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1500 questions
83
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 like: $$S_N=\sqrt{\frac1N\sum_{i=1}^N(x_i-\overline{x})^2}.$$…
shn
- 1,092
82
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
82
votes
12 answers
I lost my love of math; I'm getting it back. How can I determine if math is actually right for me?
This question has been on my mind for a very long time, and I thought I'd finally ask it here. When I was 6, my dad pulled me out of school. The classes were too easy; the professors, too dull. My father had been man of philosophy his entire life…
Chiefy
- 675
82
votes
12 answers
The limit of truncated sums of harmonic series, $\lim\limits_{k\to\infty}\sum_{n=k+1}^{2k}{\frac{1}{n}}$
What is the sum of the 'second half' of the harmonic series? $$\lim_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} =~ ?$$ More precisely, what is the limit of the above sequence of partial sums?
Daniel Pietrobon
- 4,920
82
votes
6 answers
Order of finite fields is $p^n$
Let $F$ be a finite field. How do I prove that the order of $F$ is always of order $p^n$ where $p$ is prime?
Mohan
- 15,664
82
votes
10 answers
Finding out the area of a triangle if the coordinates of the three vertices are given
What is the simplest way to find out the area of a triangle if the coordinates of the three vertices are given in $x$-$y$ plane? One approach is to find the length of each side from the coordinates given and then apply Heron's formula . Is this the…
TSP1993
- 979
82
votes
3 answers
Fourier Transform of Derivative
Consider a function $f(t)$ with Fourier Transform $F(s)$. So $$F(s) = \int_{-\infty}^{\infty} e^{-2 \pi i s t} f(t) \ dt$$ What is the Fourier Transform of $f'(t)$? Call it $G(s)$.So $$G(s) = \int_{-\infty}^{\infty} e^{-2 \pi i s t} f'(t) \ dt$$…
82
votes
6 answers
Reference request: introduction to commutative algebra
My goal is to pick up some commutative algebra, ultimately in order to be able to understand algebraic geometry texts like Hartshorne's. Three popular texts are Atiyah-Macdonald, Matsumura (Commutative Ring Theory), and Eisenbud. There are also…
82
votes
13 answers
What does "surface area of a sphere" actually mean (in terms of elementary school mathematics)?
I know what "surface area" means for: a 2d shape a cylinder or cone but I don't know what it actually means for a sphere. For a 2d shape Suppose I'm given a 2d shape, such as a rectangle, or a triangle, or a drawing of a puddle. I can cut out a 1cm…
silph
- 775
82
votes
11 answers
Infinite sets don't exist!?
Has anyone read this article ? This accomplished mathematician gives his opinion on why he doesn't think infinite sets exist, and claims that axioms are nonsense. I don't disagree with his arguments, but with my limited knowledge of axiomatic set…
Nicolas Bourbaki
- 1,706
82
votes
13 answers
What is an example of a sequence which "thins out" and is finite?
When I talk about my research with non-mathematicians who are, however, interested in what I do, I always start by asking them basic questions about the primes. Usually, they start getting reeled in if I ask them if there's infinitely many or not,…
tomos
- 1,821
82
votes
8 answers
How do I prove that a function is well defined?
How do you in general prove that a function is well-defined? $$f:X\to Y:x\mapsto f(x)$$ I learned that I need to prove that every point has exactly one image. Does that mean that I need to prove the following two things: Every element in the domain…
Kasper
- 14,040
- 18
- 71
- 129
82
votes
13 answers
Do matrices $ AB $ and $ BA $ have the same minimal and characteristic polynomials?
Let $ A, B $ be two square matrices of order $n$. Do $ AB $ and $ BA $ have same minimal and characteristic polynomials? I have a proof only if $ A$ or $ B $ is invertible. Is it true for all cases?
Andy
- 2,356
82
votes
4 answers
Differences between the Borel measure and Lebesgue measure
I'm having difficult time in understanding the difference between the Borel measure and Lebesgue measure. Which are the exact differences? Can anyone explain this using an example?
Mark Hyatt
- 905
82
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 function…
Your neighbor Todorovich
- 8,723