neurobiology - Do we understand the non-subjective mechanisms behind pleasure and pain?

Neurobiology is not my field of expertise, but this paper seems relevant:

Kent C. Berridge, Chao-Yi Ho, Jocelyn M. Richard, Alexandra G. DiFeliceantonio (2010) The tempted brain eats: Pleasure and desire circuits in obesity and eating disorders. Brain Research, 1350, 43-64.

What we eat, when and how much, all are influenced by brain reward mechanisms that
generate “liking” and “wanting” for foods. As a corollary, dysfunction in reward circuits
might contribute to the recent rise of obesity and eating disorders. Here we assess brain
mechanisms known to generate “liking” and “wanting” for foods and evaluate their
interaction with regulatory mechanisms of hunger and satiety, relevant to clinical issues.
“Liking” mechanisms include hedonic circuits that connect together cubic-millimeter
hotspots in forebrain limbic structures such as nucleus accumbens and ventral pallidum
(where opioid/endocannabinoid/orexin signals can amplify sensory pleasure). “Wanting”
mechanisms include larger opioid networks in nucleus accumbens, striatum, and amygdala
that extend beyond the hedonic hotspots, as well as mesolimbic dopamine systems, and
corticolimbic glutamate signals that interact with those systems. We focus on ways in
which these brain reward circuits might participate in obesity or in eating disorders.

You may also be interested in these two book chapters:

Smith, Kyle, Stephen V. Mahler, Susana Pecina, and Kent C. Berridge. “Hedonic Hotspots:
Generating Sensory Pleasure in the Brain.” In Pleasures of the Brain, edited by Morten
L. Kringelbach and Kent C. Berridge, 27–49. New York: Oxford University Press, 2009.

A vital question concerning sensory pleasure
is how brain mechanisms cause stimuli
to become pleasurable and liked. Pleasure
is not an intrinsic feature of any stimulus,
but instead refl ects an affective evaluation
added to the stimulus by the brain. That is,
as Frijda expresses it (Frijda, this volume;
Frijda, 2006), a pleasure gloss or hedonic
value must be actively ‘painted’ on sweet
or other sensations to make them pleasant.
Brain mechanisms of pleasure, whatever
they are, must take a mere sensory signal
and transform it into a hedonic and ‘liked’
reward.

Finding the brain mechanisms responsible
for painting a pleasure gloss is a major
challenge for affective neuroscience (Barrett
and Wager, 2006; Berridge, 2003b; Damasio,
1999; Davidson, this volume; Davidson and
Irwin, 1999; Kringelbach, 2005; Kringelbach,
this volume; LeDoux, 1996; Panksepp,
1991; Peciña et al., 2006). Fortunately,
progress on fi nding hedonic generators in
the brain is being made. In this chapter we
focus specifi cally on the neuroanatomical
hedonic hotspots in the brain where neurochemical
signals actually contribute causally
to the generation of pleasure.

Aldridge, J. Wayne, and Kent C. Berridge. “Neural Coding of Pleasure: ‘Rose-tinted Glasses’
of the Ventral Pallidum.” In Pleasures of the Brain, edited by Morten L. Kringelbach
and Kent C. Berridge, 62–72. New York: Oxford University Press, 2009.

Pleasure is not a sensation. What is it then? Nico Frijda's answer in the "pleasure questions" section of this book (which he suggested a number of years ago) epitomizes an emerging consensus among many psychologists and neuroscientists (Frijda, Chapter 6, this book). He notes that pleasure "is a 'pleasantness gloss' added to whatever is pleasant". [...]

Here we ask: how is a "pleasure gloss" encoded in brain activity? Where in the brain is this glossing operation performed and how does it work? Is it possible for neuroscientists to recognize the signature patterns of neural activity that represent a pleasure gloss? These are difficult questions that are only beginning to be addressed. The "pleasure gloss" metaphor, applied to the transformation of neural signals for a stimulus, is like a varnish that is applied on top of a dull object to transform it into a shiny one. Adding hedonic tone to the signal passed on to downstream structures, the neural gloss effectively gives the entire brain a "rose-tinted" hedonic perception of the stimulus as pleasant.

In the context of neural firing signals, our idea is that a particular pattern of neuronal spikes or action potentials in crucial neurons may apply a glaze of pleasure on what might otherwise be an ordinary sensation or action signal.

ac.commutative algebra - Is it true that, as $Bbb Z$-modules, the polynomial ring and the power series ring over integers are dual to each other?

I give this problem each year in a problem-solving seminar. Here is the solution that I wrote up. I am using $f$ instead of $varphi$ and $e_n$ instead of $x^n$.

Let $x=(x_1,x_2,dots)$. Since $2^n$
and $3^n$ are relatively prime, there are integers $a_n$ and $b_n$
for which $x_n=a_n2^n+b_n3^n$. Hence $f(x)=f(y)+f(z)$, where $y =
(2a_1, 4a_2, 8a_3,dots)$ and $z=(3b_1,9b_2,27b_3,dots)$. Now for
any $kgeq 1$ we have
$$ f(y) = f(2a_1,4a_2,dots,2^{k-1}a_{k-1},0,0,
dots) $$
$$ qquad + f(0,0,dots,0,2^ka_k,2^{k+1}a_{k+1},dots) $$
$$ qquad= 0+2^kf(0,0,dots,0,a_k,2a_{k+1},4a_{k+2},dots). $$
Hence $f(y)$ is divisible by $2^k$ for all $kgeq 1$, so
$f(y)=0$. Similarly $f(z)$ is divisible by $3^k$ for all $kgeq
1$, so $f(z)=0$. Hence $f(x)=0$.

Now let $a_i=f(e_i)$. Define integers $0< n_1 <
n_2 <cdots$ such that for all $kgeq 1$,
$$ sum_{i=1}^k|a_i|2^{n_i} < frac 12 2^{n_{k+1}}. $$
(Clearly this is possible --- once $n_1,dots,n_k$ have been chosen,
simply choose $n_{k+1}$ sufficiently large.) Consider $x=(2^{n_1},
2^{n_2}, dots)$. Then
$$ f(x) = f(a_1e_1 + cdots + a_k e_k +2^{n_{k+1}}
(e_{k+1}+2^{n_{k+2}-n_{k+1}}e_{k+2}+cdots))$$ $$ qquad=
sum_{i=1}^ka_i 2^{n_i}+2^{n_{k+1}}b_k, $$
where $b_k=f(e_{k+1}+2^{n_{k+2}-n_{k+1}}e_{k+2}+cdots)$. Thus by the
triangle inequality,
$$left| 2^{n_{k+1}}b_kright| < left| sum_{i=1}^k a_i
2^{n_i}right| + |f(x)| $$ $$ qquad <
frac 12 2^{n_{k+1}} + |f(x)|. $$
Thus for sufficiently large $k$ we have $b_k=0$ [why?]. Since
$$ b_j - 2^{n_{j+2}-n_{j+1}}b_{j+1}=f(e_{j+1}) mbox{[why?]},
$$
we have $f(e_k)=0$ for $k$ sufficiently large.}

How do ribosomes contribute to their own synthesis?

The most important parts of the ribosome are not made by other ribosomes - 5 rRNA (ribosomal RNA) of the ribosome actually do most of the direct work of creating the protein and are made by RNA polymerase ( a protein, but not the ribosome).

Then there are 92 ribosomal proteins, which as a rule bind to ribosomal RNA to support their structure and keep everything going. These are all made by ribosomes. They are thought to have appeared later in the evolution of the ribosome though I imagine that it would not be possible to constitute a working ribosome without each one of them.

these numbers are for the eukaryotic ribosome, the prokaryotic ribosome has 3 rRNA and 52 ribosomal protein components.

http://en.wikipedia.org/wiki/Ribosome

knot theory - Higher order quandle

For any codimension 2 embedding that is locally flat, there is a notion of the fundamental rack (not every element is idempotent). In Euclidean space, there is a fundamental quandle. It is defined as in the classical case: homotopy classes of paths that start on a tubular neighborhood and that end at the base point. The top of the homotopy is required to remain at the base point, but the bottom stays on the tubular neighborhood. More specifically, two paths $a_0$ and $a_1$ with $a_j(0) in partial (N(K))$ and $a_j(1)= {mbox{ pt.}}$ are equivalent if $exists: A:[0,1]times [0,1] rightarrow {bf R}^{n+2}$ such that $A(j,t)=a_j(t)$ for $j=0,1$, while $A(0,s)in N(K)$ and $A(1,s)={mbox{pt.}}$

The quandle operation is to follow $a$ to the basepoint, go down $b$ around the meridian at $b$ (in an oriented fashion), and return to the base point at $b$. See this manuscript (in particular the illustration therein) to see what is going on.

In additional to the fundamental quandle, there are quandle cocycle invariants that detect a lot of things about classical knots and knotted surfaces. For classical knots, the 2-cocycle invariant is related to a choice of longitude. See
M. Eisermann. Homological characterization of the unknot. J. Pure Appl. Algebra, 177(2):131–157, 2003 for this result.

Fenn and Rourke showed that the 3-cocycle invariant for classical knots could detect the chirality of the trefoil.

The 3-cocycle invariant for knotted surfaces (and its generalizations) are known to give strong results about non-invertibility, bounds on triple point numbers, and in the case of symmetric quandle homology, they give bounds on the triple point numbers of non-orientable surfaces.

In higher dimensions, one can prove that higher cocycle invariants exist --- there are generalizations of the Roseman moves that are given via multi-germs (due to Mond and another author), and the quandle $n$-cocycle condition corresponds to the $n$-simplex move in this context. There is a lot of work that can be done in this regard. I think most of it is straight-forward.

biochemistry - How can I measure bacterial alkaline phosphatase activity?

I want to measure alkaline phosphatase activity using PNPP in my mutant bacteria strains, but all the protocols I found involve purification of the phosphatase (which I have no need of).

Does anyone know a good protocol which involves using only the supernatant, without purification of the enzyme? What is the preferred buffer for the reaction?

Question: Annotation search in scientific literature

Well here's my take.

Too bad there is no open access to this article.

I see that they only had statistics for the top 50 annotations, which implies a fairly manual process - you are going to have to read a bunch of articles or abstracts to decide directly whether the software is right.

In this case, the authors are using a weighted latent semantic indexing method, which have started with a bunch of articles for each gene. In this case they would need to only look over the data they put into their training.

That might not answer your question though. If you wanted to search through the literature to confirm annotations generated by a different method...you could create a semi automated process perhaps like this:

Do a rough search from NCBI in an automated fashion (I use biopython) and then read every article for validation of the Gene Ontology result.

Its possible you could search for the gene and the GO term if to get a more focused result, load them into a database or just create a set of files with the data. I would build a little website that would let me scan through the abstracts with links to the complete articles (pubmed supplies DOI numbers) and just check 'yea' or 'nay' on whether confirmation happened.

There's other potential options on finding articles for search. the Genbank record for the gene often has a few references in it. One might use pubmed's referencing feature, where you can start at these confirmed articles that describes the gene in question and use pubmed to show articles that have referenced that first article. I'm not sure if this is accessible by API, but you can build an article graph by crawling the bibliographic references.

For anything that was neither directly contradicted nor confirmed you would have to go back to google and look until you found something. Clearly there are genes for which you will find no reference at all, and you will have to avoid these for confirmation.

I'm not sure if they checked for any biases here. If something like interpro2go which generates GO mappings from protein domains was used to generate the annotations, you would want to take care that you weren't just rediscovering the references which helped generate the annotations in the first place.

entomology - Are mosquitoes repelled by high frequency sound?

After seeing your question, I decided to do a bit of research on the topic.

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http://www.eurekalert.org/pub_releases/2007-04/jws-mrt041607.php

"Mosquito repellents that emit high-pitched sounds don't prevent
bites"

Some key-points from the webpage:

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A Cochrane Systematic Review of the use of electronic mosquito repellents (EMRs) failed to find any evidence that they work.

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To test these claims a team of Cochrane Researchers conducted a systematic review looking for trials conducted with EMRs. They located ten field trials that had been carried out in various parts of the world. None of these trials showed any evidence that EMRs work.

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All ten studies found that there was no difference in the number of mosquitoes found on the bare body parts of the human participants with or without an EMR

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http://en.wikipedia.org/wiki/Insect_repellent

These electronic devices have been shown to have no effect as a mosquito repellent by studies done by the EPA and many universities.

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http://www.cbc.ca/news/story/2007/04/18/mosquito-repellent.html
Key points from the article:

"There was no evidence in the field studies to support any repelling effects of EMRs, hence >no evidence to support their promotion or use"
           
-Ahmadali Enayati, researcher

This is a major point as well:

The study also said that in 12 of the 15 experiments, the landing rates of mosquitoes on subjects was in fact higher than in control groups.

The article also points out that female mosquitoes can't hear very well- Which supports the idea that high-frequency repellents are ineffective; Much more than you probably think. Why? Well, it's pretty simple:

Female Mosquitoes are the only Gender that Bite.

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Overall, when you consider the countless studies and research put in to the effects of high-frequency sounds on mosquitoes, it's pretty obvious that:

High Frequency Sounds do NOT repel mosquitoes

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oc.optimization control - Algebraic characterization of transitive spaces of matrices

The condition for $E$ to be intransitive is that the determinant form is the $0$ form somewhere other than the origin. That is, every vector $v in mathbb R^d$ gives you an alternating $d$-form on $M_d$ and on $E$ by the determinant of the images of $v$. This form is nonzero on a vector if and only if the images of the vector by $E$ are all of $mathbb R^d$.

Edit: You are right that the above was not a complete answer.

To be more explicit with bases for everything: Let $E$ have dimension $D$, with $d le D le d^2$. Let $E$ have a basis ${E_1, ... E_D }$ so that every element of $E$ is represented by a vector $(a_1,...,a_D)$. Represent every element of $mathbb R^d$ by a vector $(b_1,....,b_d)$.

Then for any vector $(b_1,....,b_d)$, the determinant form is an alternating $d$-form on $E$ identified with $mathbb R^D$. These forms have a basis of size ${D choose d}$ given by the determinants of $dtimes d$ minors, that is, project to a given $d$ coordinates, and take the determinant. To check whether the determinant form is the $0$ $d$-form, express it in terms of the basis, and see if all ${D choose d}$ coefficients are $0$. That is, check if the ${D choose d}$ determinants $det [E_{f(1)}v, ..., E_{f(d)}v]$ are all $0$ for each integer-valued function $f$ with
$1 le f(1) lt f(2) lt ... lt f(d) le D$.

As we let $v$ vary but fix a basis for $E$, the coefficients of the determinant form are homogeneous polynomials of degree $d$ in the coordinates ${b_i}$. The variety of intersections of the zeros of those polynomials on $mathbb R^d$ is the origin if and only if $E$ is transitive.

This gives you a test for a particular $E$ in terms of recognizing whether a variety is just a point. It still leaves the condition on $E$ in the Grassmannian as a projection of a variety.

ct.category theory - categorical description of elements in a direct limit

As with that previous question, I don't understand precisely what the rules of the game are when you say "without using an explicit construction". But maybe I can say something useful.

---

First I'll answer the first part of the first question. Then I'll use it to explain a little of why I don't think there's going to be a precise way to formulate the "no explicit construction" rule.

We have a sequence $A_1 to A_2 to cdots$ in the category of sets, and its colimit $A$, and we wish to show that every element of $A$ comes from an element of some $A_n$. And we wish to do it without using the explicit formula for colimits. Being a category theorist, I'll write 1 rather than $*$ for a one-element set.

I'll use the fact that the category of sets is a well-pointed topos, E say. (Those hypotheses can probably be weakened.) Take an element $a$ of $A$, that is, a map $a: 1 to A$. Since colimits are stable under pullback in any topos, pulling the colimit cocone back along $a$ gives a sequential colimit
$$
mathrm{colim}(X_1 to X_2 to cdots) = 1
$$
in the category E, together with a map $f_n: X_n to A_n$ for each $n$, making the evident square commute. (Draw a diagram!) Since any colimit of initial objects is initial, and in a well-pointed topos $1$ is not initial, at least one $X_n$ is not initial. It's a fact that in a well-pointed topos, every object is either initial or admits an element (map from $1$). So, at least one $X_n$ admits an element, $x$ say. Then $f_n x$ is an element of $A_n$, and a quick diagram chase shows that it maps to $a in A$.

So, I've answered the first question without apparently using any explicit constructions involving sets --- in the sense that I just assumed certain axioms on the category of sets and did the proof categorically. But the thing is, you can always do that. "Explicit constructions" can always be translated into categorical arguments (and though it's not apparent from the proof above, that's a totally mechanical process).

If your point of view on sets is that they are what ZFC says they are, then here's an equivalent categorical formulation: sets and functions form a well-pointed topos with natural numbers object and choice, satisfying a first-order axiom scheme of replacement. ZFC and this entirely categorical axiomatization are in a precise sense equivalent. Anything you can do in one context, you can do in the other.

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On coproducts: it can be shown that in any topos, coproducts are disjoint and the coprojections are jointly epic. I think that's in Mac Lane and Moerdijk's book Sheaves in Geometry and Logic. In a well-pointed topos, epic = surjective and monic = injective (where by sur/injective I'm referring to the elementwise notion implicit in the question). Hence your statements on coproducts hold in any well-pointed topos.

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On the question about algebraic structures:

Colimits (="direct limits") of sequences are an example of filtered colimits --- see Categories for the Working Mathematician Chapter IX, for instance. The forgetful functors Group$to$Set, Ring$to$Set, etc, all preserve filtered colimits. The jargon for this is: "the free group/ring/... monad is finitary". As the terminology hints, preservation of filtered colimits corresponds to the fact that the theory of groups, rings, etc. only involves finitary operations.

For example, the theory of groups has an operation with 2 arguments (multiplication), an operation with 1 argument (inverse), and an operation with 0 arguments (identity). The numbers 2, 1, and 0 are all finite, so the theory of groups is finitary, so the forgetful functor Group$to$Set preserves filtered colimits.

(A non-example would be the theory of ordered sets in which every countable subset had a supremum, and maps that preserved those suprema. One of the operations in the theory is "take the supremum of a countably infinite subset", so this theory isn't finitary. Correspondingly, the forgetful functor from these ordered sets to Set won't preserve filtered colimits.)

ac.commutative algebra - Computational Question about finite local rings:

Let $(A,mathfrak{m})$ be a local Artinian ring with
finite residue field, which I'm happy to assume is $mathbf{F}_3$.
(In particular, $A$ has finitely many elements.)

I would like to do some computations of the following kind, as $I$ ranges over
all of the ideals of $A$.

(0) A way to enumerate all the ideals of $A$.

(1) For an ideal $I$ of $A$, compute the length of $I/I^2$.

(2) For an ideal $I$ of $A$, compute the ideal $J = mathrm{Ann}(I)$.

(3) For an ideal $I$ of $A$, decide if $I$ is principal. (By computing the length of
$I/mathfrak{m} I$ or otherwise.)

The ring $A$ itself will be given explicitly as a quotient of a power series
ring over $W(mathbf{F}_3) = mathbf{Z}_3$. For example, $A$ might be
given as $mathbf{Z}_3[[x]]/(27,9x,x^3)$ or $mathbf{Z}_3[[x]]/(9,x^2)$.

My question: What is the computer algebra package that is best suited to carry
out these computations? (I would like something that can be semi-automated for various possible $A$.) I would be interested in even a very simple one like $mathbf{Z}_3[[x]]/(9,x^2)$

EDIT 2: There seems to be a consensus in the comments that this problem is significantly more manageable if $A$ is actually an algebra over its residue field. For example, in MAGMA, it is only possible to create ideals and quotient rings in univariate polynomial rings over fields. Other computer algebra packages have similar issues when the coefficient ring is not a field, although SINGULAR (for example) has some functionality with polynomials in several variables. As it happens, the problem I was interested in studying is still of interest for such fields.

nt.number theory - Representing numbers in a non-integer base with few (but possibly negative) nonzero digits

Background

In a recent question about Fibonacci numbers, it was claimed that

every integer can be written in the form $sum_{i=1}^6 epsilon_i F_{n_i}$ with $epsilon_i in {0,-1,1}$. The upper limit on the summation isn't a typo: every number is the sum/difference of at most 6 fibonacci's.

I believe this is false, even for larger (but still finite) values of $6$:

First of all, without loss of generality, we may assume that the representations do not repeat any Fibonacci number (i.e., the $n_i$s are distinct) and moreover, do not contain any two consecutive Fibonacci numbers (i.e., $n_i ne n_j+1$). We may arrive at such a representation by using the following simplifications repeatedly:

• If two consecutive Fibonacci numbers appear with opposite signs, simplify the expansion with the identity $F_n - F_{n-1} = F_{n-2}$.


• If two consecutive Fibonacci numbers appear with the same sign, simplify the expansion with the identity $F_n + F_{n-1} = F_{n+1}$.


• If the same Fibonacci number appears with opposite signs, simply cancel the two terms.


• If the same Fibonacci number appears with the same sign, then use the identity $F_n + F_n = F_{n-2} + F_{n-1} + F_n = F_{n-2} + F_{n+1}$ to replace them with two non-identical Fibonacci numbers.

The first three operations reduce the number of terms in the expansion and thus strictly simplify the expression (in terms of how many terms there are), but the last may need to be used several times before it "simplifies" the expression (for example, in terms of how many repeated terms there are). Nonetheless, this simplification procedure terminates, as it is impossible to get stuck in an infinite loop using the last operation alone. (Proof: we may assume that the $n_i$ are positive. Then all of the operations either reduce the number of terms, or leaves that unchanged and reduces the sum of the $n_i$.)

Now, assume we have such a representation (no identical terms, no consecutive terms) and suppose the largest Fibonacci number appearing is $F_n$. Then the next largest term (in absolute value) that may appear is $F_{n-2}$, the next largest after that $F_{n-4}$, and so on. All in all, the sum of the terms excluding $F_n$ is at most $F_{n-2} + F_{n-4} + cdots le F_{n-1}$ (proof by induction: add $F_n$ to both sides). By the triangle inequality, the sum of all the terms must be at least $F_n - F_{n-1} = F_{n-2}$. The point of this calculation is that if you want to represent a number that's less than $F_{n-2}$, you can't use terms that are $F_n$ or greater.

This leads us to our contradiction. Consider the integers between $0$ and $F_{n-2}-1$. How many possible representations are there of numbers in this range? Well, we have six terms all of which are 0 or $pm F_k$ for $klt n$ (from the above discussion), so we have at most $(2n+1)^6$ representations that could possibly fall into the range. (We're over-counting here because it won't matter and this is easier.) However, there are clearly $F_{n-2}$ different integers in the given range. Assume for contradiction that it were always possible to represent numbers as the sum/difference of at most 6 Fibonacci's. Then we would have

$$ (2n+1)^6 ge F_{n-2}. $$

Finally, because the left side grows polynomially while the right side grows exponentially, a large enough value of $n$ will produce a contradiction.

My questions

1. Is the proof above correct? (If not, and the original claim is correct, can you give me a representation of the number 5473?) Edit: Please see Michael Lugo's answer for a paper which finds the representation with the fewest nonzero digits in this "signed Fibonacci base". Please consider the following the actual question here:




2. Assuming the proof is correct, is the original claim true for other non-integer bases? What I mean is the following:

Does there exist a natural number $k$ and a real number $b>1$ such that every integer has a representation as $sum_{i=1}^k epsilon_i lfloor b^{n_i} + frac12 rfloor$? That is, does every number have a representation in "base $b$" (because $b$ is probably irrational, we round $b^n$ to the nearest integer) with at most $k$ non-zero "digits", but where the "digits" may be $pm 1$?

Note that the original claim is an instance of this: $k=6$ and $b=varphi = frac{1+sqrt 5}{2}$.

I don't think my proof works directly because I used special properties of $varphi$/the Fibonacci numbers. Is it possible to remove this reliance? In particular, the second step of the proof shows that it's not "useful" to have large Fibonacci numbers in the representation of a small number. Is the same true for every base $b$?

Note: if the digits were not allowed be negative, then my proof would go through. The main issue is whether or not $pm b^n$ for large $n$ can cancel and produce small numbers.

Thanks!

rt.representation theory - Software for Planar Algebras or Group Rings

Sage can do some things with group algebras, in particular, with group algebras for symmetric groups, but it doesn't seem to have anything about planar algebras. For example:

S = SymmetricGroupAlgebra(ZZ, 3) 
# ZZ, the integers, is the coefficient ring
# "3" means the symmetric group on 3 letters
a = S([2,1,3])  # turn the permutation [2,1,3] into an element of S
b = S([3,1,2])
(2*a + b)^2

prints out

4*[1, 2, 3] + 2*[1, 3, 2] + [2, 3, 1] + 2*[3, 2, 1]

If you'd started with a different coefficient ring:

S = SymmetricGroupAlgebra(GF(3), 3) 

then the output from the above would be

[1, 2, 3] + 2*[1, 3, 2] + [2, 3, 1] + 2*[3, 2, 1]

You can also do computations with other group algebras for other groups, but symmetric group algebras seem to be a bit better developed.

ag.algebraic geometry - Why do people think that abelian varieties are the hardest case for the Hodge conjecture?

Related to Jim Milne's answer, one might mention that Deligne proved that for abelian varieties, all Hodge cycles are "absolutely Hodge" (i.e. when you think of them embedded diagonally inside the product of the algebraic de Rham cohomology and $ell$-adic cohomology
(for every $ell$) and apply an automorphism of $mathbb C$, the resulting cycles are again diagonally embedded rational cycles, and are in fact again Hodge). Note that
if the Hodge conjecture holds, then this is certainly true (since the conjugate under any automorphism of $mathbb C$ of an algebraic cycle is again an algebraic cycle).

On the one hand, this is much more than is known about the Hodge conjecture for more general classes of varieties.

On the other hand, one can't immediately extend this to other classes of varieties
because the motives of abelian varieties don't generate all motives over a field of char. 0
(in fact, far from it, as far as I know), a fact already brought up in Donu Arapura's answer.

When is the ring of invariants of a finite group generated by symplectic reflections a complete intersection ring?

Let V be a finite dimensional symplectic vector space over $mathbb{C}$. Let $G$ be a finite subgroup of the symplectic group $Sp(V),$ which is
generated by symplectic reflections, i.e. by elements $gin G,$ such that $rank(Id_V-g)=2.$
Then it is well-known that the ring of invariants $mathbb{C}[V]^G$ is Gorenstein.

My question is assuming that V is an irreducible G-module and $dim V>2,$ when is $mathbb{C}[V]^G$ a complete intersection ring? Of course when $dim V=2$ it is a complete intersection ring (Kleinian singularities), but I don't know other examples.

neuroscience - Under what conditions do dendritic spines form?

Dendritic spines are thought to grow and recede under LTP and LTD, respectively. See (Bosch and Hayoshi 2011) for a review.

From there, much of the synaptogenesis occurs due to surface molecules present both on the dendrite and the presynaptic axon in the growth cone. Localization and guidance are achieved through gradients of growth factors in the developing nervous system See (Kolodkin and Tessier-Lavigne 2011) for a review of all of these mechanics.

How this maps back onto the human CNS and thinking/learning/memorizing is still up for debate, but some of these mechanisms must have been preserved in higher species.

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References:

Bosch M, Hayashi Y. (2012) Structural plasticity of dendritic spines. Curr Opin Neurobiol.,22(3):383-8. (Epub 2011 Sep 28).

Kolodkin AL, Tessier-Lavigne M. (2011). Mechanisms and molecules of neuronal wiring: a primer. Cold Spring Harb Perspect Biol., 3(6). [DOI]

complex geometry - Tensor product of a line bundle with a large multiple of another positive line bundle also positive?

Let us prove that for an affine variety $X$ every line bundle $E$ is "positive" according to the chosen defintion. All we need to prove is that for any hermitian metric $g$ on $E$ with curvature $w$ there is a Kahler form $w_1$ on $X$ such that $w_1>-w$. Since $X$ is affine, for any $w_1$ we have $w_1=frac{i}{2pi}partialbarpartial (f_1)$ and changing the metric $g$ on $E$ by $ge^{f_1}$ we corresponing curvature will change from $w$ to $w+w_1$, which we assume to be positive.

So we need to show the existence of arbitrary large $w_1$. Since $X$ is affine and hence admits an embedding in $mathbb C^n$, it is enough to show this for $mathbb C^n$. Moreover, since $mathbb C^n=mathbb C^1times ...times mathbb C^1$ it is enought to prove the statement for $mathbb C^1$. Now, on $mathbb C^1$ every form of the shape $w_1=h_1dzwedge dbar z$ is Kahler for $h_1>0$ and we can chose $h_1$ as large as we wish.

The conclusion is that if one choses this definition, then each line bundle on an affine $X$ is positive, which sounds strange. So I am not sure what should be a reasonable definition of positivincess in non-compact case, if it exists at all.

zoology - Small worm living in some kind of cocoon, what are these animals?

This little dude is not a Caddisfly, but a true moth, Tinea pellionella, a case making clothing moth. Confusing because they do resemble caddisfly larvae cocoons more so than those of their own family (Tineidae). Sort blurb Here:

"The brown-headed larva spins a silken case that is open at both ends. The case in the above image is covered with fine sand and debris, and superficially resembles a caddisfly case. The flattened case is about 10-11 mm long (3/8 to 1/2 inch). When crawling, the larva's head, thorax, and three pairs of legs protrude out of the case, and drag it along. According to Internet sources, the larva feeds on a variety of material, including hair, fur, silk, felt, feathers, woolen clothing, upholstered furniture and carpets. It apparently prefers darkness and soiled clothing, and is not fond of synthetic fabrics, such as nylon and polyesters."

Can you determine whether a graph is the 1-skeleton of a polytope?

A few comments:

In general, you can't tell the dimension of a polytope from its graph. For any $n geq 6$, the complete graph $K_n$ is the edge graph of both a $4$-dimensional and a $5$-dimensional polytope. (Thanks to dan petersen for correcting my typo.) The term for such polytopes is "neighborly".

On the other hand, you can say that the dimension is bounded above by the lowest vertex degree occurring anywhere in the graph.

A beautiful paper of Gil Kalai shows that, given a $d$-regular graph, there is at most one way to realize it as the graph of a $d$-dimensional polytope, and gives an explicit algorithm for reconstructing that polytope. You could try running his algorithm on your graph. (Or a more efficient version recently found by Friedman.) This algorithm will output some face lattice; that is to say, it will tell you which collections of vertices should be $2$-faces, which should be $3$-faces and so forth.

Unfortunately, going from the face lattice to the polytope is very hard. According to the MathSciNet review, Richter-Gebert has shown that it is NP-hard to, given a lattice of subsets of a finite set, decide whether it is the face lattice of a polytope. Note that this is a lower bound for the difficulty of your problem.

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Let me be more explicit about the last statement. Richter-Gebert shows that, given a collection $L$ of subsets of $[n]$, it is NP-hard to determine whether there is a polytope with vertices labeled by $[n]$ whose edges, $2$-faces and $3$-faces are the given sets. (Here $[n] = { 1,2, ldots, n }$.)

Suppose we had an algorithm to decide whether a graph could be the edge graph of a polytope. Take our collection $L$ and look at the two-element sets within it. These form a graph with vertex set $[n]$. Run the algorithm on it. If the output is NO, then the answer to Richter-Gebert's problem is also no. If the answer is YES, then we have the problem that our algorithm might have found a polytope whose $2$-faces and $3$-faces differ from those prescribed by $L$. If our graph is $4$-regular, this problem doesn't come up by Kalai's result. But, not having read Richter-Gebert myself, I don't know whether the problem is still NP-hard when we restrict to $4$-regular graphs.

However, even if Richter-Gebert's result doesn't apply directly, I find it difficult to imagine that there could be an efficient algorithm to solve the graph realization problem, since there isn't one to solve the face lattice problem.

soft question - Value of "of course" in the mathematical literature

Hello,

I agree with some of the comments above: "of course" is useful to point out that some step is trivial (e.g. direct consequence of the definition), as opposed to the rest of non-trivial parts of the proof. Sometimes, "of course" is useful just as an stylistic resource in the writing, to introduce and connect a sentence to the previous one. But it can be very frustrating for the reader if this step is non-trivial, even though the author claims it is.

I was curious about this question and decided to find some examples in the "mathematical literature", as the original poster suggested. I looked through "A Course in Arithmetic", by J-P. Serre (which many consider a very good writer of mathematics) and the expression "of course" appears exactly twice. In both cases, "of course" appears in a parenthetical remark:

1) (p.35) Corollary. - For two nondegenerate quadratic forms over $mathbb{F}_q$ to be equivalent it is necessary and sufficient that they have same rank and same discriminant.
(Of course the discriminant is viewed as an element of the quotient group $mathbb{F}_q^ast/mathbb{F}_q^{ast 2}$.)

2) (p.73) Let $A$ be a subset of $P$ [$P$ is the set of prime numbers]. One says that $A$ has for density a real number $k$ when the ratio
$$ left(sum_{pin A}frac{1}{p^s} right)/ left(log frac{1}{s-1}right)$$
tends to $k$ when $sto 1$. (Of course, one then has $0 leq k leq 1$.)

In example (1), the way the corollary is stated, a remark is needed - but (i) it is clear from the context that this is what the author means, and (ii) it is typical in this context to consider discriminants only up to squares. Here I see this "of course" as a reminder of (ii).

Example (2) is trickier, as it is not immediately obvious that the limit of the expression as $sto 1$ is between $0$ and $1$. But I do not interpret this "of course" as a "clearly" in this case, but rather a sort-of "do not worry, if you go back and check Cor 2 in p. 70, you can convince yourself that $0leq k leq 1$, and it makes sense to call this number a density".

Álvaro

PS: In "A Course in Arithmetic", the word "clearly" appears many times, while "obviously" was never used in the entire book.

at.algebraic topology - Examples of the varying strengths of topological invariants

For 1 and 2, consider the $S^2$ bundles over $S^4$ (with structure group SO(3)). Using clutching functions, one can see that there is a $mathbb{Z}$s worth of such bundles indexed by, say, k.

Using the Gysin sequence, one finds that the cohomology groups (and homology groups) are the same as those of $S^2times S^4$, mainly, a $mathbb{Z}$ in dimension 0,2,4, and 6, and 0 elsewhere.

However, for $kneq pm k'$ the bundles corresponding to $k$ and $k'$ have nonisomorphic ring structures.

By Poincare duality, the only question about the ring structure is the following: what is the square of the degree 2 generator? Turns out, the square of the degree two generator is equal to $pm k$ times the square of the degree 4 generator. Incidentally, the case $k=1$, one gets the cohomology ring structure of $mathbb{C}P^3$. In fact, the total space of the bundle is diffeomorphic to $mathbb{C}P^3$.

For your third question, I'd inspect $S^5$ bundles over $S^2$. Again, by clutching function analysis, there must be precisely two such bundles - the trivial bundle and one other. By the Gysin sequence, these must have the same cohomology groups and by Poincare duality, the ring structures must in fact agree.

However, the second Stieffel Whitney class of the trivial bundle is trivial, while the second Stieffel Whitney class of the nontrivial bundle is nontrivial, and hence the two total spaces are not homotopy equivalent. (Stieffel Whitney classes are closely related to Steenrod operations).

And just to anticipate, the spaces $S^3times mathbb{R}P^2$ and $S^2times mathbb{R}P^3$ have all the same homotopy groups, but are not homotopy equivalent (as homology will tell you).

Are there any examples of proteins with no or minimal sequence identity, but highly similar structure?

An example possibly worth taking a look at is the variable surface glycoprotein of the bloodstream form of Trypanosoma brucei.

Trypanosomes are the causative agent of sleeping sickness, and the entire cell surface is covered by a single glycoprotein called the variable surface glycoprotein, or VSG. (see here for more information)

These parasitic protozoans have the ability to 'fool' the immune response by antigenic variation. At any one time just a single VSG is expressed on the surface, but (very rarely) a trypanosome may express a different VSG gene product. Thus if the immune response succeeds in eliminating all trypanosomes expressing VSG-A, but there exists in the population just one parasite which has switched coat, this variant will poliferate as it effectively presents a different antigen to the host's defenses (and a second immune response will result).

There is litte sequence identity between different VSGs but they are homologous (descended from a common ancestor).

The crystal structure of the N-terminal domain of two different VSGs (MITat1.2 and ILTat1.24) have been solved (Blum et al. 1993; Freymann et al., 1993). Despite the low sequence identity (~16%), the structures are almost identical.

One other very interesting property of all VSGs is that they are covalently anchored to the plasma membrane via a (C-terminus) glycosylphosphatidylinositol anchor (or GPI anchor).

References

• Blum, M. L., Down, J. A., Gurnett, A. M., Carrington, M., Turner, M. J., and Wiley, D. C. (1993) Nature 362, 603-609 [pubmed]




• Freymann, D., Down, J., Carrington, M., Roditi, I., Turner, M., and Wiley, D. (1990) 2.9 A resolution structure of the N-terminal domain of a variant surface glycoprotein from Trypanosoma brucei. J. Mol. Biol. 216, 141-160 [Pubmed]

The following reference contains much useful background information, and is free to all.

• Chattopadhyay, A, Jones, N.G, Nietlispach,D, Nielsen, P.R., Voorheis, H.P., Mott H.R., Carrington, M. (2005) Structure of the C-terminal domain from Trypanosoma brucei variant surface glycoprotein MITat1.2. J. Biol. Chem., 280, 7228-7235 [Pubmed] [pdf]

Which R-algebras are the group ring of some group over a ring R?

A group algebra over the complex numbers, like any semisimple algebra, is isomorphic to a product of matrix rings,
$$R = M_{d_1 times d_1} times M_{d_2 times d_2} times cdots times M_{d_n times d_n}$$
The d_i that appear are exactly the dimensions of the irreducible representations of G. I don't know how to classify all sets of numbers that appear in this way (but the answer is not "all of them").

Over nonalgebraically closed fields of characteristic zero, nontrivial divison algebras can appear in the group algebra. I don't know if there's a restriction on the division algebras that can appear.

The situation in characteristic p is more complicated, but we can say something. A finite-dimensional algebra A has a square matrix of numerical invariants called the Cartan matrix. (I am not talking about the Lie-theoretic Cartan matrix, but I would be interested to know if they are named the same for a reason.) There is a one-to-one correspondence between simple A-modules and indecomposable projective A-modules, and the ij entry in the Cartan matrix is the Jordan-Holder index of the ith simple module in the jth projective module.

The theory of Brauer (and the subject of "part 3" of Serre's famous book on representation theory of finite groups) imposes strong conditions on the Cartan matrix when A is the group algebra of a finite group (over a large finite field). It must admit a factorization as D.D^t, where D is another matrix with nonnegative integer entries. (D is the "decomposition matrix" which describes what happens to simple modules in characteristic zero when reduced mod p.) For instance the Cartan matrix must be symmetric.

What if we work over a ring that is not a field, e.g. the integers? Here's a comment on Yemon's point that there are many pairs of groups G and H for which the group rings C[G] and C[H] are isomorphic. It is more difficult to construct such isomorphisms over smaller rings, and whether or not an isomorphism of the form Z[G] = Z[H] implied that G = H was an open problem for a long time (the "isomorphism problem for integral group rings," posed by Brauer in the 60s) A counterexample was found by Hertweck 10 years ago:

http://www.jstor.org/pss/3062112

(Pete points out above that I am assuming G is finite. When G is infinite C[G] cannot be analyzed by Wedderburn's theorem, there's no such thing as a Cartan matrix, everything breaks down. Is there a counterexample to the isomorphism problem simpler than Hertweck's if we do not require G and H to be finite?)

st.statistics - calculate percentiles from a histogram

A histogram gives you the number $n_i$ of observations between some $x_i$ and $x_{i+1}$. I'm assuming a total of $n$ observation. So, to get an approximation for the upper $p$-percentile, you want to find the maximal $j$ such that $sum_{i=j}^infty n_igeq p*n$. Then, the empirical upper $p$-percentile is between $x_j$ and $x_{j+1}$.

rt.representation theory - How to prove $U otimes Ind W = Ind(Res(U) otimes W)$

This is really a comment which got too long.

Personally, I always find this one rather confusing. If you think in terms of modules over group rings, we want to show that $U otimes (mathbb{C}[G]otimes_{mathbb{C}[H]} W) cong mathbb{C}[G] otimes_{mathbb{C}[H]} (Uotimes W)$. The $G$-equivariant isomorphism is not given by sending $uotimes (xotimes w)$ to $x otimes (uotimes w)$. There's a lot wrong with this formula, but that's not the point.

The point is that to get the right formula, one really needs to remember exactly how the universal property of induction works. I don't have Fulton and Harris in front of me to see what they say, but Serre's book has a good discussion of induction which will lead one right to the answer.

Also, unless I'm confused, this really seems to depend on the structure of $mathbb{C}[G]$ as both a ring and as a $mathbb{C}[H]$-module. One needs to know that it's a free $mathbb{C}[H]$-module, and that it has a decomposition as a $C[H]$-module into summands isomorphic to $C[H]$ that are permuted by the units of the ring $C[G]$. One could ask, for morphisms of rings $Cto Rto S$, when it's true that for an R-module M and an S-module N we have the formula $N otimes_C (Sotimes_R M) cong S otimes_R (N otimes_C M)$ (as S-modules). (Above I wrote $otimes$ instead of $otimes_{mathbb{C}}$; now C is the ground ring). I don't know how to prove this without assuming S has the sort of structure mentioned above (free as an R-module, etc.).

braided tensor categories - Quantum group as (relative) Drinfeld double?

The most elementary construction I know of quantum groups associated to a finite dimensional simple Hopf algebra is to construct an algebra with generators $E_i$ and $F_i$ corresponding to the simple positive roots, and invertible $K_j$'s generating a copy of the weight lattice. Then one has a flurry of relations between them, and a coproduct defined on the generators by explicit formulas. These are not mortally complicated, but are still rather involved. Then come explicit checks of coassociativity, and compatibility between multiplication and comultiplication. Finally, one has the $R$-matrix which is an infinite sum with rather non-obvious normalizations. Enter more computations to verify $R$-matrix axioms.

I recall learning about a nice way to construct the quantum group, which in addition to requiring less formulas has the advantage of making it clear conceptually why it's braided.

I'm hoping someone can either point me to a reference for the complete picture, or perhaps fill in some of the details, since I only remember the rough outline. That, precisely, is my question.

I include the remarks below in hopes it will jog someone's memory.

You start with the tensor category $Vect_Lambda$ of $Lambda$-graded vector spaces, where $Lambda$ is the weight lattice. We have a pairing $langle,rangle:LambdatimesLambdato mathbb{Z}$, and we define a braiding $sigma_{mu,nu}:mu otimes nu to nuotimesmu$ to be $q^{langle mu,nu rangle}$. Here $q$ is either a complex number or a formal variable. We may need to pick some roots of $q$ if we regard it as a number; I don't remember (and am not too worried about that detail). Also, here we denoted by $mu$ and $nu$ the one dimensional vector space supported at $mu$ and $nu$ respectively, and we used the fact that both $muotimesnu$ and $nuotimesmu$ are as objects just $mu+nu$.

Okay, so now we're supposed to build an algebra in this category, generated by the $E_i$'s, which generators we regard as living in their respective gradings, corresponding to the simple roots. Here's where things start to get fuzzy. Do we take only the simples as I said, or do we take all the $E_alpha$'s, for all roots $alpha$? Also, what algebra do we build with the $E_i$'s? Of course it should be the positive nilpotent part of the quantum group, but since we build it as an algebra in this category, there may be a nicer interpretation of the relations? Anyways, let's call the algebra we are supposed to build here $U_q(mathfrak{n}^+)$. I definitely remember that it's now a bi-algebra in $Vect_Lambda$, and the coproduct is just $Delta(E_i)=E_iotimes 1 + 1otimes E_i$ (the pesky $K$ that appears there usually has been tucked into the braiding data). Now we take $U_q(mathfrak{n}^-)$ to be generated by $F_i$'s in negative degree, and we construct a pairing between $U_q(mathfrak{n}^+)$ and $U_q(mathfrak{n}^-)$. The pairing is degenerate, and along the lines of Lusztig's textbook, one finds that the kernel of the pairing is the q-Serre relations in each set of variables $E_i$ and $F_i$.

Finally, once we quotient out the kernel, we take a relative version of Drinfeld's double construction (the details here I also can't remember, but would very much like to), and we get a quasi-triangular Hopf algebra in $Vect_Lambda$. As an object in $Vect_Lambda$ it's just an algebra generated by the $E_i$'s and $F_i$'s, so no torus. But since we're working in this relative version, we can forget down to vector spaces, and along the way, we get back the torus action, because that was tucked into the data of $Vect_Lambda$ all along.

So, the construction (a) gives neater formulas for the products, coproducts, and relations (including the $q$-Serre relations), and (b) makes it clear why there's a braiding on $U_q(mathfrak{g})$ by building it as the double.

The only problem is that I learned it at a seminar where to my knowledge complete notes were never produced, and while I remember the gist, I don't remember complete details. Any help?

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.

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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

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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

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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!]

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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

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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.

real analysis - When deRham curve is bijection?

Motivation: Suppose we have deRham curve. From wikipedia:

Consider some metric space $(M,d)$ (generally $R^2$ with the usual euclidean distance), and a pair of contraction mappings on M:
$d_0: M to M$
$d_1: M to M$
By the Banach fixed point theorem, these have fixed points $p_0$ and $p_1$ respectively. Let $x$ be a real number in the interval $[0,1]$, having binary expansion

$$x = sum_{k=1}^infty frac{b_k}{2^k}$$

where each $b_k$ is 0 or 1. Consider the map $c_x: M to M$ defined by
$c_x = d_{b_1} circ d_{b_2} circ ... circ d_{b_k} circ ...$
where $circ$ denotes function composition.

It can be shown that each $c_x$ will map the common basin of attraction of $d_0$ and $d_1$ to a single point $p_x$ in $M$. The collection of points $p_x$, parameterized by a single real parameter $x$, is known as the de Rham curve.

Such curve may be continuous, and we know when it happened. One of the examples of deRham curves is the Minkowski question mark function $?(x)$, which may be defined as function from Stern-Brocot tree to dyadic rationals. As a deRham curve it is defined by

$$d_0(z) = frac{z}{z+1} quad {rm and } quad d_1(z)= frac{1}{z+1}$$

$?(x)$ is continuous function of x, and even has explicit formula for inverse function, so it is bijection!

Question: Let's define deRham curve for domain $M=[0,1]times[0,1]$ with standard metric ( as a subset of $R^2$). What requirements have to be set on functions $d_0(x)$ and $d_0(x)$ defining deRham curve (or on other objects) in order to obtain deRham curve which is function and bijection from $[0,1]$ to $[0,1]$?

Remarks:

Indeed: as new user I was able to add only one link, so I decide to omit deRham and insert $?(x)$ function.

$[0,1]$ here is set of real numbers $a$ such that $0 leq a leq 1$. Yes, it was a mistake, thank You for pointing!

@Yemon Choi - As You may see, Minkowski question mark function $?(x)$ is deRham curve and normal function as well. So some deRham curves are both continuous curve defined in $M=[0,1]times[0,1]$ ( and we know when - see wikipedia definition) and good functions from $[0,1]$ to $[0,1]$. Probably other example is Cantor curve which is $M to M$ and normal function simultaneously - and its even continuous. Also Blancmange curve is deRham curve and simultaneously normal function but not has inverse since it take the same values for different argument. If You consider function definition then You may see that function F between X and Y is a relation on XxY which is functional, which means: "functional (also called right-definite or right-unique1): for all x in X, and y and z in Y it holds that if xRy and xRz then y = z." So I don't see here any conceptual problems but rather only formal ones.

Maybe I should describe my motivation: General picture is that: some deRham curves on $M=[0,1]times[0,1]$ gives a functions from $[0,1] to [0,1]$ ( when?) and some of them are even bijections. It is interesting which are bijections because such kind of curves defines in natural ( even if complicated) way some coding for representing real numbers by simple alphabet - You may see wikipedia discussion about relation between $?(x)$ and coding of the rationals on Stern-Brockot tree. Every number in Stern-Brockot binary tree may be represented by sequence of letters $LRRRLRRLLL...$ depicting the path from the root of the tree to the number. So as in this way You may wrote every rational number, and even every real if You include infinite sequences. This way You may have some "non-positional system" for representing numbers - in this place in fact You have system based on continuous fractions. Then deRham curves seems to me as natural way of generalization of such structure to other kind of "non-positional systems".It is interesting whether it is structure with many examples of such bijections or maybe $?(x)$ and Cantor curve - are the only ones. How big is that space of bijections? Are there any which allows faster computations? Or maybe more accurate or with less memory requirements to represent some kind of rationals or real numbers??

neurobiology - How does body temperature and oxygenation affect thinking ability?

Warmer temperature are shown to raise aggression level (Anderson, et al, 1995). Citing this study, DeWall, 2009 found a similar correlation between words associated with high temperatures and hostile behavior. This could be perceived as a threat to clear thinking.

Moss, 1996 shows oxygen administration increases memory. However, intermittent hypoxia on developing children led to adverse effects (both mental and physical), according to Bass, 2004. Though, I must say, you can't really determine the oxygen content indoor vs. outdoors without sensors, even if you raise in elevation and who's to say what competing particulates there are between the two environments?

It's known that kinetic motion itself increases neurogenesis, mostly in studies with exercise, like this one. There's been a lot of papers and laymen articles on the subject of exercise and neurogenesis lately. The act of walking or climbing the mountain might contribute a lot to the subjective experience of thinking.

gr.group theory - "Embedding" functions in groups

Hi all, I am looking for some help with the following question. Take a discrete bivariate function $f(x,y)$ (i.e., $x$,$y$ take values in some finite sets). Is there a way to quantify how "embeddable" this function is in Abelian groups. For example, if $x,y$ are binary and f(x,y) = x XOR y, then f is clearly embeddable in $Z_2$ (the cyclic group on 2 elements). But if f(x,y) = x AND y, f's action can only be mimicked in $Z_3$. (i.e., treat $x$ and $y$ as elements of $Z_3$, add them in $Z_3$ and map the outputs as {0,1}->0, 2->1 to mimic the AND function). Is there a formal way to characterize the smallest Abelian group in which a given f(x,y) can be embedded? I would greatly appreciate any help/reference/pointers.

Thanks,
Dinesh.

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More precisely, let $S$ be a finite set and let $f : S times S to S$ be a function. How do we determine the smallest abelian group $(G, +)$ for which there exist functions $g : S to G$ and $h : G to S$ such that $f(x, y) = h(g(x) + g(y))$?

gn.general topology - Which is the correct ring of functions for a topological space?

I thought I'd record the misgivings in my comment above as an actual "answer". I think that one needs to be clearer about just what class of topological spaces one is considering. As I said above, even if one starts by restricting attention to Hausdorff spaces, the LCHff class is atypically nice (and the CHff case extraordinarily atypically nice, as passing categorists might attest). Saying that function spaces are dual to topological spaces is a great maxim, but like all maxims needs to be wielded with a modicum of care and not always as broadly as salespeople would have you do...

In this kind of broad generality, my first instinct would be to see what Gillman and Jerison's book "Rings of continuous functions" has to say.

As several people have said above, when X is not compact then one needs to decide how interested one is at "behaviour at infinity", and choose the most appropriate algebra of functions to reflect this. If you really don't care much, then $C_0(X)$ seems natural although as I said above this might only be good for LCHff X, and there are plenty of quite interesting topological spaces which are not locally compact...

I rather suspect that when $X$ is Hausdorff, $sigma$-compact but non-compact, then $C(X)$ should have a natural Frechet algebra structure, and there might be some work done on this class of examples.

By the way, if $X$ is a metric space then there is a case that one should be looking at the algebra of Lipschitz functions on X (Nik Weaver has championed this viewpoint in the past). However, since this is not a $C^*$-algebra, it might not meet the standards of Proper Functional Analysis and should perhaps be consigned to the dustbin of history (or not, depending on your point of view).

sheaf theory - group cohomology and cohomology of classifying space

Let $G$ be a discrete group, and $BG$ is the classifying space.
It is well-known that the group cohomology of $G$-module M, is the same as the cohomology on $BG$ with coefficient in $tilde{M}$, which is the associated sheaf of $M$.

Can someone explain how these two cohomologies are related?

lo.logic - What is a logic ?

Here are three research traditions that both illustrate how the problem can be approached, and give rather different perspectives on what counts as a logic.

(this really is just to complement Dan Piponi's answer).

1. Tarski's consequence relations and abstract algebraic logic

Tarski's basic idea was to define a logic as an abstract pair of the form $langle mathcal{F},Crangle$ where $mathcal{F}$ is a free algebra of formulas and $C$ is an operator on $mathcal{P}(mathcal{F})$ [I write $mathcal{F}$ for the domain of the free algebra]. For any set of formulas $A$, the set $C(A)subseteqmathcal{F}$ is meant to represent the 'consequences' of $A$ -- so that $C$ generates a consequence relation $vdash_{C}$ defined as $Avdash_{C} B$ iff $Bsubseteq C(A)$.

Next, Tarski gave several structural conditions that the operator $C$ ought to satisfy in order for the resulting consequence relation to count as well-behaved, or logical (roughly, those conditions consist of reflexivity, monotonicity, compactness, as well as invariance under uniform substitution of variables). See here for more detail.

This view really treats logic as a (very special) branch of abstract algebra. One idea is to try to differentiate between logics, and classify them, by looking at their different algebraic properties. It was one of the earliest systematic attempts at answering the question of 'what a logic is' in such general terms.

On the other hand, this framework is generally too weak to account for quantification of any sort; the attention is almost exclusively restricted to propositional logics.

(NB. Tarski's approach eventually gave rise to some very interesting work on the general process of algebraization of a logic, under the guise of Abstract Algebraic Logic -- see, e.g. here. Interesting monographs on the topic include Rasiowa and Sikorski's An Algebraic Approach to Non-Classical Logics as well as Blok and Pigozzi's Algebraizable Logics.)

2. Model-theoretic logics, generalized quantifiers

For an introduction see this book. The model-theoretic approach studies various extensions of first-order logic: predominantly infinitary logics of the form $mathcal{L}_{alphakappa}$, where $alpha, kappa$ are ordinals (the logic $mathcal{L}_{alphakappa}$ allows conjunctions/disjunctions of less than $alpha$-many formulas, and quantification over less than $kappa$-many variables). It also covers topics like abstract characterisations of first-order logic (cf. Lindstrom's theorem mentioned in John Goodrick's answer) as well as connections with probabilistic logics.

There is related research on what makes a quantifier 'logical'. The idea is to characterise the logical quantifiers as operations of a certain type that are invariant under certain groups of transformations -- there appears to be some controversy about what exact transformations truly define the 'logical' operations (see here).

3. Applied logics

Another approach that gives a slightly different perspective on what counts as a logic is work in `applied' logic: this is a broad field of study which has at its root a dynamic view of logic (see here). Here, one uses so-called 'dynamic' modal logics to model processes that change over time, such as transitions between the states of a program (see Propositional Dynamic Logic) or informational states of agents (see Dynamic Epistemic Logic). Those logics are studied either for their intrinsic mathematical interest, or can also be applied to the study of information exchange protocols in game theory, cryptography, and various topics in formal philosophy.

The approach here is less algebraic, but focused more on model-theoretic and computational aspects. This research often bears close links to computer science and philosophy.

ag.algebraic geometry - Preschemes and schemes

This is a very minor point, but one which had been grating me for a while. I apologize for asking a relatively trivial question, but nevertheless hope that it is suitable for MO since it should have a definite answer.

In Mumford's books, for instance Curves on Surfaces or Red Book, there is thing called "prescheme" which looks like a scheme, and scheme is something else.

But this terminology does not seem to be used elsewhere, and if at all is the case, prescheme seems to be something cruder than scheme.

I will be grateful for clarifications regarding this terminology. "Curves on surfaces" is a nice book, but whenever I pick it up I find myself wondering about this without any avail.

st.statistics - Regression problem/detect outliers

RANSAC can be used to find the best fit for data which includes outliers.

---

Below is what I did to solve a line fitting problem in robotic vision using RANSAC. Dr. Mariottini was the professor who taught us the approach and provided some templates for implementation. The credit goes to him and not me. The code is for MATLAB, and the reference to Z.mat is for a file containing the detected points along the wall to be followed which also contains outliers to be excluded.

%
%%
%% CSE 4392-5369 University of Texas at Arlington  
%% Dr. Gian Luca Mariottini
%%
%% 

% Robust line fitting (using RANSAC)
close all
clear all
clc

load Z.mat

X = Z;

figure(1)
axis equal
plot(X(1,:), X(2,:), 'r+')
hold on
len=length(X(1,:));
maxp=max(X(1,:));


%% RANSAC
t=1;
T=6;
N=100;
[m_r,q_r,inliers] = f_fitlinerobust(X',t,T,N);
plot([1 maxp],[m_r*1+q_r, m_r*maxp+q_r],'b');
plot(X(1,inliers), X(2,inliers),'gO');


% Wall parameters
theta=atan2(m_r,-1);
rho=q_r * sin(theta);
m=-cos(theta)/sin(theta);
q=rho/sin(theta);
x_w=[-2,20]';
y_w=m*x_w+q;
figure(2)
hold on
plot(x_w,y_w,'r')
axis equal

% Initial robot pose
x_r(1)=0.5;
y_r(1)=7;
phi_r(1)=0%pi/100;
Dt=0.1;
% Desired pose
delta_des=2;

i=1;
for t=0:150;

    figure(1)
    hold on
    plot(x_w,y_w,'r')
    axis equal
    % Measurement model
    delta(i) = x_r(i)*cos(theta)+y_r(i)*sin(theta)-rho;% + randn(1)*0.1;
    alpha(i) = theta - phi_r(i);%+randn(1)*0.01;

    % Plot reference frame
    w_R_r = rotoz(-(pi/2-phi_r(i)));
    w_t_r = [x_r(i); y_r(i);0];
    H=[w_R_r w_t_r;
       0 0 0   1];
    f_3Dframe(H,'b',2,'_{r}');

% Control
    err([1:2],i)=[delta(i)-delta_des; -alpha(i)+pi/2];
    lambda_1=0.3;
    lambda_2=0.3;
    L=[lambda_1 0;
        0 lambda_2];
    A=[1/cos(alpha(i)) 0;
        0              1];
    U_r([1:2],i) = -A*L*(err(:,i));

%% Velocities    
v_r(i) = U_r(1,i);
w_r(i) = U_r(2,i);

    % Actuate velocities
    x_r(i+1) = x_r(i) + v_r(i)*cos(phi_r(i));
    y_r(i+1) = y_r(i) + v_r(i)*sin(phi_r(i));
    phi_r(i+1) = phi_r(i) + w_r(i);

    i=i+1;
    pause(Dt)
    clf

    end

figure
plot(err(1,:))
title('Error distance')

%figure
%plot(err(2,:))
%title('Error angle')

figure
hold on
plot(v_r,'g')
plot(w_r,'r');
title('Translational/Angular velocities')

%%
%% CSE 4392-5369 University of Texas at Arlington  
%% Dr. Gian Luca Mariottini
%%
%% See the RANSAC alg. in the slides for the parameters U, t, T and N
function [m,q,inliers] = f_fitlinerobust(U,t,T,N);

% Create a list of possible combinations of the input points (2 at a time)
n=length(U(:,1));
inliers=[];
%indexes=nchoosek([1:n], 2); % all possible pair combinations 

max_length=0;
for i=1:N,
    display(['Iteration=',int2str(i)]);
    % (i) Sample minimum number of points
    ind1=randsample(n,1);
    x1=U(ind1,1);
    y1=U(ind1,2);
    ind2=randsample(n,1);
    x2=U(ind2,1);
    y2=U(ind2,2);
    % Fit a model to these points
    [m,q]=f_fitline([x1 x2; 
                     y1 y2]);
    % (ii) Compute the distance of each point to the model (m,q) and find S
    distances = abs( m*U(:,1)+q - U(:,2) );
    indS=find(distances < t); %index of the Consensus set= inliers
    if ~isempty(indS),
      % Take the current consensus index and see if it is > than maximum so far
      if length(indS)>max_length,
        max_ind = indS;
        max_length= length(indS);
      end;
      % (iii) size(S)>T
      if length(indS)>T,
         % Final estimate
         [m,q]=f_fitline(U(indS,:));
         inliers=max_ind;
         break;
      else
          if i==N,
             [m,q]=f_fitline(U(max_ind,:)); 
             inliers=max_ind;
          end
      end
      indS=[];
    end
end
if isempty(inliers),
    display('Inlier set is empty!');
end

Does every triangle-free graph with maximum degree at most 6 have a 5-colouring?

A very specific case of Reed's Conjecture

Reed's $omega$,$Delta$, $chi$ conjecture proposes that every graph has $chi leq lceil tfrac 12(Delta+1+omega)rceil$. Here $chi$ is the chromatic number, $Delta$ is the maximum degree, and $omega$ is the clique number.

When restricted to triangle-free graphs, the equivalent question is, Does every triangle-free graph have chromatic number $leq frac Delta 2 +2$?

This is known for $Deltaleq 4$. In general for triangle-free graphs, $chi leq O(Delta/log Delta)$, so the conjecture is also true for very large $Delta$.

How about $Delta=5$? $Delta=6$? Because of parity, $Delta=6$ is the easier of these two cases (and actually easily implies the $Delta=5$ case. Can anyone prove it?

Kostochka proved that every triangle-free graph has $chi leq frac 2 3 Delta +2$. He also proved that $chileq frac Delta 2 +2$ for graphs of sufficiently large girth depending on $Delta$. Can anyone prove it for girth $geq 5$? $4$?

This would at least provide some hope for proving Reed's Conjecture for triangle-free graphs.

Does every triangle-free graph with $Deltaleq 6$ have $chi leq 5$? What about every graph with girth at least five?

zoology - Is there such thing as "biological cycles" of activity/performance/etc in mammals?

I'm looking for some information about the cycles of increased/decreased activity, mental performance or endurance that are related to a mammal's age and the time of a season.

First of all, I'm not sure what such cycles would be called. I know of circadian rhythm, which is an ~24 hour cycle. The other cycle I've heard about was circa-septan (7 days?). What keywords should I use to search for cycles that may have period of a month to several months?

The second question is - are there are episodes of increased activity/performance over a mammal's lifetime? I can think of two example - mating rituals may involve fighting and hibernation involves almost no movement at all. Are there additional events that may last more than a month?

The final question is if the age comes into equation of increased/decreased activity. Lets say a younger rat an and older rat are put into the same maze, would their ability to "solve" the maze be related to age?

This question is rather broad. Maybe there are some keywords or a science branch that deals with these questions and I can refine my search?

I appreciate your input!

nt.number theory - Can the failure of the multiplicativity of Euler factors at bad primes be corrected?

Warning: This one of those does-anyone-know-how-to-fix-this-vague-problem questions, and not an actual mathematics question at all.

If $X$ is a scheme of finite type over a finite field, then the zeta function $Z(X,t)$ lies in $1+tmathbf{Z}[[t]]$. We can calculate the zeta function of a disjoint union by the formula $Z(Xamalg Y,t)=Z(X,t)Z(Y,t)$. There is also a formula for $Z(Xtimes Y,t)$ in terms of $Z(X,t)$ and $Z(Y,t)$, but this is slightly more complicated. In fact, these two formulas are precisely the standard big Witt vector addition and multiplication law on the set $1+tmathbf{Z}[[t]]$. (Actually, there's more than one standard normalization, so you have to get the right one. I believe this ring structure was first written down by Grothendieck in his appendix to Borel-Serre, but I don't know who first made the connection with the ring of Witt vectors as defined earlier by Witt.) If we let $K_0$ be the Grothendieck group on the isomorphism classes of such schemes, where addition is disjoint union and multiplication is cartesian product, then we get a ring map $K_0to 1+tmathbf{Z}[[t]]$. We could also do all this with the L-factor $L(X,s)=Z(X,q^{-s})$ (where $q$ is the cardinality of the finite field) instead of the zeta function. This is because they determine each other.

This is all good. The problem I have is when there is bad reduction. So now let $X$ be a scheme of finite type over $mathbf{Q}$ (say). Then the L-factor $L_p(X,s)$ is defined by
$$L_p(X,s)=mathrm{det}(1-F_p p^{-s}|H(X,mathbf{Q}_{ell})^{I_p}),$$
where $I_p$ is the inertia group at $p$. (Sorry, I'm not going to explain the rest of the notation.) If $I$ acts trivially (in which case one might say $X$ has good reduction), then taking invariants under $I$ does nothing, and so as above, the L-factor of a product and sum of varieties is determined by the individual L-factors. If $I$ does not act trivially, then the L-factor of a sum is again the product of the individual L-factors, but for products there is no such formula! (The following should be an example showing this. Take $X=mathrm{Spec} mathbf{Q}(i)$, $Y=mathrm{Spec} mathbf{Q}(sqrt{2})$. The we have the following Euler factors at 2: $L_2(X,s)=L_2(Y,s)=L_2(Xtimes Y,s)=1-2^{-s}$ and $L_2(Xtimes X,s)=(1-2^{-s})^2$. So the L-factors of two schemes do not determine that of the product.) Therefore the usual Euler factor cannot possibly give a ring map defined on the Grothendieck ring of varieties over $mathbf{Q}$.

So, is there a way of fixing this problem? I would guess the answer is No, because while some people might allow you to scale Euler factors by numbers, I don't think anyone will let you change them by anything else. But maybe there is some "refined L-factor" that determines the usual one (and maybe incorporates the higher cohomology of the inertia group?) Assuming there is no known way of repairing things, I have a follow-up question: Is there some general formalism that handles this failure? And if so, how does that work?

ac.commutative algebra - Intersection of finitely generated subalgebras also finitely generated?

It is enough to show that the intersection of two finitely generated semigroups inside a finitely generated commutative semigroup is not necessarily finitely generated, for then you can consider the semigroup algebras.

So let $A$ be freely generated by ${y,z}cup{x_n:ngeq1}$ subject to the relations $yx_n=x_{n+1}$ and $zx_n=x_{n+1}$ for all $ngeq 1$, and $x_nx_m=x_{nm}$ for all $n,mgeq1$ (notice that $A$ in fact coincides with the given set of generators...). Let $A_1$ be the subsemigroup generated by $y$ and $x_1$, and let $A_2$ be the subsemigroup generated by $z$ and $x_1$. Then $A$, $A_1$ and $A_2$ are finitely generated and commutative, yet the intersection $A_1cap A_2$ is the subsemigroup of $A$ generated by ${x_n:ngeq1}$, which is isomorphic to $mathbb N$ under the product. This is not finitely generated.

Later: Yemon asks in a comment if one can change this so that the containing algebra is a domain. I think this works: let $A$ be the algebra generated by ${y,z,u}cup{x_n:ngeq2}$ subject to the relations $yx_n=x_{n+1}$ and $zx_n=x_{n+1}+u$ for all $ngeq 2$, and $x_nx_m=x_{nm}$ for all $n,mgeq1$, let $A_1$ be generated by $y$ and $x_2$, and let $A_2$ be generated by $z$, $u$ and $x_2$. (I have to remove $x_1$ for otherwise $x_1(x_1-1)=0$)

combinatorial game theory - Bipartite Nim-Geography

I guess you already know this, but I just stumbled on this paper that studies the exact same game you described:

M. Fukuyama, A Nim game played on graphs, Theoret. Comput. Sci. 304 (2003), 387–399.

$text{Section 3}$ of this paper studies the Nim game on a simple bipartite graph in which all vertices on one side are of degree (at most) $2$. At the start of the game, the token is assumed to be on a vertex on the side without the degree condition. I put the term "at most" in parentheses because this is proved to be irrelevant in the paper so that you can safely assume that all vertices on the restricted side are of degree $2$. The author gives necessary and sufficient conditions for the first player to have a winning strategy.

To decide if the first player can win by using the necessary and sufficient conditions given in this paper, if my understanding through skimming the paper is correct, you check if the following three conditions hold for each vertex $u$ on the side with the degree restriction:

1. The number $h$ of matchsticks in the nim heap on one edge with $u$ as its endpoint is different from that of the other edge with $u$ as its endpoint, (In the following, $h$ is assumed to be smaller.)




2. Split the vertex $u$ into $u_1$ and $u_2$ and w.l.o.g. assume that the unique edge of which $u_1$ is an endpoint after splitting is of weight $h$. The minimum capacity of $u_1$-$u_2$ cuts is equal to $h$.




3. For any minimum $u_1$-$u_2$ cut, the vertex the token is currently at is connected to $u_2$ (i.e., the one with more matchsticks on its edge).

Checking the first condition is trivial. The second one only needs to compute the minimum capacity, which can be done in polynomial time. For the third condition, while checking if two vertices are connected in a graph can be done quickly, we may have to list all minimum $u_1$-$u_2$ cuts. Google got me this paper that mentions an algorithm for listing all minimum $s$-$t$ cuts for fixed $s$ and $t$, where the run time depends on the total number of minimum $s$-$t$ cuts for all pair $s, t$ in the vertex set, which can be exponential... The details are relegated to another technical report by the same authors (Ref. [GN1] in the linked PDF), and there seems to be an improvement on this in said technical report. But I couldn't find it online.

I looked for a similar paper that doesn't impose the degree condition, but my google-fu failed me.

at.algebraic topology - Extending a property of commutative algebras to C infinity algebras

If A is a commutative algebra and B is an X- algebra, then the tensor product $A otimes B$ is an X-algebra (so for example, $Com otimes Lie$ is a Lie algebra). This is seen using the language of operads. Let $Com$ be the commutative operad. Since $Com(n)$ is a one dimensional vector space for every $n$, tensoring $Com$ with an operad $O$ doesn't change the operad $O$.

Does a similar thing hold true for a $C_infty$ algebra? That is, if $A$ is a $C_infty$ algebra, is $A otimes B$ an $X_infty$ algebra?

I'm still trying to familiarize myself with the language of operads, and perhaps the question can be made more precise in that language, where the infinity version of an operad is cofibrant resolution of the operad.

st.statistics - unbiased estimate of the variance of a weighted mean

First some notation. Each example is drawn from some unknown distribution $Y$ with $E[Y] = mu$ and $textrm{Var}[Y] = sigma^2$. Suppose the weighted mean consists of $n$ independent draws $X_isim Y$, and ${w_i}_1^n$ is in the standard simplex. Finally define the r.v. $X = sum_i w_i X_i$. Note that $E[X] = sum_i w_i E[X_i] = mu$ and $textrm{Var}[X] = sum_i w_i^2 textrm{Var}
[X_i] = sigma^2sum_i w_i^2$.

Generalizing the standard definition of sample mean, take
$$
hat mu({x_i}_1^n) := sum_i w_i x_i.
$$
Note that $E[hat mu({x_i}_1^n)] = sum_i w_i E[x_i] = mu = E[X]$, so $hat mu$ is an unbiased estimator.

For the sample variance, generalize the sample variance as
$$
hat sigma^2_b({x_i}_1^n) := sum_i w_i (x_i - hat mu({x_i}_1^n))^2,
$$
where the subscript foreshadows this will need a correction to be unbiased. Anyway,
$$
E[hat sigma^2_b] = sum_i w_i E[(x_i - hat mu)^2] = sum_i w_i Eleft[left(sum_j w_j (x_i - x_j)right)^2right].
$$
The term in the expectation can be written as
$$
sum_{j,k} w_j(x_i - x_j)w_k(x_i - x_k) = sum_jw_j^2(x_i - x_j)^2 + sum_{jneq k} w_j w_k(x_i - x_j)(x_i - x_k).
$$
Passing in the expectation, the first term (when $x_ineq x_j$, which would yield 0) is
$$
E[(x_i-x_j)^2] = 2E[x_i^2] - 2mu^2 = 2sigma^2,
$$
whereas the second (when $x_i neq x_j$ and $x_i neq x_k$, which would yield 0) is
$$
E[x_i^2 - x_ix_j - x_ix_k + x_jx_k] = E[x_i^2] - mu^2 = sigma^2.
$$
Combining everything,
$$
sum_i w_i left(2sigma^2sum_{jneq i}w_j^2 + sigma^2sum_{jneq kneq i} w_j w_kright)
= sigma^2( 1 - sum_j w_j^2).
$$
Therefore $E[hat sigma_b^2] - sigma^2 = -sigma^22sum_j w_j^2$, i.e. this is a biased estimator. To make this an unbiased estimator of $Y$, divide by the excess term derived above:
$$
hat sigma_u^2({x_i}_1^n)
:= frac {hat sigma_b^2({x_i}_1^n)}{1- sum_j w_j^2}
= frac {sum_i w_i(x_i - hat mu)^2}{1- sum_j w_j^2 }
$$
This matches the definition you gave (and a sanity check $w_i = 1/N$, recovering the normal unbiased estimate).

Now, if one instead were to seek an unbiased estimator of $X=sum_i X_i$, the formula would instead be $hat sigma_b^2({x_i}_1^n)(sum_j w_j^2) / ( 1 - sum_j w_j^2)$.

It is very odd for me that the documents you refer to are making estimators of $Y$ and not $X$; I don't see the justification of such an estimator. Also it is not clearly how to extend it to samples that don't have length $n$, whereas for the estimator of $X$, you simply have some number $m$ of $n$-samples, and averaging everything above makes things work out. Also, I didn't check, but it's my suspicion that the weighted estimator for $Y$ has higher variance than the usual one; as such, why use this weighted estimator at all? Building an estimator for $X$ would seem to have been the intent..

botany - If the xylem of a woody plant is composed of dead tissue, how does sapwood become heartwood?

I'm no expert, but I liked this question and did a quick literature search:

Xylem cells are certainly dead at maturity, and it is these cells that make up the majority of what we call wood. However, wood also contains other tissue types, some of which have live cells at maturity. From what I can glean here and here, it is ray parenchyma tissue that is responsible for many of the changes that take place during the transition from sapwood to heartwood. Ray parenchyma is formed at the vascular cambium, along with xylem, during stem secondary growth (wood formation). Ray parenchyma cells remain alive for years after the xylem cells in the sapwood die. Their functions are transporting nutrients radially in the stem, and storage of carbohydrates.

As the stem grows in diameter, there is a zone of transition from sapwood to heartwood where the ray parenchyma tissue begins to die. Before it dies, the cells undergo some metabolic changes and begin to synthesize secondary phenolic compounds (many different kinds) from stored starch or sugars. Then, as ray parenchyma cells die, they dump these phenolics into the surrounding xylem tissue, and the accumulation of these durable, decay-resistant compounds in the xylem leads to the formation of heartwood. In many hardwood trees this parenchyma death and phenolic accumulation is preceded by vessel-plugging, in which the ray parenchyma cells form balloon-like growths that block the flow of water through the xylem. So, once sapwood changes to heartwood there should be no living cells left, and the water transport capacity of its xylem is gone or greatly diminished.

There appears to be quite a bit of variability between species in when heartwood formation occurs and what triggers it. From what I could see in my brief literature search, it is all under fairly complex genetic and hormonal control that remains an area of active research. I would suggest (and speculate a bit) that the trigger has something to do with maintaining an optimum amount of sapwood as the tree grows in leaf area (which increases demand for water transport) and height (which increases demand for support, or stem diameter).

examples - Do good math jokes exist?

An excerpt from H. Petard, "A contribution to the mathematical theory of big game hunting," The American Mathematical Monthly, vol. 45, no. 7, pp. 446-447, 1938:

The Hilbert, or axiomatic, method. We place a locked cage at a given
point of the desert. We then
introduce the following logical
system.

• Axiom I. The class of lions in the Sahara Desert is non-void.


• Axiom II. If there is a lion in the Sahara Desert, there is a lion in the
  cage.


• Rule of Procedure. If p is a theorem, and "p implies q" is a
  theorem, then q is a theorem.


• Theorem I. There is a lion in the cage.

The method of inversive geometry. We place a spherical cage in the desert, enter it, and lock it. We perform an inversion with respect to the cage. The lion is then in the interior of the cage, and we are outside.

The method of projective geometry. Without loss of generality, we may
regard the Sahara Desert as a plane.
Project the plane into a line, and
then project the line into an interior
point of the cage. The lion is
projected into the same point.

The Bolzano-Weierstrass method. Bisect the desert by a line running N-S. The lion is either in the E portion or in the W portion; let us suppose him to be in the W portion. Bisect this portion by a line running E-W. The lion is either in the N portion or in the S portion; let us suppose him to be in the N portion. We continue this process indefinitely, constructing a sufficiently strong fence about the chosen portion at each step. The diameter of the chosen portions approaches zero, so that the lion is ultimately surrounded by a fence of arbitrarily small perimeter.

integrable systems - What is exactly the (singularity) confinement property ?

The singularity confinement property refers to a property of discrete integrable systems. I am unaware of this property in the context for continuous systems. I can understand why you might have difficulty in getting a definition, since it is rather oddly defined all too often. Since the paper of Goriely and La Fortune have been referenced I will assume that you are interested in this property for maps rather than partial difference equations.

Let me start with an example that hopefully illustrates what is going on, before generalizing: Let us take a simple second order difference equation, which we right as a sequence, satisfying the recurrence relation:

$x_{n+1}x_{n-1} = 1+ x_n$

If we let $x_1 = -1$, then $x_2 = 0$, regardless of what $x_0$ is. Then $x_3 = -1$ and $x_4 = infty$. Worst of all, you have also lost your initial conditions, namely $x_0$. Suppose, instead of $x_1 = -1$, we take $x_1 = -1+e$, where e is assumed very small. Then

$x(2) = frac{e}{x_0}$

$x(3) = frac{e+x_0}{(e-1)x_0}$

$x(4) = frac{1+x_0}{e-1}$

Notice that in the limit as $e to 0$, $x_2 to 0$, $x_3 to -1$, $x_4 to -1 -x_0$. Somehow, in the limit around a "bad" point, or "singularity", we recover initial conditions, namely $x_0$. We say that the singularity is confined because, despite a loss of information at $x_2$ and $x_3$ in the limit, we can regain the information again at $x_4$ in the same limit. We say the equation has the singularity confinement property if we can always regain the initial conditions in the limit somewhere along the line. This is the usual, however you may want something more general.

The generalization of the above is a map

$f: (x_n, x_{n-1}) to (x_{n+1},x_n) = left(frac{x_n+1}{x_{n-1}},x_nright)$,

however, we generally take a map of $R^n$ and then take the corresponding projective space, $P^n$, then we consider a map

$f: P^n to P^n$.

This function now describes a discrete dynamical system, via $y_{n+1} = f(y_n)$. A singularity in this context is where the function, $f$, fails to be invertible. For example, in the above, we had all initial conditions of the for $(x_0,-1)$ being sent to $(-1,0)$. However, we say the singularity is confined if some power of $f$ is invertible. Note that in the above example, the third power of the map can be continuously extended to the map $(x_0,-1) to (-1,-1-x_0)$, hence the singularity, $(-1,0)$ is confined. A system has the singularity confinement property if all singularities are confined.

ag.algebraic geometry - Is the mapping from a scheme to its global sections a closed map?

[Added: I misread the question, and in fact this answer does not answer the OP's question,
but rather the following question: is $phi(T)$ closed in Spec $Gamma(T)$, which is
a different question. Probably the upvotes can be attributed to the link to the stacks project!]

If $T$ is quasi-affine (i.e. admits an open immersion into affine space),
then the map $phi$ is an open immersion, and in fact Spec $Gamma(T)$
is the initial object in the category of affine schemes containing $T$
as an open subscheme.

In particular, in this case $phi$ has closed image
if and only if and only if $T$ is in fact affine. [Added: As Qing Liu points out in a comment below, in this quasi-affine situation, $phi$ is in fact a closed map onto its image.]

Thus if we take $T$ to be ${mathbb A}^2_k setminus {0}$ for some
field $k$, i.e. affine $2$-space with the origin removed,
then we get an example of $T$ where this map is open with non-closed
image (since this $T$ is quasi-affine but not affine). Note that
Spec $Gamma(T) = {mathbb A^2}_k$.

(This is a geometric analogue of Qing Lui's more arithmetic example;
what both have in common is that a closed point was removed from a 2-dimensional
affine scheme, so as to make a quasi-affine scheme that is not affine.[Added: I also misread Qing Liu's example; my remark would apply to the affine line over ${mathbb Z}$ with a closed point removed; Qing's example is more complicated, since it is actually dealing with the OP's question. One can make a geometric analogue of Qing's example by deleting a closed point from ${mathbb A}^1times {mathbb P}^1$; more geometrically still, remove one of the lines of a ruling from a projective quadric surface, and then remove an additional point.])

EDIT: In the definition of quasi-affine, one should also require that $T$
be quasi-compact. (The stacks project
is a terrific resource for these foundational definitions in scheme theory,
particularly with regard to finiteness and separation issues.)

Note that if $T$ is any quasi-compact scheme, then the map $T to$ Spec $Gamma(T)$ has dense image. (If $f in Gamma(T)$ and $D(f)$ is the usual affine open in Spec $Gamma(T)$,
i.e. Spec $Gamma(T)_f$, then if $phi^{-1}(D(f))$ is empty, it must be
that $f$ is locally nilpotent on $T$. Since $T$ is quasi-compact this implies
that $f$ is actually nilpotent, and hence that $D(f)$ is empty.) As Martin notes
in his answer, this is similarly true if $T$ is reduced.

It need not be true if $T$ is non-reduced and non-quasi-compact (since $T$
may then admit locally nilpotent sections of $mathcal O_T$ that are not
globally nilpotent, e.g. $T = coprod_n$ Spec $k[x]/(x^n)$).