Concerning the Spanier group relative to an open cover

Let $ \mathcal{U} = \{ U_i \; |\; i\in I \}$ be an open covering of $ X$ ‎. Spanier defined $ \pi (\mathcal{U}‎, ‎x)$ to be the subgroup of $ \pi_1 (X‎, ‎x)$ which contains all homotopy classes having representatives of the following type‎: $ ‎\prod_{j=1}^{n}u_j *v_j * u^{-1}_{j}‎, ‎$ ‎where $ u_j$ ‘s are paths (starting at the base point $ x$ ) and each $ v_j$ is a loop inside one of the neighbourhoods $ U_i \in \mathcal{U}$ ‎.

‎If an open cover $ ‎\mathcal{U}$ is a refinement of an open cover $ ‎\mathcal{V}$ ‎, then $ \pi (‎\mathcal{U}‎, ‎x) \subset \pi (‎\mathcal{V}‎, ‎x)$ ‎.

My question is that:

If $ [f][g]\in \pi (‎\mathcal{U}‎, ‎x)$ for $ [f],[g]\in \pi_1 (X,x)$ , then is there any refinement $ \mathcal{V}$ of $ \mathcal{U}$ so that $ [f]\in \pi (\mathcal{V},x)$ ?

A curious inequality concerning binomial coefficients

Has anyone seen an inequality of this form before? It seems to be true (based on extensive testing), but I am not able to prove it.

Let $ a_1,a_2,\ldots,a_k$ be non-negative integers such that $ \sum_i a_i = A$ . Then, for any non-negative integer $ B \le A$ : $ $ \sum_{(b_1,\ldots,b_k): \sum_i b_i = B} \prod_i \frac{\binom{a_i}{b_i}}{\binom{A-a_i}{B-b_i}} \ge {\binom{A}{B}}^{2-k}. $ $ The sum on the left is over all tuples $ (b_1,b_2,\ldots,b_k)$ of non-negative integers, with $ b_i \le a_i$ for all $ i$ , whose sum is equal to $ B$ .

Code segment concerning polymorphism. Why does the following result in a compilation error?

Why does the following code result in a compilation error? Since it is a GeeksforGeeks object, shouldn’t it use the getValue() method found in class GeeksforGeeks. I added a getValue() method to the base class and the code compiled. What is the reasoning for this?

class GFG  {      protected void getData()      {          System.out.println("Inside GFG");      }  }   class GeeksforGeeks extends GFG  {      protected void getData()      {          System.out.println("Inside GeeksforGeeks");      }       protected void getValue()      {          System.out.println("GeeksforGeeks");      }  }   public class Test  {      public static void main(String[] args)      {          GFG obj = new GeeksforGeeks();          obj.getValue();      }  } 

Another question concerning p and t

I refer to an article concerning p and t : edited Sep 14 ’17 at 2:48 / Bjørn Kjos-Hanssen answered Sep 13 ’17 at 21:50 / Mark Fischler

I already asked a question December 14th 2018 and I received among others this answer from Alex Kruckman Dec 14 at 21:50 (thank you Alex) : … you’re correct that {2 to the power of m! : m∈ℕ} is a pseudo-intersection of the family ({m to the power of k : m∈ℕ})k∈ℕ.

I have please another question, now concerning t : could someone give me an example of a family respecting the finite-intersection criteria and fully ordered by the relation ‘almost included’, and yet however having no pseudo-intersection ? Thanks in advance.

Social Media Sharing Buttons concerning the Chinese Great Firewall

We are about to build a multi-regional website that targets users in China and other places. However, a concern was raised about the use of social media sharing buttons. Since China blocks common social media websites like Facebook and Twitter, we fear that showing buttons with their icons on them will cause China to block our site. However, we do want to allow users from other countries to share our website to these social media websites.

So the questions are:

  1. Will these social media buttons cause our site to be banned in China?
  2. If they are not banned, will they affect rankings in search engines?
  3. If the localised version (zh-hans) does not have these buttons, will it helps? However, the homepage in default language (which is not Chinese) will still have them.
  4. Will there be a difference to use sub-directories or sub-domains for different languages, since sub-domains are usually viewed as a different site?

Any suggestions are welcome.