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4
votes
4answers
355 views

The set of all things. A thing itself?

If the universe is the set of all things. Does it contain itself? In other words is it a thing itself? I know its a stupid question, but it really grinds my gears. Thanks! Edit 8.12 Okey, someone ...
0
votes
2answers
95 views

Formal definition of plane

The formal definition of plane says that: A plane is a set of points such that if any two points are taken on it, all the points lying on the line joining these two points also lie on the plane. The ...
3
votes
3answers
388 views

Still struggling with proofs. [closed]

How do you construct rigorous math proofs on your own? Also how do you verify? I am finishing up my first semester of undergraduate analysis and still am struggling with writing proofs. Even though I ...
2
votes
1answer
343 views

Software for generating Cayley graphs of $\mathbb Z_n$?

Does it exist any program (for linux) which can generate a nice Cayley graph of any $\mathbb Z_n$? (If it's possible to create such a graph at all, that is.) (where perhaps $n ≤ 100$ or something ...
2
votes
1answer
173 views

Understand a weird method of calculus

I see this method of calculus on youtube and my question: is this method valid? How we can understand it? Thanks.
8
votes
3answers
347 views

Bedtime maths books?

Most of math books require you to copy proofs and do excersices to extract the content from them. Are there any good serious math books which require only reading and no writing? ADDED: One ...
3
votes
0answers
217 views

Relationship between paradoxes in logic and geometry/topology

Though I've been reading for years, this is my first question here. Believe it or not, I've tried the search feature- apologies if this is a duplicate. The main point of this post can be summarized as:...
3
votes
3answers
104 views

Why do we have $3$ Isomorphism Theorems?

This is a bit of a soft question, but I've wondered about it since being introduced to the $3$ isomorphism theorems (I'm aware of the $4^{th}$ as well, but it is not typically presented in the ...
1
vote
2answers
42 views

Easy Trig question

Easy question here. I ran across this trig identity while doing a problem: $\tan(x)\cdot\tan(y) =-1$ implies that $y = x \pm \pi/2$. Why is this?
6
votes
6answers
4k views

What is the purpose of the limit?

I haven't taken Calculas yet, but I see the use of limit (approaching zero or infinity) in other classes such as physics. I just wanted some intuitive explanation of the limit. I was thinking that ...
3
votes
4answers
526 views

Stats is not maths?

How mainstream is the claim that stats is not maths? And if it's right, how many people don't agree? Given that it's all numbers, taught by maths departments and you get maths credits for it, I ...
1
vote
1answer
72 views

Why can the limit of a sequence approach a number and converge, but the limit of the series must approach $0$ to converge?

My question may not make much sense because I'm still trying to wrap my mind around infinite sequences and series. I seem to have good working knowledge of when and why to apply a certain tests for a ...
1
vote
2answers
54 views

Improper integral: $ \int_{-1}^{\infty} \frac{dx}{x^2 + \sqrt[3]{x^4 + 1}}$

Decide if the integral $$ \int_{-1}^{\infty} \frac{dx}{x^2 + \sqrt[3]{x^4 + 1}}$$ converges. I decided to write $ \int_{-1}^{\infty} \frac{dx}{x^2 + \sqrt[3]{x^4 + 1}}$ = $ \int_{-1}^{1} \frac{dx}{...
0
votes
0answers
89 views

Nonsingular curve

I am kind of curious about the nice properties between affine nonsingular curve and projective nonsingular curve. In my feeling to define sheaf of nonsigluar curve, most of the resources are focusing ...
2
votes
4answers
1k views

Trouble understanding math proofs

*edit Even though there are already answers to my question, I appreciate anyone that offers their advice! I am not sure if this is the right place to ask this but I usually ask for help here. I am a ...
0
votes
3answers
1k views

Method of proving trignometric identities

How do we prove trigonometric identities? When my teacher did it in school today all I could see was him doing random steps. I didn't really understand his method. I am basically asking whether my ...
1
vote
0answers
89 views

Weierstrass and Borel summation

In the Wikipedia article on Borel summation, there is the following quote attributed to Gösta Mittag-Leffler: Borel, then an unknown young man, discovered that his summation method gave the '...
3
votes
2answers
2k views

How does one get better at real analysis proofs?

How does one proceed through a math proof in real analysis? My instructor always says make a diagram, but I am not a visual learner. It seems that whenever I write out the definition of an assumption, ...
2
votes
1answer
229 views

Explaining probability theory versus statistics

I'm not sure whether this question was asked before, but it's hard to search because of lots and lots non-descriptive titles like "statistics and probability". The context: There is an anecdote I ...
5
votes
1answer
451 views

How to begin doing research

I wanna start doing my own research... I've grown somewhat restless about the regular undergraduate training - study something, solve the problems, rinse and repeat. I feel like I've had enough ...
1
vote
2answers
119 views

Nice notation for projection maps

Let $X\times Y$ be a product of two object of a category, and consider the natural projections $$ X\times Y \to X \quad\text{ and }\quad X\times Y \to Y. $$ Usually I denote them by $\pi_X$ and $\pi_Y$...
5
votes
0answers
102 views

Is there a proper etiquette for asking an author for their (mathematical) software?

This may not be the correct place to ask such a question. I have read a mathematical paper on multiclass total variation clustering. I wish to use the algorithm in the contents to compare with another ...
2
votes
1answer
86 views

Flat modules on Stacks Project.

I have been reading through a bit of the material from the Stacks project, and there is a statement that I cannot make sense of. Lemma 10.36.19(7) states: Let $R$ be a ring. (7) Suppose $R\...
1
vote
1answer
90 views

differential equations and physical intuition

Often when you study differential equations, you find phenomena in nature modeled by those equations. Sometimes an insight into a physical problem can help you to solve a differential equation. My ...
1
vote
2answers
44 views

finding large primes

I was wondering if anyone proved about a specific a number that it has to have a prime factor bigger than the currently largest known prime, without specifying how to find this factor, would it be an ...
3
votes
1answer
1k views

Kolmogorov & Fomin Textbooks

There are two books by Kolmogorov & Fomin that I am interested in purchasing, namely Introductory Real Analysis and Elements of the Theory of Functions and Functional Analysis. Now, this will be ...
2
votes
1answer
161 views

Jacobi identity and Leibniz rule - the same thing?

Is there any formal connection between the Jacobi identity $$[[a,b],c] = [a,[b,c]] + [b,[c,a]]$$ and the Leibniz rule $$d(a \cdot b) \cdot c = a \cdot d(b) \cdot c + b \cdot c \cdot d(a) ~\text{?}$$
2
votes
2answers
1k views

Most important Linear Algebra theorems?

I was reading up on symmetric matrices and the textbook noted that the following is a remarkable theorem: A matrix $A$ is orthogonally diagonalizable iff $A$ is a symmetric matrix. This is ...
1
vote
0answers
52 views

A Question Related to the Divergence Test for Series

Suppose $\lbrace x_i\rbrace$ is a sequence of real numbers. If $\lim x_i=0$ then for every $\varepsilon>0$ there exists an $N$ such that $$n\ge N\Rightarrow \vert x_n\vert<\varepsilon\qquad(*)$$...
3
votes
2answers
123 views

Numerically Misleading Results

Are there any calculations or results that have similar answers and when compared numerically look the same, but in actual fact after so much precision, the answers diverge from each other? An ...
1
vote
0answers
23 views

Good notation for many random points approximating an area.

I'm trying to say that as the number of random coordinate points points you plot approaches infinity, it is equivalent to an area integral where each point is an infinitesimally small $\mathrm{d}x \...
3
votes
2answers
139 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 ...
0
votes
2answers
188 views

How was Integral Calculus discovered/derived? [closed]

Can you please explain in layman terms. I don't know if this is a duplicate, but if I find one I will delete this question. The thing I don't get the most is differentials.
1
vote
1answer
108 views

Is there a logic for recursion rules of divisibility?

I knew the divisibility rule for 7, but my sir told me that these methods are known as recursion rules for divisibility. My sir also told them for 11, 13,17,19. But is there any logic behind it? Or is ...
4
votes
2answers
132 views

Soft question. Why do concrete problems motivate abstract theory?

From what I learnt, I found that mathematicians develop abstract theory often in order to solve concrete, classical problems. For example, I read that ideals were introduced to solve problems in ...
6
votes
2answers
259 views

Zorn's Lemma and Injective Modules

In my study of injective modules over commutative rings, i noticed that Zorn's Lemma is often employed in the proofs. Here are three examples: 1) Baer's Criterion 2) the characterization of injective ...
10
votes
4answers
343 views

Mathematical notation around the world

What are the differences in mathematical notation around the world? I know that in some other countries they write 1,2 meaning 1.2, but what else can be confusing in an academic environment (when ...
6
votes
1answer
301 views

What is meant by “mathematical maturity”?

I have often heard people talk of "mathematical maturity", sometimes in the sense of the maturity required to understand an area of mathematics or in the approach to a problem or proof. However, it's ...
2
votes
3answers
196 views

Question on Rudin sequences?

In baby Rudin, Rudin shows that $$\lim_{n \to \infty}\sqrt[n]{p} = 1.$$ In the proof of limit he tries to prove that the limit is $1$. So he takes $x_n = \sqrt[n]{p} - 1$. I have never noticed this ...
18
votes
3answers
2k views

Did Gauss ever make a mistake?

I have read a bit about Gauss, who was well known for being careful in only publishing work he had perfected (or in his own words "few, but ripe"). What is interesting to me about Gauss though is that ...
1
vote
1answer
47 views

Is it generally preferred that empty products are gotten rid of where possible?

Is it generally preferred that empty products are gotten rid of where possible? For example: Stewart's structure theorem says that for a positive integer $n$, every positive integer $\leq n$ has a ...
2
votes
5answers
89 views

How to notice that $3^2 + (6t)^2 + (6t^2)^2$ is a binomial expansion.

The other day during a seminar, in a calculation, a fellow student encountered this expression: $$\sqrt{3^2 + (6t)^2 + (6t^2)^2}$$ He, without much thinking, immediately wrote down: $$(6t^2+3)$$ What ...
3
votes
0answers
55 views

Henri Poincaré writings

I have heard that Poincaré writings were very intuitive in its approach and not very formal in the arguments. I'm searching for something like this to complement my study of dynamical systems. I ...
7
votes
4answers
319 views

Formality and mathematics

Why is it important to be formal in mathematics? Is formality beneficial for students? Or is it just to scare students away from mathematics?
3
votes
2answers
136 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 set-...
6
votes
2answers
281 views

What is a (the?) good starting point for learning the modern “higher” mathematics?

As many of you know, category theorists are currently doing, among other things, a great job in advertising their modern developments. And I must say, this works for me - in particular, I find myself ...
1
vote
0answers
49 views

Covering lemmas and Differentiation Theory

On an intuitive level, why do differentiation theorems often make use of covering lemmas?
0
votes
1answer
87 views

Maple plot range

I've been searching for a while, but I can't seem to find the syntax for this (if it exists!) Is it possible to plot a range but exclude certain values? I wish to plot a single graph that ranges from $...
6
votes
3answers
261 views

How much topology for graph theory?

I am writing a thesis in the context of descriptive complexity in theoretical computer science and therefore need to study a little bit of graph theory. My background is not mathematics but computer ...
0
votes
2answers
37 views

Is this element-of_{ij} - looking symbol the Levi-Civita symbol?

I'm reading this formula: from a page Is the symbol that looks like an element-of symbol with two indices i and j the Levi-Civita symbol? Mathematics is my weak-side so I'm not sure. Actually I ...