For instance, take three coin flip variables X1,X2,X3 (outcomes {+1,-1} with equal probability) and take them to be independent. then they've all got zero mean, unit variance, and zero covariance e.g. <X1.X2> = 0
and then construct the variables Y1 = X1 - (X1+X2+X3)/3, etc.
you'll find that the new variables have zero mean, variance 1/3, and covariance -2/3
So... I am guessing, the source of the correlation is the reporting here?
> what's not obvious are again the cross-cases. suppose they all work out like this: HB, BH, HF, FH, BF, FB: ++ (12.5%) +- (37.5%) -+ (37.5%) ++ (12.5%)
I recall one explanation on Bell inequality as some pressing button to light up lamp game explains how because of the richer structure of the operators that appeared in the expectation values (compared to just scalars in the classical case), it provide the wiggle room needed to achieve lager correlations thus give rise to entanglement
Actually, I wonder if your above sports example can be used to describe Bell inequality
To get Bell's inequality, note that the outcomes of the games are intended to be some random vector $(X_1,X_2,X_3)=(\pm 1,\pm 1,\pm 1)$
in order for the outcomes to not show any bias, you need +++ and --- to show up equally often
and similarly for the pairs {+--, -++}, {-+-,+-+} , {--+,++-}
if you now consider the products, you see that the three pairwise products are +1 for +++ and ---
whereas for the rest you have one +1 and two -1
consequently when you add the products together you get +3 in the first two cases and -1 in the rest. as such, the expected value of X1.X2+X2.X3+X1.X3 has to be between 3 and -1
whereas in the previous case the expected value of the products worked out to be -1/2 for each of them
well, note that p12 = <X1.X2> in this case defines the correlation coefficient of X1 and X2. (the mean values of X1,X2 are zero and their stddev's are one, so the usual definition simplifies a bit)
so the point is to look at c12+c23+c13
however, the motivation is a bit harder to explain. I guess you can partly relate it to what I was doing earlier re: (X1+X2+X3)^2
i.e. the expected values of X1^2, X2^3, X3^2 are all 1, so the expected value of their squared sum is determined by the sum of those product-moments
<H^2>+<F^2>+<B^2> +2<HB>+2<BF>+2<FH> hmmm, it feels like the expression X1.X2+X2.X3+X1.X3 is summing up all the correlations that are present in the system
Say I have an $n \times n$ covariance matrix for a sample set of $n$ random variables. Is there any meaning if the sum of the rows of this matrix? Is it a meaningful measurement of the contribution of each random variable to the variance of sum of the random variables?
the main point is that, if you only work with sets of games whose outcomes are +1/-1, then the pairwise correlations are constrained more than they'd be otherwise
(As pointed out by helpful users, one of the bolded claims is interpretation dependent. Therefore, unless otherwise specified, Copenhagen interpretation is used throughout)
Recall that for classical correlations, the observables are already determined even before a measurement, and thus knowing ...
We have the ring $R=\mathbb{Z}_5[x]$ and the element $b=(x-4)(x-3)^2$ of $R$. Let $A$ the set of $a$ in $R$ such that $at=b$ has a solution $t\in R$. How can we determine the number of elements of $A$ ? Could you give me a hint?
hmm, so I am guessing, the extra correlation beyond entanglement in the PR box is that equation AB=(a+b) mod 2 that forces both inputs and outputs to correlate with each other even though the inputs can be randomly chosen between 0 or 1
Does it make sense to talk about: "The set of stabilisers of vectors of an irreducible subspace corresponding to a certain irreducible representation." Or is there a better way to express that?
@TobiasKildetoft: Can you specify were the uncertainty comes about exactly?
OK
@TobiasKildetoft: The point is, that the subspace should be spanned only one irrducible representation. So is it that already implied in "irreducible subspace"?
In case of higher-than one dimensional irreps, there are different possibilites which elements of $G$ are stabilized, which leads to different subgroups of $G$ their intersection is the kernel.
Remember that to a mathematician, a representation consists of two objects: A vector space (this is where the vectors I am talking about are) and an action of the group on this space.
Yes, I want to focus on one certain specific representation, like e.g. the trivial representation. But then we get only the set containing $G$ as the only element. No please no characters. One motivation for this is to get independent from characters in the formulation.
I am not sure, I am just looking for how to express that map $\phi(\rho_k): G \to \{H_j\}_{j\in S}$ in a mathematically concise way. With $\rho_k$ being a specific irrep of $G$ (like $E$) and the set of $H_j$ the stabilizers.
I need a way to define $\phi(\rho_k)$ so that every mathematician immediately understands what is meant.
Finding them is no issue - thats specific computations we don routinely. But I have conjecture which I like to express in a waterproof way (mathematically).
I know that its common, but I always thought there would be some more general principles "behind it" that can be formulated without recurring to linear algebra
There are several equivalent ways to view representations, and we just need to pick one in order to make things precise (though there can also be merit in using different ones depending on what one wishes to talk about)
Not just the peculiar $G$ we are talking about but also then also its subgroups, I suppose yes, but that is always a little bit hard to follow for me, since I cannot clearly see the connections between different groups, so to say.
So for subgroups, this is just a matter of only taking some of the symmetries. But for other groups that are unrelated to the given one, it just becomes a matter of pure abstract stuff
No, I meant given $V$, the question is whether we only want subgroups stabilizing a single vector in $V$, or if we also want those that stabilize some set of vectors
(these latter will just be intersections of the former though)
Ok, then a short formulation that I am fairly certain should be unambiguous to mathematicians would be that this is the set of point-stabilizers for $G$ acting on $V$
@TobiasKildetoft: Maybe it helps when I shortly try to explain what would be the next step.
@TobiasKildetoft: The next step would be to map the ireps in a stabilizer to the irrep(s) in $G$. And that will depend strongly on $k$, when I am not mistaken.
@TobiasKildetoft: In the sense of $\{A,B\}$ in $C_2$ came from $E$ in $D_3$. My conjecture is about some functional connection between these irreps.
@LeakyNun oh yes i don't know a clean way of doing it.. does (finite) direct product coincide with (finite) direct sum in the category of top groups? or maybe more generally do they coincide when we're looking at group objects in some category?
@Rudi_Birnbaum Ohh, now we seem to be getting into differences in terminology again. The stabilizers so far have been subgroups of $G$, so there are no irreps "in" them
@TobiasKildetoft: And clearly $E$ from $D_{3h}$ can be assigned to $A,B$ in $C_2$, in this "descent in symmetry". It seems in some cases there can be ambiguities but in point groups these can be resolved, that means you know that you end up with a set of possibilities and can name all these.
@TobiasKildetoft: A good example would be that the trivial irrep in $G$ will always stay the trivial irrep in the stabiliser but the trivial irrep in the stabilizer might come from other irreps from $G$ depending on which irrep we choose to get to our stabiliser.
@Rudi_Birnbaum Ok, I think I see what you are saying. We take our representation and restrict it to one of these stabilizers. Then we decompose this as a representation for the stabilizer?
@TobiasKildetoft: The only bit I am still missing is that you do not mention the irrep which we used to get to the stabiliser. Again in our example we use $E$ in $D_3h$ to get to $C_1$ and $C_2$ and then we check what happened to $E$. But we could also have used $A_2$ that would have led to $C_3$ as stabilizer. And then $E$ from $D_3h$ would have gotten/or stayed the irrep called $E$ in the new group $C_3$.
Well, $m$ and $\pi$ are continuous, so $\pi\circ m$ is. But then $\mu \circ (\pi \times \pi)$ is continuous and $\pi\times \pi$ is a quotient map. The universal property of the quotient thus implies that $\mu$ is continuous
(Universal property of the quotient here says that $gf$ is continuous iff $g$ is continuous if $f$ is a quotient map)
That head bang moment when you need to freeze a dihedral in 256 mlecules where the last atom in a sequence of atom numbers will vary depending on which molecule it is
Ugh, someone divide me by zero so I can stop thinking about this hellhole
@TobiasKildetoft:That is when we choose the subgroup just like that. But according to my understanding there is (at least in point groups) a possibility to get certain informations on a connection between the irreps when you do determine the stabiliser with respect to some irrep of the group, so this is not just some subgroup selection.
I guess we defined the topology on $G/H$ to be the quotient topology so really the content here was that if $f_1:X_1\to Y_1$ and $f_2:X_2\to Y_2$ are quotient maps, then $f_1\times f_2 : X_1\times X_2 \to Y_1\times Y_2$ is a quotient map, as well as the categorical characterization of the quotient
filter is a generalization of sequences and nets and things like that
you get things like
X is a Hausdorff space if and only if every filter base on X has at most one limit. X is compact if and only if every filter base on X clusters or has a cluster point. X is compact if and only if every filter base on X is a subset of a convergent filter base. X is compact if and only if every ultrafilter on X converges.
and complete lattice is saying that the possible topologies on a set S forms a complete lattice
i.e. it is closed under arbitrary supremum and arbitrary infimum
@Daminark you have the neighbourhood filter at each point, and you can express continuity in terms of filter
@TobiasKildetoft: Lets assume we have only one-dimensional real irreps with entries "-1" and "1". Then we select one irrep, and remove all $g$ with entries $-1" from the table. And we are there. Then we exactly know which irrep came from which.
So I'm vaguely familiar with nets and I definitely do think they get some good stuff done over sequences by virtue of generality, though as it stands it feels to me like if you're in a context where nets do things that sequences don't, you're in that side of topology where it's a bit more set pushing than I'd like
For the most part I've neglected point-set however in functional analysis there was a context where it mattered
It seems to me that you have sorta two main ways in which your spaces begin to go to shit
In analysis, especially functional analysis, the axioms of countability begin to matter
And in algebra it seems like separation is more what hurts you
We had a pset problem one time which said to prove that if an infinite-dimensional Banach space is reflexive or has separable dual, then you can find a sequence on the unit sphere converging weakly to 0
Now in class we already proved that the weak closure of the unit sphere is the closed unit ball so we were like, how is this not trivial, and why do you have these assumptions in particular?
The point was that we were talking about a topology which a priori wasn't first countable. The content of the problem was to say that if a space has separable dual, then its unit ball is weakly metrizable, so you can actually do things with sequences
And if a space is reflexive, you take the closure of the span of some sequence of linearly independent vectors, then that gives a separable subspace, which is reflexive as well by virtue of living in a larger reflexive space
Now, if a space is separable and reflexive, then its dual is separable and reflexive (in general separable doesn't imply separable dual), so you can apply the above business there
yeah, when I first learnt about nets from some conversation in this chat, I was discussing with user21820 and leaky about the difference between countablity and uncountability, which then the discussion leads into topologies that are not first countable, which is why I found nets more appealing since the context I am interested in are those topological spaces which cannot be adequately converged with sequences
Nets being based on directed sets which in turn are preorders imposed on sets are also mroe generalised than well orderings, thus doing problems involving this help me to build intuition on how to deal with posets that are not well ordered (and possibly don't always have sup for every subset)
so that when I get used to them enough, problems involving sequences will be easier since well orders are "nicer" than generic posets
One of the things being discussed is the relationship between uncountability and completeness of the space, that is something I want to understand more
because e.g. you cannot have a complete vector space of infinite dimensions with a hamel basis of countable cardinality (and other related things)
and for non separable hilbert spaces, even a countable orthgonal basis is not enough to complete it
Topology will thus help me to understand and comprehend the relevant proofs on how the failure of completion occurs and why ramping up the cardinality and then suddenly you can complete the space
thus give a better idea on what properties of an object is an uncountable cardinal is controlling
Say I have an $n \times n$ covariance matrix for a sample set of $n$ random variables. Is there any meaning if the sum of the rows of this matrix? Is it a meaningful measurement of the contribution of each random variable to the variance of sum of the random variables?
