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21
votes
0answers
4k views

What are some strong algebraic number theory PhD programs? [closed]

I am currently applying for PhD programs in the US. My main interests are number theory and algebra. More specifically, I am interested in algebraic number theory (number fields, Galois groups, ...
4
votes
2answers
590 views

What is the exact motivation for the Minkowski metric?

In introductory texts about Lorentz Geometry, one always learns about the Minkowski space, i.e. $R^4$ with the Minkowski metric $$ m(x, y) := -x_0 y_0 + x_1y_1 + x_2y_2+ x_3 y_3 $$ Using this ...
2
votes
8answers
487 views

Why Not Define $0/0$ To Be $0$?

For every number $x$, $x\times 0=0$, hence $\dfrac{0}{0}$ can be any number! So $\dfrac{0}{0}$ "is knows as indeterminate" [1]. But what if we define it to be $0$? I already have an answer, but ...
3
votes
1answer
199 views

motivating the conservation of symplectic area by way of general (coordinate) covariance

I'm trying to motivate why a symplectic structure captures exactly the right structure one needs to do classical mechanics. The easiest part of this story goes like this: we need a procedure for ...
1
vote
0answers
58 views

Is there a textbook treatment of Ky Fan's minimax theorem and its generalizations?

Theorem 2 in Ky Fan(1952) is a powerful tool in zero-sum games, which states: Let $X$ be a compact Hausdorff space and $Y$ an arbitary set (not topologized). Let $f$ be a real-valued function on ...
2
votes
2answers
144 views

Most unusual form of mathematical induction

After reading the algebraic proof of Fundamental Theorem of Algebra, where induction was carried out on "The highest power of $2$ dividing $n$", which I regard to be unusual and brilliant at the same ...
1
vote
0answers
66 views

Who made now part of the problem?

Who came up with the meme of putting the current year as a four digit number into exercise problems? Is there a known first historical account?
31
votes
20answers
4k views

Good math bed-time stories for children?

What are some good references/books/articles from which to derive some good bed-time math stories to pique a child's interest in math? I am fascinated by math (used to hate it as a kid) and want my ...
95
votes
28answers
8k views

What are some examples of notation that really improved mathematics? [closed]

I've always felt that the concise, suggestive nature of the written language of mathematics is one of the reasons it can be so powerful. Off the top of my head I can think of a few notational ...
2
votes
1answer
134 views

Why does $e$ seem to be an intuitive number? [closed]

I often find two numbers roughly "in the same ballpark" if they are within a factor of about $e$ of each other. For example, if I know computers generally cost upward of $\$1000$, then $\$2700$ would ...
14
votes
4answers
6k views

How to do well on Math Olympiads

I'm a high school student who really likes maths and I'm quite good at school. I want to start training maths by myself but I think I need some guidelines. I want to do well on IMO but I don't know ...
99
votes
33answers
13k views

What are the most overpowered theorems in mathematics? [closed]

What are the most overpowered theorems in mathematics? By "overpowered," I mean theorems that allow disproportionately strong conclusions to be drawn from minimal / relatively simple assumptions. ...
1
vote
1answer
187 views

Polynomials as sum of squares

Sometimes I have seen some math's competition problem solutions made by completing the expression as sum of squares. What is the intuition/computer program behind these solutions? For example, Prove ...
2
votes
1answer
81 views

Examples of a problem solved by a well-chosen derivative equaling zero

What are examples of problems which are solved by taking a derivative of a well-chosen function, and finding that it is zero, therefore the function must be constant? I can think of a few: Show ...
0
votes
1answer
56 views

We refer to X for standard notations and definitions from Y

I'm having problems with my mathematical English, so I'd like to ask for your help! Is it correct to write something like "Unless stated otherwise, we refer to [1] and [2], respectively, for standard ...
1
vote
4answers
2k views

What really is an indeterminate form?

We can apply l’Hôpital’s Rule to the indeterminate quotients $ \dfrac{0}{0} $ and $ \dfrac{\infty}{\infty} $, but why can’t we directly apply it to the indeterminate difference $ \infty - \infty $ or ...
0
votes
1answer
127 views

Quantum Mechanics-Question [closed]

I'm a 3rd year student of a Mathematic School and i'm looking what to follow after i'm done with my undergratuate studies. I'm interested in following two paths: 1)Pure Mathematics (I know that i can ...
1
vote
2answers
11k views

What are the applications of matrices in real world?

Matrices are considered very important in mathematics. What are some examples of applications of matrices to real world problems that would be understandable by a layman?
3
votes
1answer
1k views

Is it worth pursuing a statistics minor? I want to go to pure math grad school.

My school offers a minor in "quantitative methods" from the psychology department. I've taken one class from this minor (Intro to Stats) and will most likely get an A. That said, the classes in this ...
79
votes
22answers
9k views

Is math built on assumptions?

I just came across this statement when I was lecturing a student on math and strictly speaking I used: Assuming that the value of $x$ equals <something>, ... One of my students just rose ...
5
votes
5answers
364 views

Are there more real numbers than we can actually imagine?

I mean, if we could imagine all the real numbers then we could assign each number a finite sentence (or a finite book). Since the set of the finite books is countable then the set of real numbers ...
3
votes
3answers
187 views

What other definite integrals can be computed in a manner similar to $\int_{-\infty}^\infty e^{-x^2}dx$?

The technique for computing $\int_{-\infty}^\infty e^{-x^2} dx=\sqrt{\pi}$ by computing the integral squared using polar coordinates is well known. Are there any other integrals that can be computed ...
5
votes
2answers
504 views

Dogmas and Mathematics

What are the dogmas that restrict or promote the development of mathematics? I know that a dogma is a set of beliefs that is accepted by the members of a group without being questioned or doubted. ...
7
votes
1answer
496 views

Are Base Ten Logarithms Relics?

Just interested in your thoughts regarding the contention that the pre-eminence of base ten logarithms is a relic from pre-calculator days. Firstly I understand that finding the (base-10) ...
0
votes
5answers
361 views

Why doesn't $1/x=0$ have any solution?

Just out for curiosity ! Why $1/x=0$ doesn't have any solution? Or is it that the solution takes you to $1=0$ situation which would nullify mathematical principle that we stood for years Educate ...
3
votes
2answers
100 views

Additive analogy of proportionality symbol

The relation of proportionality is quite abundant, and so for convenience there exist symbols, such as "$\propto$", to denote it. I would like to know if there is likewise a symbol to denote the ...
8
votes
2answers
224 views

The 'abelian group' custom

This is just a question for fun: As far as I know, frequently it is considered to be customary to denote an additive commutative group as 'abelian group' in lowercase, although the term is named ...
1
vote
5answers
17k views

Rules for Product and Summation Notation

When we deal with summation notation, there are some useful computational shortcuts, e.g.: $$\sum\limits_{i=1}^{n} (2 + 3i) = \sum\limits_{i=1}^{n} 2 + \sum\limits_{i=1}^{n} 3i = 2n + ...
101
votes
16answers
14k views

Is 10 closer to infinity than 1?

This may be considered a philosophy but is the number "10" closer to infinity than the number "1"?
65
votes
14answers
6k views

Are all mathematicians human calculators?

I asked my dad why he did not major in math he said "because he is not good at math". I think I like math, and I think I'm ok at it, but I'm not gifted or anything like that, I just like math. I think ...
33
votes
2answers
2k views

Intuitive reasoning why are quintics unsolvable

I know that quintics in general are unsolvable, whereas lower-degree equations are solvable and the formal explanation is very hard. I would like to have an intuitive reasoning of why it is so, ...
1
vote
1answer
140 views

Analogue of prime numbers in addition? [closed]

What is the analogue of prime numbers in addition?
6
votes
1answer
161 views

How does the fundamental theorem of algebra follow from Weierstrass’s theorem.

Can anyone please explain to me how the fundamental theorem of algebra follows from or is related to the Bolzano Weierstrass’s theorem?
2
votes
2answers
219 views

What's the exact meaning of this sentence from George Peacock?

I am reading the book "A history of abstract algebra" by Israel Kleiner. The following sentence is said by George Peacock. I am not a native English speaker. So could someone translate it into plain ...
3
votes
1answer
158 views

Mnemonic for platykurtic and leptokurtic

I keep confusing terms leptokurtic and platykurtic. Is there a good mnemonic to help remember which is which? "Lepto" means "little", "platy" means "flat", and both are equally unrelated to thickness ...
9
votes
2answers
380 views

Why is the undergraph definition of Lebesgue integral so rare?

So in Pugh's Real Mathematical Analysis, the initial definition of the Lebesgue integral is as the Lebesgue measure of the undergraph of the function (where the function is nonnegative, with the usual ...
20
votes
4answers
2k views

What is a good book to study classical projective geometry for the reader familiar with algebraic geometry?

The more I study algebraic geometry, the more I realize how I should have studied projective geometry in depth before. Not that I don't understand projective space (on the contrary, I am well versed ...
0
votes
2answers
132 views

What is the most concise mathematical expression? [closed]

I was reading the introduction of a mathematics textbook and the author says "Mathematics is like a poem, it expresses a lot of meaning in as little words as possible." I am curious as to what may be ...
5
votes
0answers
145 views

Understanding Mathematics [closed]

Lately, I've been learning a lot of mathematics. But I am having a problem. Although for most part I understand in general as to how a function is derived and not so much as to understanding the proof ...
2
votes
0answers
151 views

Lattice theory in mathematics and physics

I have undertaken a project examining lattice model and trying to construct algorithm that could work on all lattice (in physical sense, or crystal structure). I notice there is a branch in ...
11
votes
4answers
964 views

Cheap online graduate math courses

I am an undergraduate math major. I want to have some math graduate courses on my transcript. I can't do this at my university. Are there online graduate math courses that I can take and receive a ...
9
votes
3answers
520 views

Proof for '$AB = I$ then $BA = I$' without Motivation?

I have read this question page (If $AB = I$ then $BA = I$) by Dilawar and saw that most of proofs are using the fact that the algebra of matrices and linear operators are isomorphic. But from a ...
8
votes
2answers
608 views

Why Goldbach's conjecture is difficult to prove?

Why Goldbach's conjecture is still non-solved and is difficult to prove? What makes the mathematicians fail when trying to prove it?
5
votes
1answer
586 views

Differentiation under (measure theoretical) integral sign

I am looking for a citable reference for the result on differentiation under the integral sign for integration against a measure. The result states that if $R \subset \mathbb R$, $(X,\mathcal F, ...
14
votes
1answer
898 views

High School Students Publishing Mathematics [closed]

Right now I'm a senior in high school and I will be submitting a number of applications for undergraduate admissions with deadlines ranging from November $30^{\text{th}}$ to early February. However I ...
4
votes
4answers
435 views

By definition, how is a prime number represented?

Even numbers can be easily represented as $2n$. Odd numbers as $2n+1$. An exactly divisible operation can be defined as $n = dq$. But, is there an specific way of representing a prime number, ...
7
votes
2answers
414 views

Logarithms of logarithms of Graham's number, is the result ever handy?

The other day I was asked how to represent really big numbers. I half-jokingly replied to just take the logarithm repeatedly: $$\log \log \log N$$ makes almost any number $N$ handy. (Assume base ...
8
votes
1answer
729 views

What's the best way to teach oneself both Category Theory & Model Theory?

I've done a bit of reading around both Category Theory & Model Theory (CT & MT) as a novice in each field. I'm interested in how they might combine, particularly when applied to Algebra. [So ...
2
votes
2answers
251 views

Undergraduate mathematics study

I don't know if this question is appropriate but I hope to get advice from people experienced in mathematics.I am currently an undergraduate and I study applied mathematics and computer science.Our ...
0
votes
1answer
103 views

ENS is an abbreviation of?… [duplicate]

In CWM Mac Lane uses the term $\mathbf {ENS}$ for a category having as objects the subsets of a given set and as morphisms the functions from these sets to these sets. What is abbreviated by the ...