Find the key points of the outer layer of n overlapping rectangles – Divide and Conquer

Given a set of $ n$ rectangles depicted as: $ [Li, Ri, Hi]$ whereby the corners of the $ i-th$ rectangle are – $ (Li, 0), (Ri,0), (Li, Hi)$ and $ (Ri, Hi)$ .

The goal is to print all of the key points of the outer layer of the $ n$ overlapping rectangles (given in a list).

Key points $ (Xj, Yj)$ are the points that collectively portray the outer layer of the rectangles – look at the example below:

Given the two blocks [4, 13, 4] and [2, 7, 10] the output should be: [2,0], [2,10], [7,10], [7,4], [13,4] and [13,0] (The points that are marked red).

This should be done in $ O(nlogn)$ time where $ n$ is the number of buildings (rectangles).

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I have tried to solve this problem by –

(1) Sorting the blocks by priority: $ Li$ then $ Ri$ then $ Hi$ .

(2) Compare any two blocks: $ [Li, Ri, Hi]$ and $ [Li+1, Ri+1, Hi+1]$ (15 different cases like Li==Li+1 and Hi< Hi+1 and stuff like this).

(3) Remove/add points according to condition (using AVL)

This works in some cases, but in others it fails miserably. I think this can be solved by Convex Hull algorithms, we have only learned the divide and conquer method so far – but I cannot link between this question and this method. I hope you can help me with this one!

Thank you

What signature food or beverage is each outer plane in the Great Wheel Cosmology known for?

Is there a list or reference somewhere showing what meals and / or beverages each outer plane in the Great Wheel Cosmology is best known for?

I am putting together a campaign idea for my PCs, a variant on Smokey and the Bandit. In that movie, there’s a bet that Bandit and Snowman can drive to Texarkana and bring back to Georgia 400 cases of Coors beer within 28 hours.

My idea is that to win a bet the PCs would travel to the four corner Outer planes in the Great Wheel Cosmology, and pick up a signature meal or beverage at each of those four planes, in a time limit, maybe 8 hours. I’m willing to Make Up Stuff, like “Angel Food Cake” from Celestia and “Extremely Hot Salsa made with Peppers from the Abyss”. However, if precedent exists, I’d like to follow it.

If you ask someone on the street which country is better known for Poutine and which for Pasta, chances are they’ll associate Pasta with Italy and Poutine with Canada. That level of association is what I’m going for.

I care more about the Great Wheel Cosmology than I do about game systems. I run a Pathfinder 1e game (which I know does not support the Great Wheel) and will bend whatever I find into Pathfinder 1e. That’s why I have not added a dnd-5e nor Pathfinder 1e tag here.

Does Blink work in the outer planes?

Does Blink work in the outer planes? The spell it states:

you vanish from your current plane of existence and appear in the Ethereal Plane

as well as:

While on the Ethereal Plane, you can see and hear the plane you originated from, which is cast in shades of gray, and you can’t see anything there more than 60 feet away

This seems to be written in particular contrast to the Etherealness spell (see also Can you become Ethereal in the Outer Planes?), which says:

This spell has no effect if you cast it while you are on the Ethereal Plane or a plane that doesn’t border it, such as one of the Outer Planes.

This seems to imply that it will work fine on the Outer Planes, but does it?

Amenability of the group of outer automorphisms of a connected compact Lie group

Forgive my ignorance on the Lie theory. I have the following questions in my current work concerning a certain property of compact connected Lie groups.

First, allow me to fix some notations. Let $ G$ be a connected compact Lie group, $ \mathfrak{g}$ its Lie algebra. Then $ \operatorname{Aut}(G)$ , the group of (smooth) automorphisms of the $ G$ , is a closed subgroup of $ \operatorname{Aut}(\mathfrak{g})$ , hence is a Lie group itself. Let $ \operatorname{ad} : G \to \operatorname{Aut}(G) \subseteq \operatorname{Aut}(\mathfrak{g})$ be the adjoint representation and denote its image by $ \operatorname{Inn}(G)$ , and needless to say, $ \operatorname{Inn}(G)$ is a normal subgroup of $ \operatorname{Aut}(G)$ . We denote the quotient group $ \operatorname{Aut}(G) / \operatorname{Inn}(G)$ by $ \operatorname{Out}(G)$ , and this is what I mean when I say the group of outer automorphisms in the title.

Question A. When $ \operatorname{Out}(G)$ is countable, is $ \operatorname{Inn}(G)$ always open in $ \operatorname{Aut}(G)$ ?

More generally, although this seems to have a higher chance to have a negative answer, one can ask the following question, which I suspect is still true:

Question B. Is $ \operatorname{Inn}(G)$ always open in $ \operatorname{Aut}(G)$ ? If Question A has an affirmative answer, this is equivalent to ask if $ \operatorname{Out}(G)$ is always countable.

Question C. When $ \operatorname{Inn}(G)$ is open in $ \operatorname{Aut}(G)$ , is $ \operatorname{Out}(G)$ (which now a countable discrete group) always amenable as a locally compact group?

A little experiment on the abelian case of the $ n$ -dimensional torus (in which case the inner automorphism group is trivial, and $ \operatorname{Aut}(T^n)$ is $ GL_n(\mathbb{Z})$ ), and the semisimple case (at least over $ \mathbb{C}$ instead of $ \mathbb{R}$ , which implies $ \operatorname*{Out}(G)$ being finite), seems to make a negative answer to the above questions not completely trivial. Although I am happy that the abelian case and the semisimple case already suffice for the purpose of my current work, which only needs $ \operatorname{Out}(G)$ to be amenable, I would be very interested if one can prove or disprove the amenability of $ \operatorname*{Out}(G)$ for a general compact connected Lie group $ G$ .

What is an example of a quasicategory with an outer 4-horn which has no filler?

A quasicategory has fillers for all inner horns $ \Lambda^i[n]$ where $ n\geq 2$ and $ 0<i<n$ , but it need not have fillers for $ i=0$ (or $ i=n)$ . In particular, for $ n=2$ and $ n=3$ there are easy counterexamples.

For n=2, let $ X=\Delta[2]$ (which is the nerve of a category): there is a map $ \Lambda^0[2]\rightarrow\Delta[2]$ that maps vertices $ 0\mapsto 1$ , $ 1\mapsto 0$ , and $ 2\mapsto 2$ , which can’t have a filler because there is no arrow from $ 1$ to $ 0$ in $ \Delta[2]$ .

For $ n=3$ , any small category $ X$ which has a morphism $ f$ which is not an epimorphism will give us an unfillable $ \Lambda^0[3]$ . For example, let $ X=N(\Delta)$ , and consider the maps $ f=d^0\colon [0] \rightarrow [1]$ and $ g=d^0\circ s^0\colon [1]\rightarrow [1]$ . Then we can construct the horn $ \Lambda^0[3]$ where the $ d_1$ -face witnesses the composition $ g\circ f = d^0 \circ s^0 \circ d^0 = d^0 = f$ , and the $ d_2$ and $ d_3$ faces witness $ \text{id}_{[1]}\circ f = f$ . If there were a filler for this horn, then we’d have $ \text{id}_{[1]}=g\circ\text{id}_{[1]}=g$ , but $ g=d^0\circ s^0$ is not the identity.

For $ n\geq 4$ , a counterexample can’t come from the nerve of a category, because nerves of categories are $ 2$ -coskeletal. I’ve tried to look at other examples of quasicategories—for example, those given in section 8 of Rezk’s notes on quasicategories—but I believe I can show the existence of outer horns for $ n\geq 4$ in every example given there.

So, are there any known examples of quasicategories with an unfillable $ \Lambda^0[4]$ ?

Tensors: representation, outer product, decomposition

it is given a tensor:

$ T=\begin{pmatrix} 1\ 1 \end{pmatrix}\circ \begin{pmatrix} 1\ 1 \end{pmatrix}\circ\begin{pmatrix} 1\ 1 \end{pmatrix}+\begin{pmatrix} -1\ 1 \end{pmatrix}\circ\begin{pmatrix} 1\ -1 \end{pmatrix}\circ\begin{pmatrix} -1\ 1 \end{pmatrix}$

1) Why is it possible to write the tensot T as: $ T=\begin{pmatrix} 2 &0 \ 0& 2 \end{pmatrix} and \begin{pmatrix} 0 &2 \ 2& 0 \end{pmatrix}$

it is given in example that I can represent the tensor T as a sum of the outer product of vector triples and as 2 matrices. I have computed the outer product of the vector triples but I can’t get the same result. Can someone provide me detailed calculation?

2) T=[[ABC]]

$ A=\begin{pmatrix} 1 &-1 \ 1& 1 \end{pmatrix}$

$ B=\begin{pmatrix} 1 &1 \ 1& -1 \end{pmatrix}$

$ C=\begin{pmatrix} 1 &-1 \ 1& 1 \end{pmatrix}$

How to compute A, B, C?

Later on the p 35 (53), example 2. or on p 36(54) 2.2.1 the vectors a,b,c are given without an explanation of how he/she competed them. In §2.2.1 it is given that “we set….” and it is all. No explanation of how they find them.

I have found examples in Analysis of 2 × 2 × 2 Tensors, p 30 (48 in pdf file) example 1,6. In this example is given a calculation of a rank of T and these decompostions without explanation.

Can someone help me to understand the examplle?

Passing data from outer preprocess to inner template

I have two templates: – a child that is printing each item in a list; – and a parent that calls it there is also an intermediate Field template that is not involved.

The parent has preprocess access to node field data that each child (node teaser) call needs to use. The child is printing items in a multivalue entity reference field.

How can the child get the data from the parent felds so it can print the right html?

Hausdorff outer measure is finite if $\sum_{j = 1}^m |f(x_i)-f(x_{i-1})|^s \le c$

Let $ f:[0,1] \to \mathbb{R}, G = graph(f)$ .

If $ \sum_{j = 1}^m |f(x_i)-f(x_{i-1})|^s \le c$ for all partitions $ 0 = x_0< \ldots < x_m = 1 $ then $ H^s(G) < \infty$

What technique can I use to prove this result?

Can it be reduced to the thorem stating that a rectifiable curve $ \Gamma$ has $ H^1(\Gamma) < \infty$ ?

In what spaces are outer product of x with itself positive semidefinite?

In the case of vectors in $ \mathbb R^n$ , it is quite simple to see that for any vector $ x$ , $ $ v^Txx^Tv = (v^Tx)^2 \geq 0$ $ so clearly the form $ xx^T$ must form a positive semidefinite matrix.

But what happens in other spaces? I guess $ x^T$ generalize to some dual vector. but what about $ xx^T$ ? That must be some sort of outer product? I have experience from Hilbert spaces in physics mostly, and there one forms such outer product often using dual vectors. But it is not clear for me that it should work ‘as usual’ in rigged Hilbert spaces…

In what spaces would this outer product make sense?