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3
votes
1answer
255 views

Is there an intuitive non-mathematical way to show $\sum_n 1/n=\infty$?

I want to show some school students about the sum of the harmonic series diverging and I need some nice interpretation for this. I'll preferably love to have a figure. Thanks in advance!
8
votes
3answers
370 views

Math under time constraints [closed]

I'm currently an undergrad math student and I have been wondering if being able to perform well under time constraints requires a deeper knowledge of the material than performing under no time ...
2
votes
3answers
134 views

teaching concept of “subspace” to linear algebra students

I'm currently a T.A. for an introductory combined linear algebra & differential equations course geared toward engineering students. One major problem I've run across is that most of my students, ...
10
votes
1answer
215 views

Independence results that cannot be established by forcing.

I read the Wikipedia article on Absoluteness recently and found mention of Shoenfield’s Absoluteness Theorem, which states that if $ \phi $ is any $ \Sigma^{1}_{2} $- or $ \Pi^{1}_{2} $-sentence of ...
3
votes
1answer
84 views

When asked to find all solutions to a differential equation, what's left implicit?

Suppose the following problem were to appear in an introductory textbook on differential equations. Find all solutions to $y'=y$. What's left implicit in such a statement? My interpretation of ...
6
votes
6answers
545 views

Understanding calculus formulas intuitively

I am currently studying calculus in Russian and my course book is very rigorous.I used to think that I understand everything but I recently noticed that I only understand the logical steps in proofs ...
10
votes
1answer
550 views

How long does it take to write a math paper? [closed]

I am currently working on an applied mathematics paper (with my supervisor, I am a student) that is long overdue: supposed to take 4-5 months but I have been working on it for 6.5 months. I was ...
12
votes
3answers
267 views

Are matrices best understood as linear maps?

Any linear map between finite-dimensional vector spaces may be represented by a matrix, and conversely. Matrix-matrix multiplication corresponds to map composition, and matrix-vector multiplication ...
6
votes
1answer
207 views

How to explain that $\Bbb{R}$ is not countable to a non-mathematician

What is the best way to explain that $\Bbb{R}$ is not countable assuming that the audience is formed of people who are not mathematicians? I ask this because these days I'm in a debate with someone ...
7
votes
1answer
509 views

Problem regarding mathematical skill of a math lover

I don’t know whether it would be a good idea to seek solution for some personal problem here. But as there’s no harm in asking I’m putting it under the soft tag. It’s about forming a career in ...
2
votes
1answer
262 views

Selecting Differential Geometry Exercises

I'm self-studying differential geometry with Do Carmo's books "Differential Geometry of Curves and Surfaces" and "Riemannian Geometry" and I find those books very good, however I feel a little ...
1
vote
1answer
87 views

“anti-Cartesian” method in plane geometry

I have got very soft question. It's true that all plane geometry problems had analytical (easy) solutions. And everyone can take Cartesian coordinate plane and count all problems for $<\infty$ ...
1
vote
1answer
164 views

Correct way of saying that some value depends on another value x only by a function of x

I would like to know what good and valid ways there are to say (in words) that some value f(x), which depends on a variable x, in fact only depends on x "through" some function of x. Example: For ...
2
votes
1answer
31 views

Is every GO-space collectionwise normal?

Is every GO-space collectionwise normal? And what is the relation between collectionwise normal and monotonically normal? I know a GO-space is always monotonically normal. Thanks.
2
votes
2answers
397 views

For homework questions what is the difference between being asked to verify something and being asked to prove something?

I've always been curious if there is a definite difference between the terms or if they just depend on the context of a problem.
34
votes
7answers
1k views

Appearance of Formal Derivative in Algebra

When studying polynomials, I know it is useful to introduce the concept of a formal derivative. For example, over a field, a polynomial has no repeated roots iff it and its formal derivative are ...
3
votes
1answer
293 views

Paris VI or ENS Lyon for master's student interested in algebra?

I have the opportunity to study in France during the forthcoming autumn, either at Paris VI (6), or at ENS Lyon, and I have trouble deciding which offer I should take. I'm mainly interested in algebra ...
16
votes
4answers
472 views

Mathematical Habit

Whenever I have to do proofs or understand a concept, I start with examples and generalize. As an aspiring mathematician, I have fears that it might hurt me, especially when we have so many ...
15
votes
0answers
232 views

Writing Hard Problems [duplicate]

Having done a fair bit of contest- type math, I was interested in trying to write similar sorts of problems, only to find myself immediately stumped. How does one write "elegant problems" of the kind ...
3
votes
0answers
738 views

Publication date for Michael Spivak - Physics for Mathematicians II?

I bought the book "Physics for Mathematicians I" by Michael Spivak (http://www.amazon.com/Physics-Mathematicians-Mechanics-Michael-Spivak/dp/0914098322), have worked through quite some chapters and ...
26
votes
3answers
803 views

Intuition on fundamental theorem of arithmetic

I'm sorry ahead if time if this is overly trivial for this site. Currently in school, much of what I enjoy is number theory - based. Currently, I lean pretty heavily on the FTA for a good deal of my ...
2
votes
2answers
976 views

What are the drawbacks of multiple-choice questions? [closed]

I can easily understand the advantage of multiple-choice questions for instance in grading and so. A drawback is that real life problem don't have multiple choice questions all the time for instance ...
6
votes
0answers
197 views

The high road to learn algebraic geometry

Suppose that a student has a basic knowledge in commmutative algebra at the level of the Atiyah-MacDonald. What do you think about the following steps to learn algebraic geometry? step 1: read ...
6
votes
1answer
497 views

How do I get into a masters course in pure mathematics?

My question is as stated in the title and to elaborate more: I would like to know if there are any standardized international exams to enter a masters course in pure mathematics (besides GRE Math) ...
3
votes
2answers
175 views

Which set of subjects has the most algebraic 'flavour' to it?

I'm finishing my undergraduate mathematics programme this summer. The graduate program at my university requires that you specify a specialization on entry. This is very difficult for me and would ...
5
votes
4answers
363 views

What pattern does set theory study?

If mathematics is the science of patterns, what pattern does set theory study? My thoughts so far: set theory studies the pattern of relationships between members of a collection and between ...
24
votes
2answers
5k views

Is there a step by step checklist to check if a multivariable limit exists and find its value?

Do we rely on certain intuition or is there an unofficial general crude checklist I should follow? I had a friend telling me that if the sum of the powers on the numerator is smaller then the ...
7
votes
3answers
293 views

Name this polytope

I was wondering what people call a certain type of shape. It is the shape formed by an orthogonal projection of a hypercube along one of its longest diagonals. In other words, fill in the missing ...
7
votes
1answer
112 views

The collection of pathological examples in one reference - Reference request

I would like to ask whether there is a reference which collects pathological examples in mathematics (in general). What I mean is that, for instance, consider Weierstrass function. It has the ...
0
votes
1answer
43 views
4
votes
1answer
8k views

What is the prerequisite knowledge for learning discrete math?

To become a better computer programmer I would like to take the time to learn discrete mathematics, but I am positive that I do not have the required existing knowledge to do so. So I would like to ...
3
votes
2answers
159 views

How is “1” defined in various branches of mathematics?

Wikipedia does not elaborate much on the concept of "One" in such branches as graph theory, ring theory, algebra, topology, measure theory, formal logic, etcetera. How can one grasp the concept of ...
4
votes
2answers
172 views

Is it necessary to know a lot of advance math to become a good junior high/high school teacher?

By "advance math" I refer to Real Analysis, Abstract Algebra and Linear Algebra (to the level of Axler). I received mainly Bs in these courses with the exception of the intro-level Linear Algebra. ...
6
votes
3answers
4k views

What on earth is the difference between Calculus and Analysis [duplicate]

I noticed that there isn't a word for Calculus in my native language, Dutch. So I just went to the English wikipedia entry on Calculus, and tried searching for the Dutch article, and as I suspected, ...
7
votes
1answer
137 views

Is this an interesting generalization of the notion of an open set?

Let $X$ denote a topological space. Some subsets $A \subseteq X$ might have the property that $\partial A = \partial(\mathrm{int}\,A).$ This is certainly true if $A$ is open (since open implies ...
3
votes
1answer
145 views

Formality fades away in the air

I'm trying to study by myself mathematics, but I realized that I have only a naive notion of certains building blocks of mathematics; certain parts of the formalism. So I tried to start with logic, ...
11
votes
4answers
626 views

What makes elementary row operations “special”?

This is probably a stupid question, but what makes the three magical elementary row operations, as taught in elementary linear algebra courses, special? In other words, in what way are they "natural" ...
2
votes
3answers
569 views

Mathematical applications of ordinary differential equations.

I'm looking for more mathematically oriented applications of ODEs (if possible of first order equations). I've browsed through several books and they are all full of physics applications and very ...
3
votes
2answers
1k views

Grad degree that mainly deals with probability/game theory/optimization?

I'm currently working but am going to take classes as a non-degree student to beef up the math part of my background. I've only taken calc 1-3, ODEs, linear algebra, logic, and decision theory so my ...
22
votes
3answers
893 views

Mathematical preparation for postgraduate studies in Linguistics

I am an undergraduate student in Mathematics and I would like to continue my postgraduate studies in the harder, more mathematical aspects of Linguistics. What exactly would that include is unknown ...
2
votes
2answers
81 views

Proving theorems about ZFC by proving them for an arbitrary model.

To prove that a statement follows from the group axioms, we typically write: Let $G$ denote an arbitrary group... Then... Thus, it s a theorem of the group axioms that... Presumably, this form ...
4
votes
0answers
95 views

Is there any equivalence between the category of schemes over $\mathbb R$ and the category real manifolds

The equivalence of the category of smooth projective curves over $\mathbb C$ and the category of compact Riemann surfaces is, I believe, well documented. For example, it is mentioned on the wiki page: ...
4
votes
0answers
129 views

Why don't we introduce the concept of base for a topology in a minimal way?

Why don't we introduce the concept of base for a topology in a minimal way exactly as we did in Linear Algebra? Edit: A topology can be obtained from a base by considering all possible unions of the ...
2
votes
1answer
112 views

An article for some undergraduate journal

I'm a student and last year I attended some courses about algebraic geometry. Referring to many books, I wrote personal notes explaining the Cech cohomology of sheaves and some of its applications in ...
16
votes
3answers
696 views

What's next for me?

I'm in my last year of undergrad, and I would like to do original research for my senior thesis. I am already published in finite group theory and am looking for a new topic to study. I have taken ...
6
votes
2answers
448 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 ...
-2
votes
2answers
287 views

Why mathematics has to be too much formal? [closed]

This is very general question I am asking here. But I think it really needs to be addressed. Whenever I come across a new concept in mathematics, I try to understand it by searching on the internet. ...
3
votes
4answers
2k views

Why do we draw the $xyz$ coordinate system like this?

Usually people (including, for instance, Calculus teachers) draw the $xyz$ coordinate system in such a way that the $y$ and $z$ axes are perpendicular to each other: Imagine I actually got three ...
2
votes
1answer
128 views

Volterra's Function - Context and History

This is a rather soft question, but I am preparing a short presentation on Volterra's function for an analysis course. Specifically, I'm wondering why was the Volterra's function a big deal during ...
3
votes
2answers
75 views

Why is $S/R$ a ring extension?

If $S$ is a ring and $R \subset S$ is a subring it's common to write that $S/R$ is an extension of rings. I frequently find myself writing this and read it quite often in textbooks and lecture notes. ...