# All Questions

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### Any material out there on *spatially* second-order cellular automata?

My model of a problem I'm working on took me to needing a spatially second-order cellular automaton (ie, x[i][t+1] is determined by x[i][t], x[i-1][t], x[i-2][t], x[i+1][t], and x[i+2][t]) (also, this ...
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### Hodge numbers of non-commutative varieties

Let $(X, \mathcal{A})$ be a non-commutative variety, by this I mean $X$ is a (smooth) algebraic variety and $\mathcal{A}$ is a sheaf of algebra on $X$. One such example in my mind is when $X$ admits ...
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### A Polynomial With Positive Prime-Density

Let $P(x)$ be a non-constant polynomial with real coefficients. Can natural density of $$\{n\ |\ [P(n)] \ \text{is prime.}\}$$ be positive?
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### Integer solutions of $x^2=4+8y^2+13z^2$

I have been looking for integer solutions of certain Diophantine equations, one of the simplest examples being $x^2=4+8y^2+13z^2$. The ideal answer would be a way to parametrize all the integer ...
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### Volume of a complex manifold

Is there a theorem which states the following? Let $\mathcal{M}$ be a $k$ -dimensional complex submanifold of $\mathbb{C}^n, \ 1 \le k \le n$. Let $V \subset M$ be open, relatively compact in $M$, so ...
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### Interpolation Operator Bounded in Sobolev Norm

Let $m\in \mathbb{N}$, $p\in [1,\infty]$, $W^{m,p}([0,1])$ the space of all functions $[0,1]\rightarrow \mathbb{R}$ which are $m$ times weakly differentiable and weak derivatives in $L^p$, ...
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### Reference request: Topological space of polygonal chains and its properties [migrated]

I'm interested in approximations of $C^1([a,b])$-functions by polygonal chains: image File:NURBstatic.svg by User:WulfTheSaxon licensed under GNU Free Documentation License A polygonal chain can be ...
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### A question on the cohomology of elliptic curves over local fields

Let $K$ be a number field,$\nu$ a nonarchimedian prime of $K$, $K_{\nu}$ the completion of $K$ at $\nu$ with maximal unramified extension $K_{\nu}^{unr}$. Let $E$ be an elliptic curve defined ...
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### Negatively curved metrics minimizing the length of a homotopy class of simple closed curves

Good afternoon everyone ! I have the following question of Riemannian geometry : Let $M$ be a smooth closed orientable manifold of dimension at least $3$, and let $\mathcal{T} = \{$ smooth ...
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### Sobolev trace map: is the fractional seminorm bounded by just the gradient?

Let $M$ be a compact Riemann manifold. Consider the trace map $T:H^1(M) \to H^{\frac 12}(\partial M)$. Is it always the case that |Tu|_{H^{\frac 12}(\partial M)} \leq C\lVert \nabla u ...
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### How do I calculate the Entropy of a vector? [on hold]

I understand the concept of entropy, I even referred to the wikipedia page but I am confused. Can anyone tell me in simple words how I could calculate the entropy of a vector, with an example please? ...
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### The mutual information rate spectrum [migrated]

Definition: $\mathbf{X}$ denotes the random vector $({X_1},{X_2},...,{X_n})$. The mutual information between $X$ and $Y$, $I(X;Y)$, is determined by the joint law of $p(X,Y)$, Given two random ...