@TedShifrin irreducible varieties can't be written is a union of 2 proper closed subsets, it's that idea but expressed in terms of open sets and generalized to spaces
Let $A \subseteq \Sigma^*$ be a set of strings. Let $B_k$ be the set of strings with $k$ prefixes in $A$, i.e. $B_k = \{s \in \Sigma^* : |\text{Pref}(s) \cap A| = k\}$. Can I obtain a generating function for $B_k$ from a generating function for $A$?
Analytic Combinatorics addresses the question of obtaining generating functions for substrings on page 211, section III.7, example III.26, but it's only for fixed strings.
what's the difference between pullback(precomposition) and composition? On wikipedia it says, "Precomposition with a function probably provides the most elementary notion of pullback: in simple terms, a function f of a variable y, where y itself is a function of another variable x, may be written as a function of x. This is the pullback of f by the function y. $f(y(x) \equiv g(x)$" Isn't this the same as the compostion of f and y?
I'm trying to show that $\gamma '(t)$ never vanishes, and my approach was to take smooth charts on $\mathbb{T}^2$ and then compute $\gamma'(t)$ in terms of local coordinates, but the author identified the torus as a subset of $\mathbb{C}^2$ so that complicates things a bit, am I using the correct approach? I can't just take usual derivative of $\gamma$ and show that it's non-zero for all $t$ since $\gamma'(t)$ doesn't really mean that here.
@TedShifrin Really it's that simple? The author defined $\gamma'(t_0)$ to be $d\gamma_{t_0}\left(\frac{d}{dt}\bigg|_{t_0}\right)$ though and since it wasn't said anywhere that this was equivalent to evaluating the derivative of $\gamma$ at $t_0$ I assumed I had to do things the hard way
No, you need to understand how to simplify things, not make them more complicated. If $\gamma\colon M\to N\subset\Bbb R^n$, then computing the derivative as a map to $\Bbb R^n$ suffices, because you know it'll land in $T_qN$.
Another one: "we only know so much". There is frequently only a short list of things one could possibly do, so it's not unreasonable to enumerate the list and decide which one.
@MikeMiller The definition of an immersion I was using was that given a smooth map $F$ between manifold $M$ and $N$, $F$ is an immersion if its differential is injective at each point. There's also a theorem that says if $\gamma : \mathbb{R} \to M$ is a smooth curve on a manifold $M$ then $\gamma$ is a smooth immersion iff $\gamma'(t) \neq 0$ for all $t$, this theorem is the main result I'm trying to use
@TedShifrin Ohh I think I get what you're saying now. so $\gamma'(t_0)$ non-zero in the sense of normal calculus means that $d\gamma_{t_0}(1)$ is non-zero in $T_{\gamma(t_0)}(\mathbb{T}^2)$
@TedShifrin Should I worry about giving a formal proof of that (non-zero derivative in terms of usual calculus passes to non-zero differential) at the moment or just accept it and move on ?
Okay so because we're working on subset of $\mathbb{R}^n$ and $\mathbb{C}^n$ essentially we can just work with usual multivariable calculus instead of relying on the abstract manifold stuff (since there's a correspondence in any case)
Wel well wel, it seems that a counter example has been found to Donkin's tilting conjecture. Good thing I am leaving academia or this would have required some serious overhaul to my research statement.
Ahh, then it does seem something it wrong (but calculatinga few terms would not have told you that it was non-zero as there will potentially be a lot of cancellation).
Anyway, the fastest way is probably to differentiate a suitable number of times
Sagan's book is fairly thorough for the classical approach using Young symmetrizers to define Specht modules
There is also another approach using induced modules, but I am not aware of a good reference for that except for more general references which really deal with Hecke algebras and Soergel bimodules in general
Then maybe reading something like the exposition in Geck-Pfeiffer on characters of Coxeter groups is a good approach
Actually, there is also a third approach using that the group algebra is positively based in the KL basis and that all the left cell modules are simple.
It states that certain indecomposable modules for reductive algebraic groups in positive characteristic remain indecomposable when restricted to a certain subgroup
(probably the vaguest statement of it I have ever written)
@Astyx Not sure what you mean. The dual is always the same as a vector space. We only change the action
Anyway, you cannot in general expect to have nice enough properties to get everything via subgroups and some extra process (at least not in any uniform way). At least outside Coxeter groups
@Astyx No, I am saying that the fact that dualizing the rep corresponds to dualizing the partition is something very special
@Astyx Anyway, $l^2(G/H)$ will almost never be the correct thing, as it will rarely be irreducible. You need some way to single out an irreducible constituent in it.
There are some different ways to do this, depending on the group, but there will rarely be any really good one (outside Coxeter groups)
That space is really just the induced rep, and we induce from a Young subgroup. The Specht module is the unique one that does not occur in any such induced modules for larger partitions (in the dominance order)
and it even occurs with multiplicity $1$ the first time it appears
I have a collection of processes that will each end at some random real-valued point in time, and at a particular time I am going to examine all the data I have collected for these processes.
There is a probability density function that governs how long these processes each last, p(X|\mu), and I want to estimate \mu. The values that have ended correspond to points in the density function, but those that have not ended correspond to points in the CDF, ergo actual probabilities. I'm not sure how to reconcile these into a single likelihood function. Is anyone knowledgable?
Sorry if I'm kinda dumping that in the middle there, I feel like it's a stupid question with an easy answer, but I'm a very stupid person, and therefore outmatched by it
It is just the natural way to associate a subgroup to a partition (to $(n_1,\dots, n_m)$ associate $S_{n_1}\times\cdots\times S_{n_m}$)
This phenomenon is really a special case of what happens for Soergel bimodules (or, since this is a Weyl type, for projective functors on Category $\mathcal{O}$).
Do we also have that the Specht representations are all the irreducible representations of any Coxeter group or is that specific to the symmetric group ?
I have to go right now, otherwise I won't be able to eat anything this lunch. Thansk a lot for your time, I'd love to continue this conversation later on ! And I'll look Geck-Pfeiffer up.
I'm reading through an opencv example that performs camera alignment.
More specifically the homography is being estimated in such code, but there are these few lines that aren't quite clear to me:
//! [compute-plane-normal-at-camera-pose-1]
Mat normal = (Mat_<double>(3, 1) << 0, 0, 1);
Mat norma...
"Thus infinity + 1 = infinity only makes sense if we understand that each infinity in the equation is a different element in the set, where their difference is exactly 1."
@ÍgjøgnumMeg just watched a movie about the chinese remainder theorem (av.tib.eu/media/19882). Is there a relation between solving ODEs and solving modulo equations?
I mean in both cases you can search for a general solution by adding the solution "homogeneous" problem to a particular solution of the inhomogeneous problem.
@Rudi_Birnbaum: The phenomenon you're observing holds whenever you have a linear problem and you compare solutions of the homogeneous and inhomogeneous problems. So it applies to linear ODEs and to linear systems of equations in general.
Given a context-free grammar $G$, how can one systematically construct a grammar $G_k$ such that
$$ L(G_k) = \{w \in \Sigma^* : |\text{Pref}(w) \cap L(G)| = k\} $$
where $\text{Pref}(w)$ is the set of $|w|+1$ prefixes of $w$? Assume $G$ is unambiguous, if needed.
Let's say I have some $R$-module $M$ which has generating set $E \subseteq M$ and another $R$-module $N$, can I define a $R$-linear map $f : M \to N$ by defining $f(e_i) = \text{something in $N$}$ for all $e_i \in E$ and then extend by linearity to get a well-defined $R$-linear map?
You need to pay attention to the relations if it's not a basis, right. But at least one can say "basis" in that case, and it's in general hard to determine the relations.
@KarlKronenfeld I just meant things get irritating if you chose too many generators because you just threw generators at a dartboard to see what sticks.
I think that's when you throw a second dart and hit the other one straight in the middle, but the other one was an exploding dart and suddenly they're both gone
Given a context-free grammar $G$, how can one systematically construct a grammar $G_k$ such that
$$ L(G_k) = \{w \in \Sigma^* : |\text{Pref}(w) \cap L(G)| = k\} $$
where $\text{Pref}(w)$ is the set of $|w|+1$ prefixes of $w$? Assume $G$ is unambiguous, if needed.
