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();      }  } 

How can I code an algorithm (concerning networks)

my question is basic but I want to know how to code algorithms like this, TY for your help !

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.