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1500 questions
164
votes
8 answers
What's the point in being a "skeptical" learner
I have a big problem:
When I read any mathematical text I'm very skeptical. I feel the need to check every detail of proofs and I ask myself very dumb questions like the following: "is the map well defined?", "is the definition independent from the…
Dubious
- 13,350
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- 142
163
votes
7 answers
Is non-standard analysis worth learning?
As a former physics major, I did a lot of (seemingly sloppy) calculus using the notion of infinitesimals. Recently I heard that there is a branch of math called non-standard analysis that provides some formalism to this type of calculus.
So, do you…
user13255
163
votes
1 answer
Pythagorean triples that "survive" Euler's totient function
Suppose you have three positive integers $a, b, c$ that form a Pythagorean triple:
\begin{equation}
a^2 + b^2 = c^2. \tag{1}\label{1}
\end{equation}
Additionally, suppose that when you apply Euler's totient function to each term, the equation…
Misha Lavrov
- 142,276
163
votes
14 answers
Why is gradient the direction of steepest ascent?
$$f(x_1,x_2,\dots, x_n):\mathbb{R}^n \to \mathbb{R}$$
The definition of the gradient is
$$ \frac{\partial f}{\partial x_1}\hat{e}_1 +\ \cdots +\frac{\partial f}{\partial x_n}\hat{e}_n$$
which is a vector.
Reading this definition makes me consider…
Jing
- 2,327
163
votes
19 answers
What actually is a polynomial?
I can perform operations on polynomials. I can add, multiply, and find their roots. Despite this, I cannot define a polynomial.
I wasn't in the advanced mathematics class in 8th grade, then in 9th grade I skipped the class and joined the more…
Travis
- 3,396
- 6
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- 44
163
votes
19 answers
Online tool for making graphs (vertices and edges)?
Anyone know of an online tool available for making graphs (as in graph theory - consisting of edges and vertices)? I have about 36 vertices and even more edges that I wish to draw.
(why do I have so many? It's for pathing in a game)
Only tool…
f20k
- 1,733
162
votes
33 answers
What are the most overpowered theorems in mathematics?
What are the most overpowered theorems in mathematics?
By "overpowered," I mean theorems that allow disproportionately strong conclusions to be drawn from minimal / relatively simple assumptions. I'm looking for the biggest guns a research…
Samuel Handwich
- 2,771
162
votes
1 answer
What functions can be made continuous by "mixing up their domain"?
Definition. A function $f:\Bbb R\to\Bbb R$ will be called potentially continuous if there is a bijection $\phi:\Bbb R\to\Bbb R$ such that $f\circ \phi$ is continuous.
So one could say a potentially continuous (p.c.) function is "a continuous…
M. Winter
- 29,928
162
votes
3 answers
Why is the eigenvector of a covariance matrix equal to a principal component?
If I have a covariance matrix for a data set and I multiply it times one of it's eigenvectors. Let's say the eigenvector with the highest eigenvalue. The result is the eigenvector or a scaled version of the eigenvector.
What does this really…
Ryan
- 5,509
161
votes
31 answers
Stopping the "Will I need this for the test" question
I am a college professor in the American education system and find that the major concern of my students is trying to determine the specific techniques or problems which I will ask on the exam. This is the typical "will this be on the test?"…
Wintermute
- 3,828
161
votes
5 answers
Can you raise a number to an irrational exponent?
The way that I was taught it in 8th grade algebra, a number raised to a fractional exponent, i.e. $a^\frac x y$ is equivalent to the denominatorth root of the number raised to the numerator, i.e. $\sqrt[y]{a^x}$. So what happens when you raise a…
tel
- 1,863
160
votes
19 answers
What is the difference between a point and a vector?
I understand that a vector has direction and magnitude whereas a point doesn't.
However, in the course notes that I am using, it is stated that a point is the same as a vector.
Also, can you do cross product and dot product using two points instead…
6609081
- 1,705
- 2
- 12
- 6
160
votes
20 answers
Are there any open mathematical puzzles?
Are there any (mathematical) puzzles that are still unresolved? I only mean questions that are accessible to and understandable by the complete layman and which have not been solved, despite serious efforts, by mathematicians (or laymen for that…
Řídící
- 3,210
160
votes
4 answers
The direct sum $\oplus$ versus the cartesian product $\times$
In the case of abelian groups, I have been treating these two set operations as more or less indistinguishable. In early mathematics courses, one normally defines $A^n := A\times A\times\ldots\times A$; however in, for example, the fundamental…
Sputnik
- 3,764
160
votes
15 answers
Are there real-life relations which are symmetric and reflexive but not transitive?
Inspired by Halmos (Naive Set Theory) . . .
For each of these three possible properties [reflexivity, symmetry, and transitivity], find a relation that does not have that property but does have the other two.
One can construct each of these…
000
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