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1
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0answers
34 views

What are some genuine ways to define the derivative of a fractal?

Seeing the success of applying measure theory to generalize integration to fractals, I wonder whether or not there is a method to generalize the derivative to a fractal. Most courses start off fractal ...
2
votes
0answers
54 views

What is the point of basis vectors?

Why do we even bother with basis vectors? Why don't we just notate an element $x$ of an $n$-dimensional vector space $V$ as an ordered set $(x_1,x_2,...,x_n)$ and go from there?
-1
votes
0answers
49 views

Why do some mathematics professors teach more/less courses than others? [migrated]

Not sure if this belongs on the Academia site, but since I'm a math major and the question is based solely on my observation of mathematics professors, I figured this site would be best. At my ...
8
votes
2answers
93 views

How would you explain a quadratic field to a beginner?

How would you explain a quadratic field to a beginner? Eg. how did the subject first start? All the modern stuff they use to explain it makes it really confusing how one should think about it in more ...
2
votes
1answer
49 views

Blow-up toric varieties.

I have to take a talk of an hour and I have to talk about blow-up of toric varieties. Can you suggest me some interesting examples that I can present? How can I find a good reference for the theory ...
3
votes
0answers
34 views

Generalizing convexity of sets

Let $X$ be a subset of some Euclidean space. We say that $X$ is convex if for any two points $p$, $q$ in $X$, the line segment joining $p$ and $q$ is also in $X$. But what if we loosen this definition ...
3
votes
0answers
40 views

What is the densest, most opaque way of saying two odd numbers add up to an even number? [closed]

Here's what I've come up with: In $\mathbb Z$, any pairwise sum of elements in the $\langle 2 \rangle + 1$ coset is in $\langle 2 \rangle$. But there's got to be a way to make this even briefer, ...
0
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3answers
38 views

Proving limits via episilon-delta definition vs. algebraic manipulation.

In real analysis textbooks, limits are proved using the epsilon-delta definition directly. However, at some point, limits start being solved using algebraic manipulation. For example: $$\lim_{x \to ...
2
votes
1answer
97 views

Amazing integrals and how is solved it [closed]

There a lot of integrals, however many people solved it in different ways, we can find interesting integrals in Table of Integrals, Series, and Products. I wonder What is the most exciting integral ...
1
vote
2answers
37 views

SDE Modeling: Ito vs. Stratonovich

In my SDE class last semester there were some hints that sometimes an SDE model makes more sense in the Ito sense, and sometimes in the Stratonovich sense. This was explained very briefly and vaguely. ...
0
votes
2answers
21 views

Given the solution to some differential equation, is the original equation necessarily unique?

For example, if I have the fundamental solution set $\{x^2\}$, such that $y(x)=Cx^2$ is the solution to some unknown differential equation, is it guaranteed that only one such equation exists with ...
1
vote
2answers
22 views

Confusion about division in rates.

I would really appreciate help with this because it's been driving me insane for a while now... I understand what "per" means in "$x$ kilometers per $y$ hours". What I don't understand is how to make ...
6
votes
1answer
81 views

Could the real numbers have been invented without the natural numbers

The real numbers are constructed from the rational numbers which are constructed from the integers which, in turn, are constructed from the natural numbers. But if we had no notion of the natural ...
-2
votes
2answers
51 views

Do you have any good iOS app suggestions for taking notes? [closed]

Thanks for stopping by my thread. I'm an engineering student with an avid interest in Maths. I hope to do some kind of research in Maths someday. I really enjoy Maths and am doing a lot of reading so ...
3
votes
3answers
145 views

What is it like to understand complicated/advanced mathematics?

Whenever I see very complex equations, they look, in a way, beautiful even though I don't understand them. This was directly taken from another question: "- Definition 1 - Given an open subset ...
2
votes
0answers
59 views

Does Fractional Calculus have a real connection with Fractals? (or is it just an extra variable trick)

The fractional derivative and integral (operators that let you differentiate or integrate a fractional number of times) have drawn a lot of attention from people outside the field. Yet, after reading ...
0
votes
0answers
41 views

intuition about calculating partial sums of series

The partial sums $$1 + 2 + 3 + \cdots + n$$ of the simple arithmetic progression can be calculated by reordering and adding. The partial sums $$1 + \frac{1}{2} + \frac{1}{4} + \cdots + ...
0
votes
0answers
19 views

Stochastic integral and usual integral addition

Let's say I have two processes and I would like to say something about their sum. In the case of deterministic functions, $\int f(t)dt + \int g(t)dt = \int f(t)+g(t)dt $, and I can then possibly say ...
10
votes
0answers
124 views

Is it a good approach to heavily depend on visualization to learn math?

I am a third year undergraduate and I am a beginner on these "real mathematics" (no pun intended). Before contacting the "real math", my math level should be considered to be "good", although I was ...
5
votes
3answers
49 views

Odd and Even Parity in Proofs

The notion of the parity is very important in a variety of branches of mathematics. Specifically, I am looking for proofs that use parity in the even-vs-odd sense to prove their points. For example, ...
1
vote
0answers
29 views

Seeking after notation for two objects equal up to a constant

Sometimes we want to express that two objects are equal up to a constant but there is no need to keep writing out the constant or constants. For example, often times the constant or constants involved ...
2
votes
0answers
31 views

Perimeters Areas and Volumes

I have to write an article for a school magazine. I thought it is better to choose a simple topic like Perimeter, Area and Volume. I am looking for historical fact and surprising facts about ...
6
votes
2answers
84 views

Exploring $ \sum_{n=0}^\infty \frac{n^p}{n!} = B_pe$, particularly $p = 2$.

I was exploring the fact that $$ \sum_{n=0}^\infty \frac{n^p}{n!} = B_pe,$$ where $B_n$ is the $n$th Bell number. I found this result by exploring the series on wolframalpha and looking up the ...
0
votes
0answers
32 views

What should I study, if I want to learn about higher dimensional spaces and objects? Also, what resources should I obtain?

I am becoming interesting in learning about higher dimensions. What are subjects I could study, and what are some good resources for those subjects?
4
votes
0answers
73 views

Proofs shorter than the statement of the theorem

In Postnikov's first book in his Lectures in Geometry Series, Analytic Geometry, he states and then proves the Desargues theorem. Then he writes (in my English translated copy) "The proof has turned ...
34
votes
2answers
346 views

When are two proofs “the same”?

Often, we find different proofs for certain theorems that, on the surface, seem to be very different but actually use the same fundamental ideas. For example, the topological proof of the infinitude ...
2
votes
2answers
79 views

How to generate complicated looking identities such as $\sqrt [3] {2 + \sqrt 5} - \sqrt [3] {2 - \sqrt 5}=1$ easily?

How to generate complicated looking identities, or even more complicated looking identies such as $\sqrt [3] {2 + \sqrt 5} - \sqrt [3] {2 - \sqrt 5}=1$ easily? I saw the identity to be shown. What is ...
7
votes
2answers
256 views

An intuitive definition of contour integration.

Recently I have been trying to learn the method of contour integration, but the Wikipedia article and others don't really help. Is there some resource which provides a definition which can be followed ...
1
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0answers
19 views

Application of theorems 'about free groups'

What consequences have theorem Any nonzero subgroup of free group is free or some another similar theorems? P.S. especially not-group-theretic applications.
2
votes
0answers
79 views

Argument for the zero vector not being defined as an eigenvector

Two days ago, my lecturer of Advanced Numerical Methods gave a review on the topic about eigenvalues and eigenvectors. Just as the lecturer presented the definition of eigenvalues and eigenvectors, a ...
1
vote
0answers
35 views

Intuition for Entropy over Fractals

Is there intuition for "mathematical" entropy. I know that physical entropy tracks the order in a dynamical system, for thermodynamics. As entropy goes up, general randomness and disorder goes up. ...
1
vote
1answer
61 views

Understanding and teaching the concept of derivative

I need to prepare an introductory lecture about derivatives and the concept of differentiation to a class of people with a general mathematical background (who have also studied calculus a few years ...
0
votes
0answers
23 views

On the hypothesis of the change of variable theorem

I´m studying the change of variable theorem for a function $f:\mathbb R^n \to \mathbb R$ and my teacher gave us the theorem as follows: Theorem: Let $f:A\subset \mathbb R^n \to \mathbb R$ be ...
1
vote
1answer
55 views

informal semantics regarding CH and AC

why is the assertion $\aleph_1=2^{\aleph_0}$ referred to as a hypothesis, whereas $$\forall \alpha( S_\alpha \ne \varnothing) \Rightarrow \prod_\alpha S_\alpha \ne \varnothing$$ is called an axiom? ...
1
vote
1answer
37 views

how to understand Taylor's inequality intuitively?

I am learning the Taylor Series at the moment and I am trying to figure out how to understand Taylor's inequality intuitively. I know you can integrate repeatedly and prove the inequality is ...
3
votes
0answers
77 views

How to understand if one is eligible for research?

What are the eligibility criteria for one to undergo research in Mathematics? Or should I place the question as what virtues of a student are given importance when one is interviewed for a PhD ...
1
vote
5answers
143 views

Why do counits go that way?

Imagine you want to motivate for an audience the definition of an adjunction in terms of unit and counit. So you can say: Often two functors $\mathcal{C} \begin{array}{c} \stackrel{\large ...
1
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0answers
41 views

Surreal numbers in set theories other than ZFC

This isn't really a question rather than my thoughts on these things; I initially had questions but believe I managed to answer them. Regardless, here goes. Feel free to correct any mistakes, as I'm a ...
1
vote
1answer
65 views

How is problem solving ability on an olympiad level relevant to mathematical graduate study/research?

I am starting on math later than many of my peers and hence have little to no experience in competitive problem solving. Is this a disadvantage during the study of the more abstract mathematics that ...
0
votes
0answers
39 views

Suggestions(Anything) regarding GRE Math Subject test [duplicate]

I am to appear for GRE Math Subject test probably this year or next year .I have basic knowledge of calculus 1,2,3 ,group theory ,Linear Algebra , Small part of real analysis . I haven't yet studied ...
0
votes
0answers
17 views

I hear that some operators don't have analytical properties. What does that mean?

The floor, ceiling, and mod functions are very useful operators, but in general discussion I've heard their usefulness called into question because of their lack of analytical properties. For instance ...
1
vote
1answer
87 views

Building a training program… for mathematics

I want to ask for your advice building my one-year training program for mathematics. Objectives: Keep 'mathematically fit' Improve for the pleasure Get competent at high-level economics and ...
1
vote
1answer
22 views

Expressing a line as a linear combination of two points on the line.

I'm currently reading Pugh's Analysis. He makes the statement that the line between two points x and y is the set of linear combinations $sx + ty$ where $s + t = 1$. I'm satisfied that this is true, ...
4
votes
3answers
105 views

Difficulty faced in solving maths problems

I am a student in 12th grade and am fond of mathematics. I enjoy reading mathematics but when it comes to problems I just get completely stuck. Its not that I don't understand the problem but often ...
1
vote
0answers
48 views

Ideas for math problem solving class for undergraduate students in university

In our university there is a huge gap between two group of students. a group of them came from Math Olympiad competitions and have a very strong background from high school but others, they have just ...
5
votes
4answers
931 views

Albert, Bernard and Cheryl popular question (Please comment on my theory)

Here is the problem, I think that there is one point that makes the question ambiguous, I think they should explicitly say the reason why Albert knows that Bernard does not know the date. Case 1: ...
0
votes
1answer
52 views

Do equations that rely on a fractional number of variables exist?

In statistics, data is usually fitted with trend lines. Usually you can get statistics back that say things pertaining to how correlated one variable is with another. For instance if a variable $x$ ...
1
vote
2answers
58 views

Numeric system without “zero”, how to explain importance of zero to average person?

As we all knew that Aryabhata (http://en.wikipedia.org/wiki/Aryabhata#Place_value_system_and_zero) invented zero ($0$) in our number system. I have few questions about it. How did the numeric system ...
3
votes
2answers
38 views

Concept of random sample? I have a truly problem understanding it.

I have to solve a probability problem and it says that we take a random sample of size 10. But I don´t understand the concept (I´m on my first course on probability). Suppose that we have a box with ...
0
votes
0answers
52 views

Collatz algorithm generalization try-out (Collatz k-algorithm)

Recently I have been reading about the Collatz conjecture here in Mathematics Stack Exchange, and also found the fantastic paper of professor Lagarias about it. Everything was so interesting (and I ...