# All Questions

106,238
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### Minimal coupling

Let $\mu,\nu$ be probability measures defined on a common measure space $(\Omega,\mathcal F)$. A coupling of $\mu,\nu$ is a probability measure $\pi$ on $(\Omega^2,\mathcal F^2)$ with marginals $\mu,\...

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14 views

### On elements of $L^{1} [0,1]$ space

As we know the elements of $L^{p}$ spaces are classes of function not merely functions. Having this in mind, let $[f] \in L^{1} [0,1]$ for some function $f:[0,1] \to \mathbb R$. Assume we know that ...

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49 views

### The Riemann zeta function and differential operators

I've revisited an old post of mine--Dirac's Delta Functions and Riemann's Jump Function J(x) for the Primes--dealing with Riemann's "jump" or "staircase" function (aka, Π(x)) that has unit steps for ...

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30 views

### Does perron vector maximize $x^TAx$ in the simplex?

Let $\mathbf{A}$ be any $n\times n$ symmetric positive matrix ($A_{ij}>0$). It is easy to show that the solution to the following optimization problem
\begin{align}
\max_{\mathbf{x}}~~\mathbf{x^...

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25 views

### Primitive action of wreath product

I say in advance that I am really new to Group Theory, so if my question is trivial I apologize in advance.
Let $A$ and $H$ be groups and $\Omega$ be a $H$-set. In this set-up, we can define the ...

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18 views

### Matrix norm question [on hold]

Please I need some help. Explain why the statement in the preceding corollary is equivalent to
the following statement. : If ||| · ||| is a matrix norm, and if |||A||| < 1, then I ? A is ...

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35 views

### Extension of a bilinear form to the exterior algebra

In Serre's Local Field, at the beginning of the section 2 chapter 3, he has wrote "it is known that $T$ extends to a non-degenerate bilinear form on the exterior algebra of $V$", where $T$ is a non-...

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21 views

### Change variable in integration with symmetry

Not sure if I can ask such fundamental problem here.
Let
$G$ be a finite group, $\sigma \in G$. Consider linear group actions fo $G$ on $\mathbb{R}^n$.
$\sigma(K)=\{x\in \mathbb{R}^n: \sigma^{-1}(...

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10 views

### Root subgroups of simply connected Chevalley groups and their generators

I'm looking for a detailed mapping of the root subgroups of the simply connected Chevalley groups of type other than $A_n$, and their generators into $\operatorname{GL}_n(\mathbb{C})$(relative to a ...

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54 views

### Is the pullback of an ample bundle minus the exceptional divisor is ample

If X is a normal projective variety with an ample line bundle $L$, and $\pi:Y\to X$ a resolution of $X$ and $E$ be the exceptional divisor, then is it true that $A\pi^{\star}L-[E]$ is always ample for ...

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24 views

### Maximum Principles in Parabolic PDE with Neumann Condition

I am looking for some maximum principles and comparison results for parabolic equations. The most complete book I've found on this subject is: Murray Protter, Hans Weinberger - Maximum Principles in ...

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33 views

### Units in the coordinate ring on a reductive group

Let $K$ be a field and $G$ a connected reductive group over $K$.
Can we describe $K[G]^{*}$?

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63 views

### Tannaka-Krein duality in Chari-Pressley's book

I am not sure that this was not discussed before, so excuse me in this case. This can be considered as a special case of my previous question here.
V.Chari and A.N.Pressley in their "Guide to Quantum ...

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21 views

### Number of connected components of the centre of a Levi subgroup

Let $G$ be a connected complex semisimple algebraic group and $T\subset B\subset G$ a choice of maximal torus and Borel subgroup. Let $\Phi$ be the root system and $\Pi\subset\Phi$ the set of simple ...

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36 views

### An irrational complete intersection surface with good reduction everywhere

Do there exist 3 absolutely irreducible homogeneous polynomials in $\mathbb{Z}[a, b, c, d, e, f]$ such that
each one defines a hypersurface in $\mathbb{P}^5_{\mathbb{Z}}$ smooth over $Spec(\mathbb{Z})...