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1500 questions
80
votes
6 answers
How to take the gradient of the quadratic form?
It's stated that the gradient of:
$$\frac{1}{2}x^TAx - b^Tx +c$$
is
$$\frac{1}{2}A^Tx + \frac{1}{2}Ax - b$$
How do you grind out this equation? Or specifically, how do you get from $x^TAx$ to $A^Tx + Ax$?
victor
- 923
- 1
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- 6
80
votes
6 answers
When writing in math, do you use a comma or colon preceding an equation?
This is a general question about mathematical writing especially for writing research papers and the like.
Question: Do you precede an equation with a comma or colon?
Example A:
The following equation is the Yosida-Hawking-Penrose-Dantzig…
Shamisen Expert
- 8,092
80
votes
2 answers
Why are the last two numbers of this sequence never prime?
I had the idea to make a script that generates a pattern like this:
1
2 3
4 5 6
7 8 9 10
...
and so on. After that, I replaced every non-prime by a '-' character and every prime number by a '|'. The output begins like that:
-
||
-|-
|---
|-|--…
Bruno Henrique
- 853
80
votes
3 answers
Mathematicians shocked(?) to find pattern in prime numbers
There is an interesting recent article "Mathematicians shocked to find pattern in "random" prime numbers" in New Scientist. (Don't you love math titles in the popular press? Compare to the source paper's Unexpected Biases in the Distribution of…
Tito Piezas III
- 54,565
80
votes
2 answers
Factorial of a matrix: what could be the use of it?
Recently on this site, the question was raised how we might define the factorial operation $\mathsf{A}!$ on a square matrix $\mathsf{A}$. The answer, perhaps unsurprisingly, involves the Gamma function.
What use might it be to take the factorial of…
Oliphaunt
- 933
80
votes
2 answers
Inscribing square in circle in just seven compass-and-straightedge steps
Problem Here is one of the challenges posed on Euclidea, a mobile app for Euclidean constructions: Given a $\circ O$ centered on point $O$ with a point $A$ on it, inscribe $\square{ABCD}$ within the circle — in just seven elementary steps. Euclidea…
PDE
- 1,467
80
votes
13 answers
Why there is no sign of logic symbols in mathematical texts?
Either in undergraduate or graduate textbooks on Mathematics (Real/Complex Analysis, General Topology, Differential Geometry, ...), I never saw symbols $\Rightarrow$, $\iff$, $\forall$, $\exists$, etc. Instead, I just see their "read as" or…
MKR
- 1,088
80
votes
7 answers
Intuition behind conjugation in group theory
I am learning group theory, and while learning automorphisms, I came across conjugation as an example in many textbooks. Though the definition itself, (and when considering the case of abelian groups), it seems pretty innocent, I have to admit that…
Nikhil
- 1,047
80
votes
5 answers
Completion of rational numbers via Cauchy sequences
Can anyone recommend a good self-contained reference for completion of rationals to get reals using Cauchy sequences?
Derek Scavo
- 1,983
80
votes
5 answers
Product of two Gaussian PDFs is a Gaussian PDF, but Product of two Gaussian Variables is not Gaussian
The Product of Two Gaussian Random Variables is not Gaussian distributed:
Is the product of two Gaussian random variables also a Gaussian?
Also Wolfram Mathworld
So this is saying $X \sim N(\mu_1, \sigma_1^2)$, $Y \sim N(\mu_2, \sigma_2^2)$ then…
Frames Catherine White
- 1,570
80
votes
8 answers
incremental computation of standard deviation
How can I compute the standard deviation in an incremental way (using the new value and the last computed mean and/or std deviation) ?
for the non incremental way, I just do something…
shn
- 1,062
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- 17
80
votes
6 answers
Why is $\omega$ the smallest $\infty$?
I am comfortable with the different sizes of infinities and Cantor's "diagonal argument" to prove that the set of all subsets of an infinite set has cardinality strictly greater than the set itself. So if we have a set $\Omega$ and $|\Omega| =…
user17762
79
votes
6 answers
Intuitive explanation of a positive semidefinite matrix
What is an intuitive explanation of a positive-semidefinite matrix? Or a simple example which gives more intuition for it rather than the bare definition. Say $x$ is some vector in space and $M$ is some operation on vectors.
The definition is:
A $n$…
user915
79
votes
8 answers
Very *mathematical* general physics book
I am searching for a book to study physics. So far, I've been suggested
Resnick, Halliday, Krane, Physics,
but it doesn't seem to be very suited for a math major. Can you suggest some more mathematical books?
By mathematical I mean:
rigorous,…
Dal
- 8,214
79
votes
4 answers
Why the emphasis on Projective Space in Algebraic Geometry?
I have no doubt this is a basic question. However, I am working through Miranda's book on Riemann surfaces and algebraic curves, and it has yet to be addressed.
Why does Miranda (and from what little I've seen, algebraic geometers in general) place…
Potato
- 40,171