Hello. I'm finding good references for abstract polytope. I choose Peter McMullen, Egon Schulte, <Abstract Regular Polytopes> and I'll read it. But, well, one review in Amazon said "there have been huge advances in the field of abstract polytopes." I suddenly want to know what is huge advances.
Prove that a function $f: V \rightarrow \mathbf{R}^{m}$ is linear if and only if each component of $f$ is linear. Determine
the form of a linear map $T: \mathbf{R}^{n} \rightarrow \mathbf{R}^{m} .$
I get the the form of a linear map $T: \mathbf{R}^{n} \rightarrow \mathbf{R}^{m} .$ which is conse...
Round Robin Tournament
I understood the hint involving the Pigeon Hole Principle but I'm confused on how to apply it to the inductive step. Any help is appreciated.
What I have so far:
We will prove by induction on $n$.
Base Case: $n=0$, so there is only 1 team. Base case holds as we can easil...
So, given a set of convex polyhedra such that these polyhedra will tessellate the plane and one of the polyhedra is a regular n-gon, can you determine the smallest cardinality that set can have in terms of n?
(Also, random-15, you'll probably want to ask Ted when he's in, as he is the resident geometer, but I'll predict that it depends very heavily on who you are as a person)
@Rithaniel oh okay. I thought you were asking from PG, and I didn't understand your question, so I thought this means I should study more before asking.
Nah, you're good. What I asked is a potentially difficult question, from what I know. Someone in here might have an easy answer, though, so I figured I would drop it in here, just in case.
Given a continuous function $f\colon\mathbb{R}^m\rightarrow\mathbb{R},\,m\ge1$ and $x,y\in\mathbb{R}^m$, s.t. $f(x)\neq f(y)$ and a $c$ between $f(x)$ and $f(y)$, can one always find a $m-1$-dimensional topological submanifold $M\subseteq\mathbb{R}^m$, s.t. $f(M)=\{c\}$ and $\mathbb{R}^m\setminus M$ consists of exactly two connected components, one of which contains $x$ and the other contains $y$?
@Thorgott Originally misread that as $f^{-1}(c)=M$
@Thorgott I'm a bit worried you could have something be 0 on the topologist's sine curve (the graph of $y=\sin(1/x)$ together with the interval $\{x=0~\&~-1\le y\le1\}$), positive above it, and negative below it
As noted here, the mean minimizes the mean squared error and the median minimizes the mean absolute error. Is there a bounded loss function whose minimizer is also the mean?
Let such a function be $f$. Then $f \circ \mathrm{sqrt}$ is also bounded, but minimizing it would be minimizing a bounded function of the absolute error, no? And by the same logic wouldn't its minimizer be the median rather than the mean?
Both. I guess I should say bounded as a function of $|\hat{x} - x|$.
Since that'll satisfy the properties I expect.
Another example is, in the limit as $p \rightarrow \infty$, $\operatorname{argmin}_{\hat{x}} \mathrm{E}_x [|\hat{x} - x|^p]$ yields the mid-range, and in the limit as $p \rightarrow 0$ it yields the mode.
Suppose we want to predict the probability of some future event. We can set up a prediction market where people trade binary options that pay $1 if the event occurs and $0 if it doesn't. The market price is then a good estimate for the probability of the event.
But what if we're interested in predicting a real-valued random variable X rather than a binary-valued one? I propose setting up a prediction market where people trade binary options of the form "X will be less than x" for arbitary x. It seems this would yield a good estimate for the *full* probability distribution (more precisely, …
The motivation for my earlier question was the case where you're just interested in the mean of the distribution and want to set up a prediction market to trade options with payoffs between $0 and $1, in such a way that the price will be a good estimate for the mean.
I don't know enough about the stock market but I think while there are mechanisms for predicting the expected value of something, I'd be surprised if there were mechanisms for predicting the probability distribution
$$X:=\Bbb H^2 \times \Bbb R$$ is one of the valid Thurston geometries. But can you do something like: $$\zeta:=\Bbb R^4\times\Bbb C^2?$$
What geometries can be combined like this and which ones cannot?
I now want to know what a Thurston geometry is defined as
Oh yeah, I had a question earlier that you might be interested in: What is the smallest set of polyhedrons containing a regular n-gon, such that the set will tessellate the plane? Can it be expressed in terms of n? @AkivaWeinberger
@Ultra: I have no idea what you're talking about, but there is never a maximal embedding space. You can always set $\Bbb R^n$ into $\Bbb R^N$ with $N>n$ by fixing the last coordinates.
@Random-15 You shouldn't just randomly pin people. What is this about?
Let $A$ be a finite Boolean ring.
Suppose we had a size function $\phi: A \to \Bbb{N}$, which means $\phi(x) = 0$ is not possible. The size function is such that $\phi(x +y) \leq \phi(x) + \phi(y)$. I'm looking for other properties required for $\phi$ to efficiently find a minimimum $x \in A$ ...
@Alessandro it's been alright so far, I get the feeling I'm gonna need to work super hard to stay on top of things because of my somewhat weak background/rusted skills in some areas lol, otherwise it's going fine :) I spoke to one of my Klassenkameraden earlier and it turns out he's not finding the exercises that easy either and I was actually able to help him with some stuff
which is nice because I thought I was just weaker than everyone lol
@Alessandro we have to hand them in in pairs, but my ANT practice session isn't until tomorrow so I don't have a partner yet, altho I know one person in the group so I'll ask her tomorrow
since approaching unknown people is of course forbidden
nah I saw some guy working on the ANT problem sheet in some open area today so I approached and asked how he was finding the exercises, but it turns out he's attending a different practice session to me
Consider the problem of finding the minimum of the quadratic form $r(\textbf{x})=x_1^2+x_2^2+x_3^3$ subject to the constraint $q(\textbf{x})=x_1^2+3x_2^2+x_3^3+2x_1x_2-2x_1x_3-2x_2x_3=1$. The diagonal representation of $q$ is $y_2^2+4y_3^2$ and that of $r$ stays the same. Solving the problem, one can simply insert $q$ in $r$ and minimise.
Though, why is this possible? Aren't the diagonal representations based on two different basis, i.e. the basis of eigenvectors of $q$ and $r$ respectively, and so represent different coordinate systems?
A lecturer once gave a very elementary proof that $*$-homomorphisms between C*-algebras are always norm-decreasing. It is well-known that this holds for a $*$-homomorphism between a Banach algebra and a C*-algebra, but all the proofs I find involve the spectral radius and so.
If I remember it w...
If $f(m)$ is the least integer so that any closed $m$-manifold embeds into $\Bbb R^{f(m)}$ then what's known is $f(2^n) = 2^{n+1}$ and $f(m) \leq 2m-1$ for $m \neq 2^n$
But the function itself is certainly not known in any explicit way
In Folland's Real Analysis, he says that
$$
g_1^{-1}((a,\infty]) = \bigcup_{j = 1}^{\infty}f_j^{-1}((a,\infty])
$$
where $g_1(x) = \sup_j f_j(x)$, but he doesn't explain why. Why is this true?
@AlessandroCodenotti In any partially ordered set $A$ we define a subset $C\subset A$ to be convex if for any two points $x,y\in C$ we have $x\leq a\leq y$ implies that $a\in C$.
I just did a little more in this topic. The reason why we have to restrict ourself on a good ordered set(like a set with linear order) is because the definition of supremum of a set of functions from $X\to Y$ where $X,Y$ are partially ordered, does not behave naturally with the initial lattice structure(that is, the complete lattice $P(X)$)...
So it should not happen in general that the supremum and inverse image functor commutes.
Where the supremum refers to either of the set Hom(X,Y) or of P(X), the union.
It's not confusing, @Sha. $(r,\theta)$ are the polar coordinates. The mapping from polar to cartesian coordinates is $f(r,\theta) = (r\cos\theta,r\sin\theta) = (x,y)$.
Hello, quick question: Can a model interpret a formula both true and false or not interpret it in first order logic? Looking at the definition, it looks like the model is inheriting the internal logic of the meta-object/model. Just one or two sentence of guidance would be nice.
If the density of a solid ball depends on the formula $k(2a-R)$ (where $k$ is a constant), is it obvious that the radius of this sphere then is $a$? The problem is related to evaluating a triple integral, and the integral w.r.t. to $R$ has limits from $0$ to $a$? (Shouldn’t $R=2a$?)
It sounds more like you're looking for a book on functional analysis. Lax's is appropriate. Ted's suggestion is a fine book and does cover Banach spaces.
@FuzzyPixelz Okay. It is not given in the problem that $a$ is the radius (in the solution it is), only that the density in the ball depends on the formula given. Looking at the formula, it would make more sense if the radius was 2a, since then anything beyond 2a of radius would have 0 density.
Round Robin Tournament
I understood the hint involving the Pigeon Hole Principle but I'm confused on how to apply it to the inductive step. Any help is appreciated.
What I have so far:
We will prove by induction on $n$.
Base Case: $n=0$, so there is only 1 team. Base case holds as we can easil...
$$X:=\Bbb H^2 \times \Bbb R$$ is one of the valid Thurston geometries. But can you do something like: $$\zeta:=\Bbb R^4\times\Bbb C^2?$$
What geometries can be combined like this and which ones cannot?
Let $A$ be a finite Boolean ring.
Suppose we had a size function $\phi: A \to \Bbb{N}$, which means $\phi(x) = 0$ is not possible. The size function is such that $\phi(x +y) \leq \phi(x) + \phi(y)$. I'm looking for other properties required for $\phi$ to efficiently find a minimimum $x \in A$ ...
A lecturer once gave a very elementary proof that $*$-homomorphisms between C*-algebras are always norm-decreasing. It is well-known that this holds for a $*$-homomorphism between a Banach algebra and a C*-algebra, but all the proofs I find involve the spectral radius and so.
If I remember it w...
In Folland's Real Analysis, he says that
$$
g_1^{-1}((a,\infty]) = \bigcup_{j = 1}^{\infty}f_j^{-1}((a,\infty])
$$
where $g_1(x) = \sup_j f_j(x)$, but he doesn't explain why. Why is this true?
