dna - Is telomere shortening consistant over consecutive cell divisions from zygote to a differentiated cell?

Considering the complexity of embryogenesis, a temporal referance would be helpful to coordinate the developmental sequences during embryogenesis and fetal development which is to be completed within the gestation period. Can telomere shortening if consistant be used as a reference by every cell to do what it is destined to do at particular time frame? This is without considering the influence of telomerase etc.

This is also without prejudice to the role of DNA directed molecular signalling.

To repeat the question is telomere shortening consistant over consecutive cell divisions from zygote to a differentiated cell? If so can it function as temporal control of embryogenesis?

microscopy - What to look for when buying a light microscope?

I am a science lover and want to buy a microscope to explore things around me. Like studying cell structures, microorganisms, blood and plants. It would be great if there is way to photograph slides. Just curious to know and explore things around me.

I want to know which features I should consider while buying it?

nt.number theory - The difficulties in proving modularity lifting theorems over non-totally real fields

Note: This is a fairly precise and detailed question about an important but technical aspect of algebraic number theory. My answer is written at a level that I think is appropriate for the question; it assumes some familiarity with the topic at hand.

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The most basic difficulty is that there is non a map $R rightarrow {mathbb T}$ in general
(i.e. one typically doesn't know how to create Galois representations attached to automorphic forms).

The second difficulty is that in the TWK method, one must argue with auxiliary primes (the primes typically labelled $Q$), and show that as you add these primes, ${mathbb T}$ grows in a reasonable way (basically, is free over $mathcal O[Delta_Q],$ where $Delta_Q$ is something like the $p$-Sylow subgroup of $({mathbb Z}/Q{mathbb Z})^{times}.)$

One shows this (or some variant of it) by considering the analogous queston about cohomology of the arithmetic quotients. Suppose for a moment we are in the Shimura variety context, or perhaps the compact at infinity context. Then it will be the middle dimensional cohomology that is of interest, and if we localize at a non-Eisenstein maximal ideal we might hope to kill all other cohomology. Then we can replace a comuptation of middle dimensional cohomology by an Euler characteristic computation, and its easy to see that the Euler char. will multiply by $|Delta_Q|$ when we add the auxiliary primes $Q$.

But in more general contexts, there won't be a single middle dimension in which the maximal ideal of interest is supported (even if it is non-Eisenstein), and computing Euler characteristics will just give $0$, which is not much use. It's not clear that it's even true that adding the auxiliary primes forces the approriate growth of cohomology, and possible torsion in the cohomology just adds to the complication.

There is much current work, by various groups of researchers, with various different approaches, aimed at breaking this barrier.

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I should add that one can now handle certain questions about non-totally real field,
say question related to conjugate self-dual Galois reps. over CM fields, because
these are still related to a Shimura variety context. This plays a role in the recent
progress on Sato--Tate for higher weight forms by Barnet-Lamb--Geraghty--Harris--Taylor
and Barnet-Lamb--Gee--Geraghty, and is also the basis for a recent striking theorem
of Calegari showing that if $rho:G_{mathbb Q} to GL_2({mathbb Q}_p)$ is ordinary
at $p$ and de Rham with distinct Hodge--Tate weights (and probably $overline{rho}$ should satisfy some technical conditions), then $rho$ is necessarily odd!

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big picture - Why is 2 so odd?

My take on this issue is that p=2 isn't really strange---all small primes are strange, it's just that the smaller you are, the earlier you become troublesome. Look at recent R=T results in the theory of automorphic representations. Nowadays people can prove these sorts of things for $n$-dimensional representations, but they need to assume $p>n+1$ or some such thing. The thing about $p=2$ is that it's so small that it's already causing problems when one is considering $GL(1)$, which is an abelian situation. Now abelian situations are so much easier to understand than the general situation that they are more prevalent in the literature. For example things like quadratic reciprocity can be viewed of as some consequence of class field theory, which is really 1-dimensional representations of Galois groups, and already $p=2$ is causing a problem. Similarly Fontaine's results on commutative group schemes of $p$-power order runs into some trouble when $p=2$ (his basic linear algebra data doesn't give you an equivalence between finite flat group schemes over ${mathbf Z}_p$ and "easy semilinear algebra" when $p=2$) and again it's because 2 is just too small. But as people formulate higher-dimensional analogues of these things, they will no doubt have to rule out more primes. So it's not that 2 is behaving badly, it's just that from 2's point of view the theory is more advanced, so you have to deal with more special cases.

ag.algebraic geometry - morphism closed + fibres proper => proper?

The answer is no. Consider an integral nodal curve $Y$ over an algebraically closed field, normalize the node and remove one of the two points lying over the node. Then you get a morphisme $f : Xto Y$ which is bijective (hence homeomorphic), separated and of finite type, and the fibers are just (even reduced) points. But $f$ is not proper (otherwise it would be finite and birational hence coincides with the normalization map).

In the positive direction, you can look at EGA, IV.15.7.10.

[Add] There is an elementary way to see that $f$ is not proper just using the definition. Let $Y'to Y$ be the normalization of $Y$. So $X$ is $Y'$ minus one closed point $y_0$. It is enough to show that the base change of $f$ to $Xtimes Y' to Y times Y'$ is not closed. Consider the closed subset
$$Delta=leftlbrace (x, x) mid xin X rightrbrace subset Xtimes Y'.$$
Its image by $f_{Y'} : Xtimes Y' to Ytimes Y'$ is $leftlbrace (f(x), x) mid xin Xrightrbrace$ which is the graph of $Y'to Y$ minus one point $(f(y_0), y_0)$. So $f$ is not universally closed, thus not proper.

gn.general topology - Unusual Space-Filling Curve

Around 1998, I encountered a (forgotten) reference to a particularly strange space-filling curve.

Consider a foliation as a collection of continuous nonintersecting curves that start at (0,0) and end at (1,1) and collectively fill the unit square, such as the graphs of functions f_t(x) = x^t where t >=0. Supposedly there exists a continuous curve G that starts at (1,0), ends at (0,1), fills the unit square, and crosses each f_t curve only once.

This initially sounds even more impossible than the Cantor curve. But intuitively a space-filling curve could trace back and forth over the f_t curves and only cross at the corners (0,0) and (1,1). Can someone please explain a construction of such a space-filling curve?

entomology - How do small animals make loud sounds?

The Cicada

A careful study of the noise-making apparatus of the cicada can be found in a 1994 paper by Young and Bennet-Clark.$^1$ The authors generated sounds at about 0-16 kHz at peaks on the order of 100 dB using cicadas in various stages of deconstruction. The cicada uses a resonant organ-system called the tymbal which buckles and unbuckles rapidly to produce sound. The buckling-in is caused by muscle contraction and is louder than the buckling-out (relaxation) phase. Air sacs (a feature of many other small noisemakers) serve to amplify the sound. The tymbal itself, for the species in this paper, has a resonant frequency of about 4kHz (Young, page 1017). The song of the cicada as modified/amplified by air sacs and other structures, is often around 10kHz.$^{2}$

Pure Tone vs. Diffuse Tone?

The vocalizations of large vertebrates are a complex superposition of waves that in the frequency spectrum are somewhat spread out. To the extent that a frog or a bird emits a pure tone, the energy will be confined to a narrow frequency range and this may be a strategy for achieving greater amplitude. Given comparable audio sensitivity, however, the intensity of pure tones will depend on amplitude (intensity), regardless of frequency.$^{3,4}$

Other Small Loud Animals

While the songs of cicadas are intense, especially in concert, on an individual level there is competition from other species. According to a Gizmodo article quoting assorted scientists, the snapping shrimp produces a transient snap that is around 200 decibels, a level that one site describes as "deafening." For perspective, dolphins can emit short chirps of 220 dB but these are outside the range of human hearing. The lion emits a roar of 115 dB which is sustained and audible 5 miles away, according to the article. Elephants also are capable of 117 dB cries, as are howler monkeys.

Both the shrimp and the cicada use non-vocal vibration to create their sounds. The shrimp uses a "spring-loaded claw" (the spring is muscle). The localized force of one part of the claw hitting the other generates a bubble (this is known as inertial cavitation). When the bubble collapses it generates a shock wave (noise) that stuns fish (prey).$^5$ The noise of frogs is produced as air passes from the lungs through the larynx, amplified by distended air sacs which resonate. Birds can produce up to 135dB (the Mollucan Cockatoo). They generally force air past (membranes and) a specialized organ called the syrinx located at the bottom of the trachea (see the Wikipedia note on bird vocalization).

Micronecta scholtzi, a 2mm-long aquatic insect, is for its size the loudest known animal. It creates a sound of 99.2 dB intensity which (despite being largely lost in transition from water to air) is audible to humans ashore. According to the Wikipedia note it creates this sound by "stridulating a ridge on its penis across corrugations on its abdomen." The area involved is about 50$\mu m$ across. Details of the mechanism are poorly understood. The article's comparison of the sperm whale's 236 (underwater) dB song gives perspective, as a sperm whale can weigh 14 metric tons.

A Common Aspect of Sound Intensity: Cavity Resonance

Descriptive studies of sound-creation by small animals (with the possible exception of the snapping shrimp) do not fully explain why a one-gram bug can make a bigger noise than a lion. Purely vocal methods of larger vertebrates produce sustained noise on the order of 100 dB but the non-vocal instruments of smaller creatures are capable of short bursts of amazing intensity.

It is difficult to generalize but because air sacs are part of many sound-making schemes, cavity resonance probably plays an important role. Like spring-mass systems or RLC circuits, cavities have resonant frequencies at which amplitude of a signal may be increased (the policeman's whistle is a familiar example). Another paper by Bennet-Clark and Young gives a sketch of a theory along these lines. At resonant frequencies the impedance of the instrument falls sharply and instead of being dissipated (generally as heat) the energy emerges as sound.$^6$

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$^1$ Bennet-Clark and Young, The Role of the Tymbal in Cicada Sound Production, J. of Experimental Biol. (1995) 198, 1001-1019.

$^{2}$ The frequency of cicadas is variable, mostly on the order of 10kHz but occasionally very low (< 1kHz), mostly due to body size. See this article.

$^3$ Sound--essentially a compression wave--diminishes with distance. So when the Wiki article on noise levels compares noise levels it includes the distance from the object. For example, 100 db (comparable to the cicada) is the level of noise associated with a jack-hammer at 1 meter away.

$^4$ Audible range for humans is roughly 15 Hz-16000 Hz. As mentioned below, the dolphin can emit very intense high-pitched sounds that we don't hear at all, so the analogy to EM waves (higher frequency = higher energy) doesn't help predict perception. See Pfaff and Stecker, Loudness and Frequency Content of Noise in the Animal House, Lab. Animals (1976) 10, 111-117.

$^5$ M. Versluis, B. Schmitz, A von der Heydt & D. Lohse (2000). "How snapping shrimp snap: through cavitating bubbles". Science 289 (5487): 2114–2117.

$^6$ Bennet-Clark and Young, Short Communication, The Scaling of Song Frequency in Cicadas, J. Exp. Biol. 191, 291-294 (1994).