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2
votes
2answers
127 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
2answers
78 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 ...
1
vote
2answers
73 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 ...
1
vote
0answers
59 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 ...
4
votes
0answers
127 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 ...
3
votes
0answers
120 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 ...
11
votes
4answers
346 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 ...
8
votes
8answers
685 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 $B$. ...
2
votes
3answers
79 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
1answer
70 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 ...
16
votes
6answers
2k views

When to use the contrapositive to prove a statment

My question tries to address the intuition or situations when using the contrapositive to prove a mathematical statement is an adequate attempt. Whenever we have a mathematical statement of the form ...
4
votes
3answers
278 views

Paradoxes in Logic

What is it that makes something a paradox? It seems to me that paradoxes are just, in many cases, misunderstandings about the properties some object can have and so misunderstandings about ...
11
votes
4answers
2k views

Topics on Number theory for undergraduate to do a project [closed]

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 ...
2
votes
1answer
236 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
447 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 ...
15
votes
1answer
980 views

Why are there so few Euclidean geometry problems that remain unsolved?

Stillwell mentions in his book Mathematics and its History that: Most of the really old unsolved problems in mathematics, in fact, are simple questions about the natural numbers... What is it ...
5
votes
2answers
503 views

What fields (and operators acting on those fields) might form the basis of alien mathematics?

Addition and multiplication, according to most histories, arose in human civilizations out of a need to count a finite number of objects, and then later on especially, to measure land. What might be ...
0
votes
3answers
68 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 ...
1
vote
1answer
58 views

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

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, ...
4
votes
1answer
155 views

Shannon's MTC as 'information theory'

I'm a little confused as to whether or not this question belongs here or on http://cstheory.stackexchange.com/, so please, bear with me. I've been reading a few books on the concept of information, ...
3
votes
3answers
481 views

What are the “real math” connections between Euclidean Geometry and Complex Numbers?

Some background: I am a high school student and I am very interested in math. I have done a lot of the extracurricular learning I have done is through doing math problems from various competitions, ...
6
votes
4answers
1k views

Some Questions regarding preparing for Math Olympiads (searched but didn't get answers)

Many questions have been asked on this site regarding preparation for olympiads like the Putnam. I've read those questions and accordingly decided to start with Engel's "Problem Solving" but I have a ...
1
vote
2answers
350 views

What problems are there left to solve? [closed]

From the ancient Greek mathematicians (Archimedes, Pythagoras) before Christ to Issac Newton to George Birkhoff, these mathematicians have made huge strides in mathematics, developing theorems and ...
114
votes
18answers
11k views

How do you describe your mathematical research in layman's terms?

"You do research in mathematics! Can you explain your research to me?" If you're a research mathematician, and you have any contact with people outside of the mathematics community, I'm sure ...
1
vote
2answers
50 views

PDE course question

What courses do I need for a course in partial differential equations? My university has a prereq of Multivariate Calculus and Ordinary Differential Equations. However, I opened up a book on pdes in ...
1
vote
1answer
135 views

Examples for Conditional Expectation (modern probability theory)

I'm in the process of learning about conditional expectation in the framework of modern probability theory. The sudden change brought about by the notion of conditional expectation being a function on ...
3
votes
3answers
188 views

How to know if I'm doing real analysis correctly?

I've just started a second year course in real analysis. This is my first proof-oriented course. Last year, our maths curriculum was introductory tertiary calculus and algebra. When I practised ...
2
votes
2answers
113 views

Why doesn't this work? — The V.I. Arnold Primary School Problem (Two women started at sunrise…)

I've got a, perhaps silly, question. I understand the various solutions to this problem: Two women started at sunrise and each walked at a constant velocity. One went from A to B and the other ...
2
votes
2answers
112 views

Information on crucial results concealed as exercises or neglected in a textbook

First, where can students find lists, information, or resources on the crucial results, inequalities, theorems, etc... which a textbook might not explictly feature or even bring up at all? Second, ...
5
votes
3answers
141 views

On trusting the mathematical process [closed]

In studying math we are, at least partially, interested in making abstraction of real world problems and solving them through rigorous techniques and methods, and then interpreting the result. Let us ...
1
vote
0answers
35 views

Elementary set theory problem - lack of working memory

Here's an example to illustrate the problem: I have this problem from a book about set theory: $\wp(\cup_{i \in I}A_i)\nsubseteq \cup_{i \in I} \wp(A_i)$ I'm supposed to translate it to pure ...
3
votes
2answers
132 views

For what categories do category algebras exist?

The monoid algebra $R[M]$, for a commutative ring $R$ and a monoid $M$, can be described as the free $R$-algebra on $M$. We think of $R[M]$ as the set of finite formal sums of elements of $M$ with ...
8
votes
1answer
673 views

What pure mathematics foundations should an applied mathematician have?

I'm studying mathematics, with some statistics also, and I've always chosen applied courses. I'm getting to the point where I'm studying 3rd year undergraduate to graduate level material. My first ...
3
votes
3answers
135 views

Explanation of “without loss of generality”

I've been doing math for some time now, but I'm starting to get into serious proofs now that I'm taking analysis and such. I'm just confused with what it actually means? Does it infer that although ...
4
votes
0answers
160 views

Largest mathematics library

Just out of curiosity, what is the largest mathematics/mathematically-oriented library in the world? Is it open to the public? I would like to visit it sometime.
42
votes
7answers
12k views

I can't do math?

So after failing another math test today I have realized that I can't do math on my own. For whatever reason when I get to a test I just can't do math without extra resources. I think this is because ...
2
votes
2answers
115 views

Where do non-associative rings appear?

When reading papers on algebraic topology, I often find the term "associative ring". The multiplication structure of a ring is normally assumed to be associative, therefore I guess that ...
5
votes
2answers
311 views

Expository texts on advanced subjects

I'm reading The Princeton companion to Mathematics and Basic Notions of Algebra by Shafarevich. Both of them are really pleasant reading, the first one treats the topics from a more elementary point ...
0
votes
1answer
83 views

Help finding an area for my Masters

Next year I will be doing my MSc. in mathematics at the University of Toronto. I, however, am not sure whether after I will want to pursue my PhD. or not. So I was wondering what branch of mathematics ...
4
votes
8answers
2k views

Studying Math, All Over Again

I am 16 and never really paid attention to math. For me it was just one more obstacle in passing the exams. Now that I study computers, mostly on my own, I find that there is indeed a need for me to ...
8
votes
6answers
642 views

Thinking of mathematics in terms of analogs

I think the way that I've come to think about mathematics is becoming problematic and I'm wondering if I should abandon it. When I study mathematics, I find myself trying to compare the mathematical ...
10
votes
4answers
790 views

Is studying mathematics chronologically a good idea or not and why?

In high school nowadays most mathematics you learn is fairly 'old'. You have your geometry, all of which (taught in high school) was known to the Greeks more than 2 thousand years ago. You have ...
2
votes
0answers
250 views

Vector-Lattices and “Approximating $\mathscr{L^1(\mathbb{R}^k)}$”.

In this question I asked whether $\mathscr{L}^1(\mathbb{R}^k)$ forms a category in any way. It was concluded that indeed it does not. I thought to myself, "well, could we at least approximate the ...
6
votes
1answer
500 views

best book for real analysis for undergraduate [duplicate]

I am in under graduate first year. I want to learn some advanced real analysis. Can you give me some suggestions? One suggestion I got is Rudin's mathematical analysis. Can I buy that?
2
votes
0answers
65 views

A Proof-Library

Is there any source, perhaps a website or a book, which keeps track of all different proofs ever announced of a theorem? For example, I have heard that there are about $80$ different proofs of the ...
15
votes
4answers
6k views

What are some classic fallacious proofs? [duplicate]

If you know it, also try to include the precise reason why the proof is fallacious. To start this off, let me post the one that most people know already: Let $a = b$. Then $a^2 = ab$ $a^2 - b^2 = ...
1
vote
2answers
96 views

Metric Topology

Suppose our topological space is $\mathbb{R}$. Why is each basis element $(a,b)$ for the order topology is a basis element for the metric topology? Munkres says, ...
4
votes
4answers
134 views

Why are roots of polynomials called geometric objects?

I read the following from the Wikipedia article about algebraic varieties: Proven around the year 1800, the fundamental theorem of algebra establishes a link between algebra and geometry by ...
8
votes
3answers
1k views

How do I “learn” more difficult algebra?

I do not really understand where I was suppose to learn this kind of stuff. I am always told that my algebra knowledge is the biggest reason I am so bad at math. But I do not understand where I was ...
4
votes
1answer
316 views

Forcing the discriminant of an integral basis to be a Carmichael number.

I was thinking about the following lemma recently. Lemma: Let $K=\mathbb{Q}(\theta)$ for some algebraic number $\theta$ and let $n=[K:\mathbb{Q}]$. If $\{\tau_1, \,\dots\,, \tau_n\}$ consists of ...