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6
votes
6answers
425 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
208 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
56 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
53 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} ...
0
votes
0answers
68 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 ...
1
vote
3answers
640 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
524 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 ...
1
vote
0answers
56 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
532 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
148 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 ...
3
votes
1answer
131 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 ...
0
votes
2answers
84 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 ...
5
votes
0answers
75 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
63 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 ...
1
vote
1answer
75 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
40 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 ...
1
vote
1answer
481 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
104 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 $$d(a \cdot b) \cdot c = a \cdot d(b) \cdot c + b \cdot c \cdot d(a) ~\text{?}$$
1
vote
2answers
610 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 ...
0
votes
0answers
195 views

study abroad math

Does anyone know about the math in moscow program and the budapest semester in math? In fact, I would like to know how competitive it is to get accepted into these programs and what is the acceptance ...
1
vote
0answers
48 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 ...
3
votes
2answers
109 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
19 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
90 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
103 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
62 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 ...
1
vote
1answer
176 views

What major to choose, out of three possible math majors?

I need help with trying to figure out which major I should try to graduate with. Currently, I am a senior undergraduate student with an actuarial science major. I have three actuarial exams finished, ...
4
votes
2answers
104 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 ...
4
votes
2answers
111 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 ...
8
votes
4answers
256 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 ...
2
votes
0answers
155 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 ...
3
votes
3answers
124 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 ...
16
votes
3answers
909 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
44 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
85 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 ...
1
vote
0answers
142 views

Transitioning From Pure Maths To Applied.

I was wondering if anyone had any experience transitioning from pure mathematics to applied mathematics? I am currently doing an Msc in pure maths and my background is mainly algebraic. I was ...
3
votes
0answers
42 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 ...
6
votes
4answers
220 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
124 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 ...
6
votes
2answers
220 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
36 views

Covering lemmas and Differentiation Theory

On an intuitive level, why do differentiation theorems often make use of covering lemmas?
0
votes
1answer
63 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
184 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
29 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 ...
1
vote
1answer
113 views

Relation between the prime density and Riemann's zeros?

Soft question: Where is the connection between the zeros of Riemann's $\zeta$-funciton and the density of prime numbers? Is there a short answer to this question, to get the overview? I once had a ...
1
vote
1answer
297 views

Proof that imaginary numbers exist? [duplicate]

How do imaginary numbers exist? I know you can't use the conventional number system, but use the complex one. But, how do you prove that the complex number system exists in the first place?
5
votes
1answer
211 views

Do Symmetric problems have Symmetric solutions?

There is a general notion of Symmetry in mathematics, that of an object being constant under some transformation. If we think of our object as being a "mathematical problem" we can see certain ...
4
votes
0answers
242 views

Mathematics felt by Srinivasa Ramanujan

At the moment I am reading the book Ramanujan's Papers by B.J. Venkatachala, V. Vinay and C.S. Yogananda; when clarifying some doubt with a professor, he told me that Srinivasa Ramanujan used Galois ...
13
votes
1answer
190 views

What's so cool about local compactness?

As I study more algebraic number theory, I hear more and more often about local compactness: locally compact fields, locally compact topological groups, Stone-Čech compactification of locally compact ...
1
vote
1answer
44 views

Can we determine which statements are incomplete due to Godel?

Due to Godel's incompleteness theorems we know that there are true statements in a system that cannot be proven with that system. My questions are 1) can we tell which statements in a system are the ...