dg.differential geometry - Principal Bundle Connection Correspondence for two descriptions of the $mathbb{CP}^2$

The answer is no. You can see that for example from the connection one forms which are Lie algebra valued. In the first case they are u(2) valued and in the second case they are u(1) valued. However, in the case of CP2 (which also generalizes to CPn), the U(1) and SU(2) factors of the isotropy group U(2) commute, this means that given a U(2) connection on U(2)-->SU(3)-->CP2, you can project it to the U(1) factor to obtain a U(1) connection on U(1)-->S^5-->CP2, such that both connections will have the same horizontal subspaces.

dg.differential geometry - Singular matrix and wedge product

Your condition on $X$ is that it has a kernel, and that by itself does not mean that
$X wedge X$ doesn't have to vanish. For instance in five dimensions, you could have
$$X = begin{pmatrix} 0 & 1 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 end{pmatrix}.$$
Then letting $t = (0,0,0,0,1)$, your condition is satisfied, but $X wedge X$ is not zero.

However, it is true in $2n$ dimensions that a non-zero $t$ exists if and only if $X^{wedge n} = 0$. That's because $X^{wedge n}$ is proportional to the Pfaffian of $X$, which is a certain square root of the determinant of $X$. (In odd dimensions, the determinant of an antisymmetric matrix is zero by calculation, while the Pfaffian is set to zero by definition. So $t$ always exists in this case.)

order theory - not sure what these basic FO symbols mean

I know some basic logic symbols, but i'm not sure what this formula means:

Fragment:
http://i35.tinypic.com/14jt1n9.png

Full paper:
http://www.newton.ac.uk/preprints/NI07003.pdf

in particular, what is the subscript "1<=i

and the N with the A superscript. the multiple typefaces of "A".

this is beyond the basic and/or/exists/forall that i know in basic logic

is there a statement of this formula and its conditions in english words?

Thanks!

Updated: oops my bad, I should've read the part where they defined it!

knot theory - Why is the Alexander polynomial a quantum invariant?

This is far from being a full answer, but I think the key is 'how do you see the skein relation from the classical definition?'. Once you know the skein relation, it's easy to show it's a quantum invariant, and the skein relation was discovered in the 1960's, long before anyone knew about quantum groups.

evolution - Are there genes in humans from the common ancestor of all organisms?

I very much agree with bitwise's answer. I just want to point out that even in terms of nucleotide sequence there are some extremely conserved genes.

The most highly conserved are ribosomal RNA genes. The image below shows the conservation of 16S rRNAs from archaea, bacteria, and eukaryotes (eukaryotes do not have 16S in their cellular genome but the gene is present in the DNA of mitochondria and chloroplasts). Residues in red are conserved across all domains of life (image source):

A very nice study from 2008 [1] identified tRNA genes as the next most conserved and ABC transporter nucleotide binding domains (NBDs) after that. The alignment below (taken from the paper) shows the conservation of NBDs from various species. Note that they include eukaryotes, archaea and bacteria:
[2]

If you look at orthologous genes in general, without specifying a specific level of sequence identity, then you can find various genes (notably those coding for the enzymes of the glycolysis pathway as mentioned in the comments) that are conserved across most, if not all, cellular life.

So, to answer your question probably yes. At least, there are genes in humans today that can be found in all or almost all other life forms. Whether there were present in LUCA we cannot know for sure, it does seem likely however given how widespread they are today.

Reference

1. Isenbarger T.A. et al, The most conserved genome segments for life detection on Earth and other planets, Orig Life Evol Biosph. 2008 Dec;38(6):517-33.

soft question - Tropicalizing the learning

Yes, it is possible. People who have done excellent tropical work with little Algebraic Geometry background include Federico Ardila, Michael Joswig and Josephine Yu. (I hope I won't insult any of these people by saying that they do not strike me as having much algebraic geometry.)

However, I have had bad luck introducing people to tropical geometry without talking about
valued fields, Grobner degenerations, toric varieties and the other algebraic technology.
I can give a nice colloquium talk or write a nice expository paper where I gloss over this material. But this leaves the reader without an intuition to figure out which questions are reasonable to ask, or any idea of where nontrivial results might come from. This is especially true because so much tropical work right now is not solving specific problems formulated by experts, but in finding the definitions and theorems to make precise the phenomena which people have observed.

ADDED I also like Ben's answer. There are parts of tropical geometry which use very serious algebraic geometry, but there are also parts where it is being used for motivation and intuition. You could probably get a lot of what you need from Cox-Little-O'Shea, Fulton's "Algebraic Curves" and some good reference on Grassmannians and hyperplane arrangements.

reference request - Books that discuss spectral graph theory and its connection to eigenvalue problems in hyperbolic geometry

Lubotzky and Zuk's book on property ($tau$) discusses expander graphs and the minimal eigenvalue of the Laplacian on covers of Riemann surfaces. See for example Prop. 2.9 in the book. There's also his book Discrete groups, expanding graphs and invariant measures, but it's not available online. I don't know of relations between eigenvalues of graphs and eigenvalues of surfaces deeper into the spectrum, since the correspondence is only coarse. The papers of Robert Brooks and his collaborators are quite readable.

mathematics education - How to study a math text

I am a big believer in learning by exploration: It builds independence and confidence... something you need if you plan to pursue further your math journey. That means when you find an interesting result or theorem in a book, explore it's consequences, see the details of the proof, etc. It helps your understanding.

Most important when learning math: do a lot of exercises. If possible, do a lot of difficult ones. Really, do them, there is no learning if don't get your hands dirty! Don't expect to solve every single difficult problem you try, but seriously, trying them will help you tremendously. As a rule of thumb, I never spend more that half an hour on a problem, unless it's really something very motivating.

And last: Life is sequence of choices, and exploration takes time, a lot of it!
Which means: you learn much faster by reading (and paying attention to what you read, they call it focus), but you don't master something unless you get your hands dirty, as I said.

algebraic number theory - Is there a notion of Galois extension for Z / p^2?

The above title is in fact a special case of what I want to ask.

Certainly we have a well defined notion of Galois extension for $ mathbb{Q}_p $. The intersections of these extensions to the ring of integer of the absolute algebraic closure of $mathbb{Q}_p$ give us a notion of Galois extensions for $mathbb{Z}_p $. ( I know that there is a notion of Galois extension for commutative rings, and I believe that it should give us this. Am I correct?)

Let's go further. Let $A_K$ be the ring of integer in a finite Galois extension $K$ of $ mathbb{Q}_p$. Let $e$ be the ramification degree of $K$ over $mathbb{Q}_p$. The injection of $ mathbb{Z}_p$ into $A_K$ will induce an injection of $ mathbb{Z} / p^n $ into $ A_K / mathfrak{p}^{en} $. In this picture, there seems to be some desire to say that $ A_K / mathfrak{p}^{en} $ is the correct notion Galois of extension of $ mathbb{Z} / p^n $. But there are problems; taking this notion of Galois extension, if $K$ is has ramification degree $e >1$, the corresponding extension $ A_K /p^e $ is not a field (it is not even an integral domain).

Question 1: Is there any notion of Galois extensions corresponding to what I desire?

Question 2: Can a class field theory (i.e a nice description of absolute abelian Galois extension) of $ mathbb{Z}/p^n$ be developed in this context? Is there any relationship between this and the local class field theory of $mathbb{Q}_p$ ( which is the same as that of $mathbb{Z}_p $)?

biotechnology - glossary of biotechnological engineering

Similar to A good book for history of biology/biotechnology for lay people

but not for lay people.

Can anyone recommend an advanced dictionary of biotechnological engineering concepts in English (contemporary, not historical)?

There doesn't seem to be anything adequate at my research institution nor even country. I have been looking for PDF's and e-books since they can be previewed more easily than ISBN #'s, but any suggestions are welcome.

human biology - What controls the feeling of discomfort/comfort before and after sleep?

I'm interested in which biological systems or hormones are involved in the following phenomenon:

Before sleep it may be difficult to find a comfortable position, and muscular aches and pains are more evident.

After 8 hours of sleep, many more positions are comfortable, and muscle aches/tension are non-existent. Some time after awakening, the feeling disappears.

What causes the difference in feeling between prior to falling asleep and after? Are there some hormones involved? Is it caused by some form of thought/feeling inhibition (e.g., part of a brain "falls asleep")?

If I remember correctly, after long running muscles accumulate some compound that is a result of metabolizing energy without enough oxygen, which causes muscles to ache. Could a similar process be involved?

genetics - Can genes that activate transcription factors also called be called transcription factors?

You need to re-write your question, it is ambiguous and your use of terms is incorrect... Assumption: by "activation" you mean "activation of transcription resulting in the expression of the transcription factor"

1) Transcription factors are proteins

2) Genes are comprised of DNA elements

A transcription factor can be involved in initiating the EXPRESSION of a transcription factor, whereafter that second distinct transcription factor initiates the EXPRESSION of another gene that encodes another transcription factor.

Answer: No, genes do not "activate" transcription factors*

*Unless you are proposing the philosophical question of whether the DNA binding domain itself, which endows the transcription factor with a state of being active duty (i.e. fulfilling its purpose as a transcription factor), and thus that purpose is fulfilled only when the DNA binds the the TF, then a DNA binding domain can indeed "activate" the TF... but I'm pretty sure this isn't what you're asking.

graph theory - Why are there 1024 Hamiltonian cycles on an icosahedron?

Fix one edge $e$ of the graph (1-skeleton) of an icosahedron.
By a computer search, I found that there are 1024 Hamiltonian cycles that include $e$.
[But see edit below re directed vs. undirected!]
With the two endpoints of $e$ fixed, there are 10 "free" vertices in the cycle.
Because $1024=2^{10}$, it makes me wonder if there might be a combinatorial viewpoint that
makes it evident that there are 1024 cycles including a fixed edge.
It could just be a numerical coincidence, but if anyone
sees an idea for an argument, I'd appreciate hearing it. Thanks!


Incidentally, this MathWorld page says there are 2560 Hamiltonian cycles all together (without
the fixed edge condition). (Thanks to Kristal Cantwell for pointing me to this page.)

Edit. I apologize for misleading! :-/ When I looked at the full output of paths more carefully,
I realize I inadvertently computed directed cycles, so each is represented twice, i.e., both
$$ lbrace 2, 7, 6, 11, 8, 9, 4, 10, 12, 5, 3, 1 rbrace $$
$$ lbrace 1, 3, 5, 12, 10, 4, 9, 8, 11, 6, 7, 2 rbrace $$
are included, etc. So there are 512 undirected cycles, 1024 directed cycles.
The paths are listed here: hpaths.html.

bioinformatics - How are Genetic Circuits Modelled?

I've read a recent Nature Methods paper by Moon T.S. et al, in which a synthetic genetic circuit consisting of layered logic gates was created. For example, the paper, a circuit is modelled in Figure 4a. How is this circuit computationally modelled? Can you refer me to any graduate level texts on the subject?

ap.analysis of pdes - Rellich-Necas identity

(Boo! I tried to post this in a comment to Ady, but the HTML Math won't parse right. So here goes. Sorry about the really long equation being broken up not very neatly.)

Googling Rellich-Necas turns up a bunch of recent papers by LUIS ESCAURIAZA in which the identities are used. But as far as I can tell the identity is just a simple differential equality obtained from symbolic manipulation of terms. The following seems to be a straight-forward version of the identity: let $A = (A_{ij})$ be a symmetric bilinear form (with variable coefficients) on R^N, $v$ a vector field, $u$ a function, and $delta$ denoting the Euclidean divergence, we have

$ delta( A(nabla u,nabla u) v) = 2 delta( v(u) A(nabla u)) + delta(v) A(nabla u,nabla u)$
$- 2A(nabla u) cdot nabla v cdot nabla u - 2 v(u) delta(A(nabla u)) + v(A)(nabla u,nabla u)$

Where $v(u)$ is the partial derivative of $u$ in the direction of $v$, and $A(nabla u)cdotnabla v cdot nabla u$ is, in coordinates, $partial_i u A_{ij} partial_j v_k partial_k u$ with implied summation, and $v(A)$ is the symmetric bilinear form obtained by taking the $v$ partial derivative of the coefficients of $A$.

Verifying that the identity is true should just be a basic application of multivariable calculus.

dg.differential geometry - tangent bundle of a fiber bundle

I just wanted to elaborate on Benoît Kloeckner's answer, so if you like what I say, please upvote his answer.

By a frame, I mean a basis of the tangent space at a point on a smooth
manifold $M$. The space $F$ of all possible frames, called the frame
bundle, is a principal $GL(n)$-bundle over the manifold, $n$ is the
dimension of the manifold. A point in $F$ is given by $(x, e)$, where
$x in M$, $e = (e_1, dots, e_n)$, and $e_i in T_xM$. Associated
with each point is the dual frame $omega^1, dots, omega^n in
T_x^*M$. Let $pi: F rightarrow M$, $pi(x,e) = x$, denote the
natural projection.

There is a natural set of $n$ $1$-forms $hatomega^1, dots,
hatomega^n$ on $F$, which are called either "tautological" or
"semi-basic" and act as follows: If $v in T_{(x,e)}F$, then
$
langle hatomega^i,vrangle = langleomega^i,pi_* v rangle,
$
where $omega^1, dots, omega^n in T^*_xM$ form a dual basis to the basis
$e_1, dots, e_n in T_xM$. These forms have the universal property
that given any section $s: M rightarrow F$, $s^*baromega^i$ are
$1$-forms on $M$ dual to the moving frame given by the $e_i$.

You can check that any connection $nabla$ on $T_*M$ determines a set
of global $1$-forms $hatomega^i_j$ on $F$, such given any section
$s = (s_1, dots, s_n): M rightarrow F$, $nabla s_j =
s_is^*hatomega^i_j$. Therefore, a connection on $F$ gives a set of global
$1$-forms $hatomega^1, dots, hatomega^n, hatomega^1_1, dots, hatomega^n_n$
that trivialize $T^*F$. The dual vector fields
trivialize $T_*F$.

Since there always exists a connection on $T_*M $, this shows that $F$
has a parallelizable tangent bundle. The same argument can be extended
to any principal $G$-bundle of tangent frames. As observed by
Hoeckner, the case $G = O(n)$ corresponds to a Riemannian structure.

This, of course, does not answer the original question, but it is a
important case where the answer is yes. These global $1$-forms are
extremely useful in many contexts; the work of Robert Bryant
illustrates this.

ct.category theory - Can adjoint linear transformations be naturally realized as adjoint functors?

There a simple way to make this work:

Say T:V->X is a map of inner-product vector spaces. You can view V as a category, where Hom(v,w) is a singleton set containing one real number, the inner product <v,w>, and similarly for X.

Composition, a binary operation, is defined (stupidly, as in any category with singleton hom-sets) as follows:

Comp_{uvw} : Hom(u,v)xHom(v,w) -> Hom(u,w)

by (<u,v>,<v,w>) |-> <u,w>

Then the adjoint T*:X->V satisfies

<Tv,x>=<v,T*x>, i.e. Hom(Tv,x)=Hom(v,T*x)

, meaning it is a right adjoint to T (in a very strong sense: we have equality of these hom-sets instead of just natural isomorphism).

The triviality of this example reflects the fact that that T and T* are called "adjoint" simply because they belong on opposite sides of a comma :)

In general, if H is any function of two variables, we can say that g is right adjoint to f "with respect to H" if H(f(a),b)=H(a,g(b)), and say that "adjoint functors" are "adjoint with respect to Hom" (up to natural isomorphism, of course).

human biology - What is the functional difference between hemoglobin and ferritin?

Hemoglobin is the protein of erythrocytes (red blood cells) which has ferrous ions (Fe2+) bound in its subunits. These are able to keep oxygen bound which enables the cell to transport oxygen through the circulation. It's not really for storage of iron, it's for using it.

Ferritin is the actual storage protein, cells express it to store iron in case of deficiency and also to regulate the amount they have in the cell. According to this, it's mainly expressed in muscle, liver and kidney cells (but I'm not too sure about the details of that study so I might have got that wrong).

Edit: just found this in one of my old lectures; unfortunately it doesn't quote sources: ferritin stores iron in liver and heart. The total iron in the body is ~3.9g, of which 2.5g are in use in hemoglobin, 500mg in stores (an additional 250mg in the liver), 150mg in bone marrow, 300mg in myoglobin and 150mg in other enzymes. The remaining 5mg are bound to transferrin in the plasma.

universal algebra - Is there a notion of congruence relation for essentially algebraic structures?

In universal algebra there is the notion of congruence relation: Consider a (1-sorted) algebraic structure, i.e. a set $A$ with a bunch of finitary operations $f_i$ satisfying equations.

A congruence relation is an equivalence relation $sim$ on $A$ such that the operations on $A$ produce well-defined operations on the set $A/sim$ of equivalence classes by applying them to representatives. I.e. for any operation $f$ on $A$ the operation $bar{f}$ on $A/sim$ given by $bar{f}([x_1],...,[x_n]):=[f(x_1,...x_n)]$ is well-defined, i.e. if $x_1 sim y_1, ... , x_n sim y_n$ then $f(x_1,...,x_n) sim f(y_1,...,y_n)$ for all operations $f$ of the given structure.
Thus these relations are the right ones to form quotients inside the given category of algebraic structures.

An essentially algebraic structure is a (if 1-sorted) or several (if many-sorted) sets with partially defined operations satisfying equational laws, where the domain of any given operation is a subset defined by equations between previously defined operations (equivalenty: it is a $Set$-model of a finite limit sketch). The standard example are categories, where one has three global operations, identity, source and target, and a partial operation, composition, defined only for certain pairs of morphisms.

My question is: Is there a notion of congruence relation for these more general algebraic structures? E.g. one equivalence relation on each set satisfying the analogous properties to the above? If so have these "congruence relations" been studied, do they e.g. form lattices?

Motivation: Just curiosity really. I asked myself this question, after reading this MO-question of Colin Tan, which might be a special case. He asks whether there is a way to collapse two objects in a category. If there was a lattice of congruence relations on a category, there might be the congruence relation generated by the relation which identifies just the two objects (this would of course mean to treat categories in an "evil", non-two-categorical way, but that was what the question sounded like to me).
Googling did reveal nothing, so I ask you people...

reference request - How to prove that a map is a Serre fibration?

I want to prove that the homotopy groups of some topological space $B$ of interest to me (not a CW complex) are trivial. I have a strategy of proof that consists in introducing another space $E$ that is contractible, and easily comes with a continuous surjection $pi :Eto B$. If I can prove that any continuous map $f:I^kto B$ lifts to a continuous map $tilde f : I^kto E$, then I'm done.

If I am not mistaken, this lifting property is true as soons as $pi$ is a Serre fibration. Here is my question: are there classical way to prove such a thing, and were can I learn them (or simply learn about Serre fibrations)?
Of course, any reference for the initial problem, which seems slightly weaker, is welcome too.

I guess that I should be able to manage my case by hand, but I think it may be an opportunity to learn more mathematics.

pr.probability - What are the big problems in probability theory?

In limit theorems, one of the biggest problem is to give an answer to Ibragimov's conjecture, which states the following:

Let $(X_n,ninBbb N)$ be a strictly stationary $phi$-mixing sequence, for which $mathbb E(X_0^2)<infty$ and $operatorname{Var}(S_n)to +infty$. Then $S_n:=sum_{j=1}^nX_j$ is asymptotically normally distributed.

$phi$-mixing coefficents are defined as
$$phi_X(n):=sup(|mu(Bmid A)-mu(B)|, Ainmathcal F^m, Bin mathcal F_{m+n},minBbb N ),$$
where $mathcal F^m$ and $mathcal F_{m+n}$ are the $sigma$-algebras generated by the $X_j$, $jleqslant m$ (respectively $jgeqslant m+n)$, and $phi$-mixing means that $phi_X(n)to 0$.

It was posed in Ibragimov and Linnik paper in 1965.

Peligrad showed the result holds with the assumption $liminf_{nto +infty}n^{-1}operatorname{Var}(S_n)>0$. It also holds when $mathbb Elvert X_0rvert^{2+delta}$ is finite for some positive $delta$ (Ibragimov, I think).

ag.algebraic geometry - Does a birational involution of C^n always have a fixed point?

The answer is no. There are plenty of counterexamples, for example the map given by "a-fortiori", which is $(x,y)mapsto (x+1/y,-y)$, which can be generalised to any dimension.

The good question is probably to look at birational involutions of $mathbb{P}^n$.

analytic number theory - Injectivity of Transfer (Verlagerung) map

I'm not sure how to answer the more philosophical question (it's likely you could encode enough of the axioms to force the purely group-theoretic version of the question to be true, but to ask whether that's what's "really" going on....), but it's certainly not true for all pairs of groups that fit in a similar commutative diagram -- in fact, it's not even true for all such pairs of groups coming from similar questions in algebraic number theory. For example, instead of taking the maximal abelian extension of $K$, take the maximal abelian extension of $K$ which is unramified outside of a set of primes, or split completely at a set of primes, or both -- and you'll pick up a kernel to you transfer map (see Gras, Class Field Theory for some specific calculations of kernels like this). A very relevant related topic worth bringing up is the theorem of Gruenberg-Weiss, which gives an impressively vast generalization of the group-theoretic (and hence the ideal-theoretic) principal ideal theorem entirely in terms of kernels of related transfer maps.

nt.number theory - Polynomial representing all nonnegative integers

This is a cute problem! I toyed with it and didn't really get anywhere - I got the strong impression that it requires fields of mathematics that I am not expert in.

Indeed, given that the problem seems related to that of counting integer solutions to the equation $f(x,y) = c$, one may need to use arithmetic geometry tools (e.g. Faltings' theorem). In particular if we could reduce to the case when the genus is just 0 or 1 then presumably one could kill off the problem. (One appealing feature of this approach is that arithmetic geometry quantities such as the genus are automatically invariant (I think) with respect to invertible polynomial changes of variable such as $(x,y) mapsto (x,y+P(x))$ or $(x,y) mapsto (x+Q(y),y)$ and so seem to be well adapted to the problem at hand, whereas arguments based on the raw degree of the polynomial might not be.)

Of course, Faltings' theorem is ineffective, and so might not be directly usable, but perhaps some variant of it (particularly concerning the dependence on c) could be helpful. [Also, it is overkill - it controls rational solutions, and we only care here about integer ones.] This is far outside of my own area of expertise, though...

The other thing that occurred to me is that for fixed c and large x, y, one can invert the equation $f(x,y) = c$ to obtain a Puiseux series expansion for y in terms of x or vice versa (this seems related to resolution of singularities at infinity, though again I am not an expert on that topic; certainly Newton polytopes seem to be involved). In some cases (if the exponents in this series expansion are favourable) one could then use Archimedean counting arguments to show that f cannot cover all the natural numbers (this is a generalisation of the easy counting argument that shows that a 1D polynomial of degree 2 or more cannot cover a positive density set of integers), but this does not seem to work in all cases, and one may also have to use some p-adic machinery to handle the other cases. One argument against this approach though is that it does not seem to behave well with respect to invertible polynomial changes of variable, unless one works a lot with geometrical invariants.

Anyway, to summarise, it seems to me that one has to break out the arithmetic geometry and algebraic geometry tools. (Real algebraic geometry may also be needed, in order to fully exploit the positivity, though it is also possible that positivity is largely a red herring, needed to finish off the low genus case, but not necessary for high genus, except perhaps to ensure that certain key exponents are even.)

EDIT: It occurred to me that the polynomial $f(x,y)-c$ might not be irreducible, so there may be multiple components to the associated algebraic curve, each with a different genus, but presumably this is something one can deal with. Also, the geometry of this curve may degenerate for special c, but is presumably stable for "generic" c (or maybe even all but finitely many c).

It also occurs to me that one use of real algebraic geometry here is to try to express f as something like a sum of squares. If there are at least two nontrivial squares in such a representation, then f is only small when both of the square factors are small, which is a 0-dimensional set and so one may then be able to use counting arguments to conclude that one does not have enough space to cover all the natural numbers (provided that the factors are sufficiently "nonlinear"; if for instance $f(x,y)=x^2+y^2$ then the counting arguments barely fail to provide an obstruction, one has to use mod p arguments or something to finish it off...)

EDIT, FOUR YEARS LATER: OK, now I know a bit more arithmetic geometry and can add to some of my previous statements. Firstly, it's not Faltings' theorem that is the most relevant, but rather Siegel's theorem on integer points on curves - the enemy appears to be those points $(x,y)$ where $x,y$ are far larger than $f(x,y)$, and Siegel's theorem is one of the few tools available to exclude this case. The known proofs of this theorem are based on two families of results in Diophantine geometry: one is the Thue-Siegel-Roth theorem and its variants (particularly the subspace theorem), and the other is the Mordell-Weil theorem and its variants (particularly the Chevalley-Weil theorem). A big problem here is that all of these theorems have a lot of ineffectivity in them. Even for the very concrete case of Hall's conjecture on lower bounding $|x^2-y^3|$ for integers $x,y$ with $x^2 neq y^3$, Siegel's theorem implies that this bound goes to infinity as $x,y to infty$, but provides no rate; as I understand it, the only known lower bounds are logarithmic and come from variants of Baker's method.

As such, a polynomial such as $f(x,y) = (x^2 - y^3 - y)^4 - y + C$ for some large constant C already looks very tough to analyse. (I've shifted $y^3$ here by $y$ to avoid the degenerate solutions to $x^2=y^3$, and to avoid some cheap way to deal with this polynomial from the abc conjecture or something.) The analogue of Hall's conjecture for $|x^2-y^3-y|$ suggests that $f(x,y)$ goes to $+infty$ as $x,y to infty$ (restricting $x,y$ to be integers), but we have no known growth rate here due to all the ineffectivity. As such, we can't unconditionally rule out the possibility of an infinite number of very large pairs $(x,y)$ for which $x^2-y^3-y$ happens to be so close to $y^{1/4}$ that we manage to hit every positive integer value in $f(x,y)$ without hitting any negative ones. However, one may be able to get a conditional result assuming some sufficiently strong variant of the abc conjecture. One should also be able to exclude large classes of polynomials $f$ from working; for instance, if the curve $f(x,y)=0$ meets the line at infinity at a lot of points in a transverse manner, then it seems that the subspace theorem may be able to get polynomial bounds on solutions $(x,y)$ to $f(x,y)=c$ in terms of $c$, at which point a lot of other tools (e.g. equidistribution theory) become available.

Another minor addendum to my previous remarks: the generic irreducibility of $f(x,y)-c$ follows from Bertini's second theorem, as one may easily reduce to the case when $f$ is non-composite (not the composition of two polynomials of lower degree).

books - Number theory textbook based on the absolute Galois group?

If you want to go further in understanding this point of view, I would advise you to begin learning class field theory. It is a deep subject, it can be understood in a vast variety of ways, from the very concrete and elementary to the very abstract, and although superficially it appears to be limited to describing abelian reciprocity laws, it in fact plays a crucial role in the study of non-abelian reciprocity laws as well.

The texts:

Ireland and Rosen for basic algebraic number theory, a Galois-theoretic proof
of quadratic reciprocity, and other assorted attractions.

Cox's book on primes of the form x^2 + n y^2 for an indication of what some of the content of class field theory is in elementary terms, via many wonderful examples.

Serre's Local Fields for learning the Galois theory of local fields

Cassels and Frolich for learning global class field theory

The standard book at the graduate level to learn the arithmetic of non-abelian (at least 2-dimensional) reciprocity laws is Modular forms and Fermat's Last Theorem, a textbook on the proof by Taylor and Wiles of FLT. But it is at a higher level again.

I don't think that you will find a single text on this topic at a basic level (if basic
means Course in arithmetic or Ireland and Rosen), because there is not much to say beyond what you stated in your question without getting into the theory of elliptic curves and/or the theory of modular forms and/or a serious discussion of class field theory.

Also, as basic suggests, you could talk to the grad students in your town, if not at your institution, then at the other one down the Charles river, which as you probably know is currently the world centre for research on non-abelian reciprocity laws (maybe shared with Paris). Certainly there are grad courses offered on this topic there on a regular basis.

physiology - How does extracellular potassium ion concentration and calcium ion concentration affect the excitability of a cell?

Well, the answer which explains the difference between calcium and potassium is quite simple. Do we agree that charge separation (between the two sides of the membrane) is the thing which creates the potential? meaning, if the charge concentrations at the two sides were the same, then the membrane potential was zero. right?
So here is the thing. You have much more Ca outside the cell than inside and you have much more potassium inside than outside. So now, if you add Ca outside, you increase the difference between the two sides (the separation of Calcium ions) so that Ca will be in a greater state of unrest, the system will be less stable and the potential will be higher. On the other hand, when you add K outside, you narrow the gap between its two concentrations on the two sides (decrease its gradient), so now it'll experience less tension, less unrest, hence contribute less potential to the membrane.
Remember that in our cells, the ion pumps are the ones which serve as the battery that keeps the potential fixed. When do you think the pumps will have to work harder and to do more work (voltage times charge is work), when you add more K outside, or when it's the Ca which is being added?

pr.probability - Peakedness of multimodal distributions

In Probability theory, does there exist some measures of peaked-ness for multi-modal distributions. I guess kurtosis as such would not be a good measure of peaked-ness for multimodal distributions. Please correct me if I am wrong. Can you point me to some of them which are simpler to compute.

ct.category theory - How cavalier can I be when demanding a category have direct sums?

I don't think your definition of direct sum is quite right (even if you add the obviously necessary condition that the isomorphisms be natural). My understanding is that a direct sum / biproduct is an object that is both a product and a coproduct in a compatible way. This is usually phrased by saying that you have coproducts and products, and the unique morphisms $0to 1$ and $Xsqcup Y to Xtimes Y$ are isomorphisms. In terms of your definition, I think this would be equivalent to saying that you have a zero object, and the composite isomorphism
$$hom(Z,Z) cong hom(X,X)times hom(Y,X)times hom(X,Y)times hom(Y,Y)$$
relates $1_Z$ to $(1_X,0,0,1_Y)$ (where $0$ is the map factoring through the zero object). It's true, but not (I think) obvious, that if you have products and coproducts and an arbitrary natural family of isomorphisms $Xsqcup Y cong Xtimes Y$, then you actually have biproducts. But I don't think this works as a definition for an individual biproduct.

As to your actual question, I don't have a complete answer, but one thing to note is that in the world of categories enriched over additive monoids (or groups), direct sums are absolute (co)limits, aka Cauchy (co)limits. That means that they are automatically preserved by any AbMon-enriched functor, and moreover the 2-category of AbMon-enriched categories with direct sums is reflective in the 2-category of all AbMon-enriched categories. Therefore, after performing any "free" or "quotient" or "colimit" construction on AbMon-enriched categories, you can always apply the reflector to add any direct sums that might be missing (and whatever direct sums you might already have had won't be changed). In particular, this provides a construction of an additive category "presented" by any notion of generators and relations: first generate the free AbGp-enriched category, then reflect into additive categories.

In general, it's not obvious to me that if you add some additional structure freely (like kernels or cokernels), then apply the above "Cauchy-completion" reflector, that the presence of the new thing you added is preserved by the reflector. But if it isn't, then perhaps some sort of sequential colimit of successive approximations could be performed. Note that of the other constructions you mentioned, splitting of idempotents is also an absolute (co)limit, so it behaves similarly to direct sums, whereas kernels and cokernels are not.

However, none of this really answers the question you actually asked, which is whether such "free" constructions in the world of AbGp-enrichment already preserve the presence of direct sums, without the need to Cauchy-complete. I would guess that in general they don't, but I don't have a counterexample.

biochemistry - Does ethanol destroy RNase?

You don't want to only inhibit or temporarily denature RNAses, if you work with RNA you have to permanently inactivate the RNAses. I work with RNA, and I haven't seen anyone use ethanol to remove RNAses, I would not trust it to work reliably. RNAses are really stable enzymes, they even survive autoclaving to some part, so I would not be suprised if they can refold after ethanol exposure.

The usual methods to remove RNAses are

• Treatment with 0.1% DEPC (heat up to at least 60 °C, or autoclave if possible to remove residual DEPC later). Cannot be used with e.g. Tris buffer or anything else that reacts with it. DEPC is carcinogenic, so you should follow the proper safety procedures.


• Anything made of metal or glass, heat up to 250 °C for around 2 hours


• 1M NaOH for at least 30 minutes (you need to wash thoroughly to remove the NaOH)

You should be aware that anything that destroys RNAses will likely do the same to your RNA.

Autoclaving does not protect completely against RNAses, but it still reduces RNAse activity significantly. So while you should not rely on autoclaving to get rid of RNAses, in my experience it is sufficient if you start from a source that is unlikely to contain a lot of RNAses. Pipette tips (not loose tips out of a bag, to minimize handling) and millipore-filtered water are pretty safe after they are autoclaved.

There are also commercial RNAse inhibitors, but I've only used them in cases where I had to mix RNA and protein, and the protein wasn't completely RNAse-free.

epidemiology - Where can I find a list of diseases and their incidence?

A couple resources that might be of use:

If you subscribe to it (annoyingly, I cannot get at my copy), the American Academy of Pediatric's Red Book may contain incidence figures. I don't recall, but it contains a remarkable amount of information in it.

For viral diseases, most if not all entries in Fields Virology have an Epidemiology section, and many of these touch on the incidence rate, especially for more common diseases.

In addition to the CDC (listed in another answer), take a look at the Agency for Healthcare Quality Research (AHRQ)'s Healthcare Cost and Utilization Project (HCUP) data. While this will only give you incidence in terms of hospitalizations rather than all cases, it may be potentially useful.

Beyond that, I'd suggest looking for particular papers, or search for "DISEASE OF INTEREST" meta-analysis in Google Scholar or PubMed for overall incidence figures.

Additionally, a useful formula:

Prevalence = Incidence * Duration. Which in turn means Prevalence/Duration = Incidence. If you can find those two pieces of information, you can calculate the incidence for what you want.

There's also a way to extract incidence from a mathematical modeling paper about a disease which reports an R0 figure, but that may be a touch technical given your stated desires.

reference request - Have this subclass of split graphs been studied before?

I am interested in the properties of the following subclass of split graphs:

The class consists of all split graphs $G=(Ccup I)$ where $C$ is a clique and $I$ an independent set, and every pair of vertices in $I$ have at least one common neighbor in $C$.

Does this class of graphs have a special name? Has this class and its properties been studied? If so, what would be some good references for this?

ds.dynamical systems - Why do dynamicists worry so much about differentiability hypotheses in smooth dynamics?

My impression always was that this is because of ingrained mathematical culture of seeking the most precise requirements for a particular theorem to hold. That way, when a non-smooth situation eventually does emerge, the theorems are already in place to deal with it. It's one of the differences in culture between pure mathematics, applied mathematics and physics.

I always wondered about looking at the problem the other way around: if you assume that stronger and stronger continuity holds, then what extra properties hold? What about $C^{infty}$ cases and analytic, is there a gap in between?

The same phenomenon appears in a lot of the optimization literature, where weaker and weaker continuity requirements are made on the functions being optimized. I find this weird -- why not instead create faster and faster optimizers that leverage the smoothness properties present in most applications?

co.combinatorics - Combinatorics journals processing time

I think for more practical purposes, the times from submission to rejection and acceptance are more useful! I'm not sure the backlog matters at all anymore (at least, not if you post your papers to the arxiv). In particular, there are some journals with notoriously large backlogs for which the time it takes to reach a decision is, in my limited experience, quite short.
– GS
Jan 25 '10 at 10:06

ecology - What does self-preservation stem from?

This is a prototypical case of evolution by natural selection. Any trait that prevents the organism from being eaten or destroyed will probably make that organism more likely to reproduce* than similar organisms that do not have that trait**. This results in self-preservation traits becoming more prevalent in the population and eventually ubiquitous.

* Or they reproduce more, or are able to provide for their offspring better, etc. Self-preservation behaviors that reduce reproductive success are not selected for and generally aren't common (unless they are a special case of some general trait that has a net reproductive benefit).

** Assuming that the trait is reproductively favorable after considering any trade-offs such as increased energy expenditure.

ac.commutative algebra - Generalizing miracle flatness (Matsumura 23.1) via finite Tor-dimension

I post this answer to give some intuition about what is really happening behind the scene in the theorem mentioned. If $f:Arightarrow B$ is flat, then obviously the image of any $A$-regular sequence under $f$ is a $B$-regular sequence. This can be seen by tensoring the $A$-Koszul complex on an $A$-regular sequence, by $B$.

Now let's ask this question: Suppose a map $f:Arightarrow B$ has the property that it maps any $A$-regular sequence to a $B$-regular sequence. Is $f$ flat then? The answer is no. As an example, you can consider the Frobenius endomorphism $F:Arightarrow A$ of a local ring of characteristic $p>0$. Obviously it maps every regular sequence to a regular sequence, but $F$ is not flat, unless $A$ is regular, by a theorem of Kunz. Another example is any endomorphism $f$ of a local Cohen-Macaulay ring $(A,mathfrak{m}_A)$ for which $f(mathfrak{m}_A)A$ is $mathfrak{m}_A$-primary. One can see quickly that the image of any regular sequence is a regular sequence, but in general $f$ need not be flat.

The reason for this failure is existence of modules of infinite projective dimension. The condition that $f$ sends any regular sequence to a regular sequence only guarantees that finite free resolutions of $A$-modules stay exact after tensoring by $B$. This quickly follows from Buhsbaum-Eisenbud exactness criterion. (cf. p. 37, Corollary 6.6 in Topics in the homological theory of modules over commutative rings, M. Hochster.)

When $A$ is regular, however, every finite $A$-module has finite projective dimension. That's why in this case the condition that every $A$-regular sequence will be mapped to a $B$ regular sequence by $f$ is equivalent to flatness! (keep in mind that flatness only needs to be checked on finite modules.) The conditions $B$ Cohen-Macaulay and $dim B=dim A+dim F$ are just meant to guarantee that any $A$-regular sequence is mapped to a $B$-regular sequence, as you can check quickly. To check this, take an $A$-regular sequence $x_1,ldots,x_t$, extend it to a maximal regular sequence $underline{x}:=(x_1,ldots,x_d)$ in $A$, then use the dimension assumption and the fact that $B$ is Cohen-Macaulat to show that $f(underline{x}):=(f(x_1),ldots,f(x_d))$ is a regular sequence in $B$.

(Note that on one hand, the inclusion $f(underline{x})Bsubseteqmathfrak{m}_AB$ gives $dim B/f(underline{x})Bgeqdim B/mathfrak{m}_AB$. On the other hand the map $A/underline{x}rightarrow B/f(underline{x})B$ gives $dim B/f(underline{x})Bleq dim A/underline{x}+dim B/mathfrak{m}_AB=0+dim B/mathfrak{m}_AB$. Hence $dim B/f(underline{x})B=dim B-dim A$.)

pr.probability - Problem with a Long Range Correlated Time Series

Consider a stochastic process $X_t$ , $t in 1,2,3,..,N $.

$X_t$ is a Bernoulli variable and $Pr(X_t=1) = p$ for all $t$.
The Autocovariance function $gamma(|s-t|)= E[(X_t - p)(X_s -p)]$ is given

$
gamma(k) = frac{1}{2} (|k-1|^{2H} - 2|k|^{2H} + |k+1|^{2H}).
$

For a constant $Hin (0,1)$ This is the same autocovariance as for fractional gaussian noise (increments of the fractional brownian motion), and give a autocovariance which falls like a power law when $k$ goes to infinity.

Let X and Y be process with the given properties, I am interested in the following probability distribution:

$
Prleft(sum_{i=0}^N X_i Y_i = kright)
$

That is the distribution of the overlap of two such processes. For $H=1/2$ the process is not correlated and I have the simple result that $Pr(X_t Y_t)=p^2$, and that

$
Prleft(sum_{i=0}^N X_i Y_i = kright) = {N choose k} p^{2k} (1-p^2)^{N-k}.
$

But for $Hneq 1/2$, I do not know how to deal with the long range correlation. Is there a way to proceed on this problem? I regret i never took a class in Stochastic Analysis, and I really hope the question makes sense. Any help or input would be highly appreciated.

mp.mathematical physics - The Quantum Operations On The Bipartite Systems

Given two distinct and noninteracting quantum mechanical
systems $mathfrak{S}_1$ and $mathfrak{S}_2$ with state spaces
$mathcal H_1$ and $mathcal H_2$, respectively, the state space
of the combined system $mathcal S_1+mathcal S_2$ is the
tensor product Hilbert space
$mathcal H=mathcal H_1otimesmathcal H_2$. Density operators
$Winmathcal D(mathcal H)$, and effects
$Finmathcal E(mathcal H)$. Similarly, there are corresponding
symbols $W_iinmathcal D(mathcal H_i),
F_iinmathcal E(mathcal H_i)$ for subsystems
$mathfrak{S}_i(i=1,2)$, respectively.

Given any quantum operation, $Phi:
mathcal D(mathcal H)rightarrow mathcal D(mathcal H)$, of
the composite system $mathcal S_1+mathcal S_2$.

Problem: (1) Do there exist whether or not two quantum
operation $phi_1$ and $phi_2$, of the subsystems $mathfrak{S}_1$
and $mathfrak{S}_2$, respectively, such that the following
diagram is commutative:

$$
begin{diagram}
node{mathcal D(mathcal H_1)} arrow[4]{e,t}{phi_1}node[4]{mathcal D(mathcal H_1)}\
node{}\
node{mathcal D(mathcal H_1otimesmathcal H_2)}
arrow[2]{n,l}{Tr_2} arrow[4]{e,t}{Phi} arrow[2]{s,l}{Tr_1}
node[4]{mathcal D(mathcal H_1otimesmathcal H_2)} arrow[2]{s,r}{Tr_1} arrow[2]{n,r}{Tr_2}
\
node{}\
node{mathcal D(mathcal H_2)} arrow[4]{e,b}{phi_2}
node[4]{mathcal D(mathcal H_2)}
end{diagram}
$$
i.e.
$$begin{eqnarray}
Tr_2(Phi(W))&=&frac{tr(Phi(W))}{tr(phi_1(Tr_2(W)))}phi_1(Tr_2(W)),\
Tr_1(Phi(W))&=&frac{tr(Phi(W))}{tr(phi_2(Tr_1(W)))}phi_2(Tr_1(W)),
end{eqnarray}
$$
where $phi_i: mathcal D(mathcal H_i)rightarrow
mathcal D(mathcal H_i)(i=1,2)$ and $Tr_i:
mathcal D(mathcal H)rightarrow
mathcal D(mathcal H_i)$ is a partial trace with respect to the subsystem $mathfrak{S}_i(i=1,2)$.

(2) If quantum operation $phi_1$ and $phi_2$ exist, give the
relationship among the quantum operations $Phi, phi_{1}$ and
$phi_2$.

Closed vs Rational Points on Schemes

The following result deals with the case of finite type affine schemes over an arbitrary field $k$.

Theorem: Let $A$ be a finitely generated algebra over a field $k$. Let $iota: A rightarrow overline{A} = A otimes_k overline{k}$.
a) For every maximal ideal $mathfrak{m}$ of $A$, the set $mathcal{M}(mathfrak{m})$ of
maximal ideals $mathcal{M}$ of $overline{A}$ lying over $mathfrak{m}$ is finite and
nonempty.
b) The natural action of $G = operatorname{Aut}(overline{k}/k)$ on $mathcal{M}(mathfrak{m})$ is transitive. Thus $operatorname{MaxSpec}(A) = G backslash
operatorname{MaxSpec}(overline{A})$.
c) If $k$ is perfect, the size of the $G$-orbit on $mathfrak{m} in operatorname{MaxSpec}(A)$ is equal to the degree of the field extension of $k$ generated by
the coordinates in $overline{k}^n$ of any $mathcal{M}$ lying over $mathfrak{m}$.

In brief, the closed points correspond to the Galois orbits of the geometric points.

This is Theorem 8 in http://www.math.uga.edu/~pete/8320notes3.pdf.

The proof is left as an exercise, with some suggestions.

Exactly where this result came from, I cannot now remember. The text for the course that these notes accompany was Qing Liu's Algebraic Geometry and Arithmetic Curves (+1!), so it's a good shot that there is at least some cognate result in there.

ag.algebraic geometry - What is the relationship between being normal and being regular?

Dear 7-adic, yes there is an implication between the two notions.

For a local ring, regular implies normal. Actually Auslander and Buchsbaum proved in 1959 that a regular local ring is a UFD and it is an easy result that a UFD (local or not) is integrally closed. Serre then gave a completely different proof. He proved that regular is equivalent to having finite global (=homological) dimension . This finiteness means that any module over the ring has a finite projective resolution. I have heard it claimed that this was the beginning of the acknowledgment of the importance of homological algebra in commutative algebra.

An example.The cone $z^2=xy$ in affine 3-space (over a field, say) is normal but not regular: its very equation suggests that we don't have the UFD property and this intuition can be converted into a rigorous proof.
Normality is a weak form of regularity. The two concepts coincide in dimension one but not in higher dimensions: the quadratic cone above shows this in dimension two.

Finally, smoothness is even stronger: it is a relative concept meaning regular and remaining regular after base change.

ag.algebraic geometry - Negative Gromov-Witten invariants

Gromov--Witten invariants are designed to count the "number" of curves in a space in a deformation invariant way. Since the number of curves can change under deformations, the Gromov--Witten invariants won't have a direct interpretation in terms of actual numbers of curves, even taking automorphisms into account.

Here is an example of how a negative number might come up, though strictly speaking it isn't a Gromov--Witten invariant. Let M be the moduli space of maps from P^1 to a the total space of O(-4) on P^1. Call this space X. Note that I said maps from P^1, not a genus zero curve, so the source curve is rigid. That's why this isn't Gromov--Witten theory. Any such map factors through the zero section (since O(-4) has no nonzero sections), so this space is the same as the space of maps from P^1 to itself. I just want to look at degree one maps, so the moduli space is 3 dimensional.

We could also compute the dimension using deformation theory: the deformations of a map f are classified by $H^0(f^ast T)$ where T is the tangent bundle of the target. The target in this case is O(-4), not just P^1, and the tangent bundle restricts to O(2) + O(-4) on the zero section. Thus $H^0(f^ast T)$ is indeed 3-dimensional, as we expected. However, the Euler characteristic of $f^ast T$ is not 3 but 0, which means that the "expected dimension" is zero.

The meaning of expected dimension is rather vague. Roughly speaking, it is the dimension of the moduli space for a "generic" choice of deformation. The trouble is that such a deformation might not actually exist. Nevertheless, we can still pretend that a generic deformation does exist and, if the expected dimension is zero, compute the number of curves that it "should" have.

What makes this possible is the obstruction bundle E on M. Any deformation of X gives rise to a section of E and the vanishing locus of this section is the collection of curves that can be deformed to first order along with X. Even though a generic deformation might not exist, the obstruction bundle does still exist, and we can make sense of the vanishing locus of a generic section by taking the top Chern class.

In our situation, the (fiber of the) obstruction bundle is $H^1(f^ast T)$. Since O(2) does not contribute to H^1, the obstruction bundle is $R^1 p_ast f^ast O_{P^1}(-4) = R^1 p_ast O_{P^3 times P^1}(-4, -4)$ where $p : P^3 times P^1 rightarrow P^3$ is the projection. By the projection formula, this is $O(-4)^{oplus 3}$ and the top Chern class is -64. This is the "Gromov--Witten invariant" of maps from P^1 to $O_{P^1}(-4)$.

Unfortunately, I don't have anything to say about what this -64 means...

ag.algebraic geometry - Sheaf isomorphism.

Suppose you have a curve $C$ such that deg$K_C =0$ and $Gamma(C,Omega_C^1) neq 0$. Does this automatically imply that $vartheta_C equiv Omega_C^1$? My thought is yes, I've seen a proposition (Stanford AG course notes) that $vartheta_C equiv Omega_C^1$ for a nonsingular plane cubic, but the proof is done in a particular case, and I must think there is easier way to show this. In particular, what is the morphism?

Thank you, just trying to make sense of this concept.

ac.commutative algebra - How much theory works out for "almost commutative" rings?

Don't get too excited about the theory of algebraic geometry for almost commutative algebras. A ring can be almost commutative and still have some very weird behavior. The Weyl algebras (the differential operators on affine n-space) are a great example, since they are almost commutative, and yet:

• They are simple rings. Therefore, either they have no 'closed subschemes', or the notion of closed subscheme must correspond to something different than a quotient.


• The have global dimension n, even though their associated graded algebra has global dimension 2n. So, global dimension can jump up, even along flat deformations.


• There exist non-free projective modules of the nth Weyl algebra (in fact, stably-free modules!). Thus, intuitively, Spec(D) should have non-trivial line bundles, even though it is 'almost' affine 2n-space.

Just having a ring of quotients isn't actually that strong a condition on a ring. For instance, Goldie's theorem says that any right Noetherian domain has a ring of quotients, and that is a pretty broad class of rings.

Also, what sheaf are you thinking of D as giving you? You have all these Ore localizations, and so you can try to build something like a scheme out of this. However, you start to run into some problems, because closed subspaces will no longer correspond to quotient rings. In commutative algebraic geometry, we take advantage of the miracle that the kernel of a quotient map is the intersection of a finite number of primary ideals, each of which correspond to a prime ideal and hence a localization. In noncommutative rings, there is no such connection between two-sided ideals and Ore sets.

Here's something that might work better (or maybe this is what you are talking about in the first place). If you have a positively filtered algebra A whose associated graded algebra is commutative, then A_0 is commutative, and so you can try to think of A as a sheaf of algebras on Spec(A_0). The almost commutativity requirement here assures us that any multiplicative set in A_0 is Ore in A, and so we do get a genuine sheaf of algebras on Spec(A_0). For D_X, this gives the sheaf of differential operators on X. Other algebras that work very similarly are the enveloping algebras of Lie algebroids, and also rings of twisted differential operators.

bacteriology - Are there any bacteria that can receive ultrasound signals?

I'm going to post a quick answer here, really a thought piece.

Usually to detect a sound wave you need a sounding board about the wavelength of the sound.

Bacteria are on the order of a few microns in length.

Ultrasound frequencies range from 2 to 200 MHz (and up I assume).

To have a wavelength on the order of 3 microns, a 100 MHz wave would be needed.

So only on the very high end of the range. If bacteria make sound though, they probably are on this frequency range.

I wonder if this has been looked at? Not sure it has. While in biology you never say never - if a bacterium really needs to pick up a wave it might have a clever adaptation to do so, but in the 100MHz + frequency range seems more likely.

dg.differential geometry - Pontrjagin numbers and exotic spheres

The Pontryagin classes of the tangent bundle are not easy to interpret (witness the Novikov conjecture), but one geometric datum that can be extracted from them is the rational cobordism class of a manifold. According to Thom, the Pontryagin numbers $p_1^{a_1}cdots p_k^{a_k}[X]$ of a closed, oriented, smooth $4n$-manifold $X$ vanish iff there's a compact, oriented $(4n+1)$-dimensional manifold bounding a disjoint union of copies of $X$. From Thom's theorem follows Hirzebruch's, expressing the (cobordism-invariant) signature $sigma(X)$ as a certain Pontryagin number $L(p_1,dots,p_n)[X]$. When $n=2$,
$$sigma(X)=(-p_1^2+7p_2)[X]/45.$$

One striking thing about this formula is that $sigma(X)$ is an integer, while $(-p_1^2+7p_2)[X]/45$ is a priori only a rational number. Milnor's observation is that an integrality theorem for a characteristic class of closed $4n$-manifolds gives rise to an invariant of those $(4n-1)$-manifolds which bound $4n$-manifolds. This is a useful principle, a variant of which also underlies Chern-Simons theory.

We can define an invariant for homotopy 7-spheres $S$ by taking $kappa(S)=45sigma(Y)+p_1^2[Y,partial Y] mod 7$; if $Y'$ is another such bounding manifold, the difference between their invariants will be $45 sigma(X)+p_1^2(X)$ for the closed manifold $X= (-Y') cup_S Y$, hence a multiple of 7 by the signature theorem. (Milnor prefers the invariant $lambda=2kappa mod 7$.)

In his later work with Kervaire ("Groups of homotopy spheres I"), Milnor identifies two different reasons why a homotopy-sphere may not be (h-cobordant to) a standard sphere: (i) it may not bound a parallelizable manifold; or (ii) it may bound a parallelizable manifold, but not one which is also contractible. A homotopy 7-sphere which bounds a parallelizable 8-manifold is in fact standard, but this is not true of homotopy 8-spheres. The invariant $lambda$ of homotopy 7-spheres is an obstruction to bounding a parallelizable 8-manifold; not a complete invariant, since Kervaire-Milnor show that there are exactly 28 h-cobordism classes.

lo.logic - How fast can the base-bumping function in Goodstein's theorem grow?

First, let me say that this is a really great question.

It seems to me that any increasing base-bumping function would give the
same Goodstein result that you eventually hit $0$. That is, I claim that for any increasing
sequence of bases $b_1$, $b_2$ and so on, if we define the
Goodstein sequence by starting with any number $a_1$, and
then if $a_n$ is defined, we write it in complete base
$b_n$, replace all instances of $b_n$ with $b_{n+1}$,
subtract $1$, and call the answer $a_{n+1}$. The theorem
would be that at some point $n$ in the construction, we
have $a_n=0$.

The proof of the original theorem proceeded by associating
any number $a$ in complete base $b$ with the countable
ordinal obtained by replacing all instances of $b$ with the
ordinal $omega$ and interpreting the resulting expression
in ordinal arithmetic. They key fact is that the ordinal
associated with $a$ in base $b$ is strictly larger than the
ordinal associated in base $b+1$ with the number obtained
by replacing all $b$'s with $b+1$'s and subtracting $1$.
If we replace $b$ with some larger $b'$ and
do the same thing, then it appears that this key fact still goes
through, since it was proved by observing what happens when the subtract-$1$ part causes a complex term to be broken up with coefficients below the new base. Thus, the newly associated ordinals would still
be descending, so they must hit $0$, but this happens only
if the numbers themselves hit $0$.

What type of cell do you start with in Meiosis?

During mitosis a diploid cell (2n = two copies of each chromosome, one from each parent) replicates its DNA so that it now has four copies of each chromosome. Then it divides, each daughter cell receives two copies of each chromosome and is again 2n.

In meiosis a diploid cell (2n) replicates its DNA so that it now has four copies of each chromosome. Then it divides, each daughter cell receives two copies of each chromosome and is again 2n. Then each of these divides once more without replicating DNA so that there are now four cells each with one copy of each chromosome (1n).

You might be tempted to think of a diploid cell which has replicated its DNA as tetraploid, but this word is not normally used in this context, since this is a transient 4n state.

This is a very broad overview. Have a look at the Wikipedia entry for meiosis to get a more detailed view and extended terminology.

@mgkrebbs (in comments):

If we are considering the meiotic divisions that create gametes, then in spermatogenesis the cell which undergoes meiosis is a primary spermatocyte, and in oogenesis it is a primary oocyte. Primary spermatocytes and primary oocytes are both diploid cells which undergo DNA replication before entering meiosis I.

RACE pcr product - Biology

Judging by your post history, it seems like you may want to pick up an introductory book on molecular biology. This question is really asking a number of questions. I will attempt to answer the difference between PCR and RACE, and I would suggest asking the others as separate questions, as @dd3 has suggested.

Polymerase Chain Reaction (PCR) is the basic method of amplifying a small amount of DNA (the template) into an exponentially large amount of DNA. This method assumes the knowledge of the sequence of (at least) the regions of DNA flanking what you wish to amplify. Using these flanking sequences, two short oligonucleotides (e.g., 18-24 nucleotides), called primers, are ordered such that one primer matches the 5' strand at the start of the sequence, while the other primer matches the 3' strand at the end of the sequence. When these primers are mixed together with the template DNA, and a DNA polymerase enzyme, and subjected to specific temperature cycles, amplification occurs.

RACE PCR is a specific type of PCR. RACE (Rapid Amplification of cDNA Ends) PCR is used when the template strand to be amplified is a specific mRNA message. In other words, this form of PCR produces a DNA copy of cDNA from an mRNA starting product. The exact details of RACE depends on which type you wish to use and can be looked up elsewhere.

Recentering a Spherical Coordinate Sytem

This is going to be unsightly...

The following Mathematica code:

Needs["VectorAnalysis`"]
Simplify@ CoordinatesFromCartesian[
 CoordinatesToCartesian[{r, theta, phi}, Spherical] 
                     + CoordinatesToCartesian[{r0, theta0, phi0}, Spherical],
 Spherical
 ] 

gives the following output (doctored so that it looks nicer):

$$ r' = sqrt{r^2+2 r_0 r left(sin (theta ) sin
left(theta _0right) cos left(phi -phi
_0right)+cos (theta ) cos left(theta
_0right)right)+r_0^2} $$

$$ theta' = cos ^{-1}left(frac{r cos (theta )+r_0 cos
left(theta _0right)}{sqrt{r^2+2 r_0 r
left(sin (theta ) sin left(theta
_0right) cos left(phi -phi
_0right)+cos (theta ) cos left(theta
_0right)right)+r_0^2}}right) $$

$$ phi' = tan ^{-1}left(r sin (theta ) cos (phi
)+r_0 sin left(theta _0right) cos
left(phi _0right),r sin (theta ) sin
(phi )+r_0 sin left(theta _0right) sin
left(phi _0right)right) $$

In this last line, there is a two-argument variant of arctan, which is explained here, for example.

ag.algebraic geometry - Given a ramified cover of a Riemann surface, is there a good choice of basis for H_1 of the source?

Here is a vague suggestion as to why there might not be a "canonical" choice of basis. I will presume that you already have in mind a choice of basis for the homology of your elliptic curve. Write Gamma_{1,2} for the group of isotopy classes of oriented diffeomorphisms of the twice-punctured elliptic curve (i.e. the mapping class group) and Gamma_1 = SL_2(Z) for the mapping class group of the unpunctured elliptic curve. Then the natural surjection Gamma_{1,2} -> SL_2(Z) has a kernel G, which can be thought of as the braid group on two strands on the torus (I'm sure Tom can say more than I can, offhand, about what this group is.)

Now G acts on H_1(X,Z), where X is your spectral curve, and I think for your basis to be "canonical" it would want to be fixed by this. But I don't immediately see why this action would be trivial. (Of course, by construction it acts trivially on the natural quotient H_1(E,Z).)

Is there a simple relationship between K-theory and Galois theory?

Perhaps the other Bloch-Kato conjecture is more relevant; it relates Milnor's higher $K$-groups and Galois cohomology.

The following text is lifted from the expository account on the arXiv.

Let $F$ be a field, $n>0$ an integer which is invertible in $F$, $bar F$ a
separable closure of $F$ and $Gamma=operatorname{Gal}(bar F|F)$. There
is an exact sequence
$$
{1}to
mathbb{Z}/nmathbb{Z}(1)to
{bar F}^timesto
{bar F}^timesto
{1}
$$
of discrete $Gamma$-modules, where $mathbb{Z}/nmathbb{Z}(1)$ is the group of $n$-th roots of $1$ in $bar F$. The associated long exact cohomology sequence and
Hilbert's theorem 90 furnish an isomorphism $delta_1:F^times/F^{times n}to H^1(Gamma,mathbb{Z}/nmathbb{Z}(1))$. Cup product on cohomology
$$
smile;:H^r(Gamma,mathbb{Z}/nmathbb{Z}(r))
times H^s(Gamma,mathbb{Z}/nmathbb{Z}(s))to
H^{r+s}(Gamma,mathbb{Z}/nmathbb{Z}(r+s))
$$
then provides a bilinear map
$
delta_2:F^times/F^{times n}times F^times/F^{times n}to
H^2(Gamma,mathbb{Z}/nmathbb{Z}(2)).
$

Lemma (Tate, 1970)
The map $delta_2(x,y)=delta_1(x)smiledelta_1(y)$ is a
symbol on $F$.

A symbol is a bilinear map $s:F^timestimes F^timesto A$ to a commutative
group such that $s(x,y)=0$ whenever $x+y=1$ in $F^times$.
There is a universal symbol $F^timestimes F^timesto K_2(F)$, giving rise
to Milnor's theory of higher $K$-groups $K_r(F)$ for every $rinmathbb{N}$,
as explained in Milnor's book.

This symbol also gives rise to a homomorphism
$$
delta_r:K_r(F)/nK_r(F)to
H^r(Gamma,mathbb{Z}/nmathbb{Z}(r)).
$$

Conjecture (Bloch-Kato, 1986)
The map
$delta_r$ is an isomorphism for all fields $F$, all integers $n>0$
(invertible in $F$) and all indices $rinmathbb{N}$.

The main theorem of Merkurjev-Suslin (1982) says that the map
$delta_2$ is
always an isomorphism ; Tate had proved this earlier (1976) for global fields.
Bloch-Gabber-Kato prove this conjecture when $F$ is a field of
characteristic $0$ endowed with a henselian discrete valuation of residual
characteristic $pneq0$ and $n$ is a power of $p$.

Somebody should ask a qustion about the current status of the Bloch-Kato
conjecture and get some experts (such as Weibel) to answer. My impression is that it is now a theorem by the work of Rost and Voevodsky, but that a proof with all the
details is not available in one place.

The Bloch-Kato conjecture makes the remarkable prediction that the graded
algebra $oplus_r H^r(Gamma,mathbb{Z}/nmathbb{Z}(r))$ is generated by
elements of degree 1. Galois groups should thus be very special among
profinite groups in this respect.

ct.category theory - What tensor product of chain complexes satisfies the usual universal property?

Recall that a chain complex is a (finite) diagram of the form
$$ V = { dots to V_3 overset{d_3}to V_2 overset{d_2}to V_1 overset{d_1}to V_0 to 0 } $$
where the $V_n$ are (finite-dimensional) vector spaces and for each $n$, $d_n circ d_{n+1} = 0$. If $V$ and $W$ are chain complexes, a chain map $f: V to W$ is a map $f_n : V_n to W_n$ for each $n$ such that all the obvious squares commute — "$[d,f]=0$" — and the pair (chain complexes, chain maps) defines a category. In fact, it is a 2-category: the 2-morphisms between $f,g : V rightrightarrows W$ are the chain homotopies, i.e. a system of maps $h_n: V_n to W_{n+1}$ such that "$[d,h] = f-g$". The category of chain complexes has a biproduct (both a product and a coproduct) $oplus$ given by the pointwise direct sum.

I thought I knew what the tensor product of chain complexes was. Namely, if $V$ and $W$ are chains, then the usual thing is to define
$$ (Votimes W)_n = bigoplus_{k=0}^n V_k otimes W_{n-k} $$
and the chain maps are the sums of the obvious tensor products of differentials, decorated with signs.

But now I'm not sure why this is the tensor product picked. Namely, if I have a linear category, I think that a tensor product $V otimes W$ should satisfy the following universal property: for any $X$, $hom(V otimes W,X)$ should be naturally isomorphic to the space of bilinear maps $V times W to X$. Now, I've never really known how to write down the word "bilinear" in a general category, without refering to individual points. But I think I do know what the "set" $V times W$ is when $V$ and $W$ are chains — it's the set underlying $V oplus W$ — and then I think I do know what bilinear maps should be.

In any case, then it's clear that the usual tensor product is not this. For example, if $V,W$ have no non-zero terms above degree $n$, then the bilinear maps $V times W to X$ I think cannot be interesting above degree $n$, whereas the above $otimes$ has terms in degree $2n$.

In any case, in HDA6, Baez and Crans consider two-term chain complexes $V_1 to V_0$ (they argue that these are the same as "2-vector-spaces"), and then construct a different tensor product, given by:
$$ Votimes W = { (V_1 otimes W_1) oplus (V_1 otimes W_0) oplus (V_0otimes W_1) to (V_0 otimes W_0) } $$
where the differential is the sum of the obvious tensor products of differentials and identity maps. They then assert that this tensor product satisfies the correct universal property, although they leave the details to the reader.

This leads naturally to:

Question: What is the precise universal property that $otimes$ ought to have, and what "product" of chain maps satisfies this universal property?

evolution - Is better healthcare a bane to the long-term survival of the human race?

The theory of natural selection has it that individuals with better genes tend to survive and reproduce, passing their genes to their offspring.

Yes, under selective pressure, that is. The stronger the selective pressure, the more the population will change. Without a selective pressure, there isn't an impetus to change.

However, due to the advancement in medical science, humans with poorer genes tend to survive and reproduce just as well.

You're making a mistake. "Poor" and "Good" are purely contextual. You might say that Sickle Cell Anemia is a poor lot to be stuck with... Unless, of course, you're in an area ravaged by Malaria. Then it's a boon.

What would be more accurate is that there aren't any strong forces acting upon humans that causes selective pressure and the proceeding adaptation. Of course, we are the result of such evolution, so by a basic standard all modern humans who reproduce are fit in a Biological sense.

For example, in the past, many people, excluding those with natural immunity due to some genetic mutations, would have succumbed to illnesses such as malaria and typhoid. But with better hygiene and medical treatment, these patients tend to survive.

The brain is the one of the finest products of evolution. Because we're no longer dying en masse at the whims of bacteria or virii (usually) does not mean we haven't adapted. Our ability to build advanced tools and perform science is an adaptation. One for which we pay heavily; we're practically helpless until our teens.

Now, imagine a future where all illnesses including cancer can be treated. How would it impact the survival of the human race?

We'd live longer, and assuming the rate of reproduction stays the same, there'd be a lot more of us.

Would we become more and more vulnerable such that our survival hinges heavily on medical technology, analogous to how an astronaut's survival is dependent on his spacesuit?

Well... Vulnerable to what? To disease? Probably not. Current Immunizations and treatments often utilize the body's own immune system; basically getting sick and all the benefits of encountering the disease without the side-effects of significant death.

To genetic diseases? No. The human population has exploded in the last hundred or so years, and our genetic diversity has been greater than at any point in the last 40,000 years when a major bottlenecking event occurred. For a genetic disease to deal significant damage on a species, it would have to be suffuse throughout the population. That's not happening.

To a post-apocalyptic world where technology breaks down and suddenly we're facing off against parasites and predators? Well, if the event ever occurs, then a lot of us won't be ready, that's for sure. The stress such a life put on us in the past limited our lifespan to nearly half what it is now, but unless there's a complete lack of knowledge many basic safety habits should persist. Cooking, boiling water, dressing wounds, how illnesses spread - these are all things which dramatically increase quality of life when utilized correctly.

Will we ever reach a point where we, quite literally, cannot live without technological intervention? No idea. Humans have very, very strong reactions to things which aren't human but appear human (The Uncanny Valley is a great example). It's just as possible that those instincts will prevent us from such a fate as the possibility that we'll be completely fine with becoming cyborgs. Nobody can even begin to give you a time-frame, though. As-is, there's no reason to become cyborgs (no Selective Pressure), and won't be for the foreseeable future.

nt.number theory - Natural models of graphs?

Motivation

I want to capture the notion of natural models of finite graphs: How can natural predicates and natural relations on a given natural base class $D$ be defined? If this succeeds the question arises if every (or which) finite graph has a natural model with respect to these "natural" ingredients.

As natural base classes I have in mind $mathbb{N}$ and sets (e.g. the finite subsets of $mathbb{N}$).

Preliminaries

Let a complete description of an (unlabeled) graph be a formula

$$(exists x_1)...(exists x_n) bigwedge_{i neq j} x_i neq x_j wedge (forall x) bigvee_i x = x_i wedge bigwedge_{i,j}[neg]?Rx_ix_j$$

The canonical model of such a description - seen as a categorical theory - consists of $[n]$ as the domain $A$ and some $R subseteq [n]^2$ as the relation. It is trivially construed from the indexes of the variables (if choosen in the canonical way as it is done above).

The canonical model is at least "natural" with respect to the domain: it's not any set but a subset of a natural base set, $mathbb{N}$, singled out by a "natural" predicate/formula $phi(x) : =x leq n$.

With respect to $R$, the canonical model isn't "natural": $R$ is just any subset of $A^2$, generally given by a formula

$$psi(x,y) := bigvee_{n_i,n_j} x = n_i wedge y = n_j$$
for appropriate $n_i, n_j$.

It is this kind of ad-hoc-formula I want to rule out. Maybe this can be done straight forwardly? Let for instance $T$ be any theory of the natural numbers and $L(T)$ be its language.

Definition: An $L(T)$-formula $Psi(x_1,...,x_n)$ is
$T$-natural iff it is
$T$-equivalent to a formula $Psi'(x_1,...,x_n)$
that doesn't contain literals of the
form $x_i = n$ with $n$ the name of an $n in
> mathbb{N}$.

(Note: I know that this definition is maybe not quite correct from the point of view of model theory, but I hope that it is sensible. If it is not: How to improve it? If it is not possible to improve it, you can stop reading here.)

Questions

The set of finite subsets of $mathbb{N}$ that can be defined by a $T$-natural formula probably depends on $T$.

Question 1: Is there a theory $T$ of
arithmetic that allows to define all
finite subsets of $mathbb{N}$ by an
$T$-natural formula.

Furthermore:

Question 2: Is there a theory $T$ of
arithmetic that allows to define all
finite subsets of $mathbb{N}^2$ by an
$T$-natural formula.

Now to the natural models of graphs:

Definition: A natural $T$-model of a
finite graph $G$ is given by a domain
$A = lbrace x in mathbb{N} |
> phi(x) rbrace$ and a relation $R =
> lbrace (x,y) in A^2 | psi(x,y)
> rbrace$ with $T$-natural
formulas $phi(x), psi(x,y)$.

Final question (maybe equivalent to Question 2):

Question 3: Is there a theory $T$ of
arithmetic that allows to find natural
$T$-models for each finite graph?

If not so: How can the finite graphs with no $T$-natural model for any $T$ be characterized?

Can anything be gained by considering set theory instead of arithmetic?

mg.metric geometry - Generalizing cosine rule to symmetric spaces

The sine and cosine rules for triangles in Euclidean, spherical and hyperbolic spaces can be understood as invariants for triples of lines. These invariants are given in terms of the distance (both lengthwise and angular) between pairs of lines. There is also a converse statement. Suppose we are living in a complete Riemannian manifold of constant curvature. If a certain sine/cosine rule is satsifies by triples of lines, then we can determine the curvature.

I'm wondering if we can generalize these sine and cosine rules to arbitrary symmetric spaces. That is, give a triple of geodesics (or parallel submanifolds, if we consider higher dimensions), are there similar invariants? Perhaps these invariants will be given in terms of representations of the coset of symmetries take a geodesic to antoher.

It would also be great if these invariants can characterize the symmetric space we are living in, just like in the case of the constant curvature case.

gn.general topology - Expressing any f(x,y) using only addition and unary functions?

This is "due in successively more exact forms to Kolmogorov, Arnol'd and a succession of mathematicians ending with Kahane", to quote T.W. Korner on the subject.

I am informed that the proof I met is prepared using:

J.-P. Kahane Sur le treizieme probleme de Hilbert, le theoreme de superposition de Kolmogorov et les sommes algebriques d'arcs croissants in the conference proceedings Harmonic analysis, Iraklion 1978 Springer 1980

G. G. Lorenz, Approximation of functions Chelsea Publishing Co. 1986 (First Ed. 1966)

A. G. Vituskin On the representation of functions by superpositions and related topics in L'Enseignement Mathématique, 1977, Vol 23, pages 255-320

[This is all from these skeleton notes (no proofs) here (Links to pdf; See Chapter 1 and Chapter 11 for references)]

gr.group theory - Decidability of conjugacy problem for finitely generated subgroups of free groups

The conjugacy problem for a free group $F_n$ on $n$ letters has an easy solution. Each element of $F_n$ is conjugate to a unique and easily computable "cyclically reduced element" (this means that if you arrange the word around a circle, then there are no cancellations), so two elements of $F_n$ are conjugate if and only if they have the same cyclically reduced conjugates.

I've been trying unsuccessfully to generalize this to solve the following problem. Let ${x_1,ldots,x_k}$ and ${y_1,ldots,y_{k'}}$ be two finite sets of elements of $F_n$. Let $G_x$ and $G_y$ be the subgroups of $F_n$ generated by the $x_i$ and the $y_i$, respectively. Is there an algorithm to decide if $G_x$ and $G_y$ are conjugate? Does anyone know how to do this? Thank you very much!

inequalities - Sum of difference moduli vs. sum of modulus differences

This is a failed attempt of mine at creating a contest problem; the failure is in the fact that I wasn't able to solve it myself.

Let $x_1$, $x_2$, ..., $x_n$ be $n$ reals. For any integer $k$, define a real $f_kleft(x_1,x_2,...,x_nright)$ as the sum

$sumlimits_{Tsubseteqleftlbrace 1,2,...,nrightrbrace ;\ left|Tright|=k} left|sumlimits_{tin T}x_t - sumlimits_{tinleftlbrace 1,2,...,nrightrbrace setminus T} x_tright|$.

We mostly care about the case of $n$ even and $k=frac n 2$; in this case, $f_kleft(x_1,x_2,...,x_nright)$ is a kind of measure for the dispersion of the reals $x_1$, $x_2$, ..., $x_n$ (more precisely, of their $frac n 2$-element sums).

Now my conjecture is that if $n$ is even and $k=frac n 2$, then

$f_kleft(x_1,x_2,...,x_nright)geq f_kleft(left|x_1right|,left|x_2right|,...,left|x_nright|right)$

for any reals $x_1$, $x_2$, ..., $x_n$.

I think I have casebashed this for $n=4$ and maybe $n=6$; I don't remember anymore - it's too long ago. Sorry. I still have no idea what to do in the general case, although my attempts at big-$n$ counterexamples weren't of much success either.

gn.general topology - Paracompact but not Hausdorff

The answer is no.
Take the "classical" example of the line with two origins. This space is non-Hausdorff, paracompact and doesn't admit partitions of unity.

EDTI: I think the question is a kind of "duplicate" .
Ok, but if you have an example for a non-Hausdorff manifold, which doesn't admit partitions of unity, you have an example for a non-Hausdorff paracompact space with the same property.

First the definition:
The line with two origins is the quotient space of two copies of the real line
$mathbb{R} times {a}$ and $mathbb{R} times {b}$.
with equivalence relation given by
$(x,a) sim (x,b)text{ if }x neq 0$.
Since all neighbourhoods of $0_a$ intersect all neighbourhoods of $0_b$, it is non-Hausdorff.
However, this space is paracompact, since $mathbb{R}$ is paracompact.

For the non-existence of a partition of unitiy: take the open covering $ U = (-infty,0) cup { 0_a } cup (0,infty)$ and $tilde{U} = (-infty,0) cup { 0_b } cup (0,infty)$. Assume, there is a partition of unity subordinate to this cover. Then the value of each origin would have to be $1$ which cannot be true. (Edit: villemoes was a little faster :-) )