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1500 questions
64
votes
14 answers
How do I motivate myself to do math again?
I have been thinking of asking for help for a few months now but posting in a public forum like this is intimidating. Still, I am currently in a university studying mathematics as an undergrad. I took quite a few knocks a few months back when I…
Dust
- 235
64
votes
5 answers
A Challenging Logarithmic Integral $\int_0^1 \frac{\log(x)\log(1-x)\log^2(1+x)}{x}dx$
How can we prove that: $$\int_0^1 \frac{\log(x)\log(1-x)\log^2(1+x)}{x}dx=\frac{7\pi^2}{48}\zeta(3)-\frac{25}{16}\zeta(5)$$ where $\zeta(z)$ is the Riemann Zeta Function . The best I could do was to express it in terms of Euler Sums. Let $I$ denote…
Shobhit
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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
- 4,867
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
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64
votes
2 answers
$x^p-c$ has no root in a field $F$ if and only if $x^p-c$ is irreducible?
Hungerford's book of algebra has exercise $6$ chapter $3$ section $6$ [Probably impossible with the tools at hand.]: Let $p \in \mathbb{Z}$ be a prime; let $F$ be a field and let $c \in F$ . Then $x^p - c$ is irreducible in $F[x]$ if and only if…
user79709
- 643
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 space?…
Narf the Mouse
- 813
64
votes
11 answers
Striking applications of linearity of expectation
Linearity of expectation is a very simple and "obvious" statement, but has many non-trivial applications, e.g., to analyze randomized algorithms (for instance, the coupon collector's problem ), or in some proofs where dealing with non-independent…
Clement C.
- 68,645
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 French:…
Martin Thoma
- 10,247
64
votes
9 answers
On the functional square root of $x^2+1$
There are some math quizzes like: find a function $\phi:\mathbb{R}\rightarrow\mathbb{R}$ such that $\phi(\phi(x)) = f(x) \equiv x^2 + 1.$ If such $\phi$ exists (it does in this example), $\phi$ can be viewed as a "square root" of $f$ in the sense of…
user1551
- 151,715
64
votes
7 answers
Does $\zeta(3)$ have a connection with $\pi$?
The problem Can be $\zeta(3)$ written as $\alpha\pi^\beta$ , where ( $\alpha,\beta \in \mathbb{C}$ ), $\beta \ne 0$ and $\alpha$ doesn't depend of $\pi$ (like $\sqrt2$ , for example)? Details Several $\zeta$ values are connected with $\pi$ , like:…
GarouDan
- 3,488
64
votes
3 answers
How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$?
How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$? Here's my attempt: Given a Cauchy sequence $\{q_n\}_{n \in \mathbb{N}}$ in $X/Y$, each $q_n$ is an equivalence class induced by $Y$, I…
Metta World Peace
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64
votes
2 answers
Is there a step by step checklist to check if a multivariable limit exists and find its value?
Do we rely on certain intuition or is there an unofficial general crude checklist I should follow? I had a friend telling me that if the sum of the powers on the numerator is smaller then the denominator, there is a higher chance that it may not…
Yellow Skies
- 1,760
64
votes
14 answers
Is there a math expression equivalent to the conditional ternary operator?
Is there a math equivalent of the ternary conditional operator as used in programming? a = b + (c > 0 ? 1 : 2) The above means that if $c$ is greater than $0$ then $a = b + 1$ , otherwise $a = b + 2$ .
dataphile
- 840
64
votes
1 answer
Which Algebraic Properties Distinguish Lie Groups from Abstract Groups?
This question is motivated by a previous one: Conditions for a smooth manifold to admit the structure of a Lie group , and wants to be a sort of "converse". Here I am taking an abstract group $G$ and looking for necessary conditions for it to admit…
Lor
- 5,788
64
votes
11 answers
Why does the discriminant in the Quadratic Formula reveal the number of real solutions?
Why does the discriminant in the quadratic formula reveal the number of real solutions to a quadratic equation? That is, we have one real solution if $$b^2 -4ac = 0,$$ we have two real solutions if $$b^2 -4ac > 0,$$ and we have no real solutions if…
user487950