genetics - Are there dog breeds that are so far apart genetically that they can't produce viable offspring?

I actually found some sort of reference for this. Apparently in the case of a beagle and irish setter pairing, they had a lot of difficulty in producing pups, but as breeds have a lot of genetic quirks, this might be due to a genetic accident; this is probably a case of mutual infertility rather than speciation. There are probably mutations in their genomes which are causing non viability in the pups the way that it does with some human parents who have difficulty having children - an immunological incompatibility or traits which cause most offspring non-viable.

In fact the discussion points out that the species designation does not always mean they cannot produce offspring, but simply do not. Coyotes and wolves will produce offspring too if they are encouraged to do so, but are competing and antagonistic in the wild and so never do mate in practice.

So the answer is probably no. (the beagle/setter pups did show up, but with a much smaller litter than usual with just 2 pups).

biochemistry - Does caffeine increase the speed at which sperm travels?

The cell response for caffeine, is at the molecular level of RyR (Ryanodine receptor), and the main effect could be a temporal rise in Ca+2 cytosolic concentration, so it can change the electrical behavior for the whole cell, and then affect their mobility. But the main area for RyR research is at spermatogenesis, there are a lot of papers like this one, for that topic

co.combinatorics - Hamilton cycle decompositions of the complete graph

Just reporting that I wrote another algorithm for this and found the following values:

3 1
5 6
7 960
9 40037760

I ran this through the superseeker on Sloane and it came up with nothing (so perhaps nobody has counted these before).

Here's my code below (it uses GAP). We generate a (n-1) x n Latin rectangle where each row is an n-cycle and the i-th and (i+(n-1)/2)-th rows are inverses.

EnumerateHamiltonDecompositionsBacktrackingAlgorithm:=function(n,L,step)
  local i,j,k,count,A;
  i:=Int((step-1)/n)+1;
  j:=(step-1) mod n+1;
  count:=0;

  if(n mod 2=0 or n<3) then return fail; fi;
  if(j=1) then A:=[Minimum(Filtered([2..n],i->ForAll([1..n-1],t->L[t][1]<>i)))]; else A:=Filtered([1..n],s->ForAll([1..n-1],t->L[t][j]<>s) and ForAll([1..n],t->L[i][t]<>s)); fi;
  for k in A do
    L[i][j]:=k;
    L[i+(n-1)/2][k]:=j;
    if((j=n and CycleLengths(PermList(L[i]),[1..n])=[n]) or j<n) then
      if(i=(n-1)/2 and j=n) then count:=count+1;
      else count:=count+EnumerateHamiltonDecompositionsBacktrackingAlgorithm(n,L,step+1); fi;
    fi;
    L[i][j]:=0;
    L[i+(n-1)/2][k]:=0;
  od;
  return count;
end;;

EnumerateHamiltonDecompositions:=function(n)
  local L;
  if(n mod 2=0 or n<3) then return fail; fi;
  if(n=3) then return 1; fi;
  L:=List([1..n-1],i->List([1..n],j->0));
  L[1]:=List([1..n],i->i mod n+1);
  L[1+(n-1)/2]:=ListPerm(Inverse(PermList(List([1..n],i->i mod n+1))));
  return Factorial(n-2)*EnumerateHamiltonDecompositionsBacktrackingAlgorithm(n,L,n+1);
end;;

The extra data point comes from assuming that (12..n) is one of the cycles, then multiplying the result by (n-2)!. This is legitimate since each decomposition contains a unique cycle with the edge 12, and by permuting the remaining n-2 edges, we generate a unique decomposition with the cycle (12..n). There are no automorphisms under this group action, so each orbit has cardinality (n-2)!.

fa.functional analysis - L_p norm balls for 12?

The L_1 ball in 2D is shaped like a diamond (L_1 is also known as the Manhattan norm). The L_∞ ball is shaped like a square (L_∞ is also known as the supremum norm). They are similar, i.e. have same shape. The L2 ball is shaped like a circle.

Hypothesis: For all p in the interval (1,2), there is q>2 such that the q-ball and the p-ball are similar. One further hypothesis is that this occurs precisely when p,q are Hölder conjugates.

I wasn't sure how to tag this.

microscopy - Online repositories of scanning electron microscope photographs?

First off I'd like to reccomend the University of Dartmouth's publicly available collection located here. They have both SEM and TEM images of a wide range of organisms and cells from algae to see urchins through everything from cholera to mammalian cells. Images are high quality, fully captioned and properly attributed.

I'm a little confused as to the problem you're having with google - see my original comment on your question. However I have looked at the photos of Wikimedia Commons in the SEM category and see that they are either images taken with, images of SEM Microscopes or technical drawings regarding their function. I assume that you're looking for the images taken with SEM.

I was able to find a great number of similar images to those in the commons with a Google Image search - I read above that you were having problems with this approach so perhaps try the search term "SEM images" in image search (or this search to avoid duplications from images you have already seen on Wiki Commons)?

Examples found on just the first page:

cell biology - Is "exhaustion" of the Hodgkin-Huxley membrane at constant stimulation a real phenomenon?

(I probably ought to have a pat answer to this on the tip of my mind, but since I don't I'm going to wing it. This is probably just an opportunity to make an utter fool of myself. Please treat everything that follows with extreme suspicion.)

I think this is effectively an artefact of the model. That may not be true in the strictest sense -- it is possible such behaviour could be produced in real experimental preparations -- but it would require driving them in drastically non-physiological ways. I am not aware of this having been done, but I'm sure that's just my ignorance -- I would be surprised if nobody has tried.

In physiological terms, though: where would such a large constant current come from in a real cell? Where would the charges go? How would that be sustained with realistic boundary conditions?

While it is possible that such an effect can come into play very transiently in living systems, it seems unlikely that a true steady state of this kind could ever be reached.

at.algebraic topology - Does homology detect chain homotopy equivalence?

Yes, this is true. Suppose $C_*$ is such a chain complex of free abelian groups.

For each $n$, choose a splitting of the boundary map $C_n to B_{n-1}$, so that $C_n cong Z_n oplus B_{n-1}$. (You can do this because $B_{n-1}$, as a subgroup of a free group, is free.) For all $n$, you then have a sub-chain-complex $cdots to 0 to B_n to Z_n to 0 to cdots$ concentrated in degrees $n$ and $n+1$, and $C_*$ is the direct sum of these chain complexes.

Given two such chain complexes $C_*$ and $D_*$, you get a direct sum decomposition of each, and so it suffices to show that any two complexes $cdots to 0 to R_i to F_i to 0 to cdots$, concentrated in degrees $n$ and $n+1$, which are resolutions of the same module $M$ are chain homotopy equivalent; but this is some variant of the fundamental theorem of homological algebra.

This is special to abelian groups and is false for modules over a general ring.