zoology - Why do animal teeth get darker if exposed?

I was wondering why the teeth of cats (And dogs, as far as I know, plus possibly some other animals or even humans) get darker if they are exposed to air / light?

Example:

(See the lower end of the tooth, which sticks out if the mouth is closed. No, the cat was not resisting, just slightly annoyed.)

So, my question is:

• Why does this happen?


• Would it happen to human teeth, if they would stick out of the mouth like the ones of cats do?

My guess is that it has something to do with the teeth reacting to air and corroding, but all my knowledge about biology comes from two years of biology in school, so I'm asking here.

ag.algebraic geometry - Spectral curve of Elliptic Calogero-Moser systems

So, the Lax operator $L(lambda)$ is given by
$$L(t,lambda)_{ij}=p_i delta_{ij}+(1-delta_{ij})f_{ij}Phi(q_i-q_j,lambda)$$
with lambda the spectral parameter, and $Phi$ the Lamé function. Using the Lax equation $dot{L}=[L,M]$, which is equivalent to $[L,frac{partial}{partial t}+M]=0$, if a matrix $A(t,lambda)$ satisfies $$left(frac{partial}{partial t}+M(t,lambda)right)A(t,lambda)=0$$ and is normalized, $A(0,lambda)=1$ it follows that
$$L(t,lambda)A(t,lambda)=A(t,lambda)L(0,lambda)$$
Hence, it is clear that $det(L-mu I)$, (and so the spectral curve) is independent of time. Now, the equation of the spectral curve is
$$Gamma:quaddet(L(t,lambda)-mu I)=0$$
Writing $$Gamma(lambda, mu)equivdet(L(t,lambda)-mu I)=sum_{i=0}^N r_i(lambda)mu^i$$
Your first question is why are the $r_i(lambda)$'s elliptic functions. Note that the matrix elements of $L$ are already doubly periodic, but they have an essential singularity at $lambda=0$. To show that the $r_i$'s are meromorphic, all you need is a gauge transformation to get rid of this singularity. Note that
$$L(t,lambda)=G(t,lambda)bar{L}(t,lambda)G^{-1}(t,lambda)$$
with $$G=left(delta_{ij}e^{zeta(lambda)q_i(t)}right)_{1le i,jle N}$$
where $zeta$ is the Weierstrass zeta function, does the job. So each $r_i(lambda)$ will be a combination of the Weierstrass $wp$ function and its derivatives, with the coefficients being integrals of the system. For each set of initial values of these integrals, the spectral curve is an $N$-sheeted covering of the base elliptic curve. The branch points will coincide with the zeros of $frac{partial Gamma(lambda,mu)}{partial lambda}$ on $Gamma$.

Look at "Introduction to classical integrable systems" by O. Babelon, D. Bernard, M. Talon, and the paper of Krichever I mention in the comments for more details.

Accurate Human Skull Model for Educational Purpose

The best way to ensure you're getting an accurate skull, near as I can tell, is striking the "model" part of your question, and go straight for a real skull.

A cursory search revealed, for instance, Skulls Unlimited International, The Bone Room, and The Evolution Store although other providers probably exist as well.

A regulatory board for model skulls probably doesn't exist, given the apparent ease with which real skulls can be obtained and kept.

derived category - determinant of a perfect complex

As I understand the construction of the determinant of a perfect complex, this definition is quite straightforward, following from the fact that in a short exact sequence, say
$$ 0rightarrow Srightarrow Erightarrow Qrightarrow 0$$
defining the determinant of the sequence to be the alternating tensor is the canonical way to make it isomorphic to $mathbb{1}$.

Also, I think good references to this may be the original paper by Knudsen-Mumford, a book by Kato, and also a paper by Kings which are listed below:

Finn Faye Knudsen and David Mumford, The projectivity of the moduli space of stable curves. I. Preliminaries on ''det'' and ''Div'' (pdf). The part about determinants appears in Chapter I, but note that there is a typo defining the determinant, namely in the map of the transposition of tensor product, there should be $alphacdotbeta$ instead the sum of these two as a power of $-1$;

Guido Kings, An introduction to the equivariant Tamagawa number conjecture: the relation to the Birch-Swinnerton-Dyer conjecture (pdf)

There is a part about determinants in lecture 1 section 5, where there are not a lot of details but it provides a good view towards the construction of determinant.

Kazuya Kato, Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via $B_{dR}$, part I Springer LNM 1553 pp 50-163 (doi:10.1007/BFb0084729), which mentions determinant in 2.1.

ct.category theory - Are abelian non-degenerate tensor categories semisimple?

OK, let me try again. First, a general construction. Let V be an object in an abelian category, A = End(V) and J a finitely generated two-sided ideal of A, with generators (j_1, ..., j_r). Define V/JV to be the cokernel of V^{r} --> V, where the map is multiplication by j_i on the ith coordinate. We need to check that this depends only on J and not on the choice of generators; I haven't actually done this. Define JV to be the kernel of V --> V/JV. So we have a short exact sequence

0 --> JV --> V --> V/JV --> 0

I claim that the action of A on V passes to an action of A/J on V/JV. Also, A acts on JV. (One can say more than this, but we don't need to.)

Using the action of A on JV, we can repeat this construction to get JV/J^2V, J^2 V/J^3 V, etcetera. All of these come with actions of A/J.

Now, suppose that all of our Hom spaces are finite dimensional. A finite dimensional algebra is semi-simple if and only if it has no nontrivial nilpotent two-sided ideal. So, suppose for the sake of contradiction that there is some (V,A,J) as above with J nilpotent. Then J^k V is eventually zero. Trace is additive in short exact sequences, so

Tr(f: V --> V) = sum Tr(f: J^k V/J^{k+1} V --> J^k V/J^{k+1} V).

If f is in J, the right hand side is 0. Also, if f is in J, so is fg for any g in A because J is an ideal. So J is in the kernel of the trace pairing, and we deduce that J=0.

So all endomorphism rings are semi-simple and, by the lemma cited above, so is the category.

Which is harder to contract - HIV or AIDS?

I am working on mathematically modeling HIV and Tuberculosis Co-infection dynamics, and am working with 12 differential equations describing the behavior of all of the compartments and parameters.

Anyway, I just have one question:

Which is harder, in general: For a susceptible person to contract HIV, or a HIV+ person to contract AIDS?

Please pardon my ignorance.

Thanks.

mathematics education - Motivation for concepts in Algebraic Geometry

I know there was a question about good algebraic geometry books on here before, but it doesn't seem to address my specific concerns.

**
Question
**

Are there any well-motivated introductions to scheme theory?

My idea of what "well-motivated" means are specific enough that I think it warrants a detailed example.

**
Example of what I mean by well motivated
**

The only algebraic geometry books I have seen which cover schemes seem to leave out essential motivation for definitions. As a test case, look at Hartshorne's definition of a separated morphism:

Let $f:X rightarrow Y$ be a morphism of schemes. The diagonal morphism is the unique morphism $Delta: X rightarrow X times_Y X$ whose composition with both projection maps $rho_1,rho_2: X times_Y X rightarrow X$ is the identity map of $X$. We say that the morphism $f$ is separated if the diagonal morphism is a closed immersion.

Hartshorne refers vaguely to the fact that this corresponds to some sort of "Hausdorff" condition for schemes, and then gives one example where this seems to meet up with our intuition. There is (at least for me) little motivation for why anyone would have made this definition in the first place.

In this case, and I would suspect many other cases in algebraic geometry, I think the definition actually came about from taking a topological or geometric idea, translating the statement into one which only depends on morphisms (a more category theoretic statement), and then using this new definition for schemes.

For example translating the definition of a separated morphism into one for topological spaces, it is easy to see why someone would have made the original definition. Use the same definition, but say topological spaces instead of schemes, and say "image is closed" instead of closed immersion, i.e.

Let $f:X rightarrow Y$ be a morphism of topological spaces. The diagonal morphism is the unique morphism $Delta: X rightarrow X times_Y X$ whose composition with both projection maps $rho_1,rho_2: X times_Y X rightarrow X$ is the identity map of $X$. We say that the morphism $f$ is separated if the image of the diagonal morphism is closed.

After unpacking this definition a little bit, we see that a morphism $f$ of topological spaces is separated iff any two distinct points which are identified by $f$ can be separated by disjoint open sets in $X$. A space $X$ is Hausdorff iff the unique morphism $X rightarrow 1$ is separated.

So here, the topological definition of separated morphism seems like the most natural way to give a morphism a "Hausdorff" kind of property, and translating it with only very minor tweaking gives us the "right notion" for schemes.

Is there any book which does this kind of thing for the rest of scheme theory?

Are people just expected to make these kinds of analogies on their own, or glean them from their professors?

I am not entirely sure what kind of posts should be community wiki - is this one of them?

photosynthesis - How does this diagram illustrate carbon cycling in lakes?

This shows the major biological transformations of carbon in any system (not just lakes).

On the Left Side:

• $GPP$ (Gross Primary Production) is the total amount of $C$ from atmospheric $CO_2$^† that is reduced into organic molecules during the calvin cycle of photosynthesis. This is the process performed by photosynthetic organisms like green plants and algae.

_{† atmospheric $CO_2$ diffuses into water and develops into several species of inorganic carbon molecules that plants and algae can use but in general you can think of this carbon as being very closely related to atmospheric $CO_2$}

• $OM$ is a pool of organic matter, which is a general term for reduced carbon molecules. $OM$ can be living organism tissues (e.g., a fish), dead tissues (e.g., senescent leaves) or biologically generated organic molecules (e.g., sugar that leached from a plant). $OM$ is the principle pool of energy and raw materials for all living things on Earth so this is a really important pool of stuff.

Since $GPP$ is taking $CO_2$ and converting it to $OM$ as the left side progresses, the amount of $CO_2$ in the atmosphere goes down and the amount of $OM$ in the biosphere goes up.

On the Right Side:

• $R$ (respiration) is the oxidation of reduced carbon-containing molecules for the purpose of extracting usable energy. This may be familiar as "cellular respiration" but there are other metabolic pathways that accomplish the same goal. In the end when the $C$ in the $OM$ is oxidized and the energy released the $C$ that was in the $OM$ gets converted into $CO_2$.

Note that this is the opposite of the Left Side, so as this occurs, the amount of $OM$ in the biosphere goes down and the amount of $CO_2$ in the atmosphere goes up.

If the Left and Right sides are balanced, then there is no net change in the size of the $OM$ pool or the amount of $CO_2$ in the atmosphere. However, note that over relatively short time scales these can be very out of balance. Any time plants or algae are growing (i.e., gaining mass as $OM$) the Left Side > the Right Side or $GPP$ > $R$. Once those plants die, they accumulate in the system as dead $OM$. The decomposition of accumulated dead $OM$ by microbes will generally make the Right Side > Left Side or $R$ > $GPP$.

This picture from Wikicommons shows the same cycle in a more realistic setting.

nt.number theory - Where can I find a comprehensive list of equations for small genus modular curves?

No, there does not exist a comprehensive list of equations: the known equations are spread out over several papers, and some people (e.g. Noam Elkies, John Voight; and even me) know equations which have not been published anywhere.

When I have more time, I will give bibliographic data for some of the papers which give lists of some of these equations. Some names of the relevant authors: Ogg, Elkies, Gonzalez, Reichert.

In my opinion, it would be a very worthy service to the number theory community to create an electronic source for information on modular curves (including Shimura curves) of low genus, including genus formulas, gonality, automorphism groups, explicit defining equations...In my absolutely expert opinion (that is, I make and use such computations in my own work, but am not an especially good computational number theorist: i.e., even I can do these calculations, so I know they're not so hard), this is a doable and even rather modest project compared to some related things that are already out there, e.g. William Stein's modular forms databases and John Voight's quaternion algebra packages.

It is possible that it is a little too easy for our own good, i.e., there is the sense that you should just do it yourself. But I think that by current standards of what should be communal mathematical knowledge, this is a big waste of a lot of people's time. E.g., by coincidence I just spoke to one of my students, J. Stankewicz, who has spent some time implementing software to enumerate all full Atkin-Lehner quotients of semistable Shimura curves (over Q) with bounded genus. I assigned him this little project on the grounds that it would be nice to have such information, and I think he's learned something from it, but the truth is that there are people who probably already have code to do exactly this and I sort of regret that he's spent so much time reinventing this particular wheel. (Yes, he reads MO, and yes, this is sort of an apology on my behalf.)

Maybe this is a good topic for the coming SAGE days at MSRI?

Addendum: Some references:

Kurihara, Akira
On some examples of equations defining Shimura curves and the Mumford uniformization.
J. Fac. Sci. Univ. Tokyo Sect. IA Math. 25 (1979), no. 3, 277--300.

$ $

Reichert, Markus A. Explicit determination of nontrivial torsion structures of elliptic curves over quadratic number fields. Math. Comp. 46 (1986), no. 174, 637--658.

http://www.math.uga.edu/~pete/Reichert86.pdf

$ $

Gonzàlez Rovira, Josep Equations of hyperelliptic modular curves. Ann. Inst. Fourier (Grenoble) 41 (1991), no. 4, 779--795.

http://www.math.uga.edu/~pete/Gonzalez.pdf

$ $

Noam Elkies, equations for some hyperelliptic modular curves, early 1990's. [So far as I know, these have never been made publicly available, but if you want to know an equation of a modular curve, try emailing Noam Elkies!]

$ $

Elkies, Noam D. Shimura curve computations. Algorithmic number theory (Portland, OR, 1998), 1--47, Lecture Notes in Comput. Sci., 1423, Springer, Berlin, 1998.

http://arxiv.org/abs/math/0005160

$ $

An algorithm which was used to find explicit defining equations for $X_1(N)$, $N$ prime, can be found in

Pete L. Clark, Patrick K. Corn and the UGA VIGRE Number Theory Group, Computation On Elliptic Curves With Complex Multiplication, preprint.

http://math.uga.edu/~pete/TorsCompv6.pdf

This is just a first pass. I probably have encountered something like 10 more papers on this subject, and I wasn't familiar with some of the papers that others have mentioned.

immunology - Do antigens protrude through the capsule/slime layer in prokaryotic organisms where these features are present?

They don't have to -- there are times in life of a bacteria (cell division, hunger, mutations, attacks of lysozyme and other enzymes, cell lysis) when any possible antigen gets less or more exposed.

About escaping phagocitosis, there are numerous strategies to achieve it -- from forming a large slime-covered colony, through killing or disabling phagocytes up to stopping digestion, escaping the phagocyte back to the environment or even living inside it (Listeria monocytogenes is a prime example -- it can even directly move from one cell to another).
EDIT: Here is a nice overview.

ct.category theory - Indecomposable objects in a category

Briefly: there's a simple difference in how they treat 0. That fixed, still neither implies the other in general. In a regular extensive category, a slight modification of the LS definition implies the Elephant one. I suspect they're not fully equivalent in anything short of a topos. As Mike Shulman points out, even in a topos they are not equivalent.

The simple difference: 0 is always indecomposable by Lambek and Scott's definition (since any map into 0 is epi), but never by the Elephant's (since the uniqueness condition won't hold; or by considering when the coproduct decomposition is empty). So, let's temporarily change one of the definitions to fix this. I'd suggest we add “…and the map $0 to X$ is not epi.” to Lambek and Scott's definition. (As you noted, their binary condition generalises to a $k$-ary one; this is just the case $k=0$.)

In eg Top, however, we can see that the Elephant def still doesn't imply the LS def. $[0,1]$ satisfies the former (it's not decomposable by an iso), but not the latter (it is decomposable by an epi). Even more, it’s decomposable by a regular epi (more on this distinction below).

Conversely, the LS definition doesn't imply the Elephant one either; it fails in eg $mathbf{Set}^mathrm{op}$, since in $mathbf{Set}$, $0$ is co-decomposable by iso ($0 cong A times 0$) but not co-decomposable by monos (for any map $(f,g) colon 0 to A times B$, not just one but both of $f$ and $g$ are mono).

When do they imply each other? If we upgrade the LS definition to involve regular epis, then in a regular lextensive category, it implies the Elephant definition, if I'm not mistaken. For this, suppose $X$ is “indecomposable by reg epis”, and suppose $X cong A + B$ — WLOG $X = A + B$. The coproduct inclusions are then jointly reg epi, so one of them is reg epi. But it's also mono (in a lextensive category, every coproduct inclusion is a pullback of $1 to 1 + 1$, so is mono); so it's iso. There's a little more fiddly stuff to check involving messing around with $0$, but it's all the same sort of thing.

Edit from Mike Shulman's comments: if moreover we're in a pretopos, all epis are regular, so there the original LS definition will imply the Elephant definition. On the other hand, the Elephant definition doesn't imply the LS even in a topos: the terminal object of $mathbf{Sh}([0,1])$ is a counterexample, essentially for the same reasons that $[0,1]$ was a counterexample in $mathbf{Top}$.

However, the two definitions are equivalent for projective objects… and I guess that's how this situation has arisen, since a common use of indecomposable objects in topos theory is the theorem that the indecomposable projectives in a presheaf category are exactly the retracts of representables. (This is useful because it lets us recover the idempotent-completion of $mathbf{C}$, which is very close to $mathbf{C}$ itself, from $[mathbf{C}^mathrm{op},mathbf{Set}]$.)

genetics - Terminology question: the scope of an allele in an organism

Let us consider a gene FOO with novel type foo.

If I were discussing an organism that has inherited foo in every cell during classical zygote formation, then I would ordinarily just say that the organism has foo.

If I were discussing a SNP of FOO that gives rise to foo in a tumor cell of an organism, I wouldn't say that the organism has foo, but I might say that the tumor has it.

If I were discussing a chimeric or mosaic organism in which some fraction of the chimera had foo and the other fraction had FOO, I would have to say something like "in foo-containing cells ..."

But what if I wanted to particularly draw attention to the scope of applicability of the allele in question? That is, I want to talk about the scope explicitly, rather than implicitly. How could I directly refer to the scope of applicability of an allele? Is there a single adjective that captures the concept of "This organism has foo in every cell except in rare ones where a point mutation may have occurred; it's probably been inherited during normal meiosis" and is there a contrasting term for "The organism only has foo in a particular subset of its cells and there is some other genetic process necessary to explain that."?

independence results - What are some reasonable-sounding statements that are independent of ZFC?

Laver tables of order $n$ provide a potential answer (although it's not proven to be independent of ZFC, and may not be independent of ZFC).

The Laver table of order $n$ is the $2^n times 2^n$ table of an operation $star$ defined recursively by
$$
p star 1 = p+1 bmod {2^n}\
p star (q star r) = (p star q) star (p star r).
$$
They arise naturally in the study of (self) left-distributive systems (systems satisfying the second axiom above).

The first row of this table ($1 star p$) is always periodic with some period $m$ dividing $2^n$. Let $p(n)$ be this periodicity. Assuming an extremely large cardinal, Laver showed that $p(n)$ increases without bound as $n$ increases. (The consistency of this large cardinal axiom, the existence of a self-embedded cardinal, is apparently in doubt.) Under this same large cardinal hypothesis, Dougherty showed, for instance, that the first $n$ for which $p(n)>m$ grows faster than any primitive recursive function of $m$.

I'm not sure what the current state of belief is on whether this statement is independent of ZFC. There were various other statements first proved by Laver using this large cardinal hypothesis that were later proved by Dehornoy; for instance, there is an algorithm to decide the word problem in a free left-distributive algebra.

I could have sworn there was a statement in this theory that was known to be independent of ZFC, but I couldn't find it when I looked.

I'm not a logician. Apologies if I'm misstating any results.

References:

http://googology.wikia.com/wiki/Laver_table

Randall Dougherty, Critical points in an Algebra of Elementary Embeddings, Ann. Pure Appl. Logic 65 (1993), no. 3, 211-241, http://arxiv.org/abs/math/9205202

human biology - How are non-glucose sugars metabolized in the body?

In my biology book's section on disaccharide metabolism and glycolysis, it states that sugars other than glucose must be acted upon to enter glycolysis. Let's take sucrose as an example. Sucrose is hydrolyzed in the small intestine by sucrase. The resulting fructose and glucose are absorbed and transported to the liver via the portal vein. My question concerns the fate of fructose.

To undergo glycolysis, the book states that fructose is converted into either fructose-6-phosphate (F6P) or fructose-1-phosphate (F1P). Let's say it is converted to F1P. Aldolase splits this into dihydroxyacetone phosphate and D-glyceraldehyde. Triose kinase then converts D-glyceraldehyde to glyceraldehyde-3-phosphate, a glycolytic intermediate. Where is this occurring in the body? Are we still in the liver? I can't imagine that all the fructose we consume is undergoing glycolysis in the liver. To leave the liver as a sugar, it would have had to been converted to glucose, right?

In classes I've taken, I've been told that sugars that enter the liver are pretty much all converted to glucose. Once they are converted to glucose, they can be distributed to the rest of the body, stored as glycogen, etc. If we are going straight from fructose to F1P to a glycolytic intermediate, we couldn't have left the liver. How is such a transformation even useful? Anyone care to shed some light on this?

gt.geometric topology - Spanning discs in contractible 2-d complexes

I'm pretty sure this has an easy solution, but I can't seem to find it.

Let $X$ be a contractible $2$-dimensional CW-complex, let $gamma$ be an embedded loop in $X$, and let $f : D^2 rightarrow X$ be an embedding of a disc in $X$ which maps the boundary of $D$ to $gamma$.

My question is the following. Let $f' : D^2 rightarrow X$ be a continuous map of a disc into $X$ which takes the boundary of $D$ to $gamma$. Must we then have $f(D^2) subset f'(D^2)$ ? I'm pretty sure that the answer is yes, but I can't seem to prove it.

Of course, this has an obvious generalization to higher dimensional complexes, and I'd be interested in that too.

oc.optimization control - Is there an "adjacency matrix" for weighted directed graphs that captures the weights?

I am currently writing up some notes on the max-plus algebra $mathbb{R}_{max}$ (for those that have never seen the term "max-plus algebra", it is just $mathbb{R}$ with addition replaced by $max$ and multiplication replaced by addition. For some reason, authors whose main interest is control-theoretic applications never seem to use the term "tropical", and I have been reading from such authors). There is a nice result which says the following:

$textbf{Theorem.}$ Let $G$ be a directed graph on $n$ vertices such that each arc $(i,j)$ in $G$ has a real weight $w(i,j)$. Define the $n times n$ matrix $A$ by $(A)_{ij} = w(i,j)$ if $(i,j)$ is an arc, and $(A)_{ij} = -infty$ otherwise. Then for each $k > 0$, the maximum weight of a path of length $k$ from vertex $i$ to vertex $j$ is given by $(A^{otimes k})_{ij}$ (here, $A^{otimes k}$ is just the $k$th power of $A$, computed using the $mathbb{R}_{max}$ operations).

This result is certainly analagous to the standard result that the $ij$-entry of the $k$th power of the adjacency matrix gives the number of walks of length $k$ from vertex $i$ to vertex $j$. When writing up my notes I found myself claiming that the above theorem provides some evidence that $mathbb{R}_{max}$ is in fact a natural setting in which to study weighted digraphs, since there is no natural definition of an ``adjacency matrix'' of a weighted digraph (in the usual setting of $mathbb{R}^{n times n}$) that gives useful information about the weights. This seemed like too strong of a claim, especially since I am no expert in networks or combinatorial optimization. This leads to the question:

$textbf{Question.}$ Is there a standard matrix (in $mathbb{R}^{n times n}$) associated with a weighted digraph that is analogous to the adjacency matrix and captures in a useful way the weights of the arcs?

$textbf{Clarification:}$ By "analogous to the adjacency matrix" I mean a matrix that is defined simply in terms of the graph (vertices, arcs, and weights). I imagine there are all sorts of matrices associated to weighted digraphs so that computers can be used to analyze networks. But I am not interested in, say, a matrix that requires a complicated algorithm to compute its entries.

What is affine invariant used in computer vision?

The ratio of areas of $ABC$ and $ACD$ is the ratio in which the line $AC$ divides the segment $BD$ (and it is the ratio of the heights of $B$ and $D$ over $AC$ respectively). This later ratio is affine invariant as affine transformations preserve length ratios on any line. Do make sure that your points don't collapse onto a single line though.

big picture - Applications of Math: Theory vs. Practice

I'm not quite sure how to answer this but I'll take a stab anyway.

Once I started working as a mathematician, I found that my grasp of probability and discrete mathematics was very weak (now it is at least adequate). It is quite rare for me to go through the details of writing a proof; instead, I code up an idea in MATLAB (which I also learned outside of academia). Once it works, then I usually have what amounts to a proof embedded in the logical structure of code. Because my initial background was not ideal, the things that I've learned professionally have tended to have direct applications to my work.

But this has still been an esoteric bag of tricks, for which I will supply a few examples from the first five years or so of my career (it has been another five years since).

One of the first things I did was to give myself a crash course (now forgotten) in algorithms, crypto and complexity theory. I learned Markov processes and queueing theory to model coarse-grained computer network traffic, and martingales to profile its behavior. I learned the rudiments of graph theory, combinatorics and information theory to develop data structures and work with statistical symmetries in finite strings. I learned about toric varieties and briefly revived my acquaintance with index theory to understand Euler-Maclaurin formulae for polytopes, which were of theoretical import for precisely enumerating/sampling from those same statistical symmetries.

The overarching theme has always been to either develop methods of my own for tackling specific problems determined by needs external to my own narrow interests or to identify if and how someone else's constructions work, as well as to find areas for improvement. In both cases the goal has not been detailed proofs but either code or an argument for doing something in a particular way.

I will say that my formal education has been of comparatively little use. The few good techniques and working habits I've developed have come from my professional work and not from school.

gn.general topology - Relating Euler characteristic, intersection product, Morse theory (plus SU(2) and 3-manifolds)

I used to understand this stuff pretty well, but it's been a long time. I think the following answer is correct, but I'm not certain.

Since M is a rational homology sphere, the irreducible points of R are separated from the reducible points, so we can treat them separately.

[EDIT: This isn't true in general (consider the case where $M$ is a connect sum). $M$ being a QHS only guarantees that the "very reducible" points of $R$, with image in the center of $SU(2)$, are isolated from the irreducibles. So the argment below only works in a special case.]

I claim that (1) the irreducible part of R contributes zero to the homological intersection number of the Q's, and (2) the contribution of the reducible part of R is $H_1(M)$. I think claim (1) follows from

(a) the answer to your question 2, which is that the contribution of a submanifold of intersection is equal to the Euler characteristic of its normal bundle (normal to both the Q's);

(b) using symplectic structure to show that the normal bundle is isomorphic to the tangent bundle in this case; and

(c) the observation that SU(2) acts freely, so the Euler characteristic of an irred. component of R is zero.

For claim (2), the idea is to show that the intersection number is equal to the number of homomorphisms $f$ from the finite abelian group $H_1(M)$ to $S^1$. If the the image of $f$ is $pm 1$, then it corresponds to a unique transverse point of $R$. Otherwise, both $f$ and $-f$ lie on a 2-sphere component of $R$. By an argument similar to the one above, this 2-sphere contributes its Euler characteristic, namely 2, to the homological intersection number.

Is this naive test to tell whether a complex elliptic curve has complex multiplication effective?

I don't know anything about hypergeometric functions, so this is not a direct answer to your question. But, I have thought a lot about the problem of detecting complex multiplication of elliptic curves (and certain higher-dimensional analogues for abelian varieties).

Suppose you are given an algebraic integer $j$, and you wish to know whether it is a CM j-invariant. Then there is a sort of night-and-day algorithm you can perform here, where by night you reduce the elliptic curve modulo various primes and keep in mind the fact that if your elliptic curve has CM, then half the time (in the sense of density) you will get a supersingular elliptic curve and the other half you will get a CM elliptic curve whose characteristic p endomorphism algebra is the same as the algebra you started with. Thus, in practice, if your curve does not have CM, you will fairly quickly be able to rule it out by finding two primes of ordinary reduction with different endomorphism algebras. So far this is just a probabilistic algorithm. Once you figure out that the CM field is either a particular quadratic field K or there is no CM at all, you compute (e.g. by classical CM theory as described e.g. in Cox's book Primes of the form...) you compute the $j$-invariants of elliptic curves with K-CM. There are infinitely many of these, because the j-invariant depends on the endomorphism ring (equivalently, the conductor of the order), but you can either just compute all of them in order of conductor or look more carefully at the mod p reductions and get a bound on what the conductor could be.

[Edit: Actually, you can figure out exactly what the CM order must be by computing the endomorphism ring at any two primes of ordinary reduction. It is a theorem that if $E$ is a curve with CM by the order of conductor $f$ in a CM field $K$ and $p$ is a prime of ordinary reduction, the conductor of the reduced endomorphism ring is $f/p^{ord_p(f)}$, i.e., you just strip away the $p$-part of the conductor.]

This is not the state of the art, though. Rather, see the paper

Achter, Jeffrey D.
Detecting complex multiplication. (English summary) Computational aspects of algebraic curves, 38--50,
Lecture Notes Ser. Comput., 13, World Sci. Publ., Hackensack, NJ, 2005.

[A copy is available via his webpage http://www.math.colostate.edu/~achter/.]

In the paper, Achter uses Faltings' theorem and the effective Cebotarev density theorem to eliminate the "day" part of the algorithm. He also gives a complexity analysis and explains why this is faster than what I sketched above.

Finally, I'm sure the questioner knows this, but others may not: for elliptic curves over $mathbb{Q}$ there's no need to do any of this. Rather you just compute the $j$-invariant and see whether it's one of the $13$ $j$-invariants of CM elliptic curves over $mathbb{Q}$ associated to the $13$ class number one quadratic orders (yes, this relies on the Heegner-Baker-Stark resolution of Gauss' class number one problem). For the list, see e.g.

modular.fas.harvard.edu/Tables/cmj.html

immunology - What is the smallest molecule that can present as an antigen to the immune system in the context of allergies?

People often claim, in a colloquial sense, that they are "allergic to everything".

Is it possible to have a full-fledged IgE mediated allergic response to very small molecules? I was always under the impression that the smallest antigen was a oligopeptide, but is it possible for someone to be allergic to something like isopropanol, glucose, or another small organic molecule?

How about allergies to metals, as I know those are fairly common, would they be mediated through a similar mechanism?

examples - On using field extensions to prove the impossiblity of a straightedge and compass construction

I think they're equivalent. Clearly 3 ==> 2. To see 2 ==> 3, say an extension E/L of fields is 2-filtered if it has a tower of subextensions each of degree 2 over the next. Then note firstly that if E/L and E'/L are 2-filtered, then so is the compositum of E and E' in any extension of L containing both (induct on the length of the tower of, say, E); and secondly for any map f:E-->E' of extensions of L, if E is 2-filtered then so is f(E). Together these two facts imply that in 2 we may assume that K is Galois over Q, say with group G. Then Q(z)/Q corresponds to a subgroup H, and by Galois theory we just need to prove the following group-theoretic lemma:

Lemma: Let G be a 2-group and H a subgroup. Then there is a chain of subgroups connecting H to G where each is index 2 in the next.

Proof: Recall that we can find a central order two subgroup Z of G. If Hcap Z=1, then HZ gets us one up in the filtration. Otherwise Hcap Z=Z, i.e. Z is in H, and we can use induction on |G|, replacing G by G/Z.

ag.algebraic geometry - Brauer-Manin obstruction and Tate-Shafarevich group of an Abelian variety

The quotient of what you called the Brauer-Manin obstruction by the closure of $A(K)$ within it is related to the divisible part of Sha. In particular, if Sha has no divisible part (e.g. if it is finite) then the Brauer-Manin obstruction
is the closure of $A(K)$. See L. Wang, Brauer-Manin obstruction to weak approximation on abelian varieties, Israel J. Math. 94 (1996), 189–200.

Note that these two groups in your question are very different, so they can't be equal. For instance, Sha is torsion, but the Brauer-Manin obstruction usually isn't.

neurotransmitter - What is the cannabinoid autoreceptor?

CB1 and CB2 are indeed particular genes which are present in neurons, but also liver and other tissues. The HGNC website is a good resource for questions like this - HGNC is the international organization that tries to unify and track gene names.

The official gene names are CNR1 and CNR2 respectively. Gene names are a bit of a mess, since many genes have been found by different experiments and so have many names.

CNR1 has other aliases, I see:
CANN6, CB-R, CB1, CB1A, and CB1K5.

CNR2 has only the one alias: CB2.

I also searched the HGNC website for those alternate gene symbols and they appear to be different genes, although CB1 and CB2 are GPCR class receptors, humans have hundreds of these genes. Most cell scientists would not be surprised if cannabinols bound other receptors - the fact is probably only that CB1 and CB2 bind strongest and were found first. Often weak binding is important to specific pharma/neuro responses.

That is to say there shouldn't be any single Cannabinoid receptor in the animal repetoire. CB1 and CB2 almost certainly have a primary function other than cannibinol binding. Primates evolved in the old world and cannabis is thought to have originated in the Himalayan regions. From an evolutionary standpoint, the activity of cannabinoids is as much coincidence as anything, so you would expect that a single receptor gene is unlikely.

polynomials - Counting lattice points on an n-simplex

From your problem description, I assume the $x_i$ from the first two paragraphs are what is called $r_i$ later. I do not yet have a complete answer, but would like to point out some observations and ideas (sorry, I'm not allowed to write comments yet):

It seems to me that we can, without loss of generality, assume the $x_i$ to be commensurable. Otherwise, split $Sinmathbb{Z}[x_1,dots,x_n]$ into a representation wrt a basis of $mathbb{Z}[x_1,dots,x_n]$.

Thus, by multiplying through with a suitable constant, we can assume that the $x_i$ are positive integers. We may also assume $gcd(x_1,dots,x_n)=1$, since otherwise, any $S$ for which the equation has a solution is also divisible by this gcd, which allows dividing the whole equation. Edit: Both of these simplifying assumptions shift the set of solutions (to solutions for some other $S$ and $r_1,dots,r_n$, but in a bijective way.

The number of solutions for any particular $S$ and $x_1,dots,x_n$ can be counted using generating functions (similar to Polya's method for counting possibilities of giving change); with your example $S=98,a_1+99,a_2$ and $0 leq a_1,a_2 leq 100$, the number of solutions for $S$ is the coefficient of $x^S$ in the polynomial $(x^{98}+x^{2cdot98}+cdots+x^{100cdot98}),(x^{99}+x^{2cdot99}+cdots+x^{100cdot99})$, whose lowest exponent with coefficient larger than $1$ is $9899$.

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I'm not sure I've got a good way of explaining this. Essentially, the first of these polynomials is the generating function for the number of solutions for $S=98,a_1$ and the second is the generating function for the number of solutions for $S=99,a_2$. Since in these generating functions, the $S$ values are in the exponents, summation of the $S$ values corresponds to multiplication.

If you wanted to write a computer program to find the smallest $S$ such that the corresponding coefficient in the generating function as given above fulfills some condition (e.g., is larger than $1$), it would probably be a good idea to use standard written multiplication and use a heap structure for carrying out the steps. Such an implementation would provide a stream of coefficient/exponent pairs and can also use such as one of its two inputs, which means that the multiplication of very many polynomials can be performed with little memory overhead, especially without needing to store all the coefficients already checked and found not interesting, and the calculation can stop almost without computing anything beyond the first “interesting” term.

bioinformatics - How can I compare rates of evolution for two sets of proteins?

I have a list of candidate proteins as the result of my analysis. I am now trying to find various characteristics that they have in common. One of the things I would like to check is if my candidates are evolving faster or slower than the rest of the proteins in my dataset.

Now, I know how to do this manually by building multi species alignments for each of my proteins and calculating ka/ks ratios for each set of alignments. This, however, is not a trivial process and I really do not want to do this manually for my ~1500 candidates.

Can anyone suggest a tool that will take two lists of proteins and return an estimate of evolutionary rate for each one?

I am primarily interested in human but also mouse, yeast, fly and worm. Ideally, I would need a tool that can take a list of proteins, assign them to an orthologous group and return a value representing the rate of evolution for that group.

pathology - What causes mutations in regulatory genes?

nothing causes mutations in any specific genes. mutations actually occur as the result of random processes and can mutate any given point in the genome. mutations may be found more often in some genes or regions of the genome more often because they have a positive selective force that gives them more staying power in the gene pool over time.

Its better to think of it this way: mutations happen everywhere, but they stay in the gene pool more often if they do something useful. Genes are the active portions of the genome so mutations in genes are more likely to stay around.

soft question - japanese/chinese for mathematicians?

This may not be the answer you're looking for, but I thought I'd share my experience as someone who was born in Japan but was transplanted quickly into the United States. My Japanese is not nearly as good as it should be, but is certainly good enough to read math.

A beautiful part of reading Japanese/Chinese math is that you can grasp the meaning without knowing how to pronounce anything. I don't know any technical Chinese, but in Japanese,

写像

is the word for "mapping" or "function", and the literal meaning of its characters hints at this. Let me explain.

The first character means to transcribe, to picture, or to give a visual form--poetically, it can mean to simply give an abstract form to something, rather than a visual one. (For instance, the word 写真 means photograph, where the second character in this particular word means "truth". It might be silly to think the word for photograph is to "picture something truly/in its true form", but that's a beautiful translation to ponder on another occasion.)

The final character in 写像 means figure, or image, or an embodiment. For instance, the word 画像 means "image" in the computer sense of file type. In fact the character 像 alone can mean "image" in the sense of mathematics, as in the image of something under a map.

In short, the word for "function" or "map" can be literally and clumsily translated back into English as "forming an image" or "creating a figure" or "realizing a form", most abstractly. I doubt any Japanese person ever thinks in these terms, no more than we think of the word "projection" deeply in terms of its Latin roots. But to harzard a guess at the meanings of these words can be a beautiful experience, and one unique to those weirdos who know the meanings of things without knowing how to say them.

So it may be a really interesting experience to simply learn the meaning of each commonly occurring (math) character---I'll list a few below---and to get a feel for the mathematical meanings of their combinations via intuition. When I've read Japanese math books, the feeling of knowing the meaning on a page without knowing how to pronounce a word has been the most rewarding and beautiful part. If you choose to do this, the best tip I have is to simply write: Make sure you copy and write the characters over and over again, so you begin to distinguish subtle differences between them.

For the enjoyment of some, here are examples of Japanese math words and the meanings of their constituent characters. I'll list some irrelevant meanings of some characters--though characters often only take on one of many meanings based on context, I still think it's fun to know their other possible meanings.

空間 (space)

空 = sky, emptiness, space, air

間 = between, the space between, an interval of time

位相 (topology)

位 = rank (as in seniority or importance in an organization), a word for counting dead souls, decimal place, position. As a verb, it can mean to locate--i.e., to determine the position of.

相 = form, shape, appearance, the relationship of one thing to another.

Strangely enough, 位相　can also mean the phase of something, as in the angle or phase of a complex number or a wave. It also mean the phase of something as in "solid/liquid/gaseous". I would assume that the term first came to use to describe the states of matter, was tangentially used to describe the phase of wave-like phenomena since the English term "phase" was used in both instances.

微分 (derivative, to take the derivative of)

微 = infinitesimal, tiny, slight

分 = to divide, an amount of something.

In learning language so much emphasis is placed on the sounds of things, rather than on the abstract units of meaning. I suppose Chinese characters were developed exactly to avoid this aural emphasis, but it is always a joy to have zero verbal understanding with a Chinese or Korean person, but to be able to communicate by writing characters in the air.

Well, perhaps this was not helpful in the least, but maybe it will at least entertain some non-Japanese-speakers. (By the way, I'd be very curious to hear if the Chinese technical terms are the same, as almost all technical terms in Japanese utilize kanji, or Chinese characters.)

behaviour - Is it possible that the recipient of a heart transplant would display some of the donor's personality traits, as if the heart has memories?

Although there is clearly no feasible mechanism for such a phenomenon, there is good evidence that transplant patients can believe in some sort of transference of qualities from the donor. See for example (my emphasis):

Inspector, Y. et al. (2004) Another Person's Heart: Magical and Rational Thinking in the Psychological Adaptation to Heart Transplantation. The Israel Journal of Psychiatry and Related Sciences 41: 161-73.

Abstract

The goal of this study was to examine heart transplant recipients' psychological adaptation to another person's heart, with particular emphasis on recipients' attitudes toward graft and donor.
Thirty-five male heart recipients were examined by: the Symptom Distress Checklist (revised) (SCL-90-R); the Depression Adjective Checklist (DACL); a Post-Traumatic Stress Disorder Questionnaire (PTSD-Q); a Heart Image Questionnaire (HIQ); and a Semi-Structured Interview (SSI), aimed at eliciting attitudes and fantasies regarding the transplanted heart.
All instruments indicated high levels of stress even several years after the transplant, but, simultaneously, 73% of recipients felt that acquiring a new heart had had a dramatic influence on their lives with a new appreciation of the preciousness of life and a shift of priorities, toward altruism and spirituality. Sixty percent returned to work after the transplant but some had to adapt to a changed attitude from those around them who regarded them as anything from mystical creatures to vulnerable or still-sick individuals. While all recipients possessed a scientific knowledge of the anatomy and physiological significance of the heart (as revealed in the HIQ), many endorsed fantasies and displayed magical thinking: 46% of the recipients had fantasies about the donor's physical vigor and prowess, 40% expressed some guilt regarding the death of the donor, 34% entertained the possibility of acquiring qualities of the donor via the new heart. When asked to choose a most and least preferred imagined donor, 49% constructed their choices according to prejudices, desires, or fears related to ethnic, racial or sexual traits attributed to the donor.
This study confirms the intuitive idea that heart transplant involves a stressful course of events that produces an amplified sense of the precariousness of existence. Simultaneously, it gives rise to rejoicing at having been granted a new lease on life and a clear sense of new priorities, especially with regard to relationships. Less expectedly, this study shows that, despite sophisticated knowledge of anatomy and physiology, almost half the heart recipients had an overt or covert notion of potentially acquiring some of the donor's personality characteristics along with the heart. The concomitance of the magical and the logical is not uncommon in many areas of human existence, and is probably enhanced by the symbolic nature of the heart, and maybe, also, by the persistent stress that requires an ongoing, emotionally intense, adaptation process.

Another study is reported in:

Bunzel B, et al. (1992) Does changing the heart mean changing personality? A retrospective inquiry on 47 heart transplant patients. Qual Life Res. 1:251-6.

To actually investigate this further I suppose it would be necessary to set up experiments in which patients had transplant operations in which (unknown to the patient) no transplant actually took place, with the expectation that these individuals would display the same magical thinking. Grant application?

Set theory in practice

If you are being, say, at least semiformal in your approach to set theory, whether or not objects which are not sets exist depends upon the particular brand of set theory you choose. The most common contemporary set theory, ZFC, is a "pure set theory", in which every object is itself a set, so the men indeed do not form a set.

But there are other set theories which allow non set elements, or urelements (what a great name!). In particular, Quine's New Foundations with Urelements is a relatively popular such theory.

So far as I know it is towards the philosophical end of the spectrum to worry about whether sets should be allowed to contain urelements or not. The mathematical justification for this is that, using the axiom of choice, any set can be put in bijection with a von Neumann ordinal, hence a pure set. But you should be able to speak of sets of men if you want to, I suppose.

Addendum: I like Sergei Ivanov's answer. He hits the following key point: if you ask a generic mathematician whether or not an object which is not a set can be an element of a set, you will not get either "yes" or "no" as an answer, but rather an explanation of why they regard the question as being a mathematically unfruitful one. When using sets for mathematical purposes, the "nature" of the objects which comprise sets is now regarded as being completely irrelevant. This is the "structuralist" approach to mathematics, which has been clarified and taken further by the more modern categorical approach.

ct.category theory - Coequalizer in the category of primitive recursive functions

For those readers unfamiliar with the class of primitive recursive functions, let me say that you may simplify things somewhat by fruitfully thinking of them as poor cousins of the computable functions. They were introduced, before Turing, as a natural class of computable total functions. But the class of primitive recursive functions is too weak to carry out the unbounded search operation that is so essential to computability (and which leads to partial functions, when the search fails). Nothing in my answer to the question will depend on any difference between the primitive recursive functions and the class of computable functions.

My answer is that neither the category of primitive recursive functions nor of computable functions admits co-equalizers. The basic reason is that for some primitive recursive functions, the smallest congruence is simply not a computable relation.

If one thinks about how one might build the congruence, this is to be expected, since there is a hidden unbounded search operation involved, and seemingly no way to know in advance whether it will succeed. Thus, we don't expect it to be computable. More specifically, if f,g are maps from A to B, then the desired congruence on B is the smallest relation making f(a) ∼ g(a) for all a. One concrete way to think about it is that two points b and c in B become equivalent with respect to ∼, if there is a zig-zag pattern a1,a2,... such that b=f(a1), g(a1)=f(a2), g(a2)=f(a3), etc. ending at g(an)=c (or swapping f and g in this pattern). Thus, to say that b ∼ c, we seem to need to discover whether one can get from b to c by following such a zig-zag pattern through the graphs of f and g. But that is an existential search, and we seem to have no computable way to determine whether we will find one or not. (The congruence, however, will always be computably enumerable, if f and g are computable.)

Let me give now a more detailed proof. Let A be a primitive recursive subset of the plane ω², such that the projection of A onto the first coordinate { a | exists b A(a,b) } is not primitive recursive. For example, let A be the set of pairs (a,b) such that a is a Turing machine program, and b is a code for a halting computation of program a on input 0. This is a primitive recursive set, since it is Delta₀ definable, but the projection of A onto the first coordinate is precisely the halting problem, the set of halting programs, and this is definitely not primitive recursive, because it is not even computable.

Now, consider the function f(a,b)=(a,b+2), if ¬A(a,b), otherwise f(a,b)=(a,b+1) if A(a,b) holds. Let i(a,b)=(a,b) be the identity function.
Suppose towards contradiction that h:ω² to N is a co-equalizer of f and i. According to my (crude) understandiing of what this means, which I just learned this afternoon at the Wikipedia page here, it means that h.f = h.i, and h has the universal property that wheneverr h'.f = h'.i, then there is u with u.h = h'. (I am using dot for composition.)

Consider the case that a is not in the projection of A, so that A(a,b) never holds. In this case, we will have f(a,b)=(a,b+2) for all b, and so h(a,b) = h(i(a,b)) = h(f(a,b)) = h(a,b+2). By the universal property, h will send all the even values h(a,2b) to one value, and h(a,2b+1) to another. In particular, in this case, h(a,0) is not equal to h(a,1).

In the other case, suppose that a is in the projection of A, so that A(a,x) for some smallest value x. Now, we will still have h(a,b) = h(a,b+2) for values of b below x. But now, once at x, we see that h(a,x) = h(i(a,x)) = h(f(a,x)) = h(a,x+1). So here, h is now agreeing at an odd and even value of the second coordinate. It follows that all the odd and even values below x must be sent to the same value by h, and so in this case, h(a,0) = h(a,1).

Thus, the function g(a) = 1 if h(a,0) = h(a,1) and g(a) = 0 if h(a,0) not= h(a,1) is precisely the characteristic function of the projection of A. But that projection was not computable! It follows that h cannot be computable either. Thus, there can be no co-equalizer of the functions f and i within the category of computable functions, and hence also no co-equalizer within the category of all primitive recursive functions. QED

Finally, let me apologize for my clumsy working with category theory here. I am a category theory newbie, coming from logic and set theory, with a little computability theory. But if I may say so, I find this question attractively on the boundary between logic and category theory. The audience for the question would seem to be two large groups of talented people, logicians and category theorists, who I believe might benefit from greater interaction.

ag.algebraic geometry - Points and DVR's

Suppose that $X$ is a projective variety, and that $v$ is a discete valuation on $K(X)$
(trivial on $k$) whose corresponding valuation ring we will denote by $R$. The valuative criterion shows that
the map Spec $K(X) rightarrow X$ extends to a map Spec $R rightarrow X$. If I have the terminology correct, the image of the closed point of Spec $R$ is called the centre of
the valuation $v$ on $X$. It has codimension anywhere between $1$ and dim $X$.
Note that if $x in X$ is the centre, then $R$ dominates $mathcal O_x$ in $K(X)$
(i.e. we have a local inclusion of local rings $mathcal O_x subset R$).

Let's suppose for a moment that $X$ is a smooth surface. If the centre $x$ is codimension 1,
then both $mathcal O_x$ and $R$ are (discrete) valuation rings. Since valuation rings
are (characterized by being) maximal for the partial order of dominance, $R$ and $mathcal O_x$ coincide, and so the discrete valuation $v$ is just that given by the divisor of which
$x$ is the generic point.

Suppose instead that $x$ is a closed point.
Now we can blow up $x$ in $X$, to get a projective variety $X_1$, and the centre of $v$ in $X_1$ will now be contained in the exceptional divisor of $X_1$ (i.e. the preimage of $x$). If it coincides with the exceptional divisor,
then we have found a curve on $X_1$ giving rise to $v$; otherwise it is a point
$x_1$, which we can blow up again.

Either we eventually obtain a divisor on some iterated blow-up of $X$, or we
obtain a sequence of points $x in X, x_1 in X_1, ldots,$ with each $X_n$ a blow-up
of the previous. In this case one sees that $R = bigcup mathcal O_{x_n}.$

There are a couple of exercises related to this issue in Hartshorne, namely II.4.5, II.4.12, and V.5.6. If I understand them correctly, any such sequence of $x_n$ gives
a valuation ring $R$ in this way, and $R$ is a discrete valuation ring unless one constructs the sequene $x_n$ in the following manner: choose an irreducible curve $C$ in $X$ and define $x_n$ to
be the intersection of the proper transform of $C$ in $X_n$ with the exceptional
divisor. For a sequence $x_n$ constructed in this latter manner, one obtains not
a discrete valuation ring, but rather a rank 2 valuation ring: the valuation is determined
by first taking the valuation at the generic point of $C$, and then (for those functions
which are defined and non-zero at this generic point) restricting to $C$ and computing
the order of zero or pole at $x$.

What is the geometric intuition for the discrete valuation rings that correspond
to an infinite sequence $x_n$ rather than to some curve on $X$? One can think of
them as a transcendental curve on $X$, passing through $x$.
Indeed, imagine you had such a curve.
Then you could restrict a rational function to it; since it is transcendental,
a non-zero rational function would not have a zero or pole along this curve, and so
would restrict to give a non-zero meromorphic function on the curve. We could then
compute the order of the zero or pole of this meromorphic function at $x$. In other
words, because the curve is transcendental, we get a rank one valuation, in contrast to the rank two valuations that arise when we apply this process with an algebraic curve
$C$ passing through $x$.

I'm not sure about the details of the higher dimensional case. (Among other things, I am worried about the possibility of the center being codim > 1, but singular, which seems like it could complicate the analysis.) Does anyone here know how it goes?

examples - When does 'positive' imply 'sum of squares'?

I think your question lives most naturally in the category of ordered rings.

Here is one example: a field can be ordered iff it is formally real: i.e., iff -1 is not a sum of squares. However, more is true: if x is any element of a field K of characteristic different from 2 which is not a sum of squares, then there exists an ordering < on K in which x is negative. Thus any field which admits more than one ordering will have positive elements which are not sums of squares. For example, in Q(sqrt{2}), with the usual convention, sqrt{2} is positive, but it is not a sum of squares, because in a different ordering (here, an adjustment of the given ordering by a field automorphism!) it is negative.

Another Example: No, a positive definite rational or integral quadratic form need not be equivalent to a sum of squares. For instance the quadratic forms x^2 + y^2 and x^2 + 2y^2
are not equivalent over Q. For one thing, the discriminant of the quadratic form (= the product of the coefficients, for a diagonal quadratic form) is well-determined up to a square in the ground field. So it comes back to the fact that in R, every positive number is a square, but not in Q.

For matrices: look at the 1x1 case!

As was alluded to before, another case of this is Hilbert's 17th problem: let K be an ordered field with real closure R. (For simplicity just take K = R = real numbers!) Let f in K(x_1,..,x_n) be a rational function such that for all (a_1,...,a_n) in R^n at which f is defined, f(a_1,...,a_n) >= 0. Then there are rational functions g_1,l..,g_m in
K(x_1,...,x_n) such that f = g_1^2 + ... + g_m^2.

differential topology - Stokes' theorem etc., for non-Hausdorff manifolds

The existence of flows in the direction of a vector field seems to require Hausdorff; indeed, consider the vector field $frac{partial}{partial x}$ on the line-with-two-origins. We have no global existence of a flow for any positive t, even if we make our space compact (that is, considering the circle-with-one-point-doubled). If the nonexistence of the flow is not visibly clear, consider instead the real line with the interval [0,1] doubled.

Also, partitions of unity do not exist; for example, in the line with two origins, take the open cover by "the line plus the first origin" and "the line plus the second origin". There is no partition of unity subordinate to this cover (the values at each origin would have to be 1).

For me, a basic example of the beauty of this function-theoretic approach is the definition of a vector field as a derivation $Dcolon C^infty(M)to C^infty(M)$. The proof that such a derivation defines a vector field hinges upon the fact that $Df$ near a point p only depends on $f$ near the point p. To prove this fact you use the fineness of your sheaf $mathcal{O}_X$, i.e. the existence of partitions of unity. (It is true though that the failure of fineness in the non-Hausdorff case is of a different sort and might not break this particular theorem.) I feel that the existence of partitions of unity, and the implications thereof, is one of the basic fundamentals of approaching smooth manifolds through their functions; more importantly, a good handle on how partitions of unity are used is important to understand the differences that arise when the same approach is extended to more rigid functions (holomorphic, algebraic, etc.).

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Now that the question has been edited to ask specifically about Stokes' theorem, let me say a bit more. Stokes' theorem will be false for non-Hausdorff manifolds, because you can (loosely speaking) quotient out by part of your manifold, and thus part of its homology, without killing all of it.

For the simplest example, consider dimension 1, where Stokes' theorem is the fundamental theorem of calculus. Let $X$ be the forked line, the 1-dimensional (non-Hausdorff) manifold which is the real line with the half-ray $[0,infty)$ doubled. For nonnegative $x$, denote the two copies of $x$ by $x^bullet$ and $x_bullet$, and consider the submanifold $M$ consisting of $[-1,0) cup [0^bullet,1^bullet] cup [0_bullet,1_bullet]$. The boundary of $M$ consists of the three points $[-1]$ (with negative orientation), $[1^bullet]$ (with positive orientation), and $[1_bullet]$ (with positive orientation); to see this, just note that every other point is a manifold point.

Consider the real-valued function on $X$ given by "$f(x)=x$" (by which I mean $f(x^bullet)=f(x_bullet)=x$). Its differential is the 1-form which we would naturally call $dx$. Now consider $int_M dx$; it seems clear that this integral is 3, but I don't actually need this. Stokes' theorem would say that

$int_M dx=int_M df = int_{partial M}f=f(1^bullet)+f(1_bullet)-f(-1)=1+1-(-1)=3$.

This is all fine so far, but now consider the function given by $g(x)=x+10$. Since $dg=dx$, we should have

$int_M dx=int_M dg=int_{partial M}g=g(1^bullet)+g(1_bullet)-g(-1)=11+11-9=13$. Contradiction.

It's possible to explain this by the nonexistence of flows (instead of $df$, consider the flux of the flow by $nabla f$). But also note that Stokes' theorem, i.e. homology theory, is founded on a well-defined boundary operation. However, without the Hausdorff condition, open submanifolds do not have unique boundaries, as for example $[-1,0)$ inside $X$, and so we can't break up our manifolds into smaller pieces. We can pass to the Hausdorff-ization as Andrew suggests by identifying $0^bullet$ with $0_bullet$, but now we lose additivity. Recall that $M$ was the disjoint union of $A=[-1,0)$ and $B=[0^bullet,1^bullet] cup [0_bullet,1_bullet]$. So in the quotient $partial [A] = [0]-[-1]$ and $partial [B] = [1^bullet]-[0]+[1_bullet]-[0]=[1^bullet]+[1_bullet]-2[0]$, which shows that $partial [M]neq partial [A]+partial [B]$. This is inconsistent with any sort of Stokes formalism.

Finally, I'd like to point out that Stokes' theorem aside, even rather nice non-Hausdorff manifolds can be significantly more complicated than we might want to deal with. One nice example is the leaf-space of the foliation of the punctured plane by the level sets of the function $f(x,y)=xy$. The leaf-space looks like the union of the lines $y=x$ and $y=-x$, except that the intersection has been blown up to four points, each of which is dense in this subset. In general, any finite graph can be modeled as a non-Hausdorff 1-manifold by blowing up the vertices, and in higher dimensions the situation is even more confusing. So for any introductory explanation, I would strongly recommend requiring Hausdorff until the students have a lot more intuition about manifolds.

fa.functional analysis - Symmetric (extended) Haagerup tensor product

Given a von Neumann algebra M, then the weak$^*$ (or extended) Haagerup tensor product of M with itself is the collection of $tauin Moverlineotimes M$ with $$tau=sum_i x_iotimes y_i$$ the sum converging sigma-weakly, where $$Big|sum_i x_ix_i^*Big|<infty, quad Big|sum_i y_i^*y_iBig|<infty$$ again, these sums of positives being in the sigma-weak sense. Let $sigma:Moverlineotimes Mrightarrow Moverlineotimes M$ be the swap map. Notice that the extended Haagerup tensor product is not symmetric under $sigma$.

However, suppose that I happen to know that both $tau$ and $sigma(tau)$ are in the extended Haagerup tensor product. Can I find a "symmetric" expression for $tau$, similar to that above (surely it is too much to hope that, say, also $sum_i x_i^*x_i$ and $sum_i y_iy_i^*$ converge, but is there something a little weaker?)

Pisier and Oikhberg studied something similar(ish) in a Proc EMS paper, but I don't know of any other sources in the literature.

Edit: I should say that I'm also interested in the case when actually $tau=sigma(tau)$.

real analysis - Integrability of derivatives

I believe this answers the question:

---

MR0425042 (54 #13000)
Goffman, Casper
A bounded derivative which is not Riemann integrable.
Amer. Math. Monthly 84 (1977), no. 3, 205--206.

In 1881 Volterra constructed a bounded derivative on $[0,1]$ which is not Riemann integrable. Since that time, a number of authors have constructed other such examples. These examples are generally relatively complicated and/or involve nonelementary techniques. The present author provides a simple example of such a derivative $f$ and uses only elementary techniques to show that $f$ has the desired properties.

---

The paper is available here:

http://math.uga.edu/~pete/Goffman77.pdf

Definition of a von Neumann algebra

Consider the monomorphism A→A** of Banach spaces.
Here A** denotes the second dual of A as a Banach space.
The Banach space A** is a von Neumann algebra with the predual being A*.
See Section 1.17 in Sakai's C*‑algebras and W*‑algebras.

We have a commutative square of Banach spaces consisting of morphisms
A→A**→B** and A→B→B**.
Thus we can pull back the ultraweak topology on A** to A
and obtain a functorial topology on C*‑algebras due to the commutativity of the square above.

Henceforth denote by A* the dual of A in the new topology and
by A** the dual of A* in the norm topology

If the canonical morphism A→A** is an isomorphism, then A has a predual,
therefore it is a von Neumann algebra.

Unfortunately, if A is a von Neumann algebra, then the functorial topology
does not coincide with the ultraweak topology
and the canonical morphism A→A** is not an isomorphism.

We can fix this problem by composing the monomorphism A→A** with the multiplication by a certain central projection.
However, the definition of this central projection relies on the fact that A is a von Neumann
algebra and I don't see any way to extend it to arbitrary C*‑algebras.

immunology - More proteins from diet when common cold and flu?

My coach says that I need to eat 1.2 - 1.5 grams of proteins per kilogram when I have a common cold and flu.
I normally eat one gram of proteins per kilogram, while double it when doing my exercise training.

He explained that the reason why you need to get more good proteins when you are sick is that the body is building antibodies by the adaptive immune system and later for innate immunity.

I think he can be right based only on my own experiences getting better from common cold with low protein diet - it just takes a lot of time, while under high protein diet, you get better faster.
I also think his explanation makes complete sense to me, since antibodies and antigens are proteins.

Why do you need more proteins when you are sick?

homework - Male behaviour during breeding season

a) stronger aggressive behaviour only during egg laying

c) stronger aggression when egg is first laid than during nest building

d) greater aggression during hatching eggs than during nest building

These all occur after the stage which determines fitness. Egg laying (a and c) and Egg hatching (d) occur after the male has won the right to mate.

b) stronger aggressive behaviour during nest building rather than
during hatching of eggs

Assuming the behavior increases mating success and fitness, the answer is B because this is the only strategy that attracts more females and gets rid of rival males before it is too late (i.e. before the female mates). This is also assuming that the female can/will only mate once.

In chronological order the answers run "B,C/A,D" (C & A are really one and the same in my eyes). Mating occurs between B and C.

Torsors in Algebraic Geometry?

So thanks to the comments of Tyler Lawson I have been able to figure out what is happening in this example, so I thought I should post it as an answer. I think this is also what Torsten Ekedahl was getting at in his comment, as well.

I think it helps to be extra clear because this example is rather confusing. For starters there is the group scheme, $$mathbb{G} to S.$$ In this example $S = mathbb{A}^1$ is the affine line. This is a group object over $S$, so it can be thought of as an $S$-family of group schemes. At the points $x_1$ and $x_2$ it is the trivial group, and at all other points it is some fixed abelian group $A$. For a concrete example we can take $A = mathbb{Z}/3$, and then $mathbb{G}$ looks something like this:

The bottom line represents $S$. Notice that there is a unique global section, the zero section. Away from the set $Y = x_1 cup x_2$, there are more sections. Associated to $mathbb{G}$ is a sheaf on the site of schemes over S. This is the same sheaf I called $A_{C_Y}$.

As outlined in the question we have that $check H^1(S; A_{C_Y}) = A$ is non-trivial. We can even construct a non-trivial cocycle using the covering consisting of the two open subsets
$$U_1 = S - x_1$$
$$U_2 = S - x_2$$
Notice that $U_{12} = U_1 times_S U_2 = C_Y$, the complement of Y in S. This is exactly the subspace that supports a section. The picture is a little misleading here as it looks like there are lots of sections over $C_Y$. However, because we are using the Zariski topology we have only $A$-many of them. Such a section over $C_Y$ has to be constant on $C_Y$.

Now each of these sections (of which there are A-many) gives rise to a Cech cocycle and so we should be able to construct a $mathbb{G}$-torsor over $S$ for each one of these. The usual construction is that this torsor is given as the coequalizer of
$$U_{12} times_S mathbb{G} rightrightarrows coprod U_i times_S mathbb{G}$$
Where one map is the usual inclusion and the other is also inclusion (the other one), but twisted using the cocycle.

Now the cocycle is only defined over $C_Y$. And over the complement of $C_Y$, namely Y, $mathbb{G}$ is trivial. It has a unique fiber. So I restricted attention to just the "interesting part", the $C_Y$ part. Then I got that the coequalizer becomes,
$$C_Y times A rightrightarrows (C_Y cup C_Y) times A$$
which has trivial coequalizer $C_Y times A$. All of these are true facts, except the part about $C_Y$ being the only interesting part. I was wrongly assuming that if the torsor was trivial over this part, then it had to be isomorphic to $mathbb{G}$.

This is not the case. Somehow Tyler's comments made me realize this. The actual full colimit looks something like this:

Notice that this space is a trivial $C_Y times A$-torsor when restricted to $C_Y$, and over $U_1$ and $U_2$ there exist unique sections. However there is no global section, so it is not a globally trivial object. Let's call this object P.

A little book keeping shows that there is an action of schemes over S,
$$mathbb{G} times_S P to P$$
making P into a torsor in the second sense.

So this is not a counter example. Both notions of torsor agree here.

But this raises the question:

Question: Do these two a priori different notions of torsor agree in Algebraic Geometry? If not what is the easiest counter example?

I don't know the answer to this.

digestive system - Do foods with preservatives become less toxic in the gut?

Do foods with preservatives take longer to digest?

Food preservatives are either antimicrobials (e.g. sorbate, sulfite) or antioxidants (free radical scavengers such as butylated hydroxyanisole).

Digestion is initiated in the stomach by hydrochloric acid and pepsin. It continues in the small intestine with the action of numerous other enzymes including amylases, lipases and proteases. There is no obvious way that these digestive processes would be inhibited by preservatives.

mg.metric geometry - Can any rectangle be inscribed in any convex figure?

Yes, this follows from a more general result in

Nielsen and Wright, Rectangles inscribed in symmetric continua. Geom. Dedicata 56 (1995), no. 3, 285–297 MR

(This is reference 4 in the Wikipedia article I quoted in my answer to your previous question.)

In their terminology, a simple closed curve $C$ is symmetric if there exists a point $Pnotin C$ such that each straight line through $P$ intersects $C$ in exactly 2 points. This condition is trivially satisfied when $C$ is a boundary of a convex region.

ct.category theory - Homotopy pullbacks and homotopy pushouts

You can think of the pushout of two maps f : A → B, g : A → C in Set as computing the disjoint union of B and C with an identification f(a) = g(a) for each element a of A. We could imagine forming this as either the quotient by an equivalence relation, or by gluing in a segment joining f(a) to g(a) for each a, and taking π₀ of the resulting space. If two elements a, a' of A satisfy f(a) = f(a') and g(a) = g(a'), the pushout is unaffected by removing a' from A. The homotopy pushout is formed by gluing in a segment joining f(a) to g(a) for each a and not forgetting the number of ways in which two elements of B ∐ C are identified; instead we take the entire space as the result. It is the "derived" version of the pushout.

In general you can think of the homotopy pushout of A → B, A → C as the "free" thing generated by B and C with "relations" coming from A. But it's important that the "relations" are imposed exactly once, since in the homotopical/derived setting we keep track of such things (and have "relations between relations" etc.)

Another, possibly more familiar example: In a derived category, the mapping cone of a morphism f : A → B is the homotopy pushout of f and the zero map A → 0. This certainly depends on A, even when B is the zero object: it is the suspension of A.

ct.category theory - Model category structure on Set without axiom of choice

For any elementary topos $T$, there is a model category structure on $T$, whose cofibrations are the monomorphisms, and whose weak equivalences are the maps $Xto Y$ such that, either $Y$ is empty, either $X$ is non-empty (the fibrations are the split epimorphisms). In a topos, there exists an internal Hom (written here $underline{Hom}$), as well as a subobject classifier $Omega$: it comes with a map $t:1to Omega$ which is the universal subobject, in the sense that, for any object $X$ in $T$, pulling back along $t$ induces a bijection
$$text{ { subobjects of $X$} }simeq Hom(X,Omega)$$
Using the universal property of $Omega$, the diagonal $Xto Xtimes X$ defines a map $Xtimes Xto Omega$, whence an embedding of $X$ into its power object: $Xto P(X)=underline{Hom}(X,Omega)$. This gives you injective resolutions. We thus have a functorial fibrant replacement $Xmapsto I(X)$, where $I(X)=X$ if $X$ is empty, and $I(X)=P(X)$ otherwise (note that an object of $T$ is fibrant if and only if it is either empty, either injective). It remains to factor maps into a trivial cofibration (resp. cofibration) followed by a fibration (resp. trivial fibration). Let $f:Xto Y$ be a map in $T$. Such a factorization for $f$ is given by the map $Xto Ftimes Y$ followed by the projection $Ftimes Yto Y$, for $F=I(X)$ (resp. $F=P(X)$). Hence we get a model structure on $T$, while we never used the axiom of choice.

Edit: and now, some genuine non sense (not abstract):

This model category structure on $Set$ is rather degenerate though. It seems that, to get model categories in general (on $Cat$ or on simplicial sets), we should change the definition of a model category by asking for lifting properties only internally: given maps $i:Ato B$ and $p:Xto Y$, $i$ has the (internal) left lifting property with respect to $p$ is the map
$$Hom(B,X)to Hom(X,A)times_{Hom(Y,A)}Hom(Y,B)$$
is an epimorphism. In this way $Cat$ and $SSet$ remain model categories without the axiom of choice.

Edit: here, it seems I have been rather optimistic (see Mike comment below).
I was thinking about working above an arbitrary topos (instead of sets), but,
if we work externally in a setting without axiom of choice, then the lifting property suggested above is just the usual one, so that my remark seems to be (and is) silly.
But what follows still makes sense.

Note that it is easy to get a structure of category of fibrant objects (in the sense of Brown, Abstract homotopy theory and generalized sheaf cohomology, Trans. Amer. Math. Soc. 186 (1974), 419–458) on $Cat$, which is sufficient to have calculus of fractions (up to homotopy), and gives you the abstract tools to explain the good behaviour of anafunctors. As for the size problems when we avoid the axiom of choice, maybe it would be worth looking for an "internal notion of size" to get that the Hom's of any homotopy category of a model category are small in some sense (but I had never thought seriously about this before).

evolution - Probability of Extinction under Genetic Drift

update

The answer is here!

---

Original comment/answer

Kimura and Ohta (1969) showed that assuming an initial frequency of $p$, the mean time to fixation $\bar t_1(p)$ is:

$$\bar t_1(p)=-4N\left(\frac{1-p}{p}\right)ln(1-p)$$

similarly they showed that the mean time to loss $\bar t_0(p)$ is

$$\bar t_0(p)=-4N\left(\frac{p}{1-p}\right)ln(p)$$

Combining the two, they found that the mean persistence time of an allele $\bar t(p)$ is given by $\bar t(p) = (1-p)\bar t_0(p) + p\bar t_1(p)$ which equals

$$\bar t(p)=-4N\cdot \left((1-p)\cdot ln(1-p)+p\cdot ln(p)\right)$$

This does not answer any of the two questions!

This answer gives...

• the average persistence time

but not...

• the probability of fixation rather than extinction if we wait an infinite amount of time

neither...

• the probability that the allele get extinct over a period of say 120 generations.

Can someone improve this answer?

zoology - Is life expectancy linked to intelligence in animals?

This is too long to be a comment. Life expectancy is apparently affected the size of an animal, and this is especially evident in mammals and supported by the heartbeat hypothesis. This hypothesis says that all mammals have a similar number of heartbeats, and that the larger an animal is, the less heart beats it has per minute. Therefore, the larger an animal, the longer it lives. A larger size is also associated with a larger brain (which is associated with higher intelligence, although this is not true in all cases since for example some dinosaurs were very large but had very tiny brains). All I'm saying is that any correlation between intelligence and life expectancy may not necessarily be brought about directly by intelligence, but by the size of the animal. Hope this helps.

co.combinatorics - Algorithmic Combinatorics resources?

Some branches of combinatorics lend themselves naturally to algorithms; graph theory is a natural example. However, straight-up enumerative combinatorics relies much more on analytic and algebraic methods. As a result, you often don't have any idea of how you could systematically generate the objects you want. In order to do so with a computer, you'd need to pick a "nice" encoding, impose a total order on the coded objects, and then generate them one-by-one, checking against previous ones for whatever equivalence you're concerned with. Burnside's or Polya's Theorem will tell you when you've found them all, but in the meanwhile, you want an encoding and an order that's easy to work with, and that generates successive objects fairly quickly.

Are there any good resources for this sort of algorithmic combinatorics? I'd like something that's not tailored to just a specific problem, unless that problem is representative of a large class of problems. Essentially, I'd like to know some particularly useful encodings and some useful generation algorithms. (For instance, for some problems, it might be more efficient to generate the objects randomly instead of using a total order).

Lebesgue measure of a set

Let m the n-dimensional Lebesgue measure on $R^n$.By definition of product measure, on each borelian set E

$m(E)=inf left(sum_{j=1}^infty m(R_j),:: Esubseteq bigcup R_j , ::R_j text{ rectangles}right)$

It is also true that lebesgue measures are regular, so
$m(E)=inf left(m(U), Esubseteq U, : U text{ open set} right)$.

Can I say that also holds
$m(E)=inf left(sum_{j=1}^infty m(B_j),:: Esubseteq bigcup B_j , ::B_j text{ balls}right)$ or not?

gt.geometric topology - What is an immersed submanifold?

Bing's house with two rooms is the image of an immersed sphere that is not in general position.

General position immersions are easy to build out of local pictures --- well sort of easy to build. Consider inside an $n$-ball

$D^n=$ ${ (x_1,ldots, x_n) : sum x_j^2 le 1 }$ all of the $k$-dimensional sub-spaces that have
$(n-k)$ of the $x_j =0$. This is the local picture for a minimal dimension multiple point. (The greater the multiplicity, the smaller the dimension of the intersection). You don't have to choose all such subspaces, but only some of them. In this way you have local pictures to patch together. Now if you know how to attach handles to spaces, you can attach handles that have immersed pieces together. One can construct Boy's surface from this point of view.

Sometimes you get stuck. For example, a figure 8 has one double point. Boy's surface has one triple point. Capping of a generic sphere eversion gives a $3$-manifold in $4$-space with one quadruple point. But if you start from the intersection
$(a,b,c,d,0)$ $cap (a,b,c,0,e)$ $cap (a,b,0,d,e)$ $cap (a,0,c,d,e)$ $cap(0,b,c,d,e)$ in the $5$-ball, there is no way to close this off to get a $4$-manifold with one quintuple point. There are plenty of codimension $1$ immersions in $5$-space, but they all have an even number of quintuple points.

You should also consider equatorial spheres in a large dimensional sphere. This is the boundary of the second example I gave. You can connect these with handles to get connected immersions.

A very cool example in 3-space (beyond Boy's surface and an acme Klein bottle) is obtained by twisting a figure 8 a full rotation. A half a twist gives a Klein bottle, a full-twist gives an immersed torus whose stable framing is induced by the Lie group structure.

Codimension $0$ examples are also very important. The standard $2$-disk with two handles that represents a punctured torus is the image of an immersion into the plane.

ct.category theory - Parametrized natural numbers object.

I give two examples of categories with finite products and simple NNO. In the first example the simple NNO is also a parameterized NNO, while in the second example it is not. Although it is difficult to understand your question, I believe the examples should clarify matters.

First, consider the category $mathcal{C}$ whose objects are the finite powers of $mathbb{N}$, namely $mathbb{N}^0$, $mathbb{N}^1$, $mathbb{N}^2$, ... and morphisms are set-theoretic functions $f : mathbb{N}^k to mathbb{N}^m$. This category clearly has finite products, is not cartesian-closed because there are too many morhisms $mathbb{N} to mathbb{N}$, and it has a parameterized NNO, namely the obvious one.

Second, consider the category $mathcal{D}$ whose objects are the finite powers of $mathbb{N}$, like before, and whose morphisms are as follows:

1. Morphisms $mathbb{N}^k to mathbb{N}^m$ with $m neq 1$ are all set-theoretic functions.


2. Morphisms $mathbb{N}^0 to mathbb{N}^1$ are all set-theoretic functions, i.e., for each natural number there is one.


3. Morphisms $mathbb{N}^k to mathbb{N}^1$ with $k neq 0$ are all set-theoretic functions $f : mathbb{N}^k to mathbb{N}$ for which there exists a projection $pi_j : mathbb{N}^k to mathbb{N}$ and $g : mathbb{N} to mathbb{N}$ such that $f = g circ pi_j$.

In other words, in $mathcal{D}$ every function into $mathbb{N}$ depends on only one of its parameters (exercise: prove that these are closed under composition.) The category $mathcal{D}$ has finite products and a simple NNO, namely the obvious one, but no parameterized NNO. If it did, we could construct addition ${+} : mathbb{N}^2 to mathbb{N}$ as a morphism in the category.

gr.group theory - Deriving a relation in a group based on a presentation

The theory (and practice) of automatic groups is the most generally useful systematic way to deal with these things. There is a nice package written by Derek Holt and his associates called kbmag (available for download here: http://www.warwick.ac.uk/~mareg/download/kbmag2/ ). There is a book "Word Processing in Groups" by Epstein, Cannon, Levy, Holt, Paterson and Thurston that describes the ideas behind this approach. It's not guaranteed to work (not all groups have an "automatic" presentation) but it is surprisingly effective.

I made up a short input file for kbmag, and it immediately came back with a "confluent" system of relations (a particular system which has a technical property that when you just do a series of string substitutions you always get the same answer no matter what order you do them in). For your edification, here they are (xi and yi are x^-1 and yi^-1 respectively, idWord is 1 [edited to show the derivations from kbmag):

#Initial equation number 1:
 #x*xi -> IdWord
#Initial equation number 2:
 #xi*x -> IdWord
#Initial equation number 3:
 #y*yi -> IdWord
#Initial equation number 4:
 #yi*y -> IdWord
#Initial equation number 5:
 #y^4 -> x^3*yi
#Initial equation number 6:
 #y*x*y -> x^2
#New equation number 7, from overlap 5, 3:
 #x^3*yi^2->y^3
#New equation number 8, from overlap 4, 5:
 #yi*x^3->y^4
#New equation number 9, from overlap 6, 3:
 #x^2*yi->y*x
#New equation number 10, from overlap 4, 6:
 #yi*x^2->x*y
#New equation number 11, from overlap 2, 7:
 #xi*y^3->y*x*yi
#New equation number 12, from overlap 2, 9:
 #xi*y*x->x*yi
#New equation number 13, from overlap 10, 1:
 #x*y*xi->yi*x
#New equation number 14, from overlap 1, 11:
 #x*y*x*yi->y^3
#New equation number 15, from overlap 11, 3:
 #y*x*yi^2->xi*y^2
#New equation number 16, from overlap 12, 1:
 #x*yi*xi->xi*y
#New equation number 17, from overlap 2, 13:
 #xi*yi*x->y*xi
#New equation number 18, from overlap 4, 15:
 #yi*xi*y^2->x*yi^2
#New equation number 19, from overlap 2, 16:
 #xi^2*y->yi*xi
#New equation number 20, from overlap 17, 1:
 #y*xi^2->xi*yi
#New equation number 21, from overlap 18, 3:
 #x*yi^3->yi*xi*y
#New equation number 22, from overlap 19, 3:
 #yi*xi*yi->xi^2
#New equation number 23, from overlap 2, 21:
 #xi*yi*xi*y->yi^3
#New equation number 24, from overlap 23, 3:
 #yi^4->xi*yi*xi
#New equation number 25, from overlap 3, 24:
 #y*xi*yi*xi->yi^3
#New equation number 26, from overlap 25, 2:
 #yi^3*x->y*xi*yi
#New equation number 27, from overlap 3, 26:
 #y^2*xi*yi->yi^2*x
#New equation number 28, from overlap 27, 4:
 #yi^2*x*y->y^2*xi
#New equation number 29, from overlap 3, 28:
 #y^3*xi->yi*x*y
#New equation number 30, from overlap 29, 2:
 #yi*x*y*x->y^3
#New equation number 31, from overlap 5, 5:
 #y*x^3->x^3*y
#New equation number 32, from overlap 5, 6:
 #y^3*x^2->x*y*x^2*y
#New equation number 33, from overlap 8, 7:
 #yi*x*y^3->x*y^2*x*yi
#New equation number 34, from overlap 7, 8:
 #y*x^2*y*x->x^2*y^3
#New equation number 35, from overlap 11, 6:
 #y*x*yi*x*y->xi*y^2*x^2
#New equation number 36, from overlap 12, 9:
 #xi*y^2*x->x*yi*x*yi
#New equation number 37, from overlap 10, 13:
 #yi*x*yi*x->x*y^2*xi
#New equation number 38, from overlap 11, 15:
 #y*x*yi*x*yi^2->xi*y^2*xi*y^2
#New equation number 39, from overlap 12, 16:
 #xi*y*xi*y->x*yi^2*xi
#New equation number 40, from overlap 17, 13:
 #y*xi*y*xi->xi*yi^2*x
#New equation number 41, from overlap 18, 15:
 #yi*x*yi^2*xi*y->x*yi^2*x*yi^2
#New equation number 42, from overlap 18, 20:
 #yi*xi*y*xi*yi->x*yi^2*xi^2
#New equation number 43, from overlap 19, 20:
 #yi*xi^3->xi^3*yi
#New equation number 44, from overlap 17, 21:
 #y*xi*yi^3->xi*yi^2*xi*y
#New equation number 45, from overlap 23, 20:
 #yi^3*xi^2->xi*yi*xi^2*yi
#New equation number 46, from overlap 22, 24:
 #yi*xi^2*yi*xi->xi^2*yi^3
#New equation number 47, from overlap 25, 19:
 #yi^3*xi*y->y*xi*yi^2*xi
#New equation number 48, from overlap 22, 26:
 #xi^2*yi^2*x->x*yi^2*xi^2
#New equation number 49, from overlap 27, 26:
 #yi^2*x*yi^2*x->y*xi*yi^2*x*yi
#New equation number 50, from overlap 49, 1:
 #y*xi*yi^2*xi*y->yi^2*x*yi^2
#New equation number 51, from overlap 7, 28:
 #x^3*y^2*xi->y^2*x^2
#New equation number 52, from overlap 27, 28:
 #yi*x*y^2*xi*y->y^2*xi*y^2*xi
#New equation number 53, from overlap 29, 12:
 #yi*x*y^2*x->y^3*x*yi
#New equation number 54, from overlap 31, 7:
 #x^3*y^2*x*yi->y*x^2*y^3
#New equation number 55, from overlap 32, 13:
 #x*y*x^2*y^2*xi->y^3*x*yi*x
#New equation number 56, from overlap 32, 14:
 #x*y*x^2*y^2*x*yi->y^2*x^2*y^2
#New equation number 57, from overlap 2, 56:
 #y*x^2*y^2*x*yi->x^2*y^2*xi*y^2
#New equation number 58, from overlap 56, 4:
 #y^2*x^2*y^3->x*y*x^2*y^2*x
#New equation number 59, from overlap 57, 4:
 #x^2*y^3*x*yi->y*x^2*y^2*x
#New equation number 60, from overlap 33, 34:
 #y^2*x^2*y^2*xi*y^2->x*y^2*x^2*y^2*x
#New equation number 61, from overlap 11, 35:
 #x^2*y^2*xi*y^2*xi->y*x^2*y^2*xi*y
#New equation number 62, from overlap 35, 25:
 #x*yi*x*yi^2*xi->xi*y^2*xi*y
#New equation number 63, from overlap 35, 32:
 #x^2*y^2*x^2*y^2*xi->y*x^2*y^2*x^2*y
#New equation number 64, from overlap 33, 35:
 #y^2*x^2*y^2*xi*y->yi*x*yi^2*x*yi^2
#New equation number 65, from overlap 64, 3:
 #y^2*x^2*y^2->yi^2*x*yi^2
#New equation number 66, from overlap 4, 64:
 #y*x^2*y^2*xi*y->xi*yi^2*x*yi^2*xi
#New equation number 67, from overlap 65, 3:
 #y*x^2*y->xi*yi^2*xi
#New equation number 68, from overlap 4, 66:
 #xi*y*xi*yi^2*xi->x^2*y^2*xi*y
#New equation number 69, from overlap 67, 3:
 #xi*yi*xi^2->y*x^2
#New equation number 70, from overlap 4, 67:
 #xi^2*yi*xi->x^2*y
#New equation number 71, from overlap 68, 2:
 #x^2*y^2*x->xi*y*xi*yi
#New equation number 72, from overlap 1, 69:
 #x*y*x^2->yi*xi^2
#New equation number 73, from overlap 69, 2:
 #x^3*y->xi*yi*xi
#New equation number 74, from overlap 70, 2:
 #x^2*y*x->xi^2*yi
#New equation number 75, from overlap 71, 1:
 #xi*yi^3->x^2*y^2
#New equation number 76, from overlap 2, 71:
 #yi*xi^2*yi->x*y^2*x
#New equation number 77, from overlap 72, 1:
 #xi^3*yi->x*y*x
#New equation number 78, from overlap 73, 3:
 #xi^3->x^3
#New equation number 79, from overlap 75, 4:
 #x^2*y^3->xi*yi^2
#New equation number 80, from overlap 1, 78:
 #x^4->xi^2
#New equation number 81, from overlap 2, 79:
 #xi^2*yi^2->x*y^3
#New equation number 82, from overlap 29, 36:
 #y^3*x*yi*x*yi->x*y^2*x*yi*x
#New equation number 83, from overlap 7, 37:
 #y^3*x*yi*x->yi^2*xi*y*xi
#New equation number 84, from overlap 4, 83:
 #xi*yi^2*xi^2->y*x*yi*x
#New equation number 85, from overlap 1, 84:
 #yi^2*xi^2->y^3*x
#New equation number 86, from overlap 9, 37:
 #yi^3*xi->y^2*x^2
#New equation number 87, from overlap 38, 8:
 #x^2*y^2*xi*y^2->xi*yi^2*x*yi^2
#New equation number 88, from overlap 11, 38:
 #xi*y^2*xi*y^2*xi*y->y^2*x*yi*x*yi^2
#New equation number 89, from overlap 88, 3:
 #xi*y^2*xi*y^2*xi->y*xi*y^2*xi*y
#New equation number 90, from overlap 38, 37:
 #y*xi*yi^2*x*yi^2*xi->yi*x*yi^2*x*yi^2
#New equation number 91, from overlap 39, 39:
 #x*y^2*x^2->yi*xi*y*xi
#New equation number 92, from overlap 41, 39:
 #yi*x*yi^2*x*yi^2*xi->x*yi*x*yi^2*x*yi^2
#New equation number 93, from overlap 42, 47:
 #xi*y*xi*yi^2*x*yi^2->x*y^2*xi*y^2*xi*y
#New equation number 94, from overlap 93, 4:
 #x*y^2*xi*y^2*xi*y^2->xi*y*xi*yi^2*x*yi
#New equation number 95, from overlap 2, 94:
 #y^2*xi*y^2*xi*y^2->x*y^2*x*yi*x*yi

#68 eqns; total len: lhs, rhs = 299, 246; 77 states; 0 secs.
max len: lhs, rhs = 8, 8.

#System is confluent.

#Halting with 68 equations.
#Exit status is 0