## All Questions

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### Is there anyway to rewrite a partial differential equation using language of differential forms, tensors,.etc

My question is: usually, a partial differential equation, for example, those coming from physics, is written in a lauguage of vector calculus in a local coordinate, is there anyway …
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### A family of words counted by the Catalan numbers

In recent work with Michael Albert and Nik Ruškuc, a family of words has arisen which is counted by the Catalan numbers. I've looked at Richard Stanley's Catalan exercises in EC2 a …
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### Permutation and Combination question… [closed]

Hello, I am currently studying Extension 1 Mathematics. I missed two classes and I figured out that tomorrow I will have a quiz. Can you help me to solve this permutation and combi …
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### Hyperbolic pair of pants.

Suppose $Y$ is a pair of pants with a hyperbolic structure and $\gamma_i; i = 1, 2, 3$ are the geodesic boundaries of length $l_i; i=1, 2, 3$ respectively. Now consider a essential …
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### An operation on binary strings

Consider the “product” $\gamma = \alpha \times \beta$ of two binary strings $\alpha$, $\beta$ $\in \lbrace 0,1\rbrace^+$ which one gets by replacing every 1 in $\beta$ …
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### On Perelman’s paper

In section 5 in "The entropy formula for the Ricci flow and its geometric applications" Grisha Perelman has written: Fix a closed manifold $M$ with a probability measure $m$, and …
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### Help me on proof of an equation.

I wanna prove following equation $\sum_{i=1}^n \prod_{k=1,k\neq i}^n \prod_{j=1,j\neq k}^{n+1}(x_j - x_k) = -\prod_{i=1}^n \prod_{j=1,j\neq i}^n (x_j - x_i)$ I have verified sev …
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### can we say that $(p^2+1)/2\ne p_0^2$ where $p$ is a Mersenne prime

Let $p=2^a-1>7$ be a Mersenne prime and so $a$ is an odd prime. Can we say that $(p^2+1)/2$ is not equal to the square of a prime number? Many thanks for your help BHZ
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### Why don’t more mathematicians improve Wikipedia articles?

Wikipedia is a widely used resource for mathematics. For example, there are hundreds of mathematics articles that average over 1000 page views per day. Here is a list of the 500 mo …
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### $f^{-1}\mathcal I \cdot \mathcal O_X$ vs $f^\ast \mathcal I$

Let $X$ ad $Y$ be (noetherian) schemes and let $\mathcal I \subseteq \mathcal O_Y$ be a sheaf of ideals on $Y$. Let $f \colon X \to Y$ be a morphism of schemes. In general the shea …
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Let $V:=\oplus_{j\in\mathbb{Z}}V_j$ be a graded $\mathbb{F}$-vector space over the field $\mathbb{F}$. The graded tensor product of graded vector spaces is given by $V \otimes W: … 3answers 149 views ### Group action on the real line Hi, I was wondering about the following question: if you have a faithful action of a group G on the real line R by orientation-preserving homeomorphisms, it is easy to construct … 2answers 186 views ### Order type of the smallest set containing the identity function and closed under exponentiation Let$E$be the smallest set of functions$\mathbb{N}^+\to\mathbb{N}^+$containing the identity function$n \mapsto n$and closed under exponentiation$(f,g) \mapsto \left(n \mapst …
In ZF we have the two relations $A \leq B$ and $A \leq^\ast B$ which relate the size of sets: the first says there is an injection from $A$ to $B$, the second that there is a surje …
Recently I posted a conjecture at Math.SE: $$\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx\stackrel{?}{=}\frac{\pi}{2}(\mu^2-\nu^2),$$ where \$J_\mu( …