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64
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4 answers
Why is the Laplacian important in Riemannian geometry?
As I've learned more Riemannian geometry, many of my teachers have said that studying the Laplacian (and its eigenvalues) is very important. But I must admit, I've never fully understood why.
Fundamentally, I would like to know why the Laplacian…
Jesse Madnick
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64
votes
5 answers
Is there a 3-dimensional "matrix" by "matrix" product?
Is it possible to multiply A[m,n,k] by B[p,q,r]? Does the regular matrix product have generalized form?
I would appreciate it if you could help me to find out some tutorials online or mathematical 'word' which means N-dimensional matrix…
danny_23
- 773
64
votes
2 answers
Conjecture $_2F_1\left(\frac14,\frac34;\,\frac23;\,\frac13\right)=\frac1{\sqrt{\sqrt{\frac4{\sqrt{2-\sqrt[3]4}}+\sqrt[3]{4}+4}-\sqrt{2-\sqrt[3]4}-2}}$
Using a numerical search on my computer I discovered the following inequality:
$$\left|\,{_2F_1}\left(\frac14,\frac34;\,\frac23;\,\frac13\right)-\rho\,\right|<10^{-20000},\tag1$$
where $\rho$ is the positive root of the polynomial…
HWﾠ
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64
votes
2 answers
Why does the spectral norm equal the largest singular value?
This may be a trivial question yet I was unable to find an answer:
$$\left \| A \right \| _2=\sqrt{\lambda_{\text{max}}(A^{^*}A)}=\sigma_{\text{max}}(A)$$
where the spectral norm $\left \| A \right \| _2$ of a complex matrix $A$ is defined as…
mathemage
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64
votes
3 answers
Why is it worth spending time on type theory?
Looking around there are three candidates for "foundations of mathematics":
set theory
category theory
type theory
There is a seminal paper relating these three topics:
From Sets to Types to Categories to Sets by Steve Awodey
But at this forum…
Hans-Peter Stricker
- 18,719
64
votes
6 answers
Do harmonic numbers have a “closed-form” expression?
One of the joys of high-school mathematics is summing a complicated series to get a “closed-form” expression. And of course many of us have tried summing the harmonic series $H_n =\sum \limits_{k \leq n} \frac{1}{k}$, and failed. But should we…
Srivatsan
- 26,761
64
votes
5 answers
Gradient of a dot product
The wikipedia formula for the gradient of a dot product is given as
$$\nabla(a\cdot b) = (a\cdot\nabla)b +(b\cdot \nabla)a + a\times(\nabla\times b)+ b\times (\nabla \times a)$$
However, I also found the formula $$\nabla(a\cdot b) = (\nabla a)\cdot…
Euler....IS_ALIVE
- 4,857
64
votes
1 answer
Is the largest root of a random polynomial more likely to be real than complex?
Posted on MO since it is unanswered in MSE
It is known that the number of real roots of a random polynomial with real coefficients is much smaller than the number of complex roots. WLOG, assume that the coefficients are uniformly random in $(-1,1)$…
Nilotpal Sinha
- 24,086
64
votes
6 answers
What are some examples of infinite dimensional vector spaces?
I would like to have some examples of infinite dimensional vector spaces that help me to break my habit of thinking of $\mathbb{R}^n$ when thinking about vector spaces.
emDiaz
- 813
64
votes
6 answers
Is $\{0\}$ a field?
Consider the set $F$ consisting of the single element $I$. Define addition and multiplication such that $I+I=I$ and $I \times I=I$ . This ring satisfies the field axioms:
Closure under addition. If $x, y \in F$, then $x = y = I$, so $x + y = I +…
Dan
- 18,262
64
votes
5 answers
How do you rotate a vector by a unit quaternion?
Given a 3-variable right-handed vector v that is a translation measured in local space and a unit quaternion representing an orientation from local to world space, how do you use the quaternion to rotate the vector from local space to world…
Narf the Mouse
- 813
64
votes
14 answers
What is a proof?
I am just a high school student, and I haven't seen much in mathematics (calculus and abstract algebra).
Mathematics is a system of axioms which you choose yourself for a set of undefined entities, such that those entities satisfy certain basic…
user78743
64
votes
8 answers
Why do the French count so strangely?
Today I've heard a talk about division rules. The lecturer stated that base 12 has a lot of division rules and was therefore commonly used in trade.
English and German name their numbers like they count (with 11 and 12 as exception), but not…
Martin Thoma
- 10,157
64
votes
3 answers
Why are certain PDE called "elliptic", "hyperbolic", or "parabolic"?
Why are the Partial Differential Equations so named? i.e, elliptical, hyperbolic, and parabolic. I do know the condition at which a general second order partial differential equation becomes these, but I don't understand why they are so named?
Does…
The_Lazy_Panda
- 759
64
votes
4 answers
A "new" general formula for the quadratic equation?
Maybe the question is very trivial in a sense. So, it doesn't work for anyone. A few years ago, when I was a seventh-grade student, I had found a quadratic formula for myself. Unfortunately, I didn't have the chance to show it to my teacher at that…
