# All Questions

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### How can two random variables are continuous infers that their jointly random variable is continuous [on hold]

We assume that $\forall a,b$ suchthat $a^2+b^2>0$, $aX+bY$ is continuous random variable. But we don't assume that $X$ and $Y$ are independent. My question is the following: Is it true that the ...
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### Cramer-Rao type bound for absolute estimation error

Let $\{X_1, X_2, \ldots, X_n\}$ be independent and identically distributed (i.i.d.) random variables sampled from a common distribution with density $f_{\theta}(x)$, where $\theta$ is an unknown ...
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### When are configuration spaces aspherical?

It is a theorem of Fox and Neuwirth that the space $C_k \mathbb R^2$ of unordered configurations of $k$ points in $\mathbb R^2$ is apsherical, i.e. has trivial higher homotopy groups. This has some ...
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### real representation of a product group

Let $G_1$ and $G_2$ be compact Lie groups. We know that each finite-dimensional complex irreducible representation of $G_1\times G_2$ is the tensor product of an irreducible representation of $G_1$ ...
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### Splines linearly independent

Let $N_1:=\chi_{[0,1]}$ be defined as this characteristic function and $N_n:=N_{n-1}*N_1$ then this leads to polynomials with support $[0,n]$. These splines are well-studied click for wikipedia My ...
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### Order of element in algebraic group [migrated]

Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an affine algebraic group over $\bar{\mathbb{F}}_q$, and let $F$ be a ...
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### Order dimension of $\omega^\omega/(fin)$

Let $\omega^\omega$ denote the collection of all functions $f:\omega\to\omega$. For $f,g\in\omega$ we say $f\simeq g$ if and only if $\exists N \in \omega$ such that $f(n) = g(n)$ for all $n\geq N$. ...
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### Find point of triangle [on hold]

There is the triangle (http://www.mathportal.org/calculators/plane-geometry-calculators/triangleRightAngle.gif) where we know: coordinates of points: A, C and all sides: a, b, c Angles are 45 and 90 ...
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### Generalization of Borsuk-Ulam to arbitrary ratio

Let $g: S^n \to R^n$ be a continuous odd function (i.e. $g(-x)=-g(x)$ for all $x$). The Borsuk-Ulam theorem implies that $g$ has a zero, i.e. there is an $x$ such that $g(x)=(0,0,...,0)$. Suppose $g$ ...
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### Presentation of the Rybnikov matroid

In this well celebrated work Gregory Rybnikov exhibit an example of two arrangements with the same underlying matroid, but with fundamental groups which are not isomorphic. This is a key ...
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### I have to show any non-invertible matrix is a reducible matrix [on hold]

Suppose that $A \in M_n(D)$ and $D$ be a division ring. An $n × n$ matrix $A = (a_{ij} )$ is called reducible if $A$ has a non-trivial invariant subspace in $D^n$. I have to show any non-invertible ...
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### Extending the Abel-Jacobi map over the DM-compactification $\overline{\mathcal{M}}_2$?

Let $\mathcal{M}_2$ be the moduli space of genus two curves and $\mathcal{A}_2$ the moduli space of principally polarized abelian surfaces. Then the Abel-Jacobi map gives an open embedding ...
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### Macaulay's example of prime ideals in $\mathbb C[X_1,X_2,X_3]$ having large number of generators

There is a famous example of Macaulay which shows that there are prime ideals of height two in $\mathbb C[X_1,X_2,X_3]$ having at least $l$ generators for any $l\ge 3$. In Macaulay's words, the ...
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### ${\frak b}$ and ${\frak d}$ defined with $\leq$ instead of $\leq^*$

Let $\omega^\omega$ denote the collection of all functions $f:\omega\to\omega$. For $f,g\in\omega$ we define $f\leq g$ if $f(n)\leq g(n)$ for all $n\in\omega$; $f\leq^* g$ if there is $N\in\omega$ ...
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### $L_{\infty}$-norm of a $\delta(t)$-“function”? [on hold]

In different contexts the $L_\infty$- norm may sometimes be defined as the essential least upper bound $\|\cdot\|_\infty=\operatorname{ess} \sup_t |\cdot |$ and sometimes as just the least upper bound ...
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### Fixed point property for intersection of spaces which are homeomorphic to a disk

The following question is question 9.8 from Miller's paper Some interesting problems '': Question Suppose $D_n$ a subset of the plane is homeomorphic to a disk and for every \$n\in \omega, ...