. The most basic set of irrefutable rules in mathematics. I assume it has something to do with number ...
2
votes
1answer
61 views

Good non-textbook math books

I'm looking into learning math partially by reading, I have and am currently reading books by Dover publishing. I like these books because they don't use the formulaic textbook layout and rather ...
3
votes
2answers
215 views

Existence of algorithm for determining if a given number is rational or not

As far as I understand, it is not necessarily a easy thing to prove that a real number is rational or not. For example, according to http://mathworld.wolfram.com/e.html "$e+\pi \in \mathbb{Q}$?" is ...
1
vote
1answer
34 views

Are there computer requirements other than basic C++ for being accepted to a graduate program in Non-Linear Dynamics?

Ladies and Gentlemen I am currently studying applied mathematics and statistics. I am more than just a little bit interested in studying non-linear dynamics at a graduate level. I was wondering if, ...
0
votes
3answers
43 views

Validity of Problem Solving Methods

I was in class the other day, and I was suddenly concerned with the idea that some methods don't work for solving certain problems. I'm working on integration right now, so some of the problem solving ...
4
votes
0answers
81 views

Earliest precursor to category theory

In the Historical notes section of the Wikipedia article on category theory, it is mentioned that in 1942-1945, Samuel Eilenberg and Saunders Mac Lane, in the course of their work in algebraic ...
0
votes
2answers
41 views

Calculator similar to Desmos but for 3D

Is there a calculator with functionality similar to Desmos but in 3 dimensions? I am looking to learn about families of quadric surfaces so I am looking for a 3D calculator with sliders.