evolution - Did we first have swimming birds or flying birds?

Flying came first, as far as we know. The earliest known bird (currently), Archaeopteryx lithographica, already had aerodynamic feathers (Feduccia and Tordoff, 1979). The Solnhofen Limestone, where it was discovered, is ~145 million years old, so we can place the "when" flight evolved to greater than or equal to that time.

Gansus yumenensis is regarded as the earliest aquatic bird and is dated from ~120 million years ago. The Hesperornithiformes are the sister taxon of Gansus (less derived) and also aquatic. However, they date from ~85 million years ago. It is not clear whether the ancestors of Hesperornithiformes were also aquatic. Because they branched off the main line of birds before Gansus, these ancestors would likely predate Gansus, but those relatives have not yet been discovered.

ag.algebraic geometry - Stacks and sheaves

Let me see if I understand your example correctly: you are fixing $X$ and $Y$, families
of curves over $S$, and now you are considering the functor which maps an $S$-scheme $T$
to the set of $T$-isomorphisms $f^*X to f^*Y$ (where $f$ is the map from $T$ to $S$).

If I have things straight, then this functor shouldn't be so bad to think about, because it is actually representable, by an Isom scheme. In other words, there is an $S$-scheme
$Isom_S(X,Y)$ whose $T$-valued points, for any $f:T to S$, are precisely the $T$-isomorphisms
from $f^*X$ to $f^*Y$. (One can construct the Isom scheme by looking inside a
certain well-chosen Hilbert scheme.)

One way to think about this geometrically is as follows: one can imagine that two
curves over $k$ (a field) are isomorphic precisely when certain invariants coincide
(e.g. for elliptic curves, the $j$-invariant). (Of course this is a simplification,
and the whole point of the theory of moduli spaces/schemes/stacks is to make it precise,
but it is a helpful intuition.) Now if we have a family $X$ over $S$, these invariants
vary over $S$ to give a collections of functions on $S$ (e.g. a function $j$ in the
genus $1$ case), and similarly with $Y$. Now $X$ and $Y$ will have isomorphic
fibres precisely at those points where the invariants coincide, so if we look
at the subscheme $Z$ of $S$ defined by the coincidence of the invariants,
we expect that $f^*X$ and $f^*Y$ will be isomorphic precisely if the map $f$
factors through $Z$. Thus $Z$ is a rough approximation to the Isom scheme.

It is not precisely the Isom scheme, because curves sometimes have non-trivial
automorphisms, and so even if we know that $X_s$ and $Y_s$ are isomorphic for
some $s in S$, they may be isomorphic in more than one way. So actually the
Isom scheme will be some kind of (possibly ramified) finite cover of $Z$.

Of course, if one pursues this line of intuition much more seriously, one will
recover the notions of moduli stack, coarse moduli space, and so on.

Added: The following additional remark might help:

The families $X$ and $Y$ over $S$ correspond to a map $phi:S to {mathcal M}_g
times {mathcal M}_g$. The stack which maps a $T$-scheme to $Isom_T(f^*X,
f^*Y)$ can then seen to be the fibre product of the map $phi$ and the diagonal
$Delta:{mathcal M}_g to {mathcal M}_g times {mathcal M}_g$.

In the particular case of ${mathcal M}_g$ the fact that this fibre product is representable is part of the condition that ${mathcal M}_g$ be an algebraic stack.

But in general, the construction you describe is the construction of a fibre product
with the diagonal. This might help with the geometric picture, and make the relationship to Mike's answer clearer. (For the latter:note that the path space into $X$ has a natural
projection to $Xtimes X$ (take the two endpoints), and the loop space is the fibre product
of the path space with the diagonal $Xto Xtimes X$.)

set theory - Does every non-empty set admit a group structure (in ZF)?

You cannot in general put a group structure on a set. There is a model of ZF with a set A that has no infinite countable subset and cannot be partitioned into finite sets; such a set has no group structure.

See e.g at http://groups.google.com/group/sci.math/msg/06eba700dfacb6ed

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Sketch of proof that in standard Cohen model the set $A={a_n:ninomega}$ of adjoined Cohen reals cannot be partitioned into finite sets:

Let $mathbb{P}=Fn(omegatimesomega,2)$ which is the poset we force with. The model is the symmetric submodel whose permutation group on $mathbb{P}$ is all permutations of the form $pi(p)(pi(m),n)=p(m,n)$ where $pi$ varies over all permutations of $omega$, (that is we are extending each $pi$ to a permutation of $mathbb{P}$ which I also refer to as $pi$) and the relevant filter is generated by all the finite support subgroups.

Suppose for contradiction that $pVdash " bigcup_{iin I}dot{A_i}=A$ is a partition into finite pieces"; let $E$ (a finite set) be the support of this partition. Take some $a_{i_0}notin E$ and extend $p$ to a $q$ such that $qVdash ``{a_{i_0},ldots a_{i_n}}$ is the piece of the partition containing $a_{i_0}$". Then pick some $j$ which is not in $E$ nor the domain of $q$ nor equal to any of the $a_{i_0},ldots a_{i_l}$. If $pi$ is a permutation fixing $E$ and each of $a_{i_1},ldots a_{i_n}$ and sending $a_{i_0}$ to $a_j$, it follows that $pi(q) Vdash " {a_j,a_{i_1},ldots a_{i_n}}$ is the piece of the partition containing a_j". But also $q$ and $pi(q)$ are compatible and here we run into trouble, because $q$ forces that $a_{i_0}$ and $a_{i_1}$ are in the same piece of the partition, and $pi(q)$ forces that this is not the case (and they are talking about the same partition we started with because $pi$ fixes $E$). Contradiction.

biochemistry - Single hormone opposite effects

An hormone is not different from most other molecules. To have an effect on a cell it binds to a (more or less specific) receptor, located either on the plasma membrane or inside the cell, and it initiates an intracellular cascade of events¹.

There are several ways an hormone can have different effects:

1. there can be multiple receptors for the same hormone. For instance, prolactin can bind to two receptors, called prolactin receptor (PRL-R) short and long form. The short form of the receptor has been shown to lack the ability to promote milk protein genes transcription (See Lesueur et al., PNAS - 1991).




2. the same receptor can be coupled to different intracellular pathways in different cell types / physiological conditions, thus resulting in different effects.




3. each cell type/tissue expresses a set of protein that will interact in a different manner with the intracellular cascade promoted by the hormone.




4. an hormone can interact with receptors for other molecules. For instance allopregnanolone, a metabolite of progesterone, is a potent agonist of the GABA-A receptor, giving it anxiolitic properties.

An interesting example is that of estrogenic compound. Several receptors exist for estradiol (E2). The "classical" receptors are called ER-alpha and ER-beta, and they are located in the cytoplasm. The binding of E2 to the ER promotes their dimerization and entrance into the nucleus where they can promote the transcription of various genes. ERs can also bind to other transcription factors and modulate their activity. So, depending on which transcription factors are present different genes will be transcribed in response to E2. Moreover, ER-beta can have opposite effects then ER-alpha (see for instance Weihua et al., PNAS - 2000). In addition, receptors like ERs can be activated also in absence of the endogenous hormone: for instance, dietary amino acids activate ER-alpha in liver by a mTOR-dependent phosphorylation (Della Torre et al., Cell Metabolism 2011)

To make things more complicated, membrane receptor for estrogen have been described, such as GPR30 (Revankar et al., Science - 2005), membrane "versions" of the classical ERs, plus various splicing variants of the ERs (mostly expressed in tumoral tissue).

GPR30 in breasts has been shown to activate certain molecular pathways (notably Erk1/2) that contribute to cellular growth (Filardo et al., Mol Endocrinol. - 2000), possibly linking it to proliferation of ER-negative breast tumours. Other tissues that do not express GPR30 will lack this E2-induced proliferative stimulus.

So, tissue- or time- dependent modulation of the receptors and of the intracellular pathways associated with it can allow the body to respond to the same hormone (or, more in general, to the same molecule) in opposite ways.

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¹ This is a simplification, not every substance (and not every hormone) acts by binding to receptors, but let's keep things simple.

gt.geometric topology - Triangulations coming from a poset. Or: What conditions are necessary and sufficient for a finite simplicial complex to be the order complex of a poset?

Here are necessary and sufficient conditions for an abstract, finite simplicial complex $mathcal{S}$ to be the order complex of some partially ordered set.

(i) $mathcal{S}$ has no missing faces of cardinality $geq 3$; and

(ii) The graph given by the edges (=$1$-dimensional simplices) of $mathcal{S}$ is a comparability.

[Definitions.
(a) A missing face of $mathcal{S}$ is a subset $M$ of its vertices (=$0$-dimensional simplices) such that
$M not in mathcal{S}$, but all proper subsets $Psubseteq M$ satisfy $Pin mathcal{S}$.
(b) A graph (=undirected graph with no loops nor multiple edges) is a comparability if its
edges can be transitively oriented, meaning that whenever edges ${p, r_1}, {r_1, r_2},ldots, {r_{u−1}, r_u}, {r_u, q}$ are oriented as $(p, r_1), (r_1, r_2),ldots, (r_{u−1}, r_u), (r_u, q)$, then there exists an edge ${p, q}$ oriented as $(p, q)$.]

This characterisation appears with a sketch of proof $-$ which is not hard, anyway $-$ in

M. M. Bayer, Barycentric subdivisions. Pacific J. Math. 135 (1988), no. 1, pp. 1-16.

As Bayer points out, the result was first observed in

R. Stanley, Balanced Cohen-Macaulay complexes, Trans. Amer. Math. Soc, 249 (1979), pp. 139-157.

@Rasmus and @Gwyn: The characterisation might perhaps disappoint you if you were expecting something more topological. However, it's easy to prove that no topological characterisation of order complexes is possible, and therefore a combinatorial condition such as the one on comparabilities must be used. For this, first check that the barycentric subdivision of any simplicial complex indeed is an order complex. Next observe that barycentric subdivision of a simplicial complex does not change the homeomorphism type of the underlying polyhedron of the complex. Finally, conclude that for any topological space $T$ that is homeomorphic to a compact polyhedron, there is an order complex whose underlying polyhedron is homeomorphic to $T$.

I hope this helps.

pr.probability - What is known about the Gaussian measure of the unit ball in a Hilbert Space?

You can't talk about "the" Gaussian measure on an infinite-dimensional Hilbert space, for the same reason that you can't talk about a uniform probability distribution over all integers. It doesn't exist; see Richard's answer. However, there are a lot of non-uniform Gaussian measures on infinite dimensional Hilbert spaces.

Consider the measure on $mathbb{R}^infty$ where the $j$th coordinate is a Gaussian with mean 0 and variance $sigma_j^2$, where $sum_{j=1}^{infty} sigma_j^2 < infty$ (and different coordinates are independent). This is almost surely bounded in the $ell_2$ metric, and any projection onto a finite-dimensional space has a Gaussian distribution. The squared length of a vector drawn from this measure is a sum of squares of Gaussians, and so follows some kind of generalized $chi$-square distribution. If I knew more about generalized $chi$-square distributions, I might be able to tell you what the measure of the unit ball was.

This kind of Gaussian distribution is very important in quantum optics. In fact, in quantum optics, a thermal state is Gaussian, so "the" Gaussian measure actually makes some sense.

lo.logic - What is Realistic Mathematics?

This is more a philosophical question, and therefore hasn't a definite answer.

But if you want to make plausible that AC isn't realistic mathematics, you might reason something like the following:

There is a set of mathematical sentences, that has a direct (so, not indirect yet) relation with physics. Call this set B (of Basic). Personally, I think they are in the following four areas:

• Computation
• Probability
• Geometry
• Topology

Note, I count arithmetic as part of computation, since numbers are not a physical entity, but computation is. But, likely many people will disagree.

Also note, that the sentences in B, might be far simpler than the mathematical sentences suggested in your question or in one of the answers.

Now, we have more complex mathematical sentences, that are still "realistic", if they can be converted or "instantiated" to sentences of B. Call this extended set be E. These more complex sentences capture a higher principle, which can be powerful in the science of physics. Still, there is no direct link with physics, the mathematic sentence first needs to be instantiated, to make a direct link with physics. Example, any sentence with a real number, is not be part of B, because we can not observe real numbers in physics, but they can be part of E.

Consider that there is a sentence s1 ∈ E and s2 ∈ E. Furthermore, that s2 follows from s1, but with a rather difficult proof. Suppose there is a proposed axiom a, such that s1 + a leads more directly to s2 (a simpler proof). However, a ∉ E. So, axiom a is independent from sentences in B.

From above concept it follows that axioms can exist that are "useful", because they make proofs shorter, but have nevertheless no "meaning". I do believe that AC is such axiom.

About CH, I think it is not useful and not having a meaning.

But again, this is more an opinion.