# All Questions

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### degree of polynomials in nullstellensatz

If $K$ is algebraically closed field, then consider $m$ polynomials $f_i$ for $i=\{1,\dots,m\}$ in $K[x_1,\dots,x_n]$ of degree $d_i$ each with no common root. We know there exists $g_i$ for ...
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### When is a topological space the homotopy colimit of an open covering?

Suppose that $X$ is a topological space and $\left(U_i \to X\right)$ is an open cover. We can associate to it the Cech diagram of this cover $$C_U:\Delta^{op} \to Top.$$ I know that for many good ...
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### Norm of swapped power series in the unit disk

Suppose $f(z)=a_0+a_1z+\cdots+a_nz^n+\cdots$ is defined in the unit disk and $\|f\|_{\infty}\leq 1.$ Lets form another series $g$ by interchanging $a_1$ and $a_k$ i.e. ...
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### Solutions for n such that the path number to the n-th node is also n in a complete, rooted, ordered k-ary tree

The path to find the $n$-th node within a complete, rooted, ordered $k$-ary tree can be represented by a number to the base $(k+1)$ that contains no zero digits. See A245905 in OEIS as an example for ...
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### Region of Convergence [on hold]

Using the Laplace transform, I got a frequency domain function, i.e 1/(s+1) + 1/(s+2), so my radius of convergence for the function on the right is Re{s} > -1 and for the left function it is Re{s} ...
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### Can all $L^2$ holomorphic functions on a domain vanish at a particular point?

Let $\Omega \subset \mathbb{C}^n$. Is it possible that there is a point $p \in \Omega$ such that every $f \in A^2(\Omega) = L^2(\Omega) \cap \mathcal{O}(\Omega)$ has a zero at $p$? The space ...
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### Robotics, Cryptography, and Genetics applications of Grothendieck's work? [on hold]

I was reading about the passing of Alexander Grothendieck, and something caught my interest: Mr. Grothendieck was able to answer concrete questions about these relationships by finding universal ...
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### Is ther a dense subset of parameter plane, which is not an interior?

In case of parameter plane of complex quadratic polynomial : is it possible to find part of parameter plane, scanned with given limited precision ( rasterised) where : every pixel contains part ...
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### An analytic characterization of eigenvalues of a Hermitian matrix

[..the following is trying to understand a certain argument of Terence Tao in a lecture notes of his..] If $A$ is a Hermitian matrix with eigenvalues, eigenvectors $\{ \lambda_i , e_i \}_{i=1}^n$. ...
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### Approximation of a degenerate continuous quadratic form

Let $a(\cdot)$ be a real positive definite function and let $b(\cdot)$ be a real function in $L^2$. Consider: $$P_T(x)=\frac{1}{\sqrt{T}}\int_0^T\int_0^T a(u-v)b(u-x)b(v-x)dudv.$$ The goal is to ...
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### Probability of close approach for multivariate normal variables

The following problem comes from a physical model of two groups of particles in three dimensions. I need to know the probability that the two groups of particles approach each other within some ...
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### Books on the analysis of hyperbolic partial differential equations

Most of the present books on pde analysis deal with the elliptic partial differential equations. Is there some book related to rigorous analysis with hyperbolic pdes, and especially hyperbolic systems ...
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### (Alternative) Presentation for the pure braid group of the sphere

First I need some notation (it's all standard I think). For a manifold $M$, let $F_nM = F_{0,n}M$ be the space of $n$-tuples of distinct points on $M$ ; let $B_nM = B_{0,n}M = F_nM / \Sigma_n$. When ...
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### functions of one complex variable: geometric theory

Can someone recommend a good textbook on functions of one complex variable which have good chapters on geometric theory, in English? When I studied complex analysis, I used two textbooks: An ...
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### Series of swapping terms

Suppose $f(z)=a_0+a_1z+\cdots+a_nz^n+\cdots$ is defined in the unit disk, $\mathbb{D}$, and $\|f\|_{\infty}\leq 1.$ Lets form a series $f_k$ by interchanging $a_1$ and $a_k$ i.e. ...
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### The error in Petrovski and Landis' proof of the 16th Hilbert problem

What was the main error in the proof of the second part of the 16th Hilbert problem by Petrovski and Landis? Please see this related post Added : According to their method, what of the following ...
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### Birkhoff Ergodic Theorem and Ergodic Decomposition Theorem for Continuous-Time Markov Processes

I have a couple of questions regarding ergodicity for Markov processes in continuous time. (In particular, the first question seems like it should be particularly basic, and yet I haven't managed to ...
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### Groups with a unique composition series

Which finite groups $G$ have a unique composition series? I don't mean in the sense of the Jordan-Holder theorem, but rather actually unique. Some examples are the cyclic groups $C_{p^n}$ and the ...
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### Model structure for cooperads and for coalgebras

I am studying the homotopy theory of (algebraic) operads and I came up with several questions I am unable to answer to. I would like to stress that I don't have applications in mind, I just would like ...

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