ag.algebraic geometry – About union of two algebraically independent sets

My question might be very simple for those that have a deep understanding for algebraically independent sets.

Let $P_1$ and $P_2$ are two algebraically independent sets and $P_1cap P_2=emptyset.$ Assume there exist $a,bin Qsetminus{0}$ such that $P^*=aP_1+bsubset J$ for some open interval $J.$ Does the following hold ?

I. $P^*$ is algebraically independent ?

II. $P^*cup P_2$ is algebraically ?

I believe that I is correct since just rescaling and shifting by rational numbers
About, II I do not find an example that shows it is incorrect or proof it.

Any help will be appreciated greatly.

randomness – Sampling from specific random distribution on sets

I have a random distribution on sets in mind, that has three parameters: $n, w, k$. The goal is to sample sets of $k$ integers from $(0, n)$ (without replacement) such that the elements within each set fit in a subrange of length $w$. That is, an outcome set $S$ must have properties:

  1. $S subset mathbb{N_0} ; wedge; |S| = k$
  2. $0leq min(S) leq max(S) < n$
  3. $max(S) – min(S) < w$

You can assume that $k leq w/2 < w ll n$.

Now there are many possible distributions possible over these sets. But I’m interested in those that have as property

$$forall x:P(x in S) = frac{k}{n};,$$

that is each integer in $(0, n)$ has an equal chance of being in a set when sampled (or as close as possible). Beyond the above requirements, it’d be ideal if the distribution is an maximum entropy one, but this isn’t as important, and something close would be fine too. As a minimum bar I do think every valid set should have a non-zero chance of occurring.

Is there a practical way of sampling from a random distribution that matches the above requirements?

I’ve tried various methods, rejection sampling, first picking the smallest/largest elements, but so far everything has been really biased. The only method that works that I can think of is explicitly listing all valid sets $S_i$, assigning a probability variable $p_i$ to each, and solving the linear system $$sum_i p_i = 1 quadbigwedgequad forall_x:frac{k}{n} – delta leq sum_{x in S_i} p_i leq frac{k}{n} + delta,$$ minimizing $delta$ first, $epsilon $ second where $epsilon = max_i p_i – min_i p_i$. However this is very much a ‘brute force’ approach, and is not feasible for larger $n, k, w$.

set theory – Is having injection to hereditarily size sets equivalent with choice over ZF?

It’s known that ZF proves that for every set $x$ there exists a set $H_x$ of all sets that are hereditarily strictly subnumerous to $x$. [see here]. Now is the following principle equivalent with choice over the rest of axioms of ZF?

For every set $x$, there exists a set $y$ such that: $x$ is subnumerous to $H_y$.

By “subnumerous to” its meant, as usual, possessing an injection towards; and “strictly subnumerous” means, as usual, existence of subnumerousity without existence of a bijection.

graphics – Plotting Minkowski product of two sets in complex 2D plane

I am trying to draw Minkowski product of two sets in complex 2D
plane in Mathematica. While I can draw the individual complex 2d plane for these sets
in Mathematica using ComplexRegionPlot, I do not know if there is
a way to draw the corresponding Minkowski product.

For example, consider the following complex 2d regions
begin{align*}
mathcal{G}_{1} & =left{ zinmathbf{C}midmathrm{Re}(z)geqvert zvert^{2}right} ,\
mathcal{G}_{2} & =left{ zinmathbf{C}midfrac{3}{2}mathrm{Re}(z)geqvert zvert^{2}+frac{1}{2}right} ,
end{align*}

where their Minkowski product is

$$
mathcal{G}_{1}cdotmathcal{G}_{2}=left{ z_{1}z_{2} in mathbf{C} mid z_{1}inmathcal{G}_{1},z_{2}inmathcal{G}_{2}right} ,
$$

and I am trying to plot the complex region associated with this Minkowski
product $mathcal{G}_{1}cdotmathcal{G}_{2}$. Any help/suggestions will be much appreciated.

gn.general topology – Isotopy Classes of Non-Connected Planar Sets

I was just looking through some of my old questions on StackExchange and noticed that this one went totally unresolved. I still think it’d be useful as a lemma for plane topology if it were true, and would be curious to see a counterexample if not.

Suppose we are given two pairs $(U, V)$ and $(Y, Z)$ of connected, simply connected, bounded, open sets in the plane, where $U cap V = Y cap Z = varnothing$. However, their boundaries may intersect. Assume their closures are also simply connected. Let $A = U cup V$ and $B = Y cup Z$. Suppose we also know the following, where ‘$simeq$‘ refers to planar isotopy:

  1. $U simeq Y$

  2. $V simeq Z$

  3. $bar{U} cap bar{V} simeq bar{Y} cap bar{Z}$

Then this is not enough to imply that $A simeq B$, due to an example at the bottom of this post.

So I’m wondering, what are some of the simplest conditions we can add on to those three above to ensure that $A simeq B$?

  1. Is it enough that $bar{A}$ is homeomorphic to $bar{B}$?

  2. Alternatively, assume $bar{U} setminus bar{V}$ is homeomorphic to $bar{Y} setminus bar{Z}$. Would assuming this, as well as its obverse for $bar{V} setminus bar{U}$ and $bar{Z} setminus bar{Y}$, be sufficient?

Counterexample, as promised: Let $U = Y$ be a ‘thickened open topologist’s sine curve’, i.e. a shrinking tubular neighborhood of the standard topologist’s sine curve with limit arc being $(-1, 1)$ on the $y$-axis (though not itself containing this arc, just the ‘wiggly part’ on $(0, pi)$). Let $V$ be the reflection of $U$ across the $y$-axis, and let $Z$ be the same as $V$ except scaled by $frac{1}{2}$ vertically. So in other words $Z$ is just a vertically-squished topologist’s sine curve coming from the other direction. Then clearly $U, V, Y, Z$ satisfy all of the above conditions (condition 3 is satisfied since each is just an arc, albeit of different lengths) but $A notsimeq B$.

Obviously, this problem could be generalized in all sorts of ways, though Lakes of Wada phenomena would be a complication if you start increasing the number of components in $A$ and $B$.

gn.general topology – Intersection of zero sets of continuous functions

Let the zero sets $F={x in mathbb{R}^n: f(x) = 0}$, $G = {x in mathbb{R}^n : g(x) = 0}$, where $f$ and $g$ are $m$-dimensional nonlinear continuous vector functions. Under some assumptions, these sets define hypersurfaces of zero measure in $mathbb{R}^n$. I was wondering:

  1. Are $F$ and $G$ always submanifolds embedded in $mathbb{R}^n$ or are there exceptions – in the latter case, are there conditions that guarantee that they are submanifolds ?
  2. What is the dimension of $Fcap G$ ? As pointed out here Measure of the intersection of two manifolds, if $F$ and $G$ are $(n-1)$-dimensional manifolds and their intersection is transversal, then $text{dim}(Fcap G) = n-2$. However, is there anything that can be said if the intersection is not transversal ? In general, I am interested in some sort of inequality $text{dim}(Fcap G) leq n-2$, assuming that $F subseteq G$, that $G subseteq F$ do not hold.

dicionário – Python : Build a Graph with maps of sets

Hello this is my first post and I’m not fluent in English, so forgive me if I do any wrong post protocol ….

so, I have this current implementation of a graph, through an adjacency matrix:

class GFG: 
def __init__(self,graph): 
    self.graph = graph

from which I start like this:

my_shape = (N,M)
bpGraph = np.zeros(my_shape)

and add edges like this:

vertex_a = 2
vertex_b = 5
bpGraph (vertex_a, vertex_b) = 1
g = GFG(bpGraph)

and check for an edge like this:

if bpGraph (vertice_a, vertice_b):
do something...

But this type of structure can take a lot of space if I have a lot of vertices, so I would like to switch to a set map

class GFG: 
def __init__(self,N,M):
    mysetdict = {
        "dummy_a": {"dummy_b","dummy_c"},
    }
    self.graph = mysetdict 

def add_edges(self,N,M):
    N_set = {}
    if str(N) in self.graph:
        N_set = self.graph(str(N))
    N_set.add(str(M))

    self.graph(str(N)) = N_set

def check_edges(self,N,M):
     return str(M) in self.graph(str(N))

When I try to add a vertex I get this error:

g = GFG (3,4)
g.add_edges (0, 0)
AttributeError: 'dict' object has no attribute 'add'
---> 16                 N_set.add(str(M))

im trying many ways to build this set and check if a entry exists before add a vertex_a to do a edge with vertex_b, but all my tryes results in a atribute error.

any suggestions?

Lifting properties of morphisms in category of sets

Let $f:Ato B,g:Cto D$ be morphisms in a category $mathcal C.$ We denote $fperp g$ if in every commutative square
$$require{AMScd}
begin{CD}
A @>>> C\
@VfVV @VVgV \
B @>>> D
end{CD}$$

there is a unique $d:Bto C$ making both triangles commute. Moreover, for a class $Xsubseteqoperatorname{Mor}(mathcal C)$ denote $X^perp={gmidforall fin X:fperp g},{}^perp X={fmidforall gin X:fperp g}$. Now,I want to show that in $mathbf{Set}$ we have $Epi^perp=Mono$. I think this should be somehow easy but I got lost in universal quantifiers. Inclusion “$subseteq$” should follow from
$$require{AMScd}
begin{CD}
A @>u>> C\
@V{text{id}}VV @VVgV \
A @>>> D
end{CD}$$

where the unique diagonal is $u$, but what about the other? I don’t even know what exactly I should prove. Given a mono $g:Cto D$ such that every commutative square
$$require{AMScd}
begin{CD}
bullet @>>> C\
@VVV @VVgV \
circ @>>> D
end{CD}$$

has a unique diagonal $circto C,$ then (every) $bullettocirc$ is an epi?

data sets – Hacking a BLE Device

Ok, I’m a techie but not a codie. I have a very simple, and small device manufactured by Company A. The device is an accelerometer that reads the angle and speed of whatever it is attached to. It then sends that data to an iPhone App. What I want to do, is use data the accelerometer collects and send it to my PC (not the iphone) at which point the data will be used in a separate, unrelated software.

Here’s the problem: I can’t get the Bluetooth device to connect to my windows 10 PC. It shows up as a device but will not pair successfully.

How can I get this device to pair to my windows PC? Is it not pairing because it’s programmed to interact with iOS and therefore not compatible with PC? I can’t imagine that being the case since it’s just Bluetooth transmission.

Thanks,
Jake

induction – Is Inductive Logic Programming approach applicable to general theories (not just sets of Horn clauses)?

Inductive Logic Programming (https://en.wikipedia.org/wiki/Inductive_logic_programming) find hypothesis theory H for background theory B and set of examples E. ILP algorithms and implementations usually expects, that H, B, E are logic programs – set of Horn clauses and not general FOL or HOL theories. This approach is generalized to the HOL logic programs as well, e.g. in http://andrewcropper.com/pubs/ijcai16-metafunc.pdf.

My question is – are there efforts to formulate induction/ILP for general theories. Apparently, the algorithm does exist and the problem is undecidable, but still – are there some heuristics, some approximate approaches, some more or less rigorous work for such generalization? Both – for full FOL and HOL?

E.g. references wiki articles mentions the method of inverse entailment – I see that that approach is general enough – it requires the computation of the most conscise (e.g. Occams razor principle – with the minimal Kolomogorov or other complexity) set of consequences in some depth.

Actually – ILP may be the Holy Grail of AI: 1) it can learn general policies from the specific policies and hence – implement generalization and transfer learning, e.g. in reinforcement learning; 2) it can learn the program which from background knowledge computes the set of input-output patterns (in more or less general form) and hence – solve the program synthesis task.