Hello everyone, I have a simple question that I've worked through in a GRE book, and I think the book is wrong with regards to their correct answer. mind if I share it and my reasoning?
actually, maybe i'll just post it and if anyone wants to take a look otherwise you can just ignore it
the question says:
x, y, and z are the lengths of the sides of a triangle.
then you compare two things and decide if one is greater than the other, if they are equal or if the answer cannot be determined
the comparisions are:
A: x + y+ z
B: 2z
the triangle inequality tells us:
x +y <= z
sorry greater than or equal to z
so that means that A can be one of two things, greater than 2z or exactly equal
so the correct answer should be that the answer cannot be determined
@atomsmasher In the abstract, you are right, but the question makes a tacit assumption that the triangle is nondegenerate. In this case, x+y > z strictly.
Is there a name for a state in a Markov chain that has no states leading to it (with nonzero probability)? I've heard such a thing being called a "Garden of Eden" in the context of cellular automata.
Something like "predecessorless"?
Nevermind, I guess it's called "inaccessible state".
@TedShifrin, In your Kepler's Laws lecture, you prove that "In the presence of a central force field, the orbit of an object lies in a plane." You are using in that proof, Newton's second law: in that lecture's notation: $\vec F(\vec g(t))=m\vec g''(t)$. I wondered why this holds 'mathematically'.
So, I googled it. And I got to know that Newton's laws can't be proved. So, while proving that theorem, do we assume $\vec F(\vec g(t))=m\vec g''(t)$, too? Or, does $\vec F(\vec g(t))=m\vec g''(t)$ hold in any central vector field?@TedShifrin
@Silent eh, you can't derive physics purely from math
Like, at some point in order to be doing physics you have to be making statements about behavior of objects. That's an empirical fact, and would need to be known via experimentation, not pure thought
@Daminark so, in that lecture video, particularly, at in this theorem, I should think that newton's second law is given as a hypothesis, right? its not a fact about any central force field?
hmm... I wonder if stone weierstrass theorem will also work for matrix functions $f(X)$
won't the matrix multiplication (which is a linear map of stuff on the left hand side taking each column vector on the right hand side as arguments) of say e.g. $X^2$ in some matrix polynomial $P(X)$ makes it harder to converge to some matrix functions?
While at first glance $M_n(\Bbb{R})$ has the same dimension as $\Bbb{R}^{n^2}$, the former has a multiplicaton structure that is absent in the latter, potentially restricting the range of $P(X)$ for all matrix polynomials $P$
I have to teach some graph theory next week (directed acyclic graph lib...), any good sources? I'm trying to iron out the terminology too...
"node": (data)point on graph (also vertex or junction); "edge": line between two nodes; "source": ...?; "sink": ...? - say we have a simple digraph with two nodes and one directional edge (e.g. a -> b), what are the nouns that refer to the elements of the relationship?
@AaronHall edges are also called arcs, typically an undirected graph is defined as an ordered pair G = (V,E), where V is the set of nodes and E is a set of unordered pairs in V. For a directed graph, E is a set of ordered pairs instead. The in-degree of a node is the number of edges pointing to it, and similarly out-degree, then you can define a source node as a node with in-degree 0.
@Silent: Newton's three laws are taken as axioms (physical axioms, not mathematical ones). The equation $F(g(t)) = mg''(t)$ is just a rewriting of Newton's second law. A central force is one in which $F(x)$ is a scalar multiple of $x$ (where that scalar can be a function of $x$).
That's Newton's second law. We're not talking about the force itself. We're talking about how the particle accelerates in the presence of the force. That equation is just $F=ma$.
@Danu: This question. He has it slightly mis-stated, but, nevertheless ... In the compact case, it's just finite-dimensionality and injectivity both ways. But how did we get it for the non-compact case?
I was just reading the book Linked, by Albert Barabasi. Great read if you're interested in a broad treatment of networks and how they interplay in different fields
@TedShifrin No I didn't, I guess there are people at his uni he should ask instead? Ironically they also have a great number theorist, Fabrizio Andreatta, in Milan
@Alessandro: It's sad how many undergraduates don't think about these things until it's too late. Maybe you should put a comment there for him, though?
@AlessandroCodenotti okay, say i want to embed a shape like a circle into 3D. Would one way to do that be to rotate a geodesic of a circle about the center point of the circle?
I want to embed a square, with some additional interior topological structure into $ R^3 $ and gain some intuition for how the interior topological structure inside the square should be embedded in $ R^3 $ , and how it changes with a continuous homeomorphism from cube to sphere to astroid etc. so first i would extend the square into a cube, which is homeomorphic to a sphere in $ R^3 $ right?
topology becomes a lot easier when you realize that a topological space is just a lax algebra of the monad β of ultrafilters on the (1,2)-category Rel of sets and binary relations.
Jokes apart as long as you know the definition of a topology, some separation axioms, what do first and second countable mean and what connected, compact and dense set mean you're going to be fine
@MatheinBoulomenos Balarka just managed to convince me that topological spaces (up to weak homotopy equivalence) are really $\infty$-groupoids and now you algebraists come up with this new devilry?
I've found variations of compactness to be more useful than the thousands of different separation axioms. From my experience, you either have Hausdorff or you don't really care (as in AlgGeo)
tychonoff's theorem is important I agree
tychonoff's theorem implies that profinite groups are compact
