I know that metaphase II ends at ampulla tuba uterina, but I am not completely sure where the telophase II ends. Is it in the triangular section of cervix uterii?
I just have an intuition that the thing is not ending at the ampulla, since there is some time to get to the uterus. The thing must happen before implantation to the uterus, since the egg is blastocyst at the stage.
2. When does meiosis II of oogenesis end exactly?
This is an interesting question. For any fixed positive integer $d geq 2$, write $T_d^{infty}$
for the complete infinite rooted $d$-ary tree (by this I mean every node has exactly $d$ children). Luczak and Winkler proved the existence of a procedure which will generate a sequence $(T_{n,d})_{n geq 0}$ such that for all $n geq 1$,
(a) the distribution of T_{n,d} is uniformly random over all $n$-node subtrees of $T^{infty}_d$ that contain the root of $T^{infty}_d$; and
(b) $T_{n,d}$ is a subtree of $T_{n+1,d}$.
It is not hard to show that (a) implies that for all $n$, $T_{n,d}$ is distributed as a Galton-Watson tree with offspring distribution $mathrm{Bin}(d,1/d)$, conditioned to have total size $n$.
Since $mathrm{Bin}(d,1/d)$ tends to a $mathrm{Poisson}(1)$ distribution as $d$ becomes large,
this means that as $d to infty$, the distribution of $T_{n,d}$ tends to that of a Galton-Watson tree with offspring distribution $mathrm{Poisson}(1)$ conditioned to have $n$ nodes (let me write $mathrm{PGW}_n(1)$ for this distribution). The latter distribution is the same as that of a uniformly random labelled rooted tree on labels $1,ldots,n$. (At least, the latter is true once we label the Galton-Watson tree uniformly at random, or alternately remove the labels of the labelled rooted tree.)
As noted by Lyons et al. (Theorem 2.1), all this implies in particular that one can define a similar sequence $(T_n)_{n geq 1}$ such that for all $n$, $T_n$ is a subtree of $T_{n+1}$ and $T_n$ has distribution $mathrm{PGW}_n(1)$.
However, the construction in the Luczak-Winkler paper uses flows, and it is not 100% obvious how it "passes through to the $d=infty$ limit." (I say this with the caveat that I didn't make any serious attempt at figuring this out.) As a consequence, while it is known that there exists a generation procedure of the type you are looking for, I am not aware of an explicit description of the actual rule for where the leaf should be attached to $T_n$ to create $T_{n+1}$. I asked Peter Winkler about this at a conference last year and he also didn't know (though I don't know whether he had thought about this specific question in depth, either).
Reading "Polyphasic Sleep: Facts and Myths (Dr Piotr Wozniak)", it is pointed out that infant humans do undergo polyphasic sleep. As this is where most of our development is obviously done, I do not know where I can further proceed with the question about how it would affect development? Perhaps the issue is more how it would effect the day to day performance of a developed individual? If this is the case then it is suggested by Dr Wozniak that this is likely to be highly disruptive to the individual
Those well-defined effects of natural sleep affecting stimuli on sleep patterns lead to an instant conclusion: the claim that humans can adapt to any sleeping pattern is false. A sudden shift in the schedule, as in shift work, may lead to a catastrophic disruption of sleep control mechanisms. 25% of North American population may work in variants of shift schedule. Many shift workers never adapt to shifts in sleep patterns. At times, they work partly in conditions of harmful disconnect from their body clock, and return to restful sleep once their shift returns to their preferred timing. At worst, the constant shift of the working hours results in a loss of synchrony between various physiological variables and the worker never gets any quality sleep. This propels an individual on a straight path to a volley of health problems...
It appears that polyphasic sleep encounters the precisely same problems as seen in jet lag or shift-work. Human body clock is not adapted to sleeping in patterns other than monophasic or biphasic sleep.
It would therefore seem that polyphasic sleep is certainly detrimental to health, if not development.
However studies into cognitive performance resulting from differing sleep patterns run by Dr Claudio Stampi (Published ISBN 0-8176-3462-2), he concluded that polyphasic sleep was more efficient than monophasic sleep. Therefore it may be possible that polyphasic sleep patterns have no detrimental effect on development.
Individuals sleeping for 30 minutes every four hours, for a daily total of only 3 hours of sleep, performed better and were more alert, compared to when they had 3 hours of uninterrupted sleep
There are a couple of theories mentioned in the above book (beginning pg 5) which support the development of monophasic sleep as evolution rather than a social convention:
I am sure that social factors would have an effect, however I would imagine the evolutionary pressures to be more significant.
I really would recommend the book (http://sleepwarrior.com/Claudio_Stampi_-_Why_We_Nap.pdf) if you have not yet encountered it as I found it very informative and packed with references to studies that you may find helpful.
What you've written down is relevant for finding rational torsion points on an elliptic curve. If that's what you want to do, Galois theory is certainly relevant. For instance, suppose you have an elliptic curve in Weierstrass form,
y^2 = f(x)
with f a cubic. Now suppose you find that f(x) has a linear factor (x-a). (I certainly take this to be a "Galois-theoretic" condition on f.) Then you've found a rational point of your curve, namely (a,0).
The relationship between Galois theory and points of infinite order is more subtle, involving Galois cohomology, and is discussed in chapter 10 of Silverman's book The Arithmetic of Elliptic Curves.
I'm going to be deliberately provocative and say that I don't really know of any use for the concept of bimorphism as such. (I also don't really like the name; it sounds to me like something that's both a morphism and a comorphism.)
One use that's been proposed is "to find situations in which bimorphism ⇒ isomorphism." Such situations may be interesting, but as far as I can tell they are rarely (if ever) used. What seems to happen much more often is that we have some factorization system (E,M) and we use the fact that E+M=iso, which is true for any factorization system. The most common case is probably (extremal epi, mono), followed perhaps by (epi, extremal mono); both of these are factorization systems as soon as the relevant factorizations exist.
It might happen, in some case, that E consists of exactly the epimorphisms and M of exactly the monomorphisms (such as when all epis, or all monos, are extremal). But as far as I can tell this fact -- especially the epi part of it -- is hardly ever relevant, because in practice it's quite hard to characterize the epis in a given category or to check that a given morphism is epi, nor is the answer often especially meaningful. Since monadic functors also create limits, and in particular monomorphisms, a morphism in a category monadic over Set is monic iff it is injective -- but this is not true for epis, and even in quite nice categories the epis can be fairly bizarre. It's usually the extremal epis which coincide with the "surjections" and form a factorization system with the monos.
For instance, Andrew cited vector spaces as an example of a balanced category. But as I pointed out in my comment, do we ever use that fact? What we actually teach our undergraduates is that injective+surjective=iso for vector spaces; we (or, at least, I) don't tell them anything about why surjective=epi, or even what epi means. And when doing linear algebra, I might occasionally use the fact that surjections are in particular epi (which just follows because the forgetful functor to Set is faithful), but never the converse. It's just as true for groups, rings, fields, monoids, etc. that injective+surjective=iso, and we use that fact in doing algebra all the time -- but does the non-surjective ring epimorphism Z → Q, showing that rings (unlike vector spaces) are not balanced, ever actually bother us in practice?
In the topological situation, it's true that the epimorphisms in Top are precisely the surjective continuous maps. But does that fact really help you when looking for conditions ensuring that a continuous bijection is an isomorphism, or using that fact in practice? Odds are the property of a continuous bijection you're going to use is that it's continuous and a bijection, not that it's monic and epic in the category Top.
The categorical version of "continuous bijection in Top" is "inverted by the forgetful functor to Set," and I think that in general the property of "being inverted by a forgetful functor" is quite interesting and important. For instance, a forgetful functor with the property that any morphism inverted by it is already an isomorphism is called conservative, and these include all monadic functors. The question about all the different topologies one can put on a given set also seems to me to really be about morphisms inverted by the forgetful functor; is it really important here that continuous surjections are the epis in Top? I expect that if you modify the definition of Top a little, then it may no longer be true that epis coincide with continuous surjections, and in that case I bet that it is the continuous surjections which are of more interest.
At this point, perhaps the most interesting thing I know about bimorphisms is that they often form the middle class of a ternary factorization system. I'll be happy to be proven wrong, however.
So with a lot of extra care about dimension/codimension it seems to be possible to define Chow groups over Spec Z if I understand the above answers correctly.
I may point out that in the book by Elman, Karpenko and Merkurjev "Algebraic and Geometry Theory of Quadratic Forms" (even though the title does not suggest so) they very carefully work out Chow groups, even some version of higher Chow groups. They begin by treating Chow groups over general excellent schemes (something you do not have written so explicitly in Fulton), so quite general and only later impose additional assumptions, like equidimensionality, being over a field, and all that. So maybe it is worth having a look at that.
On the other hand, they get a pullback along non-flat morphisms only with the typical more restrictive conditions. This however is crucial for turning Chow groups into the Chow ring.
So I think the construction of the intersection product [which uses the pullback along the embedding of the diagonal X -> X x X] is another very very critical matter over Z (but according to one of the other answers it can be done, that sounds very interesting).
Last but not least, just maybe another perspective, if one writes down the classical intersection multiplicity of two cycles, that can be done by first multiplying both cycles of complementary codim [so for this we need a ring structure, but let's just assume somebody can give such a structure even over Z, just to find out where we would actually be going], then the product lies in CH^n(X), n being the dimension of our scheme. Now to turn that into the classical intersection multiplicity one could pushforward this cycle along the structural map to the base field,
$X$ --> $Spec$ $(k)$
over a field(!) and $CH{_0}(Spec k) = mathbb{Z}$ and we get our intersection number. Voilà.
But if we are proper over Spec Z, we could at best pushforward
$X$ --> $Spec$ Z
but $CH{_0}(Spec(mathbb{Z}) = 0$, so nothing very interesting seems to result here.
[this argument however only makes sense if the dimension shifts in this Spec Z setup would be carried over analogously, which maybe is also stupid here for the reason that Spec Z is one-dimensional and Spec k zero-dimensional. I am just saying all this, because best and supercool would of course be somebody with a Spec *F*$_1$ having
CH_0(Spec *F*$_1$)= ? (....something, probably rather R than Z)
and that could then be our Spec Z intersection number by giving *F*$_1$ the role of a "virtual base field" and I guess some people say this should link to the Arakelovian one.... but well, that's very speculative]
So I think some people's expectation goes in the direction that the "interesting" way of doing intersection theory over Spec Z needs such a final *F*$_1$-twist.
Note maybe that the classical analgoue would be
P1 <-> Spec Z + (infinite place)
but CH_0(P1) = Z, whereas CH_0(Z) = 0, so we kind of miss something if we just use classical Chow groups over Spec Z. For other questions, classical methods work well even for Z without needing *F*$_1$ or so, for example the étale fundamental groups of both P1 and Spec Z are trivial. But for Chow theory some additional tricks seem to be required.
At least that is my impression. Of course this arithmetic aspect of intersection theory over Spec Z is a kind of different story and it also makes perfect sense to talk about classical Chow groups over Spec Z, so there is certainly nothing wrong in having CH_0(Z) = 0, just maybe for some sorts of questions of arithmetical content, this type of Chow theory may not be the right approach.
By far the most important part is the very beginning of the transcript, especially the positions +1 and +2. The conservered consensus sequence in class III T7 promoters is GGGAGA, any changes in the first two nucleotides severely reduce the transcription efficiency. Changes at positions +3 to +6 have much smaller effects.
Additionally, changes that put many AU base pairs in that region (e.g. GGUUU) seem to affect the trancription efficiency negatively.
Other sequences that are problematic anywhere, not only in the beginning are long stretches of uridines or adenines. Sequences with eight or more uridines or adenines can cause the polymerase to slip, which results in transcripts with more uridines or adenines than in the template.
These characteristics are detailed in the paper from Milligan and Uhlenbeck from 1989.