@BalarkaSen huh, here's something fun. in one of the appendices to his graphic novels, Alan Moore actually describes and uses the Koch snowflake as an analogy
Hi, a have a questions on paths, if anyone can help...If I have two parametrizations of the same curve in the complex plane, is it true that there is always a reparametrization between them? I wanto to prove (or disprove) that the value of the line integral is the same no matter the parametrization of the curve :s
Let $A$ be some non-empty finite set and $S_A$ be its permutation group. Fix some $B\subset A$ and define $H=\{\alpha\in S_A\, | \,\forall x\in B:\alpha(x)\in B\}$.
If $B=\emptyset$, does that mean $H=S_A$? The above does not mention $B$ nonempty, and I assume every permutation on $A$ vacuously satisfies the defining property for $H$.
And likewise if $A$ is any arbitrary non-empty set. I'm not very familiar with handling $\emptyset$ or vacuous truths :/
Suppose M1 and M2 are two TMs such that L(M1) = L(M2). Then
(A) On every input on which M1 doesn’t halt, M2 doesn’t halt too.
(B) On every i/p on which M1 halts, M2 halts too.
(C) On every i/p which M1 accepts, M2 halts.
(D) None of above.
My try :
It explained here that M2 accepts string...
Is there anything similar to this for quadratic reciprocity ?
I mean, the link there explains how you can figure out the solution of cubic equation by yourself without having a suddent flash of inspiration/ genius genes. Is there some similar guide for quadratic reciprocity ?
@BalarkaSen Ya, I perfectly agree with "pseudo-spiritual" and overrating part. Though I think the "half-cultural" part becomes prevalent after Dashami, right ?
I am wondering if there is a way to make this analogy of representation theory to physics more fitting: The simple modules are the atoms, the standard or costandard modules are the [something] molecules, and the tilting modules are the [something bigger] molecules
then restriction to a maximal torus (which is essentially what one does when considering characters) is looking at electrons and so on
this is two-dimensional and build from two copies of the trivial representation, but it is not the direct sum of those. One is a submodule and the other is a quotient
what precisely is meant then tends to vary (I usually specify at the beginning of my papers that I mean only rational ones, and sometimes only finite-dimensional ones)
Right. One wants comodules of the coordinate algebra instead
@AlexKChen here in the States, AP Physics C (but not B?) included some very basic SR and QM, but iirc at the instructor's discretion if time allowed...
@BalarkaSen ah so you're acquainted with the mathematics of it
Jeez, they don't mock about in this grant application. As part of filling out personal information it has "Give an account of your most significant contributions to science"
We have the permutation $\pi: (1, \ldots , k, k+1, \ldots , k+\ell) \mapsto (k+1, \ldots , k+\ell, 1, \ldots , k)$.
Is the following correct? Or should I use in an other way the permutation? $$\psi_1\land \ldots \land \psi_k\land n_1\land \ldots \land n_{\ell}=n_{\pi(k+1)}\land \ldots \land n_{\pi(k+\ell)}\land \psi_{\pi(1)}\land \ldots \land \psi_{\pi(\ell)} \\ =\text{sign}(\pi)\cdot n_{k+1}\land \ldots \land n_{k}\land \psi_{1}\land \ldots \land \psi_{\ell}$$
Hi. Suppose that $E[X_K]=O(K^{-\alpha})$ for some $\alpha>0$, and that $W_K$ is such random variable that $0\leq W_K \leq 1$. Is it possible to say something about the rate of convergence of $E[X_K W_K]$ in terms of $\alpha$ and $W_K$ (suppose that everything about $W_K$ is known)? Clearly $E[X_K W_K]=O(K^{-\alpha})$, but maybe something more can be said
each divisor will be of the form $p_1^{b_1} \cdots p_r^{b_r}$, where $b_i \in \{0, \dots, a_i \}$ @TastyRomeo
There are $(a_1+1)$ possible values for $b_1$,.., (a_r+1) possible values for $b_r$ and so totally there are $(a_1+1) \cdots (a_r+1)$ possible divisors of $\prod_{i=1}^r p_i^{a_i}$, right?
Suppose M1 and M2 are two TMs such that L(M1) = L(M2). Then
(A) On every input on which M1 doesn’t halt, M2 doesn’t halt too.
(B) On every i/p on which M1 halts, M2 halts too.
(C) On every i/p which M1 accepts, M2 halts.
(D) None of above.
My try :
It explained here that M2 accepts string...
Is there anything similar to this for quadratic reciprocity ?
I mean, the link there explains how you can figure out the solution of cubic equation by yourself without having a suddent flash of inspiration/ genius genes. Is there some similar guide for quadratic reciprocity ?
