linear algebra - Exact short sequences of vector spaces

Without using any basis of A, B or C specifically, but rather using the existence of bases for all vector spaces, we may observe that all vector spaces are free as modules; and all free modules are projective.

Finally, the projectivity property gives us a splitting by lifting the identity map on C along the surjection from B by projectivity of C. This gives us a splitting, thus splitting the entire sequence.

ETA With the arguments in the comments to Ben Websters answer, it is pointed out that without AC, and without at least the existence of bases, things can fail badly. Obviously, if we disallow AC, this answer fails as badly.

na.numerical analysis - Delta notation used for describing numerical stencil

While reading some papers translated from the Russian literature, I've noticed that a delta symbol can be used to denote a FDTD stencil that discretizes a PDE. For example, in [1], a fourth order mixed partial derivative term is denoted by

$
2frac{{partial ^4 u}}{{partial ^2 xpartial ^2 y}} = Delta _{xy}^4 u^{k + 1} _{i + 1,j + 1} + Delta _{xy}^4 u^k _{i - 1,j - 1}
$

where an example is given of

$Delta _{xy}^4 u_{i + 1,j + 1} = Delta _x^2 u_{i + 1,j + 2} - 2Delta _x^2 u_{i + 1,j + 1} + Delta _x^2 u_{i + 1,j}$

Notice that this example given in the paper does not have the ${ k,k + 1} $ superscipts.

Clearly ${i,j}$ are spatial indices and $k$ is the timestep. But what is being implied by the use of the delta symbol? I suspect that this is a differential, but I have never seen a differential with $u_{i,j}$ and $i,j$ indices. The author does not define the symbol in his paper, so I think that it should be implicitly understood. I am also unsure as to whether such a notation has also been used by other authors.

How would I write out $Delta _{xy}^4 u_{i + 1,j + 1}$ and $Delta _{xy}^4 u_{i - 1,j - 1}$ using a 5-point stencil or 7-point stencil? Are there any other papers which use similar notation?

[1] V. Saul'yev, “A difference method for solving parabolic equations of any order,” Computational Mathematics and Mathematical Physics, vol. 36(12), 1996, pp. 1697-1700.

mammals - How are blood vessels affected by UV and infrared radiation?

UV rays cause damage is by thinning the walls of surface blood vessels, leading to bruising, bleeding, and the appearance of blood vessels through the skin. Longwave UV radiation (UV-A) accounts for up to 95% of the UV radiation that reaches the earth's surface. Although UV-A is less intense than UV-B, it is more prevalent and can penetrate deeper into the skin layers, affecting the connective tissue and blood vessels, which results in premature aging - Photoaging.

At the same time UVB – is even more dangerous than UVA radiation and also causes skin damage and skin cancer. It affects the surface skin layer. The skin responds by releasing chemicals that dilate blood vessels. This causes fluid leakage and inflammation – better known as sunburn.

I am not quite sure what chemicals are generated by UV radiations beneath the skin surface. I guess it might be excess free radicals which accumulate and cause damage to blood vessels since they are delicate. There is one theory which talks about damages to skin due to Free Radical theory of aging

To give an example how UV radiation affects the blood vessel here:

In actinic purpura, UV radiation damages the structural collagen that supports the walls of the skin's tiny blood vessels. Particularly in older people, this collagen damage makes blood vessels more fragile and more likely to rupture following a slight impact.

Below figure to give an idea of how strong UV radiations are:

human biology - Do siblings have a bias toward believing they look different from each other?

You are not completely wrong but not completely right either. Some siblings tend to think they are less similar to each other, even twins. It may or may not have scientific reason, but it has sociological reason behind it. Every people in our society want to have their own separate identity and they do not want themselves to be compared with others even with their siblings.It might be also due to the result of fact that, siblings pass times with each other so often that they notice distinct differences in both their characteristics and look while ordinary people, looking at them for the first time, can not notice the differences. This is related to the reason why we can not sometime recognize people from any particular race or ethnicity at first look.

zoology - Are ape communities more "anarchist" or "communist" in structure?

Among the great apes, chimpanzees and gorillas live in very hierarchical, male-dominated clans that are often in violent conflict with other clans. Bonobos, on the other hand, lead very peaceful lives, and are female-dominated, using sexual contact as a manner of communication to reduce tension within and between groups. Orangutans are largely solitary animals, with heavy parental investment (7 years, longest among non-human primates), wherein the mother lives separately with her child during the early years of development. Among the lesser apes, gibbons and siamangs are also highly territorial and pair-bonded, though notably, polyandrous siamang groups have been shown to display higher degrees of cooperation.

Though Anarchism varies greatly between different definitions, the most popular form of contemporary Anarchism values non-hierarchical organization, centered around mutual aid, rather than the kind of isolated individualism that you seem to associate with the term. By this definition, bonobos would be more "Anarchist" than chimpanzees. By your definition, orangutans might be the most "Anarchist."

Your definition of "Communism" is closer to the contemporary definition of Anarchism above, due mainly to the historical trajectory of Communism, which in practice, did not work out so well, leaving contemporary utopian thinkers largely in the Anarchist camp. By your definition of "Communism," bonobos may be the most "Communist"; however, if you mean the state-based gulag Communism of popular culture (ie portrayals of the Soviet Union), then you might say that chimps were more Communist. This is because individual chimps sacrifice individual goal attainment for group ends under the coercion of the alpha male chimp. Of course, some people may call that Fascism.

You may want to take a look at primatologist Franz De Waal's research on bonobos vs. chimpanzees. As a pacifist, he has a bit of an agenda, in terms of trying to prove that female-dominated bonobo "societies" are inherently more cooperative and less hierarchical. Another primatologist that agrees with this view is Harvard primatologist Richard Wrangham, who looks at hierarchical organization and violent behaviors in chimpanzees.

Robert Sapolsky's research on baboons may also be of interest to you. He found a clan of baboons, in which all the alpha males had died from tuberculosis due to exposure to an infected meat found in a neighboring human village. He noted, then, that the remaining baboons lived much more peaceably and cooperatively than other observed clans, and furthermore, that this clan retained its "culture" and passed on its more cooperative ways of behaving to new members in the clan, therefore demonstrating that perhaps baboons (and other primates) are capable of having and passing on "culture." - Perhaps a more "Anarchist" culture than nature normally prescribes.

However, as a caveat, I must say: it is tenuous to make direct associations between ape behavior and human societies. Human culture, social intelligence, and symbolic reasoning facilitate human political systems. Though Sapolsky's work suggests primates may also be capable of some form of "culture," biological descriptions of ape social organization is insufficient to explain human behavior.

on counting of special case of trees on a graph

I'll answer a question raised in the comments:

Problem: Count the number of induced trees of size $k$.

According to this paper by Erdös, Saks and Sos, it is NP-complete to decide given a graph $G$ and an integer $k$, if $G$ contains an induced tree of size $k$. So, it's probably pretty damn hard to count them. Apparently, it remains NP-complete even for bipartite graphs.

Actually, the argument is pretty simple so I'll include it here. Given a graph $H$ and an integer $k$, it is well-known that the problem of deciding if $H$ has an independent set of size $k$ is NP-complete. Suppose that $H$ has $n$ vertices. Let $G$ be the graph obtained from $H$ by first adding a disjoint copy of $P_n$ (a path on $n$ vertices), and then connecting one end of $P_n$ to all the vertices in $H$. Clearly, $H$ has an independent set of size $k$ if and only if $G$ contains an induced tree of size $n+k$.

gt.geometric topology - Simplicial and cubical decompositions of low valence

Every surface can be triangulated in such a way that at most 7 trianlges meet at one vertex. Every surface can be decomposed in squares such that at every vertex at most 5 suqares meet. For surfaces of genus more than 1 this is the low bound.

What happen in higher dimensions, for example for 3 and 4-manifolds, ect...? It should be easy to show that for every dimension $n$ there are numbers $S(n)$ and and $C(n)$ such that every manifold $M^n$ admits a simplicial decomposition with at most $S(n)$
simplexes at every vertex and a cubical decomposition with at most $C(n)$ cubes at every vertex. The refference of Gil below confirms this for $n=3$.

Here are three questions (I suspect they are hard).

1) Can it be proven that $C(n)>2^n$?

2) Can it be proven that $S(n)>frac{Vol(S^n)}{Vol(Delta^n)}$, where $Delta^n$ is the spherical tetrahedron with edge of length $frac{pi}{3}$ in the unit sphere $S^n$.

3) Is there any reasonable estimation for $C(n)$ and $S(n)$ from above?

ct.category theory - Model Structure/Homotopy Pushouts in topological monoids?

Clark Barwick's answer is excellent and you should accept it. This is more of an addendum. The category Top is cofibrantly generated, so $mathcal{C} =$ Mon(Top) is also cofibrantly generated. The key paper is by Schwede and Shipley, and gives conditions on a model category $mathcal{M}$ such that Mon$(mathcal{M})$ is a model category. In the special case of $mathcal{M}$ cofibrantly generated it explains how to get your hands on the cofibrations of Mon$(mathcal{M})$. See Theorem 4.1 on page 8. Of course, now that you have your hands on the fibrations, trivial fibrations, cofibrations, and trivial cofibrations question (2) is also answered. A nice reference for relating the cylinder object to the functorial factorizations is Hovey page 9

Furthermore, every element in Top is fibrant, so the paper above gives you even stronger results, which may help you with your computations. See remark 4.5 on page 10.

The authors also wrote a second paper giving further results. It's here.

ca.analysis and odes - Real-analytic manifolds in real-analytic sets

Let $Usubset mathbb{R}^n$ be open, and let $f:Utomathbb{R}$ be real-analytic. We consider the zero set $Z:=f^{-1}({0})$.

For a paper I am writing, I am looking for the best reference to the following basic fact:

If $Z$ has topological dimension equal to $d$, then $Z$ contains a real-analytic manifold of dimension $d$.

---

I can get this from Lojasiewicz's theorem or similar results, but that is a slightly unwieldy reference, and something probably needs to be said about how exactly one deduces it. Given that the statement is rather simple, I was wondering if someone knows of a more direct reference to this fact.

And to add a mathematical question: This result is obviously much weaker than Lojasiewicz's theorem. Is there a proof that doesn't require developing the full structure theorem?

Many thanks for any pointers!

higher category theory - (infinity,1)-categories directly from model categories

Edit & Note: I'm declaring a convention here because I don't feel like trying to fix this in a bunch of spots: If I said model category and it doesn't make sense, I meant a model-category "model" of an (infinity,1)-category. Also, "model" in quotes means the English word model, whereas without quotes it has do do with model categories.

At the very beginning of Lurie's higher topos theory, he mentions that there is a theory of $(infty,1)$-categories that can be directly constructed by using model categories.

What I'd like to know is:

Where can I find related papers (Lurie mentions two books that are not available for download)?

How dependent on quasicategories is the theory developed in HTT? Can the important results be proven for these $(infty,1)$-model-categories by proving some sort of equivalence (not equivalence of categories, but some weaker kind of equivalence) to the theory of quasicategories?

When would we want to use quasicategories rather than these more abstract model categories?

And also, conversely, when would we want to look at model categories rather than quasicategories?

Does one subsume the other? Are there disadvantages to the model category construction just because it requires you to have all of the machinery of model categories? Are quasicategories better in every way?

The only "models" of infinity categories that I'm familiar with are the ones presented in HTT.

ct.category theory - In what cases does a Yoneda-like embedding preserve monoidal structure?

Day showed that, for suitable V, any monoidal structure on a (V-)functor category $[C^{mathrm{op}}, V]$ is essentially determined by its restriction to the representables as
$$ F otimes G = int^{A,B} F A otimes G B otimes P(A,B,-) $$
where $P(A,B,-) = C(-, A) otimes C(-, B)$ is a profunctor $C otimes C otimes C^{mathrm{op}} to V$. P (together with a unit and the usual structural isos) is said to endow C with a promonoidal structure.

If C is already a monoidal V-category, then there is a canonical promonoidal structure on it given by
$$ C(-, A) otimes C(-, B) = C(-, A otimes B) $$
In that case, the Yoneda embedding is strong monoidal by definition. In fact it is the unit for the monoidal cocompletion of C.

homotopy theory - complex cobordism from formal group laws?

As far as I know, there is still no such interpretation. The closest I've heard is some rumored (but unpublished) work in derived algebraic geometry interpreting MU as some kind of representing object.

Such a construction of MU in terms of formal group data be very welcome (probably even more now than when Ravenel wrote the green book).

EDIT: Some elaboration.

We do know a lot about MU. We know that it has an orientation (Chern classes for vector bundles), and in it's universal for this property. It's not then extremely suprising that we get a formal group law from the tensor product for line bundles, but the fact that MU carries a universal formal group law, and that MU ^ MU carries a universal pair of isomorphic formal group laws, is surprising. At this point it's something we observe algebraically. Even Lurie's definition of derived formal group laws, assuming I understand correctly, is geared to construct formal group laws objects in derived algebraic geometry carrying a connection to the formal group law data that we already know is there on the spectrum level, and hence ties it to the story we already knew for MU implicitly.

Some reasons these days we might want to know how to construct MU from formal group law data:

• Selfish, ordinary homotopy-theoretic reasons. It's very useful to be able to construct other spectra with specific connections to formal group law data (like K-theory, TMF, etc) and constructing them is generally very difficult. Things like the Landweber exact functor theorem, the Hopkins-Miller theorem, and Lurie's recent work give us a lot of progress in this direction, but they only apply to restricted circumstances. None of these general methods will construct ordinary integral cohomology, corresponding to the additive formal group law (only rational cohomology). If we understood how to build MU, we might understand how to generalize.


• Equivariant homotopy theory. I would tentatively say that we don't have nearly as good computational and "qualitative" pictures of the equivariant stable categories, because we don't have something like the startling MU-picture that relates it all to some stack like the moduli stack of 1-dimensional formal group laws. If we found MU by _accident_ then we don't really know how the analogue should play out in other, more general, stable categories.


• Motivic homotopy theory. Hopkins and Morel found that there is some data to formal group laws appearing in motivic stable homotopy theory via the motivic bordism spectrum MGL. I'm not up with the state of the art here but a better understanding of this connection would be very important too - for understanding MGL itself, but also hopefully for understanding the analogues of chromatic data in these categories related to algebraic geometry.


• (space reserved for connections to other subjects that I've forgotten)

pr.probability - State of the Art in Stable limits, embeddings, etc

This is a fairly broad request for references. I've tried a few hours of googling, but the usual process of chasing names and references doesn't seem to be converging on any must-read books or obviously state-of-the art articles/surveys, and I was hoping an expert around here might point me in the right direction.

Are there any good/state-of-the-art references for invariance principles (also called functional limit theorems and possibly other names) which relate to convergence to stable (but not Gaussian) processes? I'm particularly interested in papers that provide some grounding relative to the 'standard' $alpha = 2$ case, such as when something like the Skorohod embedding theorem can hold (I have seen it for $alpha > 1$, but am not sure what happens below there), or how methods change in the 'nicest' categories, such as iid variables, martingale difference arrays, strongly mixing stationary processes, etc. At the moment, I care most about the asymmetric, $alpha leq 1$ case but would be grateful for any reading suggestions.

lo.logic - Induction vs. Strong Induction

The terms "weak induction" and "strong induction" are not commonly used in the study of logic. The terms are commonly used only in books aimed at teaching students how to write proofs.

Here are their prototypical symbolic forms:

• weak induction: $(Phi(0) land (forall n) [ Phi(n) to Phi(n+1)]) to (forall n) Phi(n)$




• strong induction: $(Psi(0) land (forall n) [ (forall m leq n) Psi(m) to (forall m leq n+1) Psi(m)]) to (forall n) Psi(n)$

The thing to notice is that "strong" induction is almost exactly weak induction with $Phi(n)$ taken to be $(forall m leq n)Psi(n)$. In particular, strong induction is not actually stronger, it's just a special case of weak induction modulo some trivialities like replacing $Psi(0)$ with $(forall m leq 0 )Psi(m)$. Of course you can write variations of the symbolic forms, but the same point applies to all of them: "strong" induction is essentially just weak induction whose induction hypothesis has a bounded universal quantifier.

So the question is not why we still have "weak" induction - it's why we still have "strong" induction when this is not actually any stronger.

My opinion is that the reason this distinction remains is that it serves a pedagogical purpose. The first proofs by induction that we teach are usually things like $forall n left [ sum_{i=0}^n i = n(n+1)/2 right ]$. The proofs of these naturally suggest "weak" induction, which students learn as a pattern to mimic.

Later, we teach more difficult proofs where that pattern no longer works. To give a name to the difference, we call the new pattern "strong induction" so that we can distinguish between the methods when presenting a proof in lecture. Then we can tell a student "try using strong induction", which is more helpful than just "try using induction".

In terms of logical strength in formal arithmetic, as you can see above, the two forms are equivalent over some weak base theory as long as you are looking at induction for a class of formulas that is closed under bounded universal number quantification. In particular, all the syntactic classes of the analytical and arithmetical hierarchies have that property, so weak induction for $Sigma^0_k$ formulas is the same as strong induction for $Sigma^0_k$ formulas, weak induction for $Pi^1_k$ formulas is the same as strong induction for $Pi^1_k$ formulas, and so on. This equivalence will hold under any reasonable formalization of "strong" induction - I chose mine above to make the issue particularly obvious.

---

Addendum I was asked in a comment why
$$
(1)colon (forall t)[(forall m < t)Phi(m) to Phi(t)]
$$
implies
$$
(2)colon (forall n)[(forall m leq n)Phi(m) to (forall m leq n+1) Phi(m)].
$$
I'm going to give a relatively formal proof to show how it goes. The proof is not by induction, instead it just uses universal generalization to prove the universally quantified statements.

For the proof, I will assume (1) and prove (2). Working towards that goal, I fix a value of $n$ and assume:
$$
(3)colon (forall m leq n)Phi(m).
$$

I first want to prove $(forall m < n+1)Phi(m)$, which is an abbreviation for $(forall m)[m < n+1 to Phi(m)]$. Pick an $m$. If $m < n+1$, then $m leq n$, so I know $Phi(m)$ by assumption (3). So, by universal generalization, I obtain $(forall m < n+1)Phi(m)$.

Next, note that a substitution instance of (1) gives $(forall m < n+1)Phi(m) to Phi(n+1)$. I have proved $(forall m < n+1)Phi(m)$ so I can assert $Phi(n+1)$.

So now I have assumed $(forall m leq n)Phi(m)$ and I have also proved $Phi(n+1)$. Another proof by cases establishes $(forall m leq n+1)Phi(m)$.

By examining the proof, you can see which axioms I need in my weak base theory. I need at least the following two axioms:

I believe those are the only two axioms I used in the proof.

evolution - Is there any evidence that sexual selection may lead to extinction of species?

TL;DR:

• There is a dearth of actual experimental evidence. However:

  
  
  

  • there is at least one study that confirmed the process ([STUDY #7] - Myxococcus xanthus; by Fiegna and Velicer, 2003).

  
  

  • Another study experimentally confirmed higher extinction risk as well ([STUDY #8] - Paul F. Doherty's study of dimorphic bird species an [STUDY #9] - Denson K. McLain).

  
  



• Theoretical studies produce somewhat unsettled results - some models support the evolutionary suicide and some models do not - the major difference seems to be variability of environmental pressures.




• Also, if you include human predation based solely on sexually selected trait, examples definitely exist, e.g. Arabian Oryx

---

First of all, this may be cheating but one example is the extinction because a predator species specifically selects the species because of selected-for feature.

The most obvious case is when the predator species is human. As a random example, Arabian Oryx was nearly hunted to extinction specifically because of their horns.

---

Please note that this is NOT a simple question - for example, the often-cited in unscientific literature example of Irish Elk that supposedly went extinct due to its antler size may not be a good crystal-clear example. For a very thorough analysis, see: "Sexy to die for? Sexual selection and risk of extinction" by Hanna Kokko and Robert Brooks, Ann. Zool. Fennici 40: 207-219. [STUDY #1]

They specifically find that evolutionary "suicide" is unlikely in deterministic environments, at least if the costs of the feature are borne by the individual organism itself.

Another study resulting in a negative result was "Sexual selection and the risk of extinction in mammals", Edward H. Morrow and Claudia Fricke; The Royal Society Proceedings: Biological Sciences, Published online 4 November 2004, pp 2395-2401 [STUDY #2]

The aim of this study was therefore to examine whether the level of
sexual selection (measured as residual testes mass and sexual size dimorphism) was related to the risk of extinction that mammals are currently experiencing. We found no evidence for a relationship between these factors, although our analyses may have been confounded by the possible dominating effect of contemporary anthropogenic factors.

---

However, if one takes into consideration changes in the environment, the extinction becomes theoretically possible. From "Runaway Evolution to Self-Extinction Under Asymmetrical Competition" - Hiroyuki Matsuda and Peter A. Abrams; Evolution Vol. 48, No. 6 (Dec., 1994), pp. 1764-1772: [STUDY #3]

We show that purely intraspecific competition can cause evolution of extreme competitive abilities that ultimately result in extinction, without any influence from other species. The only change in the model required for this outcome is the assumption of a nonnormal distribution of resources of different sizes measured on a logarithmic scale. This suggests that taxon cycles, if they exist, may be driven by within- rather than between-species competition. Self-extinction does not occur when the advantage conferred by a large value of the competitive trait (e.g., size) is relatively small, or when the carrying capacity decreases at a comparatively rapid rate with increases in trait value. The evidence regarding these assumptions is discussed. The results suggest a need for more data on resource distributions and size-advantage in order to understand the evolution of competitive traits such as body size.

---

As far as supporting evidence, some studies are listed in "Can adaptation lead to extinction?" by Daniel J. Rankin and Andre´s Lo´pez-Sepulcre, OICOS 111:3 (2005). [STUDY #4]

They cite 3:

The first example is a study on the Japanese medaka
fish Oryzias latipes (Muir and Howard 1999 - [STUDY #5]). Transgenic males which had been modified to include a salmon growth-hormone gene are larger than their wild-type counterparts, although their offspring have a lower fecundity (Muir and Howard 1999). Females
prefer to mate with larger males, giving the larger
transgenic males a fitness advantage over wild-type
males. However, offspring produced with transgenic
males have a lower fecundity, and hence average female
fecundity will decrease. As long as females preferentially
mate with larger males, the population density will
decline. Models of this system have predicted that, if
the transgenic fish were released into a wild-type
population, the transgene would spread due to its mating
advantage over wild-type males, and the population
would become go extinct (Muir and Howard 1999).
A recent extension of the model has shown that
alternative mating tactics by wild-type males could
reduce the rate of transgene spread, but that this is still
not sufficient to prevent population extinction (Howard
et al. 2004). Although evolutionary suicide was predicted
from extrapolation, rather than observed in nature, this
constitutes the first study making such a prediction from
empirical data.

In cod, Gadus morhua, the commercial fishing of large
individuals has resulted in selection towards earlier
maturation and smaller body sizes (Conover and Munch
2002 [STUDY #6]). Under exploitation, high mortality decreases the
benefits of delayed maturation. As a result of this,
smaller adults, which mature faster, have a higher fitness
relative to their larger, slow maturing counterparts
(Olsen et al. 2004). Despite being more successful
relative to slow maturing individuals, the fast-maturing
adults produce fewer offspring, on average. This adaptation,
driven by the selective pressure imposed by
harvesting, seems to have pre-empted a fishery collapse
off the Atlantic coast of Canada (Olsen et al. 2004). As
the cod evolved to be fast-maturing, population size was
gradually reduced until it became inviable and vulnerable
to stochastic processes.

The only strictly experimental evidence for evolutionary
suicide comes from microbiology. In the social
bacterium Myxococcus xanthus individuals can develop
cooperatively into complex fruiting structures (Fiegna
and Velicer 2003 - [STUDY #7]). Individuals in the fruiting body are
then released as spores to form new colonies. Artificially
selected cheater strains produce a higher number of
spores than wild types. These cheaters were found to
invade wild-type strains, eventually causing extinction of
the entire population (Fiegna and Velicer 2003). The
cheaters invade the wild-type population because they
have a higher relative fitness, but as they spread through
the population, they decrease the overall density, thus
driving themselves and the population in which they
reside, to extinction.

---

Another experimental study was "Sexual selection affects local extinction and turnover
in bird communities" - Paul F. Doherty, Jr., Gabriele Sorci, et al; 5858–5862 PNAS May 13, 2003 vol. 100 no. 10 [STUDY #8]

Populations under strong sexual selection experience
a number of costs ranging from increased predation and
parasitism to enhanced sensitivity to environmental and demographic
stochasticity. These findings have led to the prediction that
local extinction rates should be higher for speciespopulations
with intense sexual selection. We tested this prediction by analyzing
the dynamics of natural bird communities at a continental
scale over a period of 21 years (1975–1996), using relevant statistical
tools. In agreement with the theoretical prediction, we found
that sexual selection increased risks of local extinction (dichromatic
birds had on average a 23% higher local extinction rate than
monochromatic species). However, despite higher local extinction
probabilities, the number of dichromatic species did not decrease
over the period considered in this study. This pattern was caused
by higher local turnover rates of dichromatic species, resulting in
relatively stable communities for both groups of species. Our
results suggest that these communities function as metacommunities,
with frequent local extinctions followed by colonization.

This result is similar to another bird-centered study: Sexual Selection and the Risk of Extinction of Introduced Birds on Oceanic Islands": Denson K. McLain, Michael P. Moulton and Todd P. Redfearn. OICOS Vol. 74, No. 1 (Oct., 1995), pp. 27-34 [STUDY #9]

We test the hypothesis that response to sexual selection increases the risk of extinction by examining the fate of plumage-monomorphic versus plumage-dimorphic bird species introduced to the tropical islands of Oahu and Tahiti. We assume that plumage dimorphism is a response to sexual selection and we assume that the males of plumage-dimorphic species experience stronger sexual selection pressures than males of monomorphic species. On Oahu, the extinction rate for dimorphic species, 59%, is significantly greater than for monomorphic species, 23%. On Tahiti, only 7% of the introduced dimorphic species have persisted compared to 22% for the introduced monomorphic species.

...

Plumage is significantly associated with increased risk of extinction for passerids but insignificantly associated for fringillids. Thus, the hypothesis that response to sexual selection increases the risk of extinction is supported for passerids and for the data set as a whole. The probability of extinction was correlated with the number of species already introduced. Thus, species that have responded to sexual selection may be poorer interspecific competitors when their communities contain many other species.

biochemistry - NADH vs. NADPH: Where is each one used and why that instead of the other?

The phosphate group in NADPH doesn't affect the redox abilities of the molecule, it is too far away from the part of the molecule involved in the electron transfer. What the phosphate group does is to allow enzymes to discriminate between NADH and NADPH, which allows the cell to regulate both independently.

The ratio of NAD⁺ to NADH inside the cell is high, while the ratio of NADP⁺ to NADPH is kept low. The role of NADPH is mostly anabolic reactions, where NADPH is needed as a reducing agent, the role of NADH is mostly in catabolic reactions, where NAD⁺ is needed as a oxidizing agent.

You'll find some more information about this in chapter 2 of "Molecular Biology of the Cell by Alberts et al.

pathology - Enlargement of joints in osteoarthrosis

Osteoarthritis results from the gradual degradation of articular cartilage. When the cartilage is damaged extensively, e.g., all the way through the cartilage and into the bone, an inflammatory response ensues in the bone. This often happens when bone is rubbing against bone.

One of bone's responses to injury and inflammation is the production of new bone. In this case, the bone, called osteophytes or bone spurs, is produced pathologically (i.e., not like the bone laid down normally during growth). Heberden's nodes are the names given to these osteophytes at the distal interphalangeal joints, but the process is basically the same at other joints in the digits, ankle, etc.

evolution - How many times did endosymbiosis occur?

Well, it seems quite obvious that it was not a single I-eat-you-but-you-survived act but rather a convergence of endosymbiotic and host species into a greater and greater cooperation.
Of course this leaves a question if there was one or more species of endosymbionts involved.

Mitochondria are a very primeval story forced by the oxygen catastrophe, so it is hard to say, although great majority of mitochondria seems to have a single origin.

Plastids are much more divergent, however it seems that they did originated from a single source, diverged into chloroplasts, cyanelles and rhodoplasts and were later mixed up by numerous acts of secondary and even tertiary endosymbiosis (plus a further evolution); this variety can be especially seen within Euglenas, and they are the main group investigated in this manner.

algebraic groups - Is every monomorphism of commutative Hopf algebras (over a field) injective?

It seems that Ben is nevertheless right that the answer to the first question is "NO". Let $G=SL(2,Bbb C)$, and $B$ be the subgroup of lower triangular matrices. Then the inclusion $Bto G$ is an epi, since every algebraic representation of $B$ that extends to $G$ does so uniquely (on the nose, not just up to an isomorphism!). This follows from the fact that in any finite dimensional representation $V$ of the Lie algebra $sl(2)$, the operator $e$ is determined by $f$ and $h$. Indeed, the kernel $K$ of $e$ is spanned by vectors $v$ satisfying $hv=mv$ and $f^{m+1}v=0$ for some integer $mge 0$, and since $V=Bbb C[f]K$, the operator $e$ on $V$ is uniquely determined.

So one might guess that a morphism of complex affine algebraic groups $phi: Hto G$ is an epi if and only if $G/phi(H)$ is connected and proper (but I did not check this).

ct.category theory - Prestacks and fibered categories

I don't have a reference right now, but I hope this answer is useful. If nothing else, perhaps you could comment on why this doesn't answer your question.

A pseudofunctor is exactly the same thing as a fibered category with a choice of cleavage (a cleavage is a choice of cartesian arrow over every morphism in the base category with given target in the fiber). That is, there is an isomorphism between the (2-)category of pseudofunctors and the (2-)category of fibered categories with cleavage (where the morphisms don't have to respect the cleavage).

By the axiom of choice, every fibered category has a cleavage, and any two choices of cleavage are canonically isomorphic (via the identity functor; remember that the functor need not respect the cleavage). So the category of fibered categories with cleavage is equivalent to the category of fibered categories, and this is an equivalence in the usual 1-categorical sense. That is, you have two functors (the forget-cleavage and choose-cleavage functors) whose compositions are naturally isomorphic to the the identity. I don't think you need to use any kind of 3-morphism even though you're dealing with 2-categories.

ct.category theory - Can epi/mono for natural transformations be checked pointwise?

Theo, the answer is basically "yes". It's a qualified "yes", but only very lightly qualified.

Precisely: if a natural transformation between functors $mathcal{C} to mathcal{D}$ is pointwise epi then it's epi. The converse doesn't always hold, but it does if $mathcal{D}$ has pushouts. Dually, pointwise mono implies mono, and conversely if $mathcal{D}$ has pullbacks.

The context for this --- and an answer to your more general question --- is the slogan

(Co)limits are computed pointwise.

You have, let's say, two functors $F, G: mathcal{C} to mathcal{D}$, and you want to compute their product in the functor category $mathcal{D}^mathcal{C}$. Assuming that $mathcal{D}$ has products, the product of $F$ and $G$ is computed in the simplest possible way, the 'pointwise' way: the value of the product $F times G$ at an object $A in mathcal{C}$ is simply the product $F(A) times G(A)$ in $mathcal{D}$. The same goes for any other shape of limit or colimit.

For a statement of this, see for instance 5.1.5--5.1.8 of these notes. (It's probably in Categories for the Working Mathematician too.) See also sheet 9, question 1 at the page linked to. For the connection between monos and pullbacks (or epis and pushouts), see 4.1.31.

You do have to impose this condition that $mathcal{D}$ has all (co)limits of the appropriate shape (pushouts in the case of your original question). Kelly came up with some example of an epi in $mathcal{D}^mathcal{C}$ that isn't pointwise epi; necessarily, his $mathcal{D}$ doesn't have all pushouts.

dg.differential geometry - Help me understand boundary terms in actions over nontrivial manifolds

The second expression should be correct. The Stokes theorem per se does not "know" about covariant derivatives. However, the differential forms have certain transformation properties under the changes of local coordinates. To get the boundary term, you need an exact $n$-form under the integral sign, and $left(partial_mu A^muright) sqrt{|g(x)|}$ just does not transform the right way (assuming that $A_mu$ are components of a vector field), so in the first case the expression under the integral sign can't be an exact $n$-form while in the second case it is if $nabla$ is the Levi-Civita connection for $g$, and that's that.

Namely, in the second case the integral can (up to an inessential constant factor) be rewritten as $$int_M partial_mu left(sqrt{|g(x)|} A^muright) mathrm{d} x,$$ and you can use the Stokes theorem.

Important warning: If $nabla$ is a completely generic connection rather than the Levi-Civita connection, our $n$-form is not exact, and the argument fails because the Stokes theorem does not apply anymore.

Now, if $nabla$ is compatible with $g$ and ${}^gnabla$ is the Levi-Civita connection for $g$, then $nabla g$=0 and it can be shown that there exists a (1,2)-tensor field $T$ such that $$nabla={}^gnabla+T.$$.

In particular, we have $$nabla_mu A^mu={}^gnabla_mu A^mu+T_{rhonu}^nu A^rho.$$

While ${}^gnabla_mu A^mu$ is proportional to $partial_mu left(sqrt{|g(x)|} A^muright)$, and we can apply the Stokes theorem to see that the contribution of this term to the integral vanishes, we get
$$int_M nabla_mu A^mu sqrt{|g(x)|} mathrm{d} x=int_M T_{rhonu}^nu A^rho sqrt{|g(x)|} mathrm{d} x.$$

However, the algebraic conditions on $T$ that follow from compatibility of $nabla$ with the metric $g$ appear to yield $T_{rhonu}^nu=0$ (have no time to write this out in detail, sorry), so the above integral vanishes and the above arguments work even if $nabla$ is just compatible with $g$. Apologies for not pointing this out in the earlier version of my answer.

evolution - Is it possible to make bacteria vulnerable to antibiotics it's resistant to?

Perhaps it is useful to first consider how resistance is gained:

If you treat a population of bacteria with an antibiotic, some may die and some may live.

If none die, they are obviously resistant.

If all of them die, then all of them were sensitive, meaning that the size of the population and the variation were not large enough in order for some of the bacteria to be resistant. One approach would be to try to get larger genetic variation by inducing mutations with some mutagen. In addition, you can take a larger population. This would in effect let you sample a larger part of the genetic space to see if you can find a resistant strain. Of course it is possible that this will not work.

If some die and some are resistant, the situation is slightly more complicated. One possibility is that those which lived are genetically resistant (their DNA is different). To see if this is the case, you take those which lived and treat them again with the antibiotic. If they all live, they are resistant. Another possibility is that they are scholastically resistant. This phenomenon is known as bacterial persistence. What happens is that there is a genetic switch that randomly bestows resistance in some fraction of the population, let's say 10 percent. If you now take this population and reapply the antibiotic, only 10% of the population will survive. This interesting mechanism allows the bacteria adapt more quickly, since they do not have to make modifications to their DNA.

I am guessing that if the bacteria are genetically resistant, they will stay resistant. If the bacteria are scholastically resistant, it might be possible to affect the genetic switch which causes the persistence. This, in addition to reducing the size of the population, might cause a situation where none of the bacteria in the population are resistant long enough to survive.

ag.algebraic geometry - Semiring of algebraic vector bundles on projective space

Let $K$ be a field and $n geq 1$. Then the set of isomorphism classes of vector bundles over $mathbb{P}^n_K$ is a semiring (i.e. almost a ring, but no additive inverses are possible). By introducing additive inverses and quotienting out exact sequences, we get the $K$-theory of $mathbb{P}^n_K$, which is known to be $mathbb{Z}^{n+1}$. But is it also possible to compute exactly the semiring?

For $n=1$, there is a result by Dedekind-Weber (1892) which proves that the semiring is $mathbb{N}[x,x^{-1}]$, where $x=mathcal{O}(1)$ (related topic). Some months ago, I was told that the structure is far more complicated for $n>1$. Can anybody elaborate this or even give a presentation of the semiring?

If necessary, you may assume $K = mathbb{C}$.

gr.group theory - dual of Z^I for uncountable I

With regard to Mariano's answer, I believe some clarification is in order. A closely related question was asked by Michael Barr and answered by user Ralph here. In brief, the homomorphism named in Martin's question is in fact an isomorphism, provided that $I$ has cardinality less than the first measurable cardinal.

Shelah and Strüngmann (accessible here) refer to this result as well, using the same source given by Ralph, just before Definition 2.1:

For generalizations to products of larger cardinalities and the resulting definition of slenderness for abelian groups we refer to [EM] or [F1]

where [EM] is the text by Eklof and Mekler. It seems that Shelah and Strüngmann are talking about something slightly different: homomorphisms out of free complete products (but using a notation which could unfortunately suggest direct products).

ra.rings and algebras - Good lattice theory books?

I agree with Gerhard. Imho, "Algebras, Lattices, Varieties I" is the best book on universal algebra and lattice theory (perhaps the best math book ever ;) Ironically, it's out of print. However, Burris and Sankapanavar is also great and is free.

As far as sharing examples of the utility of lattice theory, personally, I don't know how I got through my comps in groups, rings, and fields before I learned about lattice theory. Now the only way I can remember many of the theorems is to picture the subgroup (subring, subfield) lattice!

Professor Lampe's Notes on Galois Theory and G-sets are great examples of how these subjects can be viewed abstractly from a universal algebra/lattice theory perspective. The Galois theory notes in particular distil the theory to its basic core, making it very elegant and easy to remember, and highlighting the fact that the underlying algebras need not be fields.

There is still the question of what results are truly universal algebra results, rather than old results couched in universal algebra language? That is an interesting question, and maybe should be the subject of a different mathoverflow post...

Updates: See also this post.

homework - Why do we squint when tasting very sour things?

Short answer
Adverse stimuli in general are accompanied by contraction of facial muscles. It is part of our repertoire of emotional expressions.

Background
A disgusted expression and the expression of pain are both accompanied by squinting. Disgusting stimuli include your example of a sour taste, but also include other adverse tastes (bitter) or visual stimuli such as a gross-looking objects. In other words, squinting is likely a common way of expressing adverse stimuli. So the "sour look" may not be easily explained on its own. Instead, it is part of a huge emotional expressive repertoire that makes us human beings. To be able to read these expressions allows us to be empathetic towards others. People with autism and Asperger's syndrome may not be able to do so and are disadvantaged in their social behaviors.

To illustrate the commonalities between sour (#1), disgust (#2) and pain (#3) I would like to share the following web finds.

^{Sour taste. Source: Getty Images}

!
^{Disgust. Source: Berkeley Emotional IQ test}

^{Pain. Source: Berkeley Emotional IQ test}

PS: I am not a psychologist - Consider my contribution as an educated guess.

botany - Is Schoons Hard Shell muskmelon a hybrid?

Goldman surmises that Schoon's Hard Shell is a variant of the Bender, so would have been bred from the Surprise, Irondequoit, and Tip Top, all of which were from the Sill's Hybrid.

In that sense, I suppose it's a hybrid. I'm guessing that you're asking because you want to know if you can save the seed; that is, if Schoon's Hard Shell will breed true or will lose its characters in subsequent generations.

I can't find any specific detail on that, but given that it's sold by Seed Savers Exchange, I'd guess that it is a good candidate for seed saving, as long as you make sure it's not crossing with related species in your garden. Seed to Seed has detailed info on melon seed saving.

---

lo.logic - Which graphs are elementarily equivalent to their own disjoint sums?

In Stefan Geschke's recent
question,
one of the solutions observed that the graph consisting of
a single infinite beaded chain, a $mathbb{Z}$-chain where
each integer is connected to its nearest neighbors, is
elementarily equivalent to the disjoint sum of any number
of such chains. That is, a single chain has all the same
first order properties in the language of graph theory as
two chains, or as any number of such chains.

(The reason was that all these graphs are cycle-free and
have every element with degree $2$, but the theory
asserting this is already complete. This can be seen by
observing that every model of this theory having
uncountable size $kappa$ consists of $kappa$ many
$mathbb{Z}$-chains, and all such models are
isomorphic---in other words, the theory is
$kappa$-categorical---and so the theory is complete, since
otherwise it would have non-isomorphic models of size
$kappa$.)

My question here is about the extent to which this
phenomenon generalizes to other graphs.

Question. Which graphs $G$ are elementarily
equivalent to $Gsqcup G$? And how about $delta$ many
copies of $G$ with itself $bigsqcup_delta G$?

Let's introduce some terminology and say that a graph $G$
is 2-self-similar if $G$ is elementarily equivalent to
$Gsqcup G$, and more generally $G$ is
$delta$-self-similar if $G$ is elementarily equivalent
to $delta$ many copies of $G$.

Further questions: If $G$ is 2-self-similar, does this
imply that it is $delta$-self-similar for every $delta$?
For which $delta,gammageq 2$ does
$delta$-self-similarity imply $gamma$-self-similarity? If
$G$ is 2-self-similar, does this imply that each copy of
$G$ is an elementary substructure of $Gsqcup G$? And
similarly for $bigsqcup_delta G$?

On the one hand, the argument about $mathbb{Z}$-chains
easily generalizes to many other graphs, such as the
connected graph tree $T$ in which every vertex has degree
$3$. That is, the theory of cycle-free graphs with every
vertex of degree $3$ is $kappa$-categorical for
uncountable cardinals $kappa$ and hence complete, and so
$T$ is elementarily equivalent to any number of disjoint
copies of $T$. And we can clearly use trees of any finite
uniform degree in this argument. Also, there are
non-uniform graphs with self-similarity, such as the graph
tree where vertices alternate degree 2, degree 3, etc., and
any other definable pattern. And cycle-freeness is not
required, since one could add loops of any length to every
vertex in a $mathbb{Z}$-chain, for example, and the
original argument would still work fine.

In addition, trivial instances of self similarity arise
when $G$ is outright isomorphic to $Gsqcup G$, such as
with the infinite edgeless graph, or when $G$ is any
infinite sum of a fixed graph (and this is equivalent to
$Gcong Gsqcup G$). But the example of $mathbb{Z}$-chains
shows that this isomorphism version of self similarity is
not a necessary property for 2-self-similarity, since one
$mathbb{Z}$-chain is obviously not isomorphic to two, even
though they are elementarity equivalent.

Meanwhile, there are some easily observed obstacles to
$delta$-self-similarity:

• If $G$ has definable elements, then 2-self-similarity will fail, since every point has
  automorphic images in $Gsqcup G$.




• Similarly, if $G$ has nonempty finite definable subsets,
  then it will not be $n$-self-similar for large enough $n$,
  since again there will be too many automorphic images.
  (Perhaps this argument can be improved to show $G$ is
  not 2-self-similar; for example, this is easy to see when the copies
  of $G$ are elementary substructures of $Gsqcup G$.)




• If $G$ has finite diameter, then again self-similarity will fail, since
  multiple copies of $G$ will not be connected and hence not
  have that diameter. (Thus, for example, the countable random graph
  is not 2-self-similar.)

Finally, it seems that many similar questions can be asked
about other mathematical structures.

• Which partial orders $P$ are elementarily equivalent to
  $Poplus P$? Or to $oplus_delta P$?


• Which groups $G$ are elementarily equivalent to $Goplus
  G$? Or to $oplus_delta G$?


• Same for rings or whatever structure for which direct
  sum makes sense.

I am wondering whether there might be a general
model-theoretic characterization of self similarity.

ac.commutative algebra - Lifting results from smooth maps to essentially smooth maps.

Recall that a morphism of rings $Rto S$ is called (essentially) smooth if it is formally smooth and (essentially) finitely presented.

(Note: $Rto S$ is essentially finitely presented provided that $S$ is the localization of some finitely
presented $R$-algebra $T$ at some multiplicative system $A subset T$, that is, $S=A^{-1}T$.)

In class, our professor said that working with smooth or essentially smooth morphisms yields an effectively equivalent theory. This motivates my question: Is there a general technique to lift results from the smooth case to the essentially smooth case?

Edit: According to Mel, every essentially smooth morphism is a localization of a smooth morphism. However, this direction is much more involved than the other direction, which is immediate from the definitions. Anyway, this would be the answer to the question.

Breathing water vapour - Biology

Normal air consist of oxygen, CO₂ and nitrogen, with traces of water etc.

Now imagine displacing all gasses except oxygen with water vapour. By water vapour I don't mean hot steam or fog. Just gaseous water.

The question is would it be beneficial or harmful to human, and how much so?
And assuming it's not deadly, in which circumstances could it be useful?

One thing I am wondering in particular is deep dive and high altitude applications. Having no/low content of nitrogen in such an artificial air mixture would help to avoid decompression sickness. And one would think water vapour is at least cheaper than pure oxygen (also might be safer and have less side effects).

rt.representation theory - Proof of Steinberg's tensor product theorem

The 1980 CPS paper is short but not easy to read without enough background.
They gave the first conceptual alternative to Steinberg's somewhat opaque
and computational proof of the tensor product theorem in 1963 (which built
on the 1950s work of Curtis on "restricted" Lie algebra representations
coming from the algebraic group plus the older work of Steinberg's teacher
Richard Brauer on rank 1). Steinberg relied quite a bit on working with
covering groups and projective representations.

CPS already realized the importance of
getting beyond the Lie algebra by using Frobenius kernels. The best modern
source is the large but well-organized 1987 book by J.C. Jantzen,
Representations of Algebraic Groups (expanded AMS edition in 2003). Here
the foundations are worked out thoroughly and the CPS proof is given an
efficient treatment in part II, 3.16-3.17. While CPS had in mind the
analogy with Clifford theory for finite groups, Jantzen gives a self-contained
treatment avoiding use of projective representations or Skolem-Noether.

Apart from sources, the essential goal is to single out the finitely many
"restricted" simple modules for the Lie algebra among the infinitely many
simple (rational) modules for the ambient algebraic group, then realize the
latter modules as twisted tensor products of the former. This requires a notion of
Frobenius morphism for each power of the prime, Frobenius kernels being
infinitesimal group schemes. The Lie algebra just plays the role of
first Frobenius kernel (a normal subgroup scheme), so the Clifford theory
analogue developed by Ballard and CPS makes sense here.

pr.probability - Reference request for a "well-known identity" in a paper of Shepp and Lloyd

So one of the approaches to proving the equality in the question is via the following three steps:
First differentiate both sides of the equation to see that they agree up to a constant. This reduces to showing the case of $x = 1$, for which $log x = 0$.

Next we apply integration by parts to get
$$
int_1^{infty} exp(-y)/y dy - int_0^1 frac{1-exp(-y)}{y}dy = int_0^{infty} exp(-y) log y dy
$$

Finally observe that $Gamma'(1)$ equals the RHS, by differentiating under the integral sign, valid because things are decaying fast enough at infinity.

So it remains to show $Gamma'(1) =gamma$. I saw a soft argument (i.e., without using infinite product) in the link scipp.ucsc.edu/~haber/ph116A/psifun_10.pdf
This is re-exposed below:

first we establish that for $Psi(x) = log Gamma(x)$,
$$
Psi'(x+1) = Psi'(x) + 1/x
$$
This is easy enough since we have we have the functional equation $Gamma(x+1) = xGamma(x)$.
Next using stirling approximation we get

$$
Psi(x+1) = (x+1/2)log x -x + 1/2 log 2 pi + O(1/x)
$$
and then they differentiate this and claim that $O(1/x)' = O(1/x^2)$, which is clearly false (take $f(x) = 1/x cos(e^x)$). But I found in Wikipedia another formula that gives the precise error term in terms of an integral of the monotone function $arctan(1/x)$. So this is enough to establish $O(1/x^2)$ for the error term in the derivative of $Psi$. So we get the asymptotics $lim_{x to infty} Psi'(x+1) = log(x)$, from which we get $Psi'(1) = gamma$. Now notice $Psi'(x) = Gamma'(x)/ Gamma(x)$, and $Gamma(1) = 1$, so $Gamma'(1) = gamma$ also.

ra.rings and algebras - Quartic form which is irreducible but not geometrically irreducible

A bit more is true when $n=4$. Suppose $F$ is any perfect field, not necessarily finite. Then, for $n=4$, the form $q$ is, up to a scalar multiple, the norm form of a quartic extension of $F$. (Proof: Suppose that $V$ is the projective $F$-scheme defined by $X_1X_2X_3X_4=0$ and that $overline F$ is an alg. closure of $F$. Then the projective automorphism group of $V$ is a split extension of $T=mathbb G_m^4$ by the symmetric group $S_4$. Since $H^1(F,T)=0$, the set of isomorphism classes that we seek is $H^1(F,S_4) =Hom(Gal_F,S_4)$, as stated.) For finite $F$ there is only one such extension, of course.

For $n=3$ the configuration of $4$ lines might have triple, or quadruple, points, but if not then again there is only one isomorphism class over $overline F$. Its automorphism group is $S_4$, and the same argument shows that $q$ is unique up to scalars (you can describe it as a linear section of the quartic norm form.) For $n=2$ I have nothing to add.

ag.algebraic geometry - How is this action of monoidal derived category induced?

I am reading a paper concerning the action of monoidal category to another category.
Let $k$ be a commutative ring, $R$ is a k-algebra. $A=R-mod$, $B=R^{e}-mod=Rbigotimes _{k}R^{o}-mod$.

Consider the action:

$Btimes Arightarrow A,(M,N)mapsto Mbigotimes _{R}N$ is an action of monoidal category of $R^{e}-mod=B^{~}=(B,bigotimes _{R},R)$ on A.

The paper said this action induces the action

$Phi : D^{-}(B)times D^{-}(A)to D^{-}(A)$ of the monoidal derived category $D^{-}(B)$ on $D^{-}(A)$

I know this action should be $(M,N)mapsto Mbigotimes_{R}^{L}N$.

But I do not know how is this action of monoidal derived category on the other derived category induced by the action of monoidal abelian category. Is there a canonical way(A natural transformation)to get this action?

Notice that the action of monoidal abelian category is defined as follows

$Psi:=(Phi ,phi ,phi _{0})$

$Phi :B=(B,bigotimes _{R},R)rightarrow End(A)$

$Phi (V)cdot Phi (W)overset{phi }{rightarrow}Phi (Vbigotimes _{R}W)$

The back ground of this question is localization of differential operator in derived category, so I added the tag"algebraic geometry"

This paper is "Differential Calculus in Noncommutative algebraic geometry I" which is available in MPIM

gr.group theory - Orders of field automorphisms of algebraic complex numbers

This amounts to the Artin-Schreier theorem, which has come up several times already on MO (c.f. Examples of algebraic closures of finite index):

if $K/F$ is a field extension with $K$ algebraically closed and $[K:F] < infty$, then
$[K:F] = 1$ or $2$, and in the latter case, $F$ is real-closed.

Thus the answer here is that $n$ can be $1$, $2$ or $infty$, and all possibilities occur: the field of real algebraic numbers gives an index $2$ subfield of $overline{mathbb{Q}}$.

(Also, just to be sure, there are elements of infinite order! E.g., if not then every element would have order $1$ or $2$, so the absolute Galois group would be abelian, and thus every finite Galois group over $mathbb{Q}$ would be abelian, and this is certainly not the case.)

Why are relations of degree 3 or less enough in a presentation of the polynomial current Lie algebra g[t]?

Let $mathfrak{g}$ be a finite dimensional simple Lie algebra over $mathbb{C}$.
The polynomial current Lie algebra $mathfrak{g}[t] = mathfrak{g} otimes mathbb{C} [t]$
has the bracket
$$[xt^r, yt^s] = [x,y] t^{r+s}$$
for $x,y in mathfrak{g}$. It is graded with deg$(t) = 1$.

If we set $h=0$ in Drinfeld's first presentation of the Yangian (given in Theorem 12.1.1 of Chari and Pressley's Guide to Quantum Groups) then we get a presentation of $U(mathfrak{g}[t])$
where the generators are the elements $x in mathfrak{g}$ and $J(x) = xt$ of $mathfrak{g}[t]$ with degree $=0,1$, and the relations all have degree of both sides less than $3$.

---

Specifically we require that all the relation in $mathfrak{g}$ are satisfied for the elements with degree 0, and
(for all $x,y, x_i, y_i, z_i in mathfrak{g}$ and complex numbers $lambda, mu$):

$$lambda xt + mu yt = (lambda x + mu y)t$$
$$[x, yt] = [x,y]t,$$
$$sum_i [x_i, y_i] = 0 implies sum_i [x_i t, y_i t ] = 0$$
$$ sum_i [[x_i, y_i], z_i] = 0 implies sum_i [[x_i t, y_i t], z_i t]=0$$
Then assuming that all the relations of degree less than or equal to $3$ hold is enough to get the remaining ones.
The elements $xt^2, xt^3, ldots$ are defined inductively.
This can be proved by induction, using the Serre presentation of the finite-dimensional Lie algebra and then checking all the required relations in several cases.
But even in the $mathfrak{sl}_2$ case the argument is laborious.

Is there a better way of seeing that one needs only relations of degree less than three in order to get the rest?

mp.mathematical physics - Noether's Theorem in Quantum Mechanics

In hindsight, Noether's theorem is a dramatic hint of quantum mechanics. Mariano is completely correct in his comment that the conserved quantity is $A$ itself, but it deserves a bit of explanation.

A classical probabilistic system is characterized by an algebra of random variables. You could consider the Boolean random variables, in which case the algebra is a $sigma$-algebra $Omega$. Or you could consider real or complex random variables; if you take the bounded ones then the algebra is $L^infty(Omega)$. In quantum probability, you have the same sort of thing, except that the algebra of bounded complex random variables is a non-commutative von Neumann algebra. One choice with special properties is the algebra $mathcal{B}(mathcal{H})$ of all bound operators on a Hilbert space $mathcal{H}$.

The special property of $mathcal{B}(mathcal{H})$ is that all automorphisms are inner, so that any symmetry $A$ of a quantum dynamical system is necessarily also a random variable that you can measure. This does not happen classically, nor even for other non-commutative von Neumann algebras. Even without writing down a time-independent Schrodinger equation, it makes Noether's theorem trivial, because the symmetry $A$ must be conserved if you interpret it as a quantum random variable. Unlike in the classical case, $A$ doesn't even need to generate or come from a continuous group action.

For example, the parity operator (which negates all three coordinates of space) is a conserved quantity of electromagnetism, so it leads to a (two-valued) conserved quantity in quantum electrodynamics which is also called parity. The discrete symmetry also exists classically as a symmetry of Maxwell's equations (if you are careful to negate the magnetic field vectors twice), but the classical Noether's theorem doesn't apply.

Anyway, the identity operator is the trivial random variable that is always 1, as Aaron says.

ag.algebraic geometry - Is line bundle determined by the parameter space and fiber?

Yes, you can. This follows from the see-saw principle for instance. You can also argue directly as follows. The spaces of global sections $H^{0}(Y,f_*L)$ and $H^{0}(X,L)$ are naturally isomorphic. Since $f_*L$ is a trivial line bundle we can choose a global nowhere vanishing section $e$ of $f_*L$. Let $s in H^{0}(X,L)$ be the section of $L$ corresponding to $e$ under the above isomorphism. To show that $L$ is trivial it suffices to check that $s$ does not vanish anywhere. But if $x in X$ is a closed point where $s$ vanishes, then if we restrict $s$ to the fiber $X_{f(x)}$, we will get a section of the structure sheaf of integral projective variety which vanishes at a point. Thus the restriction of $s$ to $X_{f(x)}$ must be identically zero. This shows that $e$ vanishes at $f(x)$ which is a contradiction.

ra.rings and algebras - What is an algebraic group over a noncommutative ring?

Let $R$ be a (noncommutative) ring. (For me, the words "ring" and "algebra" are isomorphic, and all rings are associative with unit, and usually noncommutative.) Then I think I know what "linear algebra in characteristic $R$" should be: it should be the study of the category $Rtext{-bimod}$ of $(R,R)$-bimodules. For example, an $R$-algebra on the one hand is a ring $S$ with a ring map $R to S$. But this is the same as a ring object in the $Rtext{-bimod}$. When $R$ is a field, we recover the usual linear algebra over $R$; in particular, when $R = mathbb Z/p$, we recover linear algebra in characteristic $p$.

Suppose that $G$ is an algebraic group (or perhaps I mean "group scheme", and maybe I should say "over $mathbb Z$"); then my understanding is that for any commutative ring $R$ we have a notion of $G(R)$, which is the group $G$ with coefficients in $R$. (Probably there are some subtleties and modifications to what I just said.)

My question: What is the right notion of an algebraic group "in characteristic $R$"?

It's certainly a bit funny. For example, it's reasonable to want $GL(1,R)$ to consist of all invertible elements in $R$. On the other hand, in $Rtext{-bimod}$, the group $text{Aut}(R,R)$ consists of invertible elements in the center $Z(R)$.

Incidentally, I'm much more interested in how the definitions must be modified to accommodate noncommutativity than in how they must be modified to accommodate non-invertibility. So I'm happy to set $R = mathbb H$, the skew field of quaternions. Or $R = mathbb K[[x,y]]$, where $mathbb K$ is a field and $x,y$ are noncommuting formal variables.

ag.algebraic geometry - The algebro-geometric counterpart of the Dijkgraaf-Witten model

This has been done, in a variety of related ways. A lot of the difficulty is in defining an appropriate notion of a "stable" map to [pt/G].

The earliest mathematical work I know of is Chen & Ruan's "orbifold cohomology", which is done in the symplectic category. (Caveats: Abramovich's lecture notes on orbifold GW theory quote a 1996 letter from Kontsevich, who outlines a lot of the basic ideas in 2 pages. Also, string theorists were looking at non-topological sigma models to orbifolds at least as far back as Dixon, Harvey, Vafa, & Witten's 1985 papers.)

In algebraic geometry, this stuff has been studied by Jarvis, Kaufmann, & Kimura, who focused on G-bundles, and by Abramovich, Graber, & Vistoli, who figured out how to deal with D-M stacks.

(You can also carry out these constructions in K-theory for finite-dimensional Lie groups. See, for example, Frenkel, Teleman, & [cough].)

algebraic number theory - Are there any Hecke operators acting on an elliptic curve with additive reduction that I don't know about?

I could have made this question very brief but instead I've maximally gone the other way and explained a huge amount of background. I don't know whether I put off readers or attract them this way. The question is waay down there.

Let $f$ be a cuspidal modular eigenform of level $Gamma_0(N)subseteq SL_2(mathbf{Z})$ (for example $f$ could be the weight 2 modular form attached to an elliptic curve) and let $p$ be a prime. In the theory of modular forms, one Hecke operator at $p$ is singled out, namely $T_p$, sometimes called $U_p$ if $p$ divides $N$, and defined by the double coset attached to the matrix $left(begin{array}{cc}p& 0\ 0&1end{array}right)$. Now $f$ is an eigenform for $T_p$, and $f$ has an eigenvalue for this operator---a Galois-theoretic interpretation of this eigenvalue is that it is (modulo fixing embeddings of $overline{mathbf{Q}}$ in $overline{mathbf{Q}}_ell$ and $mathbf{C}$) the trace of the geometric Frobenius on the inertial invariants of the $ell$-adic representation attached to $f$, for $ellnot=p$ a prime.

Now here is a very naive question that I don't know the answer to, and I really should, and I'm sure it's very well-known to people who do this sort of stuff. Say $N=p^rM$ with $M$ prime to $p$. One can approach the theory of Hecke operators entirely locally. Let $K:=U_0(p^r)$ denote the subgroup of $GL_2(mathbf{Z}_p)$ consisting of matrices whose bottom left hand entry is $0$ mod $p^r$. Now there is an "abstract Hecke algebra" of locally left- and right-$K$-invariant complex-valued functions on $G:=GL_2(mathbf{Q}_p)$ with compact support. As a complex vector space this algebra has a basis consisting of the characteristic functions $KgK$ as $KgK$ runs through the double cosets of $K$ in $G$. But this space also has an algebra structure, given by convolution.

If $r=0$ then $K$ is maximal compact, and the structure of this Hecke algebra is well-known and easy. Via the Satake isomorphism, the abstract Hecke algebra is isomorphic to $mathbf{C}[T,S,S^{-1}]$, with $S$ and $T$ independent commuting polynomial variables. The interpretation is that $T$ is the usual Hecke operator $T_p$ attached to the matrix $left(begin{array}{cc}p& 0\ 0&1end{array}right)$ and $S$ is the matrix attached to $left(begin{array}{cc}p& 0\ 0&pend{array}right)$. One doesn't always see this latter Hecke operator explicitly in elementary developments of the theory because it acts in a very dull way---it acts by scalars on forms of a given weight and level $Gamma_0(N)$, typically (depending on normalisations) as the scalar $p^{k-2}$ on forms of weight $k$. In particular the "abstract Hecke algebra" doesn't give us any more information than that which classical texts explain, as it's generated by $T_p$, $S_p$ and $S_p^{-1}$.

The next case is $r=1$ and this case I also understand. The abstract Hecke algebra now is non-commutative, "because of oldforms": I don't think the operators attached to $left(begin{array}{cc}p& 0\ 0&1end{array}right)$ and $left(begin{array}{cc}0& p\ 1&0end{array}right)$ (that is, the operators corresponding to these double coset spaces) commute, but if $f$ has level $Mp$ and is old at $p$ then we should be working at level $M$, and if it's new at $p$ then we get two invariants---the $T_p$ (or $U_p$) eigenvalue, which is classical, and the $w$-eigenvalue, which is the local sign for the functional equation. Again both of these numbers are classical and a lot is known about them. I am pretty sure that the abstract Hecke algebra in this case is generated by these operators $T_p$, $w$, and the uninteresting $S_p$ and $S_p^{-1}$, the latter two still acting by scalars on forms of a given weight. Am I right in thinking that these operators generate the local Hecke algebra? I think so.

The next case is $r=2$ and this I am not 100 percent sure I understand. The classical theory gives us $T_p$, $S_p^{pm1}$ and $w$. Note that on a newform of level $Gamma_0(Mp^2)$, $T_p$ is zero in this situation, $S_p$ acts by a scalar, and $w$ is some subtle sign which people have clever ways of working out.

---

Finally then, the question! Let $K$ be the subgroup of matrices in $GL_2(mathbf{Z}_p)$ consisting of matrices for which the bottom left hand corner is $0$ mod $p^2$. Let $H$ denote the abstract double coset Hecke algebra of compactly supported $K$-bivariant functions on $GL_2(mathbf{Q}_p)$.

Is this abstract Hecke algebra generated (as a non-commutative algebra) by the characteristic functions of $KgK$ for $g$ in the set {$left(begin{array}{cc}p& 0\ 0&1end{array}right)$, $left(begin{array}{cc}0& p^2\ 1&0end{array}right)$, $left(begin{array}{cc}p& 0\ 0&pend{array}right)$, $left(begin{array}{cc}p^{-1}& 0\ 0&p^{-1}end{array}right)$}?

In the language I've been using in the waffle above: modular forms of level $p^2$ have an action of the Hecke operators $T_p$, $w$, and the invertible $S_p$. Are there any more, lesser known, Hecke operators that we're missing out on?

antibiotics - By what mechanism does penicillin resistance usually develop in Streptococcus pneumoniae?

Maybe you are asking where these genes came from and how they became so common?

It turns out that antibiotics are very common - many free living plants and microorganisms make antibiotics. Penicillin, the first beta lactam antibiotic found is synthesized by a fairly common mold, they kill bacteria by inhibiting the synthesis of the cell wall.

All three of these classes of defense have evolved over the hundreds of millions of years of evolution since antibiotics like penicillin started showing up (probably more like billions of years). They have all probably been with us for quite a long time.

Beta-lactamases, which enzymatically neutralized the beta lactam ring of penicillins and related compounds seem to have evolved three times and its not hard to imagine why the genes have stuck around for so long.

dd-transpeptidase, the 'penicillin binding protein' is actually an enzyme that helps to synthesize the peptidoglycan of the cell wall. It can also mutate such that it tends not to bind penicillin so well, creating a second sort of penicillin resistant bacterium.

Other genes such as ABC drug transporters can kick out strange molecules before they can cause any damage.

These and all other sorts of mutants and genes that convey resistance, once created through selection are around in the huge bacterial gene pool. Bacteria can pick them up through lateral gene transfer, plasmids (sex), phages and by taking in DNA from the environment. Most of these genes and traits were sitting around for millions of years, but the increased use of antibiotics has increased the selection pressure to retain these traits in many more bacteria. That's really where bacterial antibiotic resistance comes from.

ag.algebraic geometry - is the preorder of locally closed immersions complete?

The maximum may not exist in general. Take X=Spec k[T,U] the affine plane, A the complement of the vertical line L passing through the origin (A=Spec k[T,U,1/T]) and B the origin (Spec k[T,U]/(T,U)). Then, the maximum C of A and B in the ordered set of subschemes of X does not exist. If it was the case, then C should be some subscheme of X. Such are usually described as closed subschemes in some open subset of X, but can also be described as open subsets in some closed subschemes Z of X (I cannot find the reference in general, but it is certainly true when the schemes are noetherian). As U is (schematically)-dense in X, this Z can be nothing else as X itself, then C should be an open subset of X containing A and B, that is the open complement of a finite set of closed points of the line L except the origin, in which case it is clear that we could find an open subset of X containing A and B and strictly contained in C, which would give a contradiction.

gr.group theory - Leech lattice decomposition

Hi again,

Now I think I could perform decomposition using Wilson definition of Leech lattice using octonions (2008). I this definition Leech lattice is easily seen as union of 819 E8 sublattices;
819 = 3*(1+16+16*16).
Having this we can decompose each E8 lattice into crosses. Last step (little vague) would be to find decomposition of 819 E8 lattices into 273 triples.

But now I am struggling with following problem. Consider 2A class in Co1 having 819*759*75 elements. Each element a from 2A have two representatives in Co0. Element a corresponds to E8 sublattice in Leech defined as {v: av=-v} where I call by a also proper preimage in Co0. Now the opposite having E8 sublattice L in Leech I want to find element a(E8) in 2A class. My straightforward function build c*d*c^-1 where d is diagonal matrix changing sign in octad [1..8]. But I have not obtained Co0 element.

My goal is to find relation between Order(ab) for a,b in 2A and corresponding geometry of two E8 sublattices. The Order(ab) can be 2,3,4,5,6.

Take any other sporadic group g and certain conjugacy class cg of order 2 elements. Is it known possible values of Order(ab) for a,b in cg ?

Regards,
Marek

co.combinatorics - Graphs preserved under the Hamiltonian path operator

Other examples include $K_{a,1,1,dots,1}$ with $a$ 1's. Together with $K_a$ and $K_{a,a}$ these are the only complete $k$-partite graphs that are fixed points of the "Hamiltonian path operator". But we can do better:

(Edited to include what I have so far)

Lemma 1 If $Gcong HP(G)$ then either $G$ is empty or it has a Hamiltonian cycle.

Proof goes by showing that if $G$ doesn't have a Hamiltonian cycle then it has more edges than $HP(G)$ (Induction)

Let $V(G)=v_1,v_2,dots,v_n$ with $v_i\leftrightarrow v_{i+1}$ (where indices are $pmod{n}$) In fact we can prove:

Lemma 2 Let $1 le kle n-1$ If $v_{i}leftrightarrow v_{i+k}$ for some $iequiv alphapmod{2}$, then this is true for all $iequiv alphapmod{2}$ in particular when $n$ is odd, it is true for all $i$.

The proof is based on the observation that if $v_{i}leftrightarrow v_j$ in $G$ then $v_{ipm 1}leftrightarrow v_{jpm 1}$ in $HP(G)$

Lemma 3 If there are $aneq bin mathbb{Z}/nmathbb{Z}$ so that $forall i, v_{i}leftrightarrow v_{i+a}$ and $v_ileftrightarrow v_{i+b}$ then $G$ is the complete graph.

Corollary You can prove that if $n=|V(G)|$ is odd and $Gcong HP(G)$, then $G$ is the $n$-cycle or the complete graph $K_n$.

When $n$ is even there are non-trivial examples such as the families already mentioned. At least we know that all Graphs that are fixed points of your operator are so that $v_{i}to v_{i+2}$ describes an automorphism of $G$.

ag.algebraic geometry - Why does finitely presented imply quasi-separated ?

One of the main interests in finitely presented morphisms comes from the various theorems in EGA IV,8. They show that for many questions about morphisms of schemes and sheaves on them, the condition of finite presentation allows one to reduce to a noetherian situation. For these theorems the assumption of quasi-separatedness is crucial.

Let me quickly try to explain why. The heart of the reduction to the noetherian case are theorems like the following: Let X over Spec A be a finitely presented scheme. Then there is a subring $A_0$ of $A$ which is a finitely generated $mathbb{Z}$-algebra (and in particular noetherian) and an $A_0$-scheme $X_0$ of finite presentation such that $X$ arises from $X_0$ via the base-change $A_0to A$. If $X$ is affine, this is pretty clear, as $X$ is definied by finitely many equations in an affine space over $A$. In order to pass from the affine case to the general case, it does NOT suffice to know that we can cover $X$ by finitely many affine pieces (which would be the assumption of quasi-compactness), but we also need that the glueing data for the affine pieces are somehow described by a finite number of equations. This is ensured by the assumption that the intersection of two affine pieces is quasi-compact which corresponds precisely to the assumption that $X$ is quasi-separated over A.

I guess that these theorems were the reason for Grothendieck to include this condition in the definition of finitely presented.

gr.group theory - Orbit maximin problems

When n=2q is twice an odd prime power q, I think G = AGL(1,q) wr Sym(2) (amongst others) has the largest minimum orbit length on 2-sets. If G acts on Z = X ∪ Y, each of size q, then G has two orbits on Z⁽²⁾, { {a,b} : a,b in X } ∪ { {a,b} : a,b in Y } of size 2⋅Binomial(q,2), and { {a,b} : a in X, b in Y } of size q². When q is even, those are still the orbits, but of course AGL(1,2q) has only a single orbit.

When n=3q is three times a prime power q coprime to 3, then again AGL(1,q) wr Sym(3) looks reasonable, with only two orbits, one of size 3⋅Binomial(q,2), one of size 3⋅q². At any rate, if it is not maximal it has a fairly large minimal orbit.

Obviously you can't keep taking Sym(k) for k=2,3,4,... forever, but I think actually it suffices to use the regular permutation representation Cyclic(k) of the cyclic group of order k. The largest orbit often splits into more than one orbit, but they are all so large it appears not to matter.

In other words, you might try AGL(1,q) wr Cyclic(k) for fairly general k and q, when n=k⋅q. I think the minimum orbit size will always be the silly disjoint union one of size k⋅Binomial(q,2), which is pretty large as long as you hold k constant. You can probably replace AGL(1,q) with your paper's Γ(q,d) without too much trouble.

I am under the impression that in asymptotic group theory, it is often quite hard to find exact sharp bounds, so just having "large" examples like these might be sufficient, at least for numbers with a very large prime power factor (so that k is kept small).

epigenetics - Can rats pass on memories of a maze to their offspring?

The phenomenon you're talking about was a fad in the 60's, called 'interanimal memory transfer'. It started out when James McConnell performed a later-discredited experiment in which he found that if you chopped up flatworms which had been exposed to some stresses, and fed them to other unexposed flatworms, the unexposed worms became wary of the source of stress quicker after eating their dead companions. He jumped to the conclusion that a 'memory molecule' was being transferred, and that the cannibal worms gained the food worms' memories of the stress.

People then started looking to see if they could:

1. repeat the experiments


2. find the same phenomenon in other animals

In the first case, nobody could replicate the experiments in worms, but because McConnell was such a PR genius he managed to convince the public that his results were valid (see Rilling, 1996 for more on this).

In the second case, Frank et al. (1970) and others tried working with rats - I think this is the experiment you're talking about in the question. They found various interesting results including that if you trained rats to run through a maze by using particularly stressful negative reinforcement (like electrocution), then those rats' children would be able to learn the new maze much faster. However, Frank et al. didn't make the same mistake as McConnell - first of all they wondered if the parent rats might be leaving a scent trail. So they used duplicate mazes with the exact same design, putting the children into clean mazes. The children of adults who had already learned the maze continued to outperform the control rats - the explanation was not scent trails.

Next they wondered whether it might be that the second generation rats had been born with a higher wariness as a result of the stress their parents suffered; i.e. it could be a hormonal transfer from mother to child (e.g. cortisol, the stress hormone).

Frank et al. tested their hypothesis by torturing some rats for a while (rules about animal welfare were not strict in the 70's). They would lock some rats in a small jar and bash them about for a long time, then kill them, chop them up, and take out their livers. They fed the livers to other rats, and found that after eating the livers the other rats learned the maze much faster. They interpreted the results in what now seems a sensible light: the stressed rats were producing high concentrations of a stress-signalling molecule. When those rats either had children or were fed to other rats, they passed on high doses of the stress molecule. This raised the alterness and wariness of the receipient rats so that they were much quicker to learn which parts of the maze were dangerous.

There is no evidence that the child rats actually 'remembered' the maze - they still had to find their way around, but they were extremely wary of the electrocution plates and so avoided them, finding the safest way to the end. This is not a case of genetic memory.

cv.complex variables - Explicit Spin Structures on the Torus

Other posters explain some of the topology of spin structures. Here's a differential-geometric answer relevant to Dirac operators. The exercise you have set yourself, of understanding Dirac operators on the 2-torus, is a good one. Rather than trying to do it for you, I'll instead discuss the 3-torus (cf. Kronheimer-Mrowka, "Monopoles and 3-manifolds").

So: a spin-structure on a Riemannian 3-manifold $Y$ can be understood in the following workmanlike way: we give a rank 2 hermitian vector bundle $S to Y$ (the spinor bundle); a unitary trivialization of $Lambda^2 S$; and a Clifford multiplication map $rho colon TYto mathfrak{su}(S)$, such that at each $yin Y$ there is some oriented orthonormal basis $(e_1,e_2,e_3)$ for $T_y Y$ such that $rho(e_i)$ is the $i$th Pauli matrix $sigma_i$. More invariantly, one can instead say that $rho$ is an isometry (with respect to the inner product $(a,b)=tr(a^ast b)/2$) and satisfies the orientation condition $rho(e_1)rho(e_2)rho(e_3)=1$.

If we have two spin-structures, with spinor bundles $S$ and $S'$, we can look at the sub-bundle of $mathrm{SU}(S,S')$ consisting of those fibrewise special isometries that intertwine the Clifford multiplication maps. This bundle has fibre ${ pm 1 }$: it is a 2-fold covering of $Y$. As such it is classified by a class in $H^1(Y;mathbb{Z}/2)$, whose non-vanishing is clearly the only obstruction to isomorphism of the two spin-structures. Conversely, by tensoring everything by real orthogonal line bundles (work out what this means concretely!), you can construct all spin structures, up to isomorphism, from a chosen one.

On flat $T^3$, all the data can be taken translation invariant. The Dirac operator is then $D = sum_i{sigma_ipartial_i}$. Tensoring with an orthogonal line bundle $lambda$ (constructed, if you will, from a character $pi_1(T^3)to O(1)$) the formula becomes $D_lambda =D otimes 1_lambda$.

In 2 dimensions, the story will be similar; the new feature is that the spinor bundle splits into two line bundles. The translation-invariant Dirac operator is nothing but the Cauchy-Riemann operator $partial/partial x + i partial/partial y$.

nt.number theory - citation for first statement of the Re(s) = 6 conjecture on zeros of Ramanujan L function

Perhaps not an answer to your question, but certainly related.

In his Twelve Lectures, Hardy discusses the "Ramanujan hypothesis" to the effect that
$$
|tau(p)|le2p^{11over 2}
$$
for every prime $p$, and says that

this is the most fundamental of the unsolved problems
presented by the function.

He must have been talking about Ramanujan's 1916 paper in which he also conjectured the multiplicativity and the congruences for the $tau$-function.

The multiplicativity was established by Mordell (1918) using what we would today call Hecke operators, the congruences were studied by Swinnerton-Dyer and Serre in the early 70s, ultimately leading Serre to his modularity conjecture as proved recently by Khare--Wintenberger (2009), and the estimate $|tau(p)|le2p^{11over 2}$ followed from Deligne's proof (1973) of the Weil conjectures.

Not bad as far as the mathematics inspired by a single paper goes.

Addendum. The estimate $|tau(p)|le2p^{11over 2}$ appears as formula (104) on page 153 of Ramanujan's Collected Papers as being ``highly probable''.

mrna - Do gene expression levels necessarily correspond to levels of protein activation?

I have seen a lot of research into molecular mechanisms of diseases/phenotypes use measures of RNA as a 'proxy' for the level of protein available in the cell. Is this actually valid?

My problem with the assumption that RNA levels correlate with that of the active product (i.e. the protein) is that a lot of post translational regulation occurs, including co-factor binding and phosphorylation, to name but 2. Does anyone know of any studies that have looked into the correlation between RNA levels and protein levels, and separately into the correlations between RNA levels and active protein?

It makes sense to me that RNA would correlate with protein certainly, but whether this relates to the proteins active function is what I wonder - i.e. there could be a pool that is replenished as and when the protein levels drop, but the proteins are only actually active for short periods in response to specific stimuli. So, does anyone know of any studies that have looked into the correlation between RNA levels and protein levels, and separately into the correlations between RNA levels and active protein?

---

Update (04.07.12)

I have not accepted any answers as yet because none address my question about levels of protein activation, but I concede to Daniel's excellent point that proteins are not all activated in the same way; some are constantly active, some require phosphorylation (multiple sites?), some binding partners... etc! So a study looking at 'global' activation is not yet possible. Yet I was hoping that someone may have read some specific examples.

I today found an unpublished review by Nancy Kendrick of 10 studies that have looked at the correlation between mRNA and protein abundance - still not relating to activation. However she finishes the paper as follows;

The conclusion from the ten examples listed above seems inescapable: mRNA levels cannot be
used as surrogates for corresponding protein levels without verification.

If this is her conclusion about protein levels, then any correlation between protein activation and mRNA abundance seems unlikely (as a rule. Some protein levels do correlate with the RNA - see the paper).

I am still interested in any answers that give any information about specific examples of protein activation and mRNA levels - it seems highly unlikely there are no such studies, but I have been as yet unable to find any!

biochemistry - How do multiple replication forks function without 'colliding', and what is the benefit of this method?

I'm currently reading a little about DNA replication, and have come accross the following statement;

Replication starts from a fixed point and is bi-directional ... In Eukaryotes, there are multiple replication forks, each progressing in a bi-directional fashion.

If there is a single, long strand of DNA in a Eukaryotic cell, I see potential problems with this:

These forks involve opening up a section of double-stranded DNA, and each strand becoming a double strand in a newly synthesised piece of DNA. At some point, before any single fork has become two new double-stranded molecules, another fork could 'collide' with this, causing it to attempt to replicate the non-finished section.

Simply, how can one replication fork meet another without either exponentially increasing the number of strands being replicated?

Also, on a more general level, I would be quite interested to know the actual benefit of this, when, typically, only a single copy of the double-stranded molecule needs making.

biochemistry - What causes adenosine build up in the brain when awake?

Adenosine causes humans to become sleepy. But how ?

• During day time we consume food which is broken down into glucose. This glucose is broken down by "Glycolysis" in cell's cytoplasm during which ATP is produced. This produced ATP is is then used by body as an energy supplier. ATP breaks down into ADP and then AMP with the release of energy which our body consumes for doing work.

(Fig 1- Structure of Adenosine to ATP)

Adenosine is produced continuously and starts to accumulate around and in Adenosine receptors such as A1, A2a, A2b and A3 (as shown in fig below).Adenosine inhibit signal propagation and energy production in at least A1. (Not all the receptors induce sleep when they receive Adenosine.) "A1 receptor" receives adenosine and induces sleep. A1 receptors are implicated in sleep promotion by inhibiting wake-promoting cholinergic neurons in the basal forebrain.

Continuous inhibition of A1 receptor by Adenosine slowly induces sleep by relaxing the brain and muscles but without substantially weakening their abilities!

How is Adenosine metabolized ?

During sleep the accumulated Adenosine is metabolized by Adenosine deaminase enzyme which catalyzes the irreversible deamination of 2'-deoxyadenosine and adenosine to deoxyinosine and inosine. With the reduction in the Adenosine content the body is excited from sleep slowly.

So in short summarize:

The accumulation of adenosine during waking periods is thus associated
with the depletion of the ATP reserves stored as glycogen in the
brain. The increased adenosine levels trigger non-REM sleep, during
which the brain is less active, thus placing it in a recovery phase
that is absolutely essential—among other things, to let it rebuild its
stores of glycogen.

Because adenosine is continuously metabolized by the enzyme adenosine
desaminase, the decline in adenosine production during sleep quickly
causes a general decline in adenosine concentrations in the brain,
eventually producing conditions more favourable to awakening.

(Refer this page: MOLECULES THAT BUILD UP AND MAKE YOU SLEEP)