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8
votes
2answers
477 views

Suggestions for further topics in Commutative Algebra

I am currently taking a semester long course in Commutative Algebra. We have covered a lot of dimension theory, and today finished proving Zariski's Main Theorem, which was the professor's original ...
8
votes
3answers
754 views

Are all numbers real numbers?

If I go into the woods and pick up two sticks and measure the ratio of their lengths, it is conceivable that I could only get a rational number, namely if the universe was composed of tiny lego ...
8
votes
2answers
1k views

When can one expect a classical solution of a PDE?

When solving a PDE, there may be a classical solution or a weak solution (or distribution solution). But I am wondering that when people talk about "finding a solution" to some PDE, what do they refer ...
7
votes
1answer
163 views

How to properly use technology for back-of-the-envelope calculations?

I'm usually quite eager on using technology wherever sensibly applicable, however whenever I make some calculations I still end up using a pen and paper, by now resulting in an entire pile of sheets ...
7
votes
2answers
1k 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 ...
7
votes
1answer
604 views

I feel that (physics) notation for tensor calculus is awful. Are there any alternative notations worth looking into?

I am reading through Fung and Tong's "Classical and Computational Solid Mechanics", and feel that the Einstein summation convention saves a few symbols, at the expense of a lot of clarity. Along with ...
7
votes
1answer
241 views

Is it possible to avoid redundancy in a foundational work?

Imagine we're developing all of mathematics from scratch. We settle on using a set-theoretic foundation. Early on, we assert that an ordered pair $(x,y)$ can be abbreviated $xy$ whenever there is no ...
7
votes
2answers
322 views

A Question on RH relating to Prime Number theorem

Well, in a previous post regarding the explanation of Riemann Hypothesis Matt answered that: The prime number theorem states that the number of primes less than or equal to $x$ is approximately ...
7
votes
2answers
548 views

A question on FLT and Taniyama Shimura

Sometime back i watched the documentary of Andrew Wiles proving the Fermat's Last theorem. A truly inspiring video and i still watch it whenever i am in a depressed mood. There are certain ...
6
votes
2answers
566 views

Advice: Modern vs. Classics

First of all, my apologies if (well, I know I am but I don't know where to put it) I am posting this in the wrong place. So please feel free to move it to someplace else or to tag it differently if ...
6
votes
3answers
624 views

Why is Kunen inconsistency at the top of Cantor's upper attic?

Motivation: I have reproduced part of page 396 and 397 from Handbook of Mathematical Logic below: So if we start with a concept of number and play the game of naming the largest one, does Kunen ...
5
votes
1answer
236 views

Guide to mathematical physics?

I am currently a math phd student specializing in algebraic geometry aspiring to work at the boundaries of the the fields of mathematics and physics and so, was looking into the field of mathematical ...
5
votes
4answers
350 views

Are problems about finding the next term of a sequence mathematical? [closed]

I have run through many questions like "given the few first terms of a sequence, find the next one". My question is "are such problems mathematical?". This is something like given $f(1), f(2), f(3)$, ...
5
votes
1answer
84 views

Have arrows in a category with this property a special name?

Studying posets I encountered the notation $a\prec b$. It means that $a<b$ and no $c$ exists with $a<c<b$. If $a\prec b$ then in words $a$ is covered by $b$. Looking at a poset $P$ as a ...
5
votes
4answers
370 views

Would nonmath students be able to understand this?

For a course, I am required to do a presentation. The topic could either be something mundane, like a career strategy report, or something more interesting, such as a controversial topic, or an ...
5
votes
0answers
202 views

Convex Hulls vs Shrink Wrap

I was recently explaining to a friend what the convex hull of a set of points is using the analogy of an elastic band around a set of nails hammered into a board. I was about to say that we can ...
5
votes
1answer
1k views

Can I go through Hartshorne without knowing much analysis?

I know intro abstract algebra and some real analysis. Is this enough to study algebraic geometry from the book of Hartshorne?
4
votes
1answer
138 views

What is the best way to tell people what Analysis is about?

What is the best way to tell people what Analysis is about? I am currently taking Analysis course. However, I am really having a big difficulty explaining to people what Mathematical Analysis is ...
4
votes
3answers
177 views

Multiplication, What is It?

What is multiplication? Upon review logarithms, and square roots, I realized that I have no intuitive grasp of multiplication-well no more so than I have for addition. Is it simply another thing we ...
4
votes
2answers
239 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 ...
4
votes
2answers
155 views

Intuitive explanation for $\mathbb{E}X= \int_0^\infty 1-F(x) \, dx$

I can see by manipulating the expression why $\mathbb{E}X$ works out to be $\int_0^\infty 1-F(x)\,dx$, where $F$ is the distribution function of $X$, but what is an intuitive explanation for why that ...
4
votes
4answers
291 views

Is there a symbol for the idea of the smallest value greater than zero?

I know that it isn't actually a number but I do think it's a concept in mathematics. So the question is, is there a symbol representing this concept? I thought maybe it was Phi but I couldn't find it ...
4
votes
1answer
196 views

Beginner material for mathematical logic

I am looking for study and beginner material to study mathematical logic. I understand that it is a very broad topic but I would like to know what the best path there is to learning mathematical ...
4
votes
5answers
1k views

Is it possible to practice mental math too often?

I'm far from a mathematician, but the field I'm trying to break into (management consulting) requires a fair amount of mental arithmetic. I'm okay, but I'm not even close to as good as I need to be in ...
4
votes
2answers
459 views

Good reference texts for introduction to partial differential equation?

As the title, are there any good reference texts for introduction to partial differential equation?
4
votes
2answers
582 views

What is importance of the Bunyakovsky conjecture?

Bunuyakovsky conjecture states that: An irreducible polynomial $f(x)$ of degree two or higher with integer coefficients and property that $\gcd(f(1),f(2),......)=1$ generates for natural ...
3
votes
1answer
80 views

Software for math sketching

Usually when you're writing in LaTeX you want some pretty illustrations. Right now for geometry figures I use GeoGebra, which is easy enough; but I usually see better figures on other papers. Plus, ...
3
votes
2answers
127 views

Should axioms be viewed as part of the signature?

I included category theory in the tags in order to get feedback from the categorial logic community. It goes without saying that this isn't really category theory. A semigroup can be defined as a ...
3
votes
4answers
984 views

Can I skip the first chapter in Rudin's Principles of Mathematical Analysis?

I am a statistician who wishes to learn real analysis in order to better understand the foundations of statistics. With that aim in mind I plan to go through Rudin's classic on "Principles of ...
2
votes
1answer
116 views

How can we draw a Blaschke $3$ ellipse?

Today I read the article Ellipses and Finite Blaschke Products Ellipses and Blaschke Ellipses by Ulrich Daepp, Pamela Gorkin, and Raymond Mortini. In there they have proved very nice geometric results ...
2
votes
1answer
332 views

Analogy of ideals with Normal subgroups in groups.

I've started with Ideals in ring theory but still not comfortable with the analogy it has with normal subgroups in group theory.Like we can visualize normal subgroups as Is there some good intutive ...
2
votes
2answers
253 views

I am going to learn these math topics , please suggest me where to start?

I always did poor in mathematics and i even quit my mathematics from 10th grade but since I was good in programming ( C++ and Java) I took course related to computers in my college where I am going to ...
2
votes
1answer
710 views

Prerequisite for Differential Topology and/or Geometric Topology

What are the prerequisites to learning both or one of the items? Consider that one will have done some of the "core" classes like Differential Geometry, Real Analysis, Abstract Algebra and POint-Set ...
2
votes
2answers
464 views

Intuition for the Product of Vector and Matrices: $x^TAx $

When I took linear algebra, I had no trouble with the mechanical multiplication of matrices. Given the time to write things out and mumble a bit about ith and jth rows, I can do the products no ...
1
vote
2answers
122 views

Elementary Applications of Cayley's Theorem in Group Theory

The Cayley's theorem says that every group $G$ is a subgroup of some symmetric group. More precisely, if $G$ is a group of order $n$, then $G$ is a subgroup of $S_n$. In the course on group theory, ...
1
vote
2answers
133 views

Algebraic solution to the Broken Weight Problem

Here is a problem I was sent, which it turns out was first posed by Claude Gaspard Bachet de Méziriac in a book of arithmetic problems. The problem is as follows: A few years ago, a King's ...
1
vote
3answers
317 views

Have any one studied this binomial like coefficients before?

Consider the following identities. $\dfrac{n}{n-r}\dbinom{n-r}{r}=\dfrac{n}{r}\dbinom{n-r-1}{r-1}$ ...
16
votes
5answers
4k views

Is 'no solution' the same as 'undefined'?

Today in class my teacher wrote something along the lines of: $6^x = 0$ And proceed to heed a response from the class. A few people shouted undefined. So the teacher then writes: no solution ...
10
votes
5answers
2k views

Common misconceptions about math

YARFMO (Yet another reposting from Mathoverflow) ;-) The more you know about math the more you find conceptions previously thought correct to be false: 1.) math is not as exact as many believe - in ...
8
votes
4answers
371 views

What's the deal with integration?

So at uni we learned tricks and techniques for integration until cows came home. But to what end? Any/All definite integrals can be evaluated using numerical methods. Most integrals in application can ...
8
votes
4answers
823 views

How To Reach The “Next Level” of Mathematics

I am a junior-high pre-algebra student. I feel that my class is holding me back, so I wanted to learn "higher-level math". So what should I learn now? What do you believe is a "next step"?
7
votes
2answers
198 views

Reference request regarding calculus exam

I'm currently a first year computer science student and I'm deeply interested in calculus . That being said, what we studied so far consists of: Cantor sets, sequences and a brief introduction to ...
6
votes
2answers
426 views

Filling the gap in knowledge of algebra

Recently, I realize that my inability to solve problems, sometimes, is because I have gaps in my knowldge of algebra. For example, I recently posted a question that asked why $\sqrt{(9x^2)}$ was not ...
6
votes
1answer
252 views

Idea behind the factorization of the matrix $\operatorname{diag}(a,a^{-1})$ in algebraic K-Theory

If $a \in S$ is some invertible element in a ring $S$, then a computation shows $$\pmatrix{a & 0 \\ 0 & a^{-1}} = \pmatrix{1 & a \\ 0 & 1} \pmatrix{1 & 0 \\ -a^{-1} & 1} ...
6
votes
1answer
419 views

Mathematics needed for higher dimensional category theory?

I'm a undergrad(third year, Manchester uni) that is thinking of doing a PhD in this area or category theory in general. Just wondering, what branches of Maths should I focus on? As I've been told ...
6
votes
6answers
2k views

Looking for Proofs Of Basic Properties Of Real Numbers

I have just begun my study of complex numbers and I learned where imaginary numbers came from and their importance. However there's one thing that I need to clarify and that is the properties of real ...
5
votes
1answer
156 views

Intuition for “the existence of a basis for every vector space is equivalent to the Axiom of Choice”?

Is there a intuitive way to understand "the existence of a basis for every vector space is equivalent to the Axiom of Choice"?
5
votes
6answers
701 views

How to make sure a proof is correct

If you come up with a proof of a mathematical proposition, how do you verify the proof is correct? Put it another way, how do you avoid a wrong proof? I guess there is no definitive answer to this. ...
5
votes
3answers
318 views

Is there a branch of mathematics that requires the existence of sets that contain themselves?

I notice that Russell's paradox, Burali-Forti's paradox, and even Cantor's paradox, all depend on our tolerance of sets that contain themselves (at one level of depth or another). Thus, I was thinking ...
5
votes
2answers
376 views

Map of Mathematical Logic

My undergraduate University does not offer advanced courses on logic, I know truth tables, Boolean algebra, propositional calculus. However I want to pursue Mathematical Logic on the long term as a ...