# All Questions

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### A question on recursion in Kripke-Platek set theory with infinity and $\Sigma_{3}$-separation and $\Sigma_3$-collection

$\Sigma_{3}KP\omega$ be Kripke-Platek set theory with infinity and $\Sigma_{3}$-separation and $\Sigma_3$-collection. What strengthening of Barwise's Definition by $\Sigma$ Recursion (Theorem 6.4 on ...
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### a question about connected open sets in $R^2$

Let $U,V$ be two nonempty connected open sets in $R^2$ and $U\subsetneqq V$.I want to ask if there must exist an open ball $B\subset V$ such that $B\not\subset U$ and $B\cap U$ is a nonempty connected ...
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### Does the associated Lie algebra determine a group?

Let $G$ be a group and let $\Gamma_G(k)$ be the $k$th term of the lower central series of $G$. For each $k\geq 1$, set $\mathcal{L}_G(k)=\Gamma_G(k)/\Gamma_G(k+1)$ and ...
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### Realizing a monoid as $\mathrm{End}(G)$ for some graph $G$

This question relates to Realizing groups as automorphism groups of graphs. Given a monoid $M$, is there a graph $G$ such that the endomorphism monoid $\textrm{End}(G)$ is isomorphic to $M$?
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### Find all solutions in positive integers of the diophantine equation $w^2+x^2+y^2=z^2$ [migrated]

It's an exercises of the text book Elementary Number Theory and It's Applications 6th Edition by Kenneth H.Rosen. I wanted to solve it using the method in solving the diophantine equation ...
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### Random Matrix Theory: question on a quantity on page 87 in book of Bai and Silverstein 2010

Anyone else is reading carefully the book of Bai and Silverstein 2010, titled "Spectral Analysis of Large Dimensional Random Matrices"? On page 87 of this book, when they state the final step in the ...
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### Mayer-Vietoris sequence for topological K-theory

I'm reading the paper Loop groups and twisted K-theory I by Freed, Hopkins, and Teleman. They give some examples of computing (twisted) K groups using the Mayer-Vietoris sequence. I'm a bit ...
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### A decomposition of incompressible vector fields

In Andrew, Majda- Vorticity and incompressible flow page 93, there is a theorem which is not proved: Take a smooth incompressible (free divergence) vector field $v$ in $\mathbb{R}^2$. Call $w$ its ...
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### Example of joint cyclic and separating vector

Let $\mathcal{H}$ be a separate Hilbert space and $\mathcal{B}(\mathcal{H}) \subset \mathcal{B}(\mathcal{H}) \otimes M_2(C)$ be a W$^*$-inclusion pairs. It is known that this pair share a joint cyclic ...
In most references from dynamic system theory, the following linear continuous dynamic system is considered. $$\frac{\text{d}x(t)}{\text{d}t}=Ax(t)+Bu(t)+Dd_{1}(t)\quad (1)$$ $$y(t)=Cx(t)+Ed_{2}(t) ... 1answer 75 views ### Axiomatization of Degree Theory I am reading Ruiz's Mapping Degree Theory, and find an axiomatization of degree theory of \mathbb R^n in P38. It says that there exists a unique map d(f,D,y)\in\mathbb Z satisfies Normality ... 0answers 61 views ### Estimating the moments of a random variable Suppose i wanted to estimate the expectation and variance of a random variable X. More over suppose i could write a variable X as a sum of indicator random variables X=\sum_{i=1}^{k} X_{i}. Are ... 1answer 97 views ### Averaging maps of Riemannian manifolds Let M be a compact Riemannian manifold. We know how to average functions f\colon M\to {\mathbb R}; the integral \frac{\int_M f}{\int_M 1} returns a value in {\mathbb R}. If intead f\colon ... 1answer 77 views ### Resolvent of a triangular matrix Suppose A is a triangular matrix. What is the most efficient known algorithm to compute the polynomial (in x) matrix (xI-A)^{-1}? Of course, (xI-A)^{-1}= N(x)/p_A(x), where p_A is the ... 0answers 35 views ### Behaviour of Markov type under uniform homeomorphism of spheres A metric space (X,d_X) has Markov type p (with p \in [1,2]), if, for every stationary Markov chain \{Z_n\}_{n=0}^\infty on Y (a finite space) and every mapping f:Y \to X, one has$$ ...
Suppose I have some random variable $X$ taking values in $[a, b]$ with unknown distribution (I am happy to assume the distribution is smooth, though it would be nice to not have to). I have a ...