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differential equations – Solving a big matrix DE system using NDSolveValue

I know there are similar questions, but I was unable to understand the solution for my case in any of them.

I have the following:

Commutator(a_,b_) := a.b - b.a;

initialstate = {0,1,0};
pin = KroneckerProduct(Conjugate(initialstate),initialstate);

pI = NDSolveValue({
p'(t) == - Commutator(p(t).F1(t), L(t)) + Commutator(ConjugateTranspose(F1(t)).p(t), ConjugateTranspose(L(t)))
- Commutator(p(t).F2(t), ConjugateTranspose(L(t))) + Commutator(ConjugateTranspose(F2(t)).p(t), L(t)),
p(0) == pin
},
p,{t,0,1})

It might be relevant to say that F1(t) and F2(t) are both numerical 3×3 functions and L(t) is a big analytic 3×3 function.

I have seen in other posts that the sum will act before the dot operators causing Mathematica to misinterpret what I am trying to calculate. It indeed returns me an error if I try to execute it, so what the correct syntax for this problem is? Where I have dot operators not only in the DE system but also inside the Commutator function.

I would really appreciate it if, besides the solution, you could provide me an explanation of what is happening.

abstract algebra – The big picture regarding products and coproducts in categories

I am new to category theory. I have been reading about products and coproducts in Chapter 0 of Topology: A Categorical Approach by Bradley, Bryson and Terilla.

Is it fair to say that products and coproducts are really just methods for putting two (or more) objects from a category together to create a new object within the same category?

I am trying to grasp the essence of what’s going on before worrying about all the details (Cartesian product, direct product, direct sum, free product, disjoint union, etc.). I appreciate any help.

big data – Semi-streaming algorithm for $s$-$t$ connectivity

Let $G=(V,E)$ be an undirected graph. Given a pair of vertices $s,t in V$, how can we construct a semi-streaming algorithm which determines is $s$ and $t$ are connected? Is there any way to construct such an algorithm which scans the input stream only once?

Note that a semi-streaming algorithm is presented an arbitrary order of the edges of $G$ as a stream, and the algorithm can only access this input sequentially in the order it is given; it might process the input stream several times. The algorithm has a working memory consisting of $O(nlog^{O(1)}n)$ bits.

big o notation – How to calculate the runtime of a following code?

Could someone explain how to calculate the Big O notation for a runtime of a snippet of a code?

for (int i=0; i<list.length; i++) {
    for (int j=0; j<list.length; j++) {
        for (int k=0; k<list.length; k++) {
            if (k%2==0) {
               list(i) += list(j);
            } else {
               list(i) += list(k);
            }
       }
   }
}

From my understanding, I’m thinking it’s something like O(n^3) because the statement is executive n times for every i?

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big o notation – What is the complexity of $i^i$?

What is the complexity of the following algorithm in Big O:

for(int i = 2; i < n; i = i^i)
{
    ...do somthing
}

I’m not sure if there is a valid operator to this type of complexity.
My initial thought was as follows:

After $k$ iterations we want: (using tetration?)

${^{k}i} = n implies k=logloglog…_klog{n}impliesmathcal{O(logloglog…_klog{n})}$ (where we have k times the log function) but i’m not sure if this is evan a valid way of writing this.
Anyway, we have a complexity that that includes $k$, which does not seems right to me.

big o notation – Time complexity about Maximum subarray

I recently came across a function called the strawman algorithm which the pseudo code looks like this:

StrawmanSubarray(A):
  Initialize bestsum = A(0)  
     For left=0 to n-1:
        For right = left to n-1:
           Compute sum A(left) + . . . + A(right)
           If sum > bestsum: bestsum = sum

The time complexity is Θ (n^3), and I don’t quite get where is the 3rd n comes from to get Θ (n^3)?

big sur – macOS Big Sur not smooth

I’ve just installed macOS Big Sur 11 beta and whenever I open the notifications sidebar or any directory opening/closing effects are not smooth at all also whenever I scroll down in a folder.

I’ve checked my login items and activity monitor and nothing seems to stress the CPU

spec