homepage – If my blog on the side has no traffic is it still contributing to my home page rank?

In other words, does my blog confer some domain authority to my website, despite the fact that it receives no traffic?

Note that my rank is very good, so I’m just trying to determine why.
My home page is by far my main source of traffic, and that’s what I want.

If the answer was “no”, then logically could I delete my blog without negatively affecting my rank? If “yes”, then I can only improve things by continuing to blog and update that content, correct?

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probability – Average rank of a sequence of iid N(0,1) random variables.

Suppose we have $X_1, …, X_n$ are iid N(0,1), and the rank is defined as below
$$
r(X_i) = sum_{i neq j} I(X_j geq X_i)
$$

What is the expected rank?

I get that it is $(n-1)/2$ but Mood in “On the Asymptotic Efficiency of Certain Nonparametric Two-Sample Tests” states it to be (n+1)/2.

Am I missing something here?

matrices – Design an orthogonal basis for rank $2$ matrix in a scalar convex function

Consider the following function $$f(Q) = |Q-Q_d|_F^2,$$ where

  1. $Q = qq^T$ and $Q_d = q_dq_d^T$ with $q, q_dinmathbb{R}^4$.
  2. Obviously, $f(Q)$ is convex in $Q$.
  3. So $f(Q_d) =0$, which is minimum.

We know, $Q-Q_d$ is of rank $2$ with the basis for its column space ${q,, q_d}$.

Goal: Design a $f(Q)$ such that the matrix in $|cdot|_F^2$ is of rank $2$. Meanwhile, it has an orthogonal basis. On the other hand, $f(Q)$ has to have the properties 2. and 3.

For example, $$f(Q)=|Omega Q+QOmega^T|_F^2$$ where $Omega$ is a skew symmetric matrix. The above has an orthogonal basis ${Omega q,, q}$, since $q^TOmega q=0$.

What I try is $$Omega Q_dQ-Q_d,$$ where $Omega$ is skew-symmetric. this is because $$Omega Q_dQ-Q_d=Omega q_dq_d^Tqq^T-q_dq_d^T$$ So basis ${Omega q_d, q_d}$. However, the minimal value of $|Omega Q_dQ-Q_d|_F^2$ may not appear when $Q=Q_d$.

Any nice suggestions? I appreciate it!

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pagerank – Is domain rank capable of passing more seo than category/page ranks?

Say we have two identical websites, one with backlinks and traffic on the homepage which has categories x y z and one with a category which includes the x y z categories. The navigation menu shows the subcategories under the category. Would the first case get more views? A SEO setup that I’m working on requires me to do something like the second case and I’m not entirely sure whether it’ll work or not.

Keep in mind I’m asking the view case for both the higher and lower categories in the hierarchy.

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Topological stable rank of $C_0(mathbb{Z} times K)$

I was trying to calculate the topological stable rank of $C_0(mathbb{Z} times K)$ where $K$ is Cantor set. My idea: Since $C_0(mathbb{Z} times K)$ is not unital if we some how make it unital then $spec (C_0(mathbb{Z} times K))$ is homeomorphic to Cantor set and $C_0(mathbb{Z} times K)$ became AF-algebra. but I can’t make the algebra unital. Does anybody have any idea to make $C_0(mathbb{Z} times K)$ unital?

Real rank and suspension

Is there any relation between real rank of a $C^*$algebra and its suspension?
Like for example, what is the real rank of $C_0(mathbb{R}) otimes mathbb{K}(l^2)$