general topology – How do I show that the definition of boundary in a metric space is equivalent to the definition of boundary in a topological space?

Show that the the definition of the boundary of a subset E
of metric space X, $partial S : = {p in X: forall r>0, E cap
> B_r(p) neq emptyset text{ and } E^C cap B_r(p) neq emptyset }$

is equivalent to the definition of the boundary in case of a set E in
the topological space $(X, mathcal{T})$, $partial S := overline{E} setminus E^0$

Attempted solution:

Let $C := {p in X: forall r>0, E cap B_r(p) neq emptyset text{ and } E^C cap B_r(p) neq emptyset }$ and $D := overline{E}setminus E^0$. In order to show $C = D$, we need to first show that for any $x in C text{ implies } x in D$ and then show that for any $x in D text{ implies } x in C$.

Let $x in C$. Then, $E cap B_r(x) neq emptyset$ and $E^C cap B_r(x) neq emptyset$. Consider an arbitrary set $F_alpha$ such that $F_alpha$ is a closed subset in $X$ and $E subseteq F_alpha$. Now , I need to show that $x in F_alpha$ , which implies that $x in bigcap_{alpha} F_alpha = overline{E}$, but I have no idea how to do this.

Next, I need to show that $x notin E^0$. I am assuming we do a proof by contradiction but I am not really sure how to proceed.

Also, I have no idea on how to proceed in the reverse directions. Can someone point me in the right direction? Thanks!

functional analysis – A cone in the “boundary” of a convex cone, without topology

In a real vector space, I have two nonempty disjoint cones $A,B$, such that:

  1. $A$ is a convex cone.
  2. $A cup B$ is a convex cone.
  3. for all $a in A$ and $b in B$, we have $a+b in A$.

So you see from (3) that $B$ is in some sense contained in the “boundary” of $A$. But I don’t want to assume a topology on the vector space. My question is, has this phenomenon been discussed in any references? Is there a name for it? Are there any relevant tools or implications on $A$ and $B$?

8 – How to recenter a geolocation google common map in views after an ajax boundary filter execution?

On Drupal 8.97 with all updated modules, I have a View that uses a Geolocation Google Common Map as an attachment. The map acts a boundary filter, but after the filter executes Drupal.ajax(ajaxSettings).execute() via /modules/contrib/geolocation/modules/geolocation_google_maps/js/geolocation-common-map-google.js, I am unable to re-center the map to ANY coordinates, including the user’s location.

Before the ajax call, map.getCenter() works, and so does manually inputting coordinates, like map.getCenter({ lat: -34.397, lng: 150.644 }), but afterwards, no dice. The custom re-center buttons I created still register clicks, and if console.log the map object afterwards, I can’t see any changes to the map object…and indeed some of my other custom map controls (like zoom in and out) still work after the ajax execution. I’m baffled.

I have looked at the ajaxSettings object too, and have attempted to change some of those center related properties before executing the Drupal.ajax(ajaxSettings).execute() command as above, but I still can’t get the map center to change (though I don’t quite know if I’m looking at the right properties…there’s hundreds of them). The map can still be panned around with a mouse of course, but I ultimately would like for users to be able to re-center to their location…and I would like to programmatically recenter and execute a search using the center of city locations without having to reload the page, so it’s kind of critical that I get this sorted.

Thank you in advance for your help.

InitializeBoundaryConditions::fembdnl: The dependent variable in `1` in the boundary condition `2` needs to be linear

Checking the documentation on the error were of not that much help to me. Can someone please suggest how to deal with it?

rhop = 10922; rhon = 10922; Cp = 200; Cn = 200; Lan = 1.5*10^(-3); Lap = 1.5*10^(-3); Lbn = 1.5*10^(-3);
Lbp = 1.5*10^(-3); kp = 1.8; kn = 2.2; sigmap = 1/(1.2*10^(-5)); sigman = 1/(10^(-5)); taup = 0.00027; 
taun = -0.000156; L = 2.325*10^(-3); ha = 10;

Tinf = 298; Th = 300; Tc = 298; A = 2.325*10^(-6); Ic = 0.5;

PDE1 = rhop*Cp*D(Tp(x, y, z, t), t) == kp*(D(D(Tp(x, y, z, t), x), x) + D(D(Tp(x, y, z, t), y), y) + 
       D(D(Tp(x, y, z, t), z), z)) + 1/sigmap*(Ic/A)^2 - 
    taup*Ic/A*D(Tp(x, y, z, t), x);

PDE2 = rhon*Cn*D(Tn(x, y, z, t), t) == kn*(D(D(Tn(x, y, z, t), x), x) + D(D(Tn(x, y, z, t), y), y) + 
       D(D(Tn(x, y, z, t), z), z)) + 1/sigman*(Ic/A)^2 + 
    taun*Ic/A*D(Tn(x, y, z, t), x);

Bc1 = kp*Derivative(0, 1, 0, 0)(Tp)(x, 0, z, t) == ha*(Tp(x, 0, z, t) - Tinf);

Bc2 = kn*Derivative(0, 1, 0, 0)(Tn)(x, 0, z, t) == ha*(Tn(x, 0, z, t) - Tinf);

Bc3 = -kp*Derivative(0, 1, 0, 0)(Tp)(x, Lap, z, t) == ha*(Tp(x, Lap, z, t) - Tinf);

Bc4 = -kn*Derivative(0, 1, 0, 0)(Tn)(x, Lan, z, t) == ha*(Tn(x, Lan, z, t) - Tinf);

Bc5 = kp*Derivative(0, 0, 1, 0)(Tp)(x, y, 0, t) == ha*(Tp(x, y, 0, t) - Tinf);

Bc6 = kn*Derivative(0, 0, 1, 0)(Tn)(x, y, 0, t) == ha*(Tn(x, y, 0, t) - Tinf);

Bc7 = -kp*Derivative(0, 0, 1, 0)(Tp)(x, y, Lbp, t) == ha*(Tp(x, y, Lbp, t) - Tinf);

Bc8 = -kn*Derivative(0, 0, 1, 0)(Tn)(x, y, Lbn, t) == ha*(Tn(x, y, Lbn, t) - Tinf);

Bc9 = DirichletCondition(Tp(x, y, z, t) == Tc, x == 0);

Bc10 = DirichletCondition(Tn(x, y, z, t) == Tc, x == 0);

Bc11 = DirichletCondition(Tp(x, y, z, t) == Th, x == L);

Bc12 = DirichletCondition(Tn(x, y, z, t) == Th, x == L);

sol = NDSolve({PDE1, PDE2, Tp(x, y, z, 0) == 0, Tn(x, y, z, 0) == 0, 
   Bc1, Bc2, Bc3, Bc4, Bc5, Bc6, Bc7, Bc8, Bc9, Bc10, Bc11, 
   Bc12}, {Tp, Tn}, {t, 0, 10}, {x, 0, L}, {y, 0, Lap}, {z, 0, Lbp})

integration – What is the derivative w.r.t. the boundary in the following integral?

I want to solve the following derivation problem:

$ frac{d}{dc}int_{0}^{c-z}(c-x)dG(x) $

I know that in general, $frac{d}{dc}int_{0}^{g(c)}f(x)dx=f(c)times g'(c)$. Here, however, both the boundary and the integrand is a function of $c$. Furthermore, what is the implication of having $dG(x)$ instead of just $dx$?

In advance – thank you for any help!

Best,
Fredrik

machine learning – What does the decision boundary of XOR problem look like?

My textbook walks through an example of solving the XOR problem in machine learning using a two-dimensional RBF network. It does this by setting the centers for the two basis functions at (0,0) and (1,1). Afterwards, it challenges us and asks to imagine what the decision boundary for an RBF would look like if the two centers were instead at a sample of each class (class 0 could be (0,0) and class 1 could be (1 0)). Since there is only two classes, wouldn’t this result in essentially a straight line, or would it look like something else? Is there any reason why we would want to do this?

PDE with boundary condition (differential equation)

I am trying to find a solution to this boundary value problem

h_t= 1/2 sigma^2 x^2 h_xx + rx h_x s.t.
Ah(b,t)+Bh_x(b,t)=g(t),
where A,B,sigma,r,b are constants and g(t) is a given function of time (h_x is the partial derivative w.r.t. the first component of h(x,t)).

I need an explicit solution for it, whatever it is (any solutions satisfying the PDE and the boundary condition will be fine). I tried to use these commands, but it did not give anything:

eqn = 1/2 x^2 (Sigma)^2 D(h(x, t), {x, 2}) + r x D(h(x, t), x) – D(h(x, t), {t}) == 0;
ibc = {A h(b, t) + B Derivative(1, 0)(h)(b, t) == g(t)};
sol = DSolveValue({eqn, ibc}, h(t, x), {t, x}) // FullSimplify

I think I need to use something else, but I am just a beginner in Mathematica.
Thank you in advance!

convex polytopes – Distance to the “boundary” of a polyhedral complex

Suppose I have a polyhedral complex ${P_1, ldots, P_k}$ and let $S := cup_{i = 1}^k P_i$. I am interested in a function which measures the distance from a point $x in S$ to the “boundary” of my polyhedral complex, in other words the lower dimensional faces. More precisely, if $x in text{int}(P_i)$ for some $i$ then my function should return the distance from $x$ to the boundary of $P_i$. Otherwise, $x$ lies on a lower dimensional face and the function should return $0$.

Is there a name for this function? Has it appeared in the literature and been studied before? I’m also wondering if it has any special properties beyond simply being a distance function to a non-convex set. Thanks in advance!

What is sequential boundary of a $delta$-hyperbolic space and how is the Gromov product extended to the boundary?

I have been reading up on $delta$-hyperbolic spaces. But I am not getting a clear idea of sequential boundary of $delta$-hyperbolic spaces and how the Gromov product is extended to it. Could somebody please explain it to me?

complex analysis – Prove that this function is constant if it satisfies this condition on boundary

The following question is part of an assignment which I am trying to solve.

Question: Let $f in C(bar U) cap H(U) $and f is real valued on T=dU then f must be constant.

Condition of Maximum modulus principle are satisfied and it implies that max |f| is real. On the contrary, if I assume that f is not constant then the condition derived using maximum principle seems not to give any contradiction.