finite automata – Why are Regular sets not closed under infinite unions and intersections?

Look at $$ell={a^pmid ptext{ is prime}}.$$

This language obtain from infinite union of
$$bigcup_{igeq 2, itext{ is prime}}^{infty}L_i$$
Where each $L_i={a^imid itext{ is prime}}$ that have one word.

Another example is ${a^nb^nmid ninmathbb{N}}$
That isn’t regular and we can describe it by infinite union of regular languages
$$bigcup_{igeq 1}^{infty}a^ib^i=a^1b^1cupdots.$$ Each $a^ib^i$ is language that have one word.

For intersection, i recommend you read the following link.

sg.symplectic geometry – Some doubts regarding the paper Floer cohomology of lagrangian intersections and pseudo-holomorphic disks II

I have been reading and trying to understand the paper Floer cohomology of lagrangian intersections and pseudo-holomorphic disks II , but at this point there are some parts that the author claim that I can’t quite understand why they are true.

$1)$ In page $1001$ the author claims that due to the choice of $phi $, we will have that $phi_1circ sigma circ phi_1^{-1}$ will be again an anti-holomorphic involutive isometry , is this since the complex structure of $mathbb{C}mathbb{P}^n$ is just multiplication by $i$?

$2)$Then in page $1002$ it’s mentioned that since $u_{2k}$ and $u_{2k+1}$ are $J-$holomorphic and $u_{2k}(tau,1)=u_{2k+1}(tau,0)$ then they can be smoothly glue together to a construct a $J$-holomorphic map from $mathbb{R}times (0,2^N)rightarrow mathbb{C}mathbb{P}^n$. Does anyone know a reference where I could try and understand why this is true?

$3)$ In page $1003$ they claim that $phi_1^{2^{N-1}}u in mathcal{M}_{J,phi}(x,y)$, and so it will be $J-$holomprhic but this will come from the fact that $phi_1^{2^{N-1}}$ is holomorphic? Also then they claim that if $phi_1^{2^{N-1}}uneq u$ then it can’t be a translation of $u$, and I think this makes sense due to the transversality , however I am not quite seeing why we could have then that $phi_1^{2^{N-1}}=u$, as they claim that it is a possibility.

$4)$ In page $1004$ where they go an compute the linearization of the section $bar partial_J: mathcal{P}rightarrow mathcal{L}$, I am a bit confused with the notation , they say that $E_u(xi)=nabla_{s}|_{s=0}bar partial_J(u_s) $, however I am confused with this covariant derivatice notation, is it supposed to be just the derivative of the family of vector fields $bar partial_J (u_s)$, i.e.,$ frac{partial}{partial s}|_{s=0} bar partial_J(u_s)$? Or is some sort of a generalization of a covariant derivative?

Any help is appreciated, these are alot of questions , but I am fairly new to the subject and I am trying to understand things clearly.
Thanks in advance.

complexity theory – What is the closure of context-free languages under finite intersections?

Famously the intersection of context-free languages need not be context-free. On the other hand the intersection of context-sensitive languages is context-sensitive.

So this leads to the question: what is the closure of context-free languages under finite intersections? Does this class of formal languages have a name? Do we get all context-sensitive languages this way?

permutation – How to get the complete list of subsets the pairwise intersections of which are empty

Given the list Range(6). I want to get the sublists of length 2 where each element has length 2 and the pairwise intersections are empty. So I am looking for:

  {{{1, 2}, {3, 4}}, {{1, 3}, {2, 4}}, {{1, 4}, {2, 3}}}

Switched elements like {{1,2},{4,3}} should not appear. My code works well but when Range and lengths get bigger it consumes a lot of space and time. For sublists of length 3 with two elements the result
would look like:

    {{{1, 2}, {3, 4}, {5, 6}}, {{1, 2}, {3, 5}, {4, 6}}, {{1, 2}, {3, 
   6}, {4, 5}}, {{1, 3}, {2, 4}, {5, 6}}, {{1, 3}, {2, 5}, {4, 
   6}}, {{1, 3}, {2, 6}, {4, 5}}, {{1, 4}, {2, 3}, {5, 6}}, {{1, 
   4}, {2, 5}, {3, 6}}, {{1, 4}, {2, 6}, {3, 5}}, {{1, 5}, {2, 3}, {4,
    6}}, {{1, 5}, {2, 4}, {3, 6}}, {{1, 5}, {2, 6}, {3, 4}}, {{1, 
   6}, {2, 3}, {4, 5}}, {{1, 6}, {2, 4}, {3, 5}}, {{1, 6}, {2, 5}, {3,
    4}}}

Is there a function (maybe in Combinatorica) for this problem or a smarter way to do it? I am sure this is a standard problem and there must be a name for this kind of sublist. I would be grateful for further hints.

Here is my code:

k = 3;
t1 = Partition(#, {2}) & /@ Permutations(Range(2 k))
t2 = Map(Sort, t1, {2})
t3 = Map(Sort, t2, {1})
t4 = DeleteDuplicates(t3)

set theory – Set intersections

We have three sets A, B and C.

A has 5 values

B has 3 values

C has 3 values

Also, we know the intersections (repeated values):

A⋂B = 2 values

A⋂C = 2 values

C⋂B = 1 value

Is it possible to know (by math) how many unique values do we have, only with the information above?

Relationship between intersections

Given $N$ sets $X_1, dots, X_N$ and two definitions of intersection $cap’$ and $cap”$, is it possible to show that
$$
vert X_i cap’ X_j vert le vert X_i cap” X_j vert,
quad
forall i,j in { 1, dots, N }
$$

implies
$$
vert X_{i_1} cap’ dots cap’ X_{i_K} vert
le
vert X_{i_1} cap” dots cap” X_{i_K} vert,
quad
forall X_{i_1} dots X_{i_K} in mathcal{P}({ X_1, dots, X_N })?
$$

graphics – RegionIntersection: extra intersections

I have got an array of points, which can be represented as a ListLinePlot. Then I am trying to find the number of intersections of this ListLinePlot with a certain line. Obviously, in the demonstrated case the number of intersections should be two, but I get 8 points as a result. RegionIntersection gives the same result. How can I fix that? I can’t create and solve the system of equations, representing an intersection condition, because there is a large number of arrays to be investigated and their general appearance is unknown.Thanks in advance.

My code is (sol is an above-mentioned array of points):

lst1 = Sort(sol);

lst2 = Table({x, Max(Sort(sol((All, 2)))) - (Max(Sort(sol((All, 2)))) - Min(Sort(sol((All, 2)))))/5}, 
{x, Min(Sort(sol((All, 1)))), Max(Sort(sol((All, 1)))), 0.5});

plot = ListLinePlot({lst1, lst2}, PlotRange -> All)

l = Graphics`Mesh`FindIntersections@plot

example