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0
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0answers
50 views

Application Question - American universities strong in Differential Geometry?

Can anyone recommend some American universities (except those top 10 ones such as Harvard, Princeton, SUNY and Umichgan etc. ) which have departments with a solid focus on Geometry and Topology, ...
1
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0answers
12 views

relationship between Minkowski addition and the trajectory of a numerical controle machine?

This is a very naive question about the Minkowski addition. I hope not to be off-topic. I read in the Wikipedia's article dedicated to it : In numerical control machining, the programming of the ...
3
votes
1answer
50 views

Cut Mobius Band

$$\text{Cut a Mobius band from its center line, and then what do we get?}$$ Someone may find it's not easy to imagine without a paper in hand. However, if we cut a square paper from center line at ...
1
vote
2answers
55 views

Topics for a Ted-Talk-like presentation! (Topology/Non-Euclidean Geometry)

I'm looking for a good topic to base my presentation on (length: 15-20 mins). I'm a freshman mathematics student, and my audience won't be skilled in mathematics beyond high-school maths. I've been ...
3
votes
2answers
70 views

Are there any disadvantages to working in the category of k-spaces as opposed to Top?

Unlike the category Top of topological spaces with continuous maps as the arrows, the full subcategory of compactly generated spaces (k-spaces) is Cartesian closed. It seems like a very nice ...
2
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2answers
347 views

Is Physics really a rigorous subject? [closed]

Though I can't give a precise definition of the term rigor (or better to say mathematical rigor) but intuitively in case of mathematics one may note that when we say that 'the proof is rigorous' we ...
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10answers
351 views

Real analysis book suggestion

I am searching for a real analysis book (instead of Rudin's) which satisfies the following requirements: clear, motivated (but not chatty), clean exposition in definition-theorem-proof style; ...
0
votes
1answer
63 views

How to make a mathematical text more concise

My PhD thesis has 175 pages A4 and I shall provide also a compilation that will summarize the main results on 20 pages A5. My thesis contains algorithms and theorems examining their properties. After ...
1
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0answers
38 views

Is there are “sphere” associated to any topological vector space?

If I have a topological vector space that is not locally compact, is it still possible to associate to it some natural "sphere" like object? For locally compact Hausdorff spaces, the my first guess ...
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0answers
25 views

To understand mathematics [duplicate]

How do I achieve a good understanding of university level mathematics in order to do research in ? How do I know that the piece of math is understood and that I can go ahead?
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0answers
44 views

Geometric Meaning of Luna's Slice Theorem

I found the Luna's Slice Theorem very Technical. It will be helpful if someone illustrates the geometry involved in the theorem. Also why this theorem so useful? This is Luna's Slice theorem from a ...
63
votes
8answers
8k views

Will it become impossible to learn math? [closed]

I was thinking about this today and it seems like a good question. Assuming mathematics will keep on expanding, do you think it will ever become impossible for a beginner to learn all the known ...
3
votes
1answer
80 views

Advice for Math Majors -What to do if you come into college with a lot of college credit? [closed]

In high school I was a good maths student and took AP Calculus BC my freshman year and got a 5 and then took Multivariable Calculus, Linear Algebra, Differential Equations, Introduction to ...
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0answers
16 views

Does this matrix have a name: $x=(x_0, \ldots, x_n)$, $S(x)_{i,j} = x_{i+j \mod(n)}$. Can you help me generalise the idea?

Apologies if this question is vague/unclear (especially the title). Given a vector $x=(x_0, \ldots, x_n)$ define the matrix $S(x)_{i,j} = x_{i+j \mod(n)}$. $$ S(x) = \left( \begin{array}{rrrr} ...
0
votes
0answers
21 views

Physical interpretation of the integral formula for the solution of Laplace equation with Dirichlet/Neumann boundary condition

Suppose we have a bounded domain $D$ with smooth boundary, with $G(x,y)$ being the Green's function for the Poisson equation on $D$, i.e. $G(x,\cdot)=0$ on $\partial D$ and $\Delta_y ...
7
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0answers
98 views

Is there really anything wrong with Bourbaki's Set Theory?

Recently I have started reading Bourbaki's Theory of Sets on my own. Regarding one of the explanations of a concept when I went to a Professor of our college, he asked me why I was wasting my time ...
4
votes
2answers
88 views

where to start if you lack basics? [closed]

I've joined this forum for few weeks now and it amazes me that so many people are good at math. For me, I'm more of a history person rather than math person, so I hated math since I was young, not ...
5
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0answers
66 views

Apparent Arbitrariness in Mathematics

Something about definitions in mathematics has always interested – confused? - me, I call it “arbitrariness in Mathematics” - it's a bad name, but I don't know a better one. Let me explain: 1st - ...
5
votes
1answer
99 views

understand mathematics [closed]

Please, i need some clarifications and general remarks. I am a grad student in mathematics, once, our teacher give us a simple problem (involving vector spaces) and no one was able to solve it and he ...
1
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0answers
91 views

What is a Toy Model for the mathematician's practice? Definition and examples

Wikipedia says Toy model (physics): "In physics, a toy model is a simplified set of objects and equations relating them so that they can nevertheless be used to understand a mechanism that is also ...
2
votes
0answers
38 views

Prerequisite to study smooth 4-manifolds

I am quite interested in understanding smooth 4-manifolds. What are the necessary prerequisites in order to start my study? Also can you please suggest me some good books from where I can start? ...
1
vote
0answers
28 views

Project-Math-Computing

Im an undergraduate student and me and my roomate who is an undergraduate also would like to do a project together. Im on the pure and some aplied math branch and he is on the computing branch.We are ...
3
votes
1answer
176 views

Linear algebra and geometric insight: a rigorous approach to vector spaces, matrices, and linear applications

Could you point out some references (undergraduate level) that give a geometric understanding of vector spaces, matrices, and linear applications? As far as I know, many textbooks start with an ...
4
votes
0answers
54 views

For cardinals, if $\mathfrak{a}\ne\mathfrak{b}$ then $2^\mathfrak{a}\ne 2^\mathfrak{b}$

In the usual ZF (or ZFC) set theory, let $\mathfrak{a}$ and $\mathfrak{b}$ be cardinal numbers. Is it correct that one can neither prove nor disprove the statement: $$\mathfrak{a}\ne\mathfrak{b} ...
2
votes
1answer
51 views

Undergraduate Project Suggestions

A student of mine has expressed interest in doing an independent project next quarter with me. This would not be for credit and it is purely for her own educational stimulation. She wants to study ...
1
vote
1answer
29 views

lim inf and lim sup convergence/divergence

Assume that the following sequences are positive, then the statements: If $\lim \inf a_{n} > 1$, then $\sum_{k = 1}^{\infty} a_{k}$ diverges. If $\lim \sup a_{n} = 0$, then $\sum_{k = 1}^{\infty} ...
2
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0answers
59 views

Undergraduate Research Ideas?

I am just finishing off my final exam in my 2nd year of my degree in Mathematics & Applied Mathematics and have quite a long break coming up until I start my final year of undergrad next year. I ...
3
votes
0answers
33 views

Paper accepted in journal, should I remove style file for arXiv preprint? [migrated]

I recently sent a paper to a journal (which I put on arXiv beforehand) which has now been accepted with some corrections. I want to put the corrected version on arXiv. THe corrected version is written ...
4
votes
1answer
97 views

How to prepare myself for an advanced trignonometry exam

I'm gonna have a trigonometry/general algebra exam soon. My teacher has told us about some trignometric proofs, and we defined the $\sin$ and $\cos$ int he right way, doing all formal proofs for the ...
0
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0answers
40 views

Reference in other languages

In the reference of many papers there are usually papers/books which are not written in English; some German, some French, some Russian, etc. (I seldom see Chinese or Arabic ones, although there are ...
12
votes
2answers
237 views

What does it mean to say that a particular mathematical theory is a foundation for mathematics?

We usually hear that set theory is a foundation for contemporary mathematics. Category theory is also another foundation of maths. There are other theories which deemed to be a foundation for maths. ...
5
votes
1answer
73 views

What are some good references on how probability theory got mathematically rigorous?

I am working on a term paper for an analysis course and I thought it would be interesting to talk about the connection between analysis and probability theory. Honestly, it would also benefit me a lot ...
6
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4answers
293 views

Motivation for natural deduction

I've been learning natural deduction recently. I've seen many problems and am starting to be able to solve problems more easily. For some reason I feel the need to ask what high school math students ...
0
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0answers
32 views

Fun Lagrange multiplier problem?

Do any of you have a fun or interesting Lagrange multiplier problem that would be suitable for undergraduate calculus students? I'm planning on working through a standard Lagrange multiplier problem ...
0
votes
2answers
54 views

Why do we count in two ways?

To prove combinatorial identities, I've always been taught to count in two ways. Why do we do this, rather than just use algebraic manipulations to go from one side to the other?
1
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0answers
21 views

When setting up a probability problem, when is it appropriate to use conditioning?

I understand the principles of conditioning and its rules, but when do I decide if a problem will be easier using conditioning versus determining through other methods? I'm teaching myself probability ...
1
vote
0answers
37 views

Did Hamilton have a proof that $\mathbb{R}^3$ is cannot be turned into an $\mathbb{R}$-division algebra?

It is well-known that $\mathbb{R}^n$ cannot be made into a non-commutative $\mathbb{R}$-division algebra if $n\ne 4$. My question is whether there is a (slick) proof of this for $n=3$; in particular, ...
8
votes
1answer
123 views

How to draw greek letters on paper / blackboard?

For $\gamma$ (gamma), I've noticed people doing a sort of $\alpha$ (alpha) rotated by ninety degrees, which seems to be the standard on-paper-or-blackboard equivalent of $\gamma$. But for letters ...
4
votes
0answers
51 views

Are inequalities harder to prove than equalities?

Browsing through the inequalities tag, I see a lot of straightforward-looking arithmetic statements that I nevertheless have no idea how to prove (and apparently I'm not alone). With equalities it's ...
9
votes
2answers
132 views

Is $\sqrt2$ a tricky notation?

When someone asked me how to solve $x^2=9$,I can easily say, $x=3$ or $-3$. But what about $x^2=2$? There is NOT any "ordinary" number to solve this question. It's an irrational number. So we say ...
5
votes
2answers
135 views

The two distinct cultures in mathematics

I once read an article about two distinct culture within mathematics, "analysts" and "algebraists" when I was in high school. I am not still a graduate student, but I really want to hear what it feels ...
0
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0answers
26 views

removable singularity and injective function

Let $U \subset \mathbb{C} $ a conected open subset, $ a \in U $ and $ f:U- \{a\} \to \mathbb{C}$ a holomorphic function such that $ V=f (U-\{a\}) $ is a open bounded subset. (A) Show that $ f $ has a ...
3
votes
1answer
43 views

Are there ways of telling whether or not an integral will have a closed expression?

I was looking at some of the great integral posts that have graced this website and I am wondering if there is a way to tell whether or not a definite integral will have a closed form. If there is no ...
17
votes
6answers
361 views

Open source lecture notes and textbooks

This question is inspired by the popular "Best Sets of Lecture Notes and Articles". Indeed, I would like to collect a "big-list" of open source (that is, with $\LaTeX$ code available) high-quality ...
0
votes
1answer
35 views

What is the difference between Difference equations and Recurrence relations?

Is there any difference between Difference equations and Recurrence relations? Some people are use them as difference equations and some are use as recurrence relations. I couldn't find in anywhere. ...
3
votes
1answer
46 views

Soft Question: Linear Algebra Textbook to serve as a good foundation for Functional Analysis?

I want recommendations for an advanced linear algebra textbook that focuses on theory and provides adept background to support an advanced undergraduate or beginning graduate course Functional ...
0
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1answer
33 views

Reference for Engineering Mathematics

I am a graduate Engineer looking to qualify a post graduate entrance examination (for Master's degree) where I have 'Engineering Mathematics' as one of my Subjects.I hereby paste my course syllabus: ...
2
votes
0answers
42 views

Requesting information on constructed discontinuous functions (from any perspective)

Suppose $f:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous function. Define the function $F:\mathbb{R}\rightarrow \mathbb{R}$ as $$F(x)=f(x)\prod_{n=1}^\infty\frac{x-\frac{1}{n}}{x-\frac{1}{n}}$$ I ...
1
vote
2answers
41 views

Is it true that any quotient group G/N is abelian if N contains the commutator subgroup? [duplicate]

Is it true that any quotient group G/N is abelian if N contains the commutator subgroup(including the case when N is the commutator subgroup)? I think this is true but I just want to confirm.
0
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0answers
35 views

Good resources to learn Hopf algebras

I am finding it hard to find resources that can educate me in this subject. I have looked up this and I can't find anywhere to start. Any advice?