# All Questions

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### Isogenies of complex multiplication elliptic curves

This is a slight continuation of a previous question of mine. Given an elliptic curve $E$ over $\mathbb{Q}$ which has complex multiplication. How would one find each $p$ such that $E$ admits a $p$-...
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### Is anything written about winning the “Dollar Game” in the minimal number of moves?

I run some Master's projects on Chip-Firing games, using the Holly Krieger's Numberphile video on the topic as an initial motivation, and going on to prove the main theorem stated there (that you can ...
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### Proof that any hyperbolic group has Rapid Decay property

A classic result that states that any hyperbolic group in the sense of Gromov has Rapid Decay property in the sense of Jolisaint. But the original proof of that fact is contained in an old Ph.D. ...
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### Folding Polygons into 'Vessels'

This question is based on http://www.science.smith.edu/~jorourke/Papers/FoldingPP.pdf Define an vessel as a convex polyhedron with one face removed - in other words, a vessel can be converted into a ...
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### What is the difference between Gegenbauer and Legendre polynomials

I'm trying to find the Spherical Harmonics decomposition on a function defined on the hypersphere. I wanted to ask what are the main differences between the Gegenbauer and Legendre polynomials?
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### A question about a truncated object

I was hoping someone could help me with the understanding of a particular truncated object. Here are some background: For any object $A$ in an abelian category $\mathcal{A}$, we can view $A$ as an ...
1answer
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### Asymptotic behavior of infinite product of cosines

Consider the function $$F(z) := \cos(z) \cos(z/3) \cos(z/5) \cos(z/7) \cdots$$ Note that $\cos(z/n) = 1 - \frac{z^2}{n^2} + \cdots$ so the product is absolutely convergent to an entire function. I ...
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### What is the precise definition of “Hypergeometric motives over $\mathbb{Q}$”?

The question is as in the title, but here is some background: Section 4 of this paper by Beukers, Cohen and Mellit is called "Hypergeometric motive over $\mathbb{Q}$" but no actual (pure) ...
1answer
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### Is the fixed subring a symmetric algebra?

Let A be a finite dimensional symmetric k-algebra over some field k. The set of units of A is denoted by U(A). Suppose G is a cyclic group of prime order which acts via inner algebra automorphism on A,...
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### How to estimate the spectral radius of a matrix series?

$\{A_k\}$ is a uniformly bounded sequence of matrices whose eigenvalues are in $(0,\rho]$, $\rho<1$. Let $\Phi(k+1,j)= A_kA_{k-1}\dotsm A_j$ and $Q_k=\sum_{j=k}^{\infty}\Phi^T(j,k)\Phi(j,k)$, how ...
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### Monoidal functors and their projection functors

I first posted this post as a math I as I am instructions proceed. Suppose $M$ and $N$ are monoidal categories and let $M\times N$ denote the associated product category. $M\times N$ comes equipped ...
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### Choice of splitting in domain decomposition algorithms

When solving a PDE numerically by domain decomposition methods, what is the "optimal way" to split the domain? Are there any results stating that a particular partition of the domain yields &...
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### The solutions of a system of differential equations

Let $P(x,y) = \frac{x}{y}^{\frac{x^2}{y-x}}$ for $x \neq y$ and using the proper limits $P(x,y)=e^{-x}$ for $x=y$, $P(x,y)=0$ for $x\neq0, y=0,$ and $P(x,y)=1$ for $x=0, y\neq0.$ Consider this system ...
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### Smooth interpolation of values

Consider a sequence of points $(x_n)_{n \in \{0,\ldots,N\}^2}.$ The finite element method tells us how to find for example a piecewise linear function $f$ on $[0,1]^2$ such that $f(1/n_1,1/n_2) = x_n$ ...
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The Reed–Muller $RM(m, m)$ code sends the message $p(X_0, \ldots , X_{m - 1}) = \sum_{S \subset \{0, \ldots , m - 1\}} \alpha_S \cdot X_S$ to its set of evaluations $\left\{ p(x_0, \ldots , x_{m - 1}) ... 0answers 43 views ### PDE for the area-preserving non-parametric curve shortening flow? In dimension$1$, it is well-known that the motion of a (unparametrized) plane curve$\{(x,f(x,t);\,x \in I \subset \mathbb R)\}$under the curve shortening flow can be encoded by the PDE$$\partial_t ... 0answers 72 views ### Show that$ \sum_{i=0}^{2k} {n \choose i+1} u(2k,i) +\sum_{i=0}^{2k} {-n-1 \choose i+1} u(2k,i)=0. $Let$u(k,j) = 1$if$j=0$,$0$if$j > k$, or else it is$j*u(k-1,j-1) +(j+1)*u(k-1,j) $. Prove that$ \sum_{i=0}^{2k} {n \choose i+1} u(2k,i) +\sum_{i=0}^{2k} {-n-1 \choose i+1} u(2k,i)=0. $... 1answer 290 views ### Largeness of the set of zeroes of a Brownian motion Definitions: A measurable subset$S$of$\mathbb R$is said to be mesoscopic if there exists a continuous function$f: \mathbb R \to \mathbb R$such that$f(S)\$ is Lebesgue measurable and has nonzero ...

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