The question is rather straight forward: I have always been curious as to why, but cannot find an explanation online.
I can imagine that the mechanism is different for each, but why does brain tissue and red blood cells use specifically and only glucose for energy metabolism?
Suppose I have a surjective homomorphism of topological groups $f:E to G$. Let K be the kernel of f. The topological group K acts on E in an obvious way. When is this a fiber bundle over G? (It will then be a K-principal bundle over G).
I'm writing a paper where I make a claim about when this holds. I thought I had a reference but when I went looking for it, my claim was not in the reference.
I don't want to consider examples that are too pathological so lets assume everything is Hausdorff and paracompact. (However if people are familiar with the more general setting, I'd be curious about that too!).
Clearly a necessary condition is that $G cong E/K$, so let's assume this is the case. By homogeneity it is enough to show that f admits a local section in a neighborhood of the identity element of G. So my question is equivalent to asking if there are conditions I can impose on E and G which will ensure that f admit local sections near the identity of G.
I know by work of G. Segal ("Cohomology of Topological Groups" Symposia Math. Vol IV 1970 pg 377, in the appendix) that if G is abelian and locally contractible then the sequence $$G to EG to BG$$
is of this kind.
I want to know:
I'd also like to know some reasonable examples where this fails to be a principal bundle (if there are any).
Hi,
I am doing some Kernel density estimation, with a weighted points set (ie., each sample has a weight which is not necessary one), in N dimensions.
Also, these samples are just in a metric space (ie., we can define a distance between them) but nothing else. For example, we cannot determine the mean of the sample points, nor the standard deviation. The Kernel is just affected by this distance, and the weight of each sample:
f(x) = 1./(sum_weights) * sum(weight_i/h * Kernel(distance(x,x_i)/h))
In this context, I am trying to find a robust estimation for the kernel bandwidth 'h', possibly spatially varying, and preferably which gives an exact reconstruction on the training dataset x_i. If necessary, we could assume that the function is relatively smooth.
I tried using the distance to the first or second nearest neighbor but it gives quite bad results. I tried with leave-one-out optimization, but I have difficulties finding a good measure to optimize for in this context in N-d, so it finds very bad estimates, especially for the training samples themselves. I cannot use the greedy estimate based on the normal assumption since I cannot compute the standard deviation. I found references using covariance matrices to get anisotropic kernels, but again, it wouldn't hold in this space...
Someone has an idea or a reference ?
Thank you very much in advance!
I think your observation is a very good one, but this phenomenon is limited to classical logic and does not continue to hold when we move to intuitionistic or substructural logics.
One way of understanding the role of syntax is to take the connectives of logic as explaining what counts as a legitimate proof of that proposition. So a conjunction $A land B$ can be proven with a proof of $A$ and a proof of $B$, and an implication $A implies B$ can be can be proven with a proof of $B$, assuming a hypothetical proof of $A$, and so on. Conversely, we also give rules explaining how to use true propositions -- e.g., from $A land B$, we can re-derive $A$, and we can also rederive $B$.
If you work this out formally, you get Gentzen's system of natural deduction. The natural deduction systems for good logics admit a normal form theorem for proofs. The normalization procedure also gives us an equivalence relation on proofs (two proofs are equivalent if they have the same normal form), and it so happens that for classical logic, all proofs are equivalent. (This is a small lie: we can give more refined accounts of equivalence of classical proofs which don't equate everything, but the right answer here is still not entirely settled....)
The equivalence of proofs means that we can take the view that the meaning of a classical proposition is its truth value -- i.e., its provability -- and so algebraic models of classical logic contain all the information contained in a classical proposition. We don't need the proofs, and so syntax takes a secondary role.
But in intuitionistic and substructural logics like linear logic, not all proofs are equated. This means that we can't take the view that all the relevant information about a proposition is contained in its truth value, and so syntax retains a more important role.
Let $G$ be the graph whose nodes are the points of
$mathbb{Z}^d$ in the nonnegative orthant (i.e., all
coordinates are $ge 0$), with edges connecting each
pair of points separated by unit distance.
So the degree of each node not on the boundary is $2d$.
Now delete each node with probability $delta$,
except always retain the origin $o=(0,0,ldots,0)$.
Let $G_delta$ be the component connected to $o$.
Question 1. Does $G_delta$ contain a simple path
from the origin of infinite length?
The length of a path is its number of edges.
A simple path does not cross itself.
For $d{=}1$, the answer is 'No' for any $delta > 0$,
because eventually the run of nodes connected to $o$ will
be broken. So, almost surely every path is of finite length.
The situation is less clear to me for $d ge 2$.
Some experimentation tentatively suggests that for $d{=}2$ and $delta=frac{1}{2}$, the answer is again 'No.'
When the answer to Question 1 is 'No,'
let $r(d,delta)$ be the radius of $G_delta$, defined to be the expected length of the longest of the shortest paths
from $o$ within $G_delta$.
Question 2. What is $r(d,delta)$?
For $d{=}1$, I believe the radius is
$$sum_{k=1}^{infty} k (1-delta)^k delta = (1-delta)/delta ;.$$
For example, $r(1,frac{1}{4})=3$.
The $d{=}2$ example below shows a shortest path of length 18 connecting
$(0,0)$ to $(10,4)$. (Yellow=deleted nodes, green=component connected to origin,
blue=undeleted but disconnected from origin.)
I produced this example with $delta=0.55$.
I suspect these questions have been addressed in the literature
on random graphs, with which I am not so familiar.
Any references, reformulations, proof ideas, or partial solutions ($d{=}2$ and $d{=}3$
are of special interest to me), would be appreciated.
Thanks!
Edit. Thanks for all the references. From what I have learned so far, the model I defined is known as site percolation in the literature (in contrast to bond percolation).
My restriction to the positive orthant is not generally followed in the literation, but that
aside, there is much known, and much unknown. In general there is a critical probability
$delta_c$ for each dimension $d$ that answers my first question: for $delta < delta_c$,
the origin belongs to an infinite component, and for $delta > delta_c$, it belongs to a finite
component. Remarkably, exact values for $delta_c$ for site percolation on $mathbb{Z}^d$ for
$d ge 2$ are not known. For $d=2$, it is estimated via numerical simulations to be 0.59;
for $d=3$, it is about 0.31.
I don't know how extensive. Let's run a simple data query and find out: Go to GEO at NCBI. In the "Gene profiles" window, type CD47, and hit enter to launch the query. At the top of the resulting page, use the link labeled "Limits" to restrict the 7000+ results to human by entering the term "human" in the "DataSet organism" window. So, there are 3700+ results. Look through them to get an idea of which cell types and under which conditions CD47 is expressed.
Let's try a second method. Go to the BioGPS portal. Enter CD47 in the appropriate window and run the query. From the results, select the row with ID # 961 as this represents the human gene CD47. The resulting gene expression/activity chart for Hs (human) will show you in more general terms where CD47 is expressed.
This is a followup to Analog to the Chinese Remainder Theorem in groups other than Z_n. . I shouldn't have used the comments to ask a new question, in fact...
Here is the statement of the Chinese Remainder Theorem, as it occurs in most books and websites:
(1) Let $R$ be a commutative ring with unity, and $I_1$, $I_2$, ..., $I_n$ be finitely many ideals of $R$ such that ($I_i+I_j=R$ for any two distinct elements $i$ and $j$ of $leftlbrace 1,2,...,nrightrbrace$). Then, $I_1I_2...I_n=I_1cap I_2cap ...cap I_n$, and the canonical ring homomorphism $R/left(I_1I_2...I_nright)to R/I_1 times R/I_2 times ... times R/I_n$ is an isomorphism.
But there seems to be another, even more general form of (1) which doesn't get even half of the attention:
(2) Let $R$ be a commutative ring with unity, and $I_1$, $I_2$, ..., $I_n$ be finitely many ideals of $R$ such that ($I_i+I_j=R$ for any two distinct elements $i$ and $j$ of $leftlbrace 1,2,...,nrightrbrace$). Let $A$ be an $R$-module. Then, $I_1I_2...I_ncdot A=I_1Acap I_2Acap ...cap I_nA$, and the canonical $R$-module homomorphism $A/left(I_1I_2...I_ncdot Aright)to A/I_1A times A/I_2A times ... times A/I_nA$ is an isomorphism.
I am wondering: is (2) a trivial corollary of (1)? Because otherwise I don't see any reason why (2) shouldn't appear in literature as "the" Chinese Remainder Theorem, with (1) being but a corollary. Or is (2) wrong? The only way I see to get (2) from (1) is to apply (1) to the ring $Roplus A$ (with multiplication on $Roplus 0$ inherited from $R$, multiplication between $Roplus 0$ and $0oplus A$ given by the $R$-module structure on $A$, and multiplication on $0oplus A$ given by $0$), which seems quite artificial to me. Am I missing something very obvious?
