## 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.

## 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.

## 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?

## unity – Solving the problem of thick colliders at intersections?

While I somehow solved my collider issue in about 99% cases, there are cases where the solution simply doesn’t work.

This is an intersection and the logic of thick colliders simply don’t work here:

Other examples, in red is ideally how it should be:

Any ideas?

## 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?

## raycasting – How to calculate ray unit grid intersections?

By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.

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

## plotting – How can I mark the intersections on the x axis of the following graphs?

Im trying to reproduce the following graph:

I have the parabolas graphed, but I want to indicate the intercepts with the x axis by some point or cross, also I want to eliminate the numbers and the little lines that mark them. I have this:

Thanks in advance. Im new to mathematica so…

## 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
``````