general topology: number of topologies in a finite set with open sets of $ k $ up to homeomorphisms

Leave $ X $ Being a set of three elements. For each of the following numbers n, determine the number of distinct homeomorphism topology classes in $ X $ with exactly $ n $ open subsets (including the empty set and the entire set).
$ 1) 3 \ 2) 4 \ 3) 5 \ 4) 7 $

observation:
Assume $ tau_1 $ Y $ tau_2 $ are two homeomorphic topologies in $ X $, then for each open set $ U $ in $ tau_1 $ there is an open set $ U ^ {& # 39;} $ such that $ | U | = | U ^ {& # 39;} | $.
Using this observation, I have solved the case when there is $ 3 $ Open sets.
This topology should be $ { phi, X, U } $ Y $ | U | = $ 1 or $ 2 $. Depends on the cardinality of $ U $, the class of homomorphism determined and, therefore, there are two classes of homomorphism.

Then the $ n = 7 $ The case is simple. Because there are no topologies with seven open sets. So here the answer is $ 0 $.

I need some intuition to make the cases. $ n = 4 $ Y $ 5 $. Kindly share your thoughts. Thank you.

real analysis: finite cover of clopen sets with maximum diameter $ epsilon. $

Here is the problem:

Leave $ X $ be a compact metric space that is totally disconnected and leave $ epsilon> 0. $

(a) Show that $ X $ it has a finite cover $ mathcal {A} $ clopen sets with a maximum diameter $ epsilon. $

My judgment

With the help of many people here on this site, I was able to demonstrate that:

Yes $ X $ is a compact metric space that is totally disconnected, then for each $ r> 0 $ and every $ x in X, $ there is a clopen set $ U $ such that $ x in U $ Y $ U subseteq B_ {r} (x). $

I feel this can help me in testing the first question, but I don't know how, could someone please clarify this for me?

Also, I got hints here from a finite deck of clopen sets. but still, I can't write the solution thoroughly. Any help would be appreciated.

finite element method – FEM: electric field between two arbitrarily defined forms

I was wondering how to do the following:
I would like to calculate the electrostatic field between two ways using the FEM method.

(*Define Boundaries*)
air = Rectangle({-3, -3}, {3, 3});
object1 = Disk();
object2 = Rectangle({2, 0}, {2.5, 2});
Show(Graphics({Blue, air}), Graphics({Magenta, object1}),Graphics({Green, object2}))

enter the image description here

Calculation of the electric field at each point {x, y} in 2D space:

enter the image description here
$ r_i $ is the vector of the point charge; $ r $ is the vector to the point in 2D space (or also 3D) where we want to calculate the electric field.

I do a Mathematica function (for the moment I omit the constant term):

eField(x_, y_) := q Sum(({x, y} - pts((i)))/Norm({x, y} - pts((i)))^3, {i, n})

where pts((i)) are the limit points of the loaded object and x Y y are coordinates of the "air" object.

How it would proceed:

  1. I calculate the electrostatic field of object 1 -> $ E_1 $

  2. I calculate the electrostatic field of object 1 -> $ E_2 $

  3. I use superposition to get the resulting electric field: $ E_ {Total} = E_1 + E_2 $

I would really appreciate if someone could show me how to do it in Mathematica using Finite Element (FEM).

combinatorial: color a finite set of sticks placed on a line in red or blue

Consider a set of many finite sticks that have a finite length (all sticks are not necessarily the same length) that are placed on an infinitely long line, so that these sticks can overlap partially, completely, or not at all. Is it possible to color each stick in this red or blue set so that at each point $ p $ of the line, the number of red sticks and blue sticks differs depending on $ -1 $, $ 0 $ or $ 1 $?

Theory of the ct category: complete and finite category with nice setbacks that are not locally presentable

I have a result that is true for complete and complete finite categories, so that pullbacks retain directed collimits, by which I mean $ A times_B ( operatorname {colim} _ {i in I} C_i) = colim_ {i in I} (A times_B C_i) $, where $ I $ It is a directed category.

Thanks to a result from Adamek and Rosicky (locally presentable and accessible categories, Proposition 1.59), I know this is true in any presentable (finite) local category.

Are there any other categories that satisfy those conditions?

ag.algebraic geometry: a Lefschetz-style formula for twisting $ ell ^ infty $ of an abelian variety over a finite field

Leave $ A / mathbb F_q $ be an abelian variety on a finite field. Define $ A_ ell = A ( ell ^ infty) ( mathbb F_q) $, he $ ell ^ n $ ($ n geq 0) $ defined torsion points on the base field. I can assume $ ell neq 0 $ in $ mathbb F_q $ if necessary.

Is there a related Lefschetz-style trace formula $ A_ ell $ to the trail of Frobenius in some cohomology groups? If we leave $ a_ ell = | A_ ell | $, then note that:

$$ | A ( mathbb F_q) | = sum _ { ell} (a_ ell – 1) $$
where the right side adds up all the prime numbers and only finitely many terms are not zero. The left side has an interpretation in terms of the action of the Frobenius in the Etale cohomology groups and it would be great if any of those relationships also rose to the level of the cohomology groups.

finite element method: analysis of stress (displacement) 1-D in the ring when considering viscoelasticity

I am a new Mathematica user, and I WAS trying to solve a stress field (or displacement) problem in a ring.
Here's my problem: the inner radius of the ring is b, and the outer radius of the ring is c. The limiting condition is that there is an applied pressure p (t) on the inner surface. The outer surface is free of traction. The material is memory foam. Since the ring stress is not a function of (Theta), the stresses are functions of r and t only, making this problem a 1-D problem.

The governing equation: {D ((Sigma) r, r) – (Sigma) (Theta) / r} == 0
Constitutive equations: {(Sigma) r = Integrate (D ((Epsilon) r (r, t & # 39;), t & # 39;) * Eg (t & # 39;), {t & # 39 ;, 0, t}), (Sigma)
(Theta) = Integrate (D ((Epsilon) (Theta) (r, t & # 39;), t & # 39;) * Eg (t & # 39;), {t & # 39 ;, 0, t}) }
Stress-stress relationships: (Epsilon) r = D (u (r, t), r), (Epsilon) (Theta) = u (r, t) / r

To simplify the calculation, suppose that the module Eg (t & # 39;) = 1. The dimensions of the ring: b = 1, c = 2. The pressure: p (t) = 0.1t
My code is here

    b = 1;
    c = 2;
    p(t_) = 0.1 t;
    eqn1 = {(Epsilon)r -> D(u(r, t), r), (Epsilon)(Theta) -> u(r, t)/r};
    eqc = {(Sigma)r = Integrate(D(((Epsilon)r /. eqn1), t), {t, 0, t}), (Sigma)(Theta) = 
    Integrate(D(((Epsilon)(Theta) /. eqn1), t), {t, 0, t})};
    sys = {D(eqc((1)), r) + eqc((2))/r}; 
    uu1 = NDSolve({sys((1)) == 0, (eqc((1)) /. r -> c) == 0, (eqc((1)) /. r -> b) == -p(t), (u(r, t) /. t 
    -> 0) == 0}, u, {r, b, c}, {t, 0, 30})

I was unable to execute this code in Mathematica. Can anybody help me please? I think the problem is the constitutive equations: they contain t & # 39; and t, but I just used t for the integral.

Usual distances in DFA (deterministic finite automata)?

I've been looking in the literature for examples of distances defined in the set of DFAs (or in the set of minimum DFAs) that are defined in a given sigma alphabet.

Since the languages ​​they describe (normal languages) can be infinite in size, defining a distance is not a trivial matter.

However, having a distance over these objects can be useful for adjusting them in metric spaces, which allows for a variety of things (in my case, evaluating the performance of an algorithm).

My only consistent idea so far is to create a distance similar to the edit distance on tagged graphics in minimized DFAs.

Has anyone heard of other distances?

finite element method: Poisson equation and FEM

I'm trying to solve $ u_ {xx} + u_ {aa} = – ((3x + x ^ 2) y (1-y) + (3y + y ^ 2) x (1-x)) e ^ {x + y}, $ for $ 0 <x <1.0 <y <1, $ and homogeneous Dirichlet boundary conditions $ u (x, y) = x (1-x) y (1-y) e ^ {x + y} $ using an analytical method, a FEM and NDSolve method and then plotting all the solutions on a common graph. But for the analytical method I use:

NDSolveValue({!(
*SubsuperscriptBox(((Del)), ({x, y}), (2))(u(x, 
 y))) == ((3 x + x^2)*y*(y - 1) + (3 y + y^2)*x*(x - 1))*
Exp(x + y), 
DirichletCondition(u(x, y) == x*(1 - x)*y*(1 - y)*Exp(x + y), 
True)}, u, {x, 
  y} (Element) Disk())
Plot3D(%(x, y), {x, y} (Element) Disk())

it doesn't work I tried to do a FEM method but it didn't work.

finite groups: could we construct a finitely additive invariant measure of translation for the requirements below? If so, how?

Consider a real value function $ P $ defined by parts in a domain $ A $ as follows: $ A $ is divided into non-overlapping subsets $ A_1, ldots, A_n $and there is a function $ P_i: A_i rightarrow mathbb {R} $ defined in each region. The general function $ P $ is defined by $ P (x) = P_i (x) $ when $ x in P_i $, which specifies exclusively $ P $ throughout $ A $.

(Please note I want $ P $ ALWAYS have an average between the infimum and the supremum of $ P $range of.)

Now consider $ lambda $ like the Lebesgue measure. Leave $ A = bigcup A_i subseteq mathbb {R} $. You want a finite additive invariant measure of translation $ m $ in $ mathbb {R} $ so that

Case 1

$ m (A cap (a, b)) = lambda (A cap (a, b)) $ when $ A cap (a, b) $ It has a full measure of lebesgue.

Case 2

$ m (A cap (a, b)) = b-a $ when $ A $ it is dense in $ (a, b) $; Y, $ m (A_1 cap (a, b)) = b-a $ Yes $ A_1 $ is the only dense partitioned subset in $ (a, b) $.

Case 3

Yes $ lambda (A cap (a, b)) $ it is between $ 0 $ Y $ b-a $; Y, $ A $ it's finite then $ m (A cap (a, b)) = lambda (A cap (a, b)) $


Yes $ lambda (A cap (a, b)) = 0 $ Y $ A $ it is dense as a finite set $ T $ in $ (a, b) $; what if $ A_1 $ it is dense as a finite set $ T_1 $, $ A_2 $ it is dense as a finite set $ T_2 $, etc. then yes $ bigcup T_i subseteq T $

$ m (A_i cap (a, b)) = left | T_i cap (a, b) right | / left | T cap (a, b) right | $

Case 5

Yes $ lambda (A cap (a, b)) = 0 $ Y $ A $ is finite then $ m (A_i cap (a, b)) = left | A_i cap (a, b) right | / left | A cap (a, b) right | $

Case 6

Yes $ A cap (a, b) $ is positive but not full, so it should (somehow) partition $ (a, b) $ in subintervals where one of the first four conditions MUST apply for each subinterval. (If not, see case 6).

If the first condition applies to any of the subintervals, take the measure of those subintervals and give the remaining subintervals the measure zero. If the first condition does not apply to any subinterval; but, the second yes, takes the measure of those subintervals and gives the remaining subintervals the measure zero. If the first and second conditions do not apply to any subinterval; but, the third one does, it takes the measure of those subintervals and gives the remaining subintervals the zero measure. If the first, second, and third conditions do not apply to any of the subintervals, and the fourth condition do, take the measure of those subintervals and give the remaining subintervals the measure zero.

Case 6

Any case that does not match the previous cases can have a measure between 0 and the length of the subinterval where case 6 is kept.

However, we must have a measure where the first six cases are true.

If such a measure exists, use it to find the "average" value of some function $ P $ defined in each $ A_i $.


Before saying that such a measure does not exist (remember that it is finitely additive, it does not have to be accountingly additive), read the following:

EXTENSIVE DENSITIES AND MEASURES IN GROUPS AND SPACES G
AND ITS COMBINATORY APPLICATIONS

SOLECKI'S SUB-MEASURES AND DENSITIES IN GROUPS

DENSITIES, MEASURES AND GROUP PARTITIONS