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About the cohomology of $BG^\delta$. Making a Lie group discrete

Let $G$ be a connected Lie group. Recall that the topological group $G^\delta$ is $G$ endowed with the discrete topology. The inclusion $G^\delta \to G$ induces a map between the classifying spaces $\eta: BG^\delta\to BG$. Question 1 Let $\eta^*:H^*(BG,\mathbb{Z})\to H^*(BG^\delta,\mathbb{Z})$ be the induced map in integral cohomology. By Corollary 1 in Milnor, On the homology […]

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How can I determine how many $B$-smooth numbers are below a given number?

Suppose I have a factor Basis with primes $F =\{2, 3, \dots, B \}$ and I want to know how many numbers exist in the range $1 \leq 2^{a_2}3^{a_3} \dots B^{a_{B}} \leq N$ for a natural number $N$ with exponents $a_2,a_3,\dots,a_{B}$. Such numbers are called $B$-smooth numbers. In the Special case where $B=5$ we have […]

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Prime Ideals of $\mathbb{C}^{n}$

What are the prime ideals of the ring $\mathbb{C}^{n}$? I was thinking that it is $e_{i} : i \in [n]$, where $I = e_{i} = \mathbb{C}^{(i)}= (\mathbb{C} , \dots , 0 , \dots , \mathbb{C})$, where the $i^{th}$ copy of $\mathbb{C}$ is not in $e_{i}$ because taking quotients yields $\mathbb{C}$ which is a domain? Is […]

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O(1) or o(logn) discrepancy for multiples of an irrational for at least one sub interval

O(1) or o(logn) discrepancy for multiples of an irrational for at least one sub interval. Using $\{x\}$ to denote the fraction part of $x$ we can define for any $I\subset [0,1]$, $$E(n,\theta, I) ={ \left|\{\,\{\theta\},\{2\theta\},\dots,\{n\theta\} \,\} \cap I \right|}-n|I|$$ $ $ $$\Delta_{sup}(n,\theta)=\sup_I |E(n,\theta,I)|$$. The Equidistribution Theorem says that the sequence $a_i=(i\theta)$, $i\in\mathbb{Z}_{\geq1}$ is equidistributed modulo […]

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MLE of random samples

If $V_1$, $V_2$, $\dots$ , $V_n$ and $W_1$, $W_2$, $\dots$ , $W_n$ are independent random samples of size $n$ from normal populations with the means $\mu_1 = \alpha + \beta$ and $\mu_2 = \alpha – \beta$ and the common variance $\sigma^2 = 1$, find maximum likelihood estimators for $\alpha$ and $\beta$. Is it enough to […]

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Distance to the big cone and Seshadri constant on Fano manifolds

I would like to know something about the following two questions. Given $X$ Fano manifold and $L$ an ample line bundle on $X$, we define \begin{gather} \sigma(L,x):=\sup\{t>0\, :\, \mu^{*}L-tE \,\, \mbox{is big}\, \},\\ \end{gather} for any $x\in X$ where $\mu:Y\to X$ is the blow-up at $x$ and $E$ is the exceptional divisor. Is there any […]

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Applying the Abramov-Rokhlin skew product entropy formula to a bounded-to-one factor

Let $(X, \mathcal{B}, \mu, S)$ and $(Y, \mathcal{C}, \nu, T)$ be invertible probability-measure-preserving systems, with a measurable factor map $\pi: X \to Y$, i.e. $\pi \circ S = T \circ \pi$. Suppose that there is some $N \in \mathbb{N}$ such that $\nu$-almost every $y \in Y$ has at most $N$ preimages under $\pi$, i.e. $\# […]

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Simplify a series involving binomial coefficients

For $n=2,3,\dots$ and $k=1,2,\dots,n$, I would like to prove that $$ \sum_{\ell=k+1}^{n}\frac{2^{2\ell}}{\ell}{2n-2\ell\choose n-\ell}\left(1+\frac{2n-2\ell+1}{\ell-1}\right)=\frac{2^{2k}(2n-2k+1){2n-2k\choose n-k}}{k}. $$ I have checked it for small $n$ and $k$. However, I am unable to come up with a method to prove it. I am wondering if it is possible to (simplify) deal with $$ \sum_{\ell=k+1}^{n}\frac{2^{2\ell}}{\ell}{2n-2\ell\choose n-\ell}. $$ I would be […]

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Maximum of sum of exponential function

Let $x_1,\dots,x_n$ be a set of given vectors in $\mathbb{R}_{+}^d$. Let $c_1,\dots,c_n$ be given positive constants. I am interested in finding the vectors $w_1,\dots,w_n$ in $\mathbb{R}_{+}^d$ that solves the optimization problem \begin{align} \max_{w_1,\dots,w_n}\sum_{i=1}^{n}c_i\frac{\exp(w_i^Tx_i)}{\sum_{j=1}^{n}\exp(w_j^Tx_i)} \end{align} I am not even sure how to start around this and I currently use a out-of-the-box optimization algorithm. Is this type […]

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Obtaining a positive faithful functional on finite dimensional C*-algebra

Suppose we are given a finite-dimensional $C^*$-algebra $A$. If we want to avoid the classification theorem that says that $A$ is isomorphic to a finite direct sum of matrix algebras, is it possible to construct a faithful, positive linear functional on $A$? Using the classification, we could simply add the trace-functionals to obtain our result. […]