# All Questions

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### Weights in a mixture distribution

I'm trying to come up with a reasonable way to determine the weights in a mixture distribution. Let us consider the following example: There are two districts ($i=1,2$) in a city, both of which using ...
53 views

### Determining polynomial approximations of piecewise constant functions

Let $t_1 < t_2 < \cdots <t_m$ be real, and $X = \cup_{i=1}^{m-1} (t_i, t_{i+1})$ be a union of real open intervals. Let $f:X \rightarrow \{-1, 1\}$ be any piecewise constant function of form ...
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### Different Bialgebra/Hopf algebra structures on coalgebras

Given a coalgebra $C$, can there exist more than one algebra structure on $C$ giving it the structure of a bialgebra? I will also ask the same question for Hopf algebras.
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### Is the Poisson formula valid when the boundary condition is $L^2$?

Dirichlet problem for Laplace equation as follows \begin{eqnarray} \Delta{u}&=&0\text{ in }B_r(0)\\ u&=&g\text{ on }\partial B_{r}(0), \end{eqnarray} where $g$ is continuous. It is ...
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### Non-singular variety covered by pairwise disjoint singular subvarieties

Can you cover a non-singular algebraic variety by pairwise disjoint singular closed subvarieties? Varieties are over an algebraically closed field of characteristic other than $2$ and $3$. In ...
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342 views

### What is a function field analog of Giuga's conjecture?

Giuga's conjecture (1950), which is still open and has strong numerical support, reads : Let $n$ be a positive integer. If $1+\sum_{k=1}^{n-1}k^{n-1} \equiv 0\pmod{n}$ then $n$ is prime. What would ...
100 views

### Counting $\mathrm{mod}\:p$ solutions of Diophantine equation in two variables taking $O(p^2)$ time

Are there Diophantine equations in two variables such that counting solutions $\mathrm{mod}\:p$ requires $O(p^2)$ time? Geometrically this means we have to sort through a positive proportion of the ...
122 views

### Random walk always stays below a level $a$

Suppose we have a random walk $S_n$ with i.i.d. steps $X_i$. We assume that $$\mathbb{E}[X_i] = -\mu, \text{Var}[X_i] = 1,$$ where $\mu$ is close (or going) to zero. We also assume that the moment ...
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