Interview questions

Pocket Ace pair or loaded dice Bayesian analysis
Heads up all-in call/fold strategy - also to get to the distribution curve
Silkworm/leaf trading strategy (leaves may be perishable or non-perishable)
Typewriting monkey (or HHH in coin toss)
Logistic map
(tbc)

Survival...

is to make yourself irreplaceable. What does this mean? Be the keeper of the core knowledge, expertise, and relationships. In the mean time, make sure what you have are transferable. It is all about bargaining power vis-à-vis the organization.

I have a dream and I need a plan

So I was still right about Apple/Verizon...

My previous blog post almost a year ago...

And ...
Verizon reportedly set to sell iPhone in 2011

Conspiracy to discourage gold hoarding?

I have long heard the conspiracy theory on central banks trying to depress gold price so to promote their fiat-currency, reserve banking system. It was all talks and hypotheses, but I was shocked today to see that RSF (Required Stable Funding) factor for gold is 50% in BIS's consultant paper on liquidity management. What does it mean? RSF is basically the haircut to asset value at one-year liquidation horizon.

As a reference, RSF for cash is 0%, for HG corporate bonds is 20%. It is however as high as 50% for equities and gold. Wouldn't gold be more valuable and easier to sell at times of stress?

The message: should this be adopted by worldwide regulators, from a liquidity perspective, banks will be severely penalized for hoarding gold instead of cash. I am very intrigued and I hope someone can enlighten me why this is not a conspiracy.

The Chinese version of Monroe Doctrine

Western media have mostly portrayed recent assertiveness of the Chinese government on a range of territorial disputes as some sort of power play and a threat to the regional security. Not so regarded by the Chinese government or people themselves, who felt that China had been very consistent and it was the other side, be it Japan, Vietnam or the US that up-ed the ante.

Not an expert on the pending issues, I want to point out a few similarities to history one to two centuries back.

As China grows stronger, it is only natural that it want to enhance its edge to become a regional hegemon and ensure that its neighbors not in a position to contest its ascendancy, be it economy or military. On the other hand, US, historically having played the role of a regional hegemon of the American continents and a powerful offshore balancer on a global scale, is strongly disturbed by the idea that another regional hegemon could emerge. It has to go all out to contain China - but with the Pacific Ocean in between, it cannot achieve this militarily, at least not without the help from other regional players. To counter this possibility, China has kept playing the card of asserting bilateral talks (with its neighbors) and deflecting US influence. By claiming that South China Sea is a "core interest" China effectively stated its own Monroe Doctrine.

The crucial difference, however, is the responses from the regional players. Unlike Spain, US is not trying to colonize the region - in fact, US is very welcome in the area. To a degree ASEAN was set up because its members feared a power vacuum after US started gradually exiting Asia. Consequently, should China look to enforces its Monroe Doctrine, it would almost certainly face unanimous animosity among its neighbors, which will definitely and easily be exploited by its biggest rival, the United States.

Another comparison would be the depiction of Germany by US propaganda in the years leading to the First World War. First regarded as the beacon for economic development, then as a military threat (which turned out to be true).

While it can be argued that Chinese show of power was a bit premature, it is not necessarily an unwise move. First, it is quite probable that US has been behind the hard balls played by regionals, and China's national interest would be severely hurt if China chose not to fight back. Second, the power balance between China and US is significantly skewed toward China compared to even five years ago, with US in a lackluster "recovery" and two wars to fight. Third, aggression has not always been countered in history (not a good comparison, but US had been consistently trying to distant itself from WW1 and WW2), and it would serve China's long-term interest if it plays hard balls soon rather than being successfully contained.

Who will benefit most from this power play and emerge as the winner? It all depends on the next steps of various parties. And it probably will benefit nobody. An arms race in Asia or between US and China will do nobody any good, not to mention a modern warfare that involves China. One possible (and best) scenario is that if China will grow at its current pace for another ten or fifteen years, and no major armed conflict happened in the region, the presence of US in Asia will be gone by itself. Can China continue to grow without being disrupted or disruptive? I can only hope for the best.

Obviously my forecast went wrong...

iPhone 4 is out, and still exclusive to AT&T!

Verizon and iPhone

Guess lots of people have run out of patience for the Verizon version of iPhone. And then comes the Droid campaign that vehemently and directly bashes iPhone (Everything iDon’t Droid does), which makes people wonder: will there be an iPhone for verizon customers even in 2010, or worse - ever? This is further complicated by the recent rumor that Verizon is closely cooperating with Apple on releasing the 4G version of iPhone.

Actually if you look at all these together it kind of makes sense, and seems the rumor (of 2010 4G iPhone) could well be true. It is obvious that Verizon wants some bargaining power with Apple in the months to come, and seriously, it is Apple that stands to lose if they don’t get Verizon as a partner.

It is all business. What would benefit Apple most? Not existing sales/relationships, but new sales. It is actually to Apple’s interest if people break contract with AT&T only to buy another iPhone with Verizon (don’t forget that those phones are not interchangeable). This is a very likely scenario given the new regulations on termination fees, and grudges about AT&T. The worst thing for Apple is that people stick with their old iPhone on AT&T. How would that happen? If Verizon and Apple are still “in talks” by next year.

True that Droid’s ad directly points out the weakness of iPhone, but that will only hurt sales of Apple, not of Verizon, because there isn't any worthwhile contestant to iPhone to date. It will hurt the relationship between Verizon and Apple, but does Apple really have a choice? A few billion dollars of new sale - I doubt Apple would forgo that however they hate Verizon.

Short comment on some analyses of this financial crisis

There have been numerous analyses of the current (or “past”) credit crunch, comparing or contrasting it with the 1930 crash. The approach was often putting the DOW JONES time series on top of that of 1930 and discuss the similarities or dissimilarities between the two.

The problem with this approach was - I am shocked that nobody pointed it out - that the fundamental time scales have been changed. The time taken to disseminate market information has been shortened significantly, while other time scales (e.g. intervals between earnings reports, time taken to finalize an M&A deal) have been changed by a more moderate degree, although it is almost certain that all time scales have been shortened. It is hard to fathom which time scales here play a major role in today’s world of finance, but it is understandable that things ought to happen faster now - thus the comparison of stock indexes can not be apple-to-apple.

There is more to the problem. The behavior of dynamic systems is heavily affected by the lineup of its spectrum of modes, and a momentary resonance can lead to a quick breakdown of a recently stable system (to be fair - mathematically it has always been inherently chaotic). The question is, could the evolution of different time scales be the ultimate reason of a breakdown of the financial system?

MORE ON TIME SCALE ANALYSIS

In near-integrable systems, or near-integrable areas, the stability of the system depends heavily on the absence of resonances. In finance, we have this imperfect system in which resonances are multi-dimensional and impossible to capture accurately. However we may indirectly verify this claim by examing the ratio of key time scales - every time they are aligned with small-integer ratios chaos is imminent.

Note that there are two kinds of time scales: dissipative (e.g. time to absorb information) and cyclic; imaginary and real.

MORE ON THE CHAOS MODEL

I am tempted to describe the financial world by a series of elements: resources, assets, and consumption. Denomination is a problem. It is also uncertain whether we should describe the state in a classical or quantum manner. If there are N assets, is the financial space R^N or R^N×R^N? I suspect that the valuation method (i.e. value = dividend/(r-g)) also plays a role in the instability.

Men's Room Problem

There are N urinals; men prefer to stand away from others and they come in following a Poisson process and occupy one urinal for a fixed amount of time. How many flushes each of the urinals would get for each day?

COMMENT: this problem gets interesting when N is larger than 3 and the restroom is relatively crowded (say always slightly more than half occupied). Say N=2^n+1, the # of flushes is not a monotonous function from one side to the other; instead as n goes to infinity, it could very well have an asymptotic form like the Weierstrass function (not proved)...

Optimal trading strategy for raising silkworms

Jingles mentioned that when she was a child, she used to trade silkworms for Mulberry leaves because otherwise her worms would starve. Here is the question: what is the optimal trading strategy to have the most living silkworms by day T?

Some basic assumptions:
1, trade is one way only.
2, each silkworm consumes 1 unit of leaf every day.
3, leaves have a half life of k days (2 leaves from k days ago is equivalent to 1 leaf from today).
4, the price formula (# of leaves traded for one worm) is P(t) on day t, for starters, assume P(t) is a constant function.
5, silkworm will starve to death if not fed for one day, however, one can choose which worms to feed (so if leaves are insufficient, at least some of the worms can survive).

Stock price a chaotic revelation?

For a process x(t), define survivability ratio P(t) = chance of finding x(t) within a predetermined range of x(0); Brownian motion has P(t) = t^(-1/2). For chaotic systems with Cantori trapping mechanisms, P(t) also follows a power law P(t) = t^(-p), p=1+α, with 0<α<1.

Now, assume s(t) is the drift-adjusted part of the stock price while S(t) is the unadjusted (original) stock price. In other words:

d(lnS(t)) = μ dt + ds(t),

we might be able to check the index α for the process s(t), by sampling a lot of (t=0).

Also Re: self-criticality; people have discussed (with mixed opinions) the relationship between financial markets and earthquakes (Gutenberg-Richter law). It might be worthwhile to check the survivability function of fBm. See also 1/f noises.

Party is over

First thing first, I want to say that I really should have graduated earlier. The past year unfolded a big scene that I could not be in, and today officially announced was the news equivalent to saying “babies, party is over.”

Goldman Sachs and Morgan Stanley, the last two standing i-banks on the street, will no longer be i-banks on Monday. See the news here. Everything is to be regulated and Uncle Same says no more laissez-faire banking. Maybe it was the repeal of Glass-Stegall that should be blamed, after all.

妈的我还真是适合考试

刚才有人发给我一个据说是今年很难的一道高考数学题：
证明对任意n，存在一个长度为n的非平凡等差数列，其中任意三项不构成等比数列。
我10s就做出来了；仰天长叹一声，我这种人才真是太适合高考了。

Things to do regarding last project

1, for the zero momentum surface - I'd better do the sampling on two leaves separately.
2, for the high density along orbits - I don't think wavepacket is sufficient to explain everything (the 2nd-order approximation is very inaccurate, and base everything on an expansion-contraction balance sounds shaky to me). I think some diffusion argument can be made - so we know how to calculate the relative intensity.
3, for the fraction of regular trajectories - that result should be double-checked. I need to look into the high energy limits, and plot a few fLI vs. time to see if it is logrithmatic or linear. If no concrete results can be found, I will drop it.

I am proud of myself today

This afternoon my college classmate called me on MSN and asked if I had any ideas that can help with the current rescue work in Sichuan. He told me that he had connections so my ideas can be actually passed to people on the frontier. I supplied mine: to use truck tire tubes or air bags and either compressed air cans or even manual pumps to elate them underneath the fallen concretes for gradual lifting. I knew some basic facts about tires and I did ballpark calculations to find this idea actually quite feasible.

Later this evening when I got back on MSN my classmate told me that his connection thought highly of my idea and it was already passed to the people in charge.

I literally burst into tears, for that my ideas might be able to actually save lives. I never felt so proud of myself.

Update on May 19, I am really really happy today:
[01:51:29] 子之魂魄兮，为鬼雄 says:
你的气囊救到了人命
[01:51:41] 子之魂魄兮，为鬼雄 says:
李承光师长已经表扬了
[01:52:01] 子之魂魄兮，为鬼雄 says:
成都前指的反馈
[01:52:14] 子之魂魄兮，为鬼雄 says:
自从传达到了以后部队已经开始用高压气囊
[02:06:20] 子之魂魄兮，为鬼雄 says:
别张扬出去，咱们自己乐就可以了，毕竟出点子的人还是很多的

Possible thesis title?

The dynamics to localization in high-dimensional multi-well systems: a comparative study

Why being patient or why not?

Our subject is a wolf. He is thrown into a world comprised of sites. There are two kinds of sites: easy sites, where a rabbit is coming every T1 hours, and hard sites, where a rabbit is coming every T2 hours, T2>T1. The wolf knows that the density of easy sites is p, 0<p<1. How long should the wolf wait at one site before he decides to move around? Some extreme cases: if T2=infinite, the wolf should wait no longer than T1, and if T2=T1, he should probably just keep waiting. This decision also clearly depends on the value of p. For example, if p is 0, the wolf is better off just waiting. Note that the same strategy doesn't apply to p=1. This can explain why different animals have different behavioral patterns in terms of patience; it is an adaptation to their individual environments.

Some preliminary results show that, after waiting time of T1, the wolf is better off moving if p>T1/(T2-T1), and he should keep waiting otherwise.

Level repulsion, Landau levels, and quantum dot

SAR sent me a paper (Science, 320, p.356, 2008) which told a very intriguing story (if not conceptually new). Basically they can actually see the quantum chaos of Dirac billiard by the Coulomb blockade measurements. A relevant question would be: how to test the quantum chaos in our system?

As described here, we can make a strong analogue of our system, at the vicinity of wells, to swing spring plus a kinetic coupling. It would be interesting to study the swing spring system under magnetic field, and see how the levels quantize. As a started, there have been work done on the Diamagnetic hydrogen... Is it possible to employ similar approaches to actually identify the states in experiments?

A possible way to simplify (in steps) the potential surface

V = V(r,θ,φ), exact, useless
V = V(r) + V(θ,φ) + δV(r,θ,φ), separation + coupling, not very representative
V = V(θ,φ) + ½k[r-r₀(θ,φ)]², closest to reality, note that k, if treated as a variable of (θ,φ), has a variance less than 6%.

A possible approach to the third potential: V = ½k[r-r₀(θ,φ)]² + ε*V(θ,φ). The first part resembles a study of the Monodromy problem (of course it is much more complicated, because now we have 3 DOF, however locally we can approximate r₀(θ,φ)≈r₀(φ), for example, and from there we can further separate the phase space into tiles and continue the study.

Notes on integrators

R-K 4th order: about 4.5 times of computation time of the same step V-V.

However, it has about the same conservation of energy (a gauge of accuracy) as V-V with 1/50 time step. Overall speaking, it's about 10 times more efficient. RK4 for E = 1000 cm^-1, dt = 0.005 t₀ for t=100 ps integration gives ~ 10^-11 eV of total energy shift, for E = 500 cm^-1, it is even better (~3×10^-12 eV). Doubling dt (to 0.01 t₀) leads to ~35 times higher E_shift.

We still ought to check the symplecticity of the integrator, i.e. whether it preserves the two-form:
ω² = ∑ dp_i ^ dq_j. PS: as it turns out, the SIA4 integrator proposed by J. Candy in J. Comput. Phys, 92, 230 (1991) is a better (and symplectic) one than the plain RK4 method.

Quantum control, diffusion

It might be worthwhile to look into H=H₀+μ·E. The driving term cause the diffusion perpendicular to energy surface, and H₀ has diffusion by itself on the energy surface. If D_∥»D_⊥, it can be argued that the quasienergy (Floquet energy) eigenfunctions will be localized exponentially (see D.L. Shepelyansky, Physica D, 28, 103-114 (1987)). Also read Wigner distribution, Arnol'd diffusion.

Flashback: in the XCN isomerization studies (also between-tile hopping in the current study), everything might be broiled down to the comparison between diffusion rates along different directions (in the phase space). If D_∥»D_⊥, then the particle has a good chance to "isomerize" than to "dissociate".

For d>2 degrees of freedom systems, the invariant tori in phase space, being d-dimensional, can not slice the 2d-1 dimensional space of the surface of constant energy. This means for (d>2)-DOF systems the phase space is essentially "ergodic" (Arnol'd diffusion).

swing spring, 1:1:2 resonance, etc

It can be argued that although the radial mode is not exactly double the doubly degenerate peripheral mode frequency, a substantial coupling term (manifested by the strong overtone peak) could make an analog to the 1:1:2 resonance of swing spring valid. It would be interesting to actually observe the pulsation and precession effects described in Lynch's paper.

Regarding the polygon (Sinai) billiard analog - basically we treat the whole motion as motion in 'triangles' that tiled the icosahedron, with a reflection rule at the boundaries. However, since the triangle is actually on the S² surface, special treatment is needed; for example, a rotation around the center of a pentagon should be mapped as a "bouncing" motion between adjacent two boundaries - which is quite ill-defined in the planar case. In fact, how much efforts are warranted to just carry out this simulation is unknown, as we can always do the whole simulation and mapped it back to the sub-region. The question is more interesting, only in analytical terms (since it reflects both the symmetry and/or interconnectivity).

Lastly, a measurement of 1/20 sub-region "revisit" frequency could be highly valuable. Say we have N trajectories originating from inside the boundaries, we want to calculate the number of trajectories that remain in the boundary as a function of time t. To check if this follows a polynomial or exponential decay law can be very enlightening on the stickiness of the phase space.

PS: assuming equal 'diffusion' toward the three adjacent tiles, in the long run, the decay should follow an exponential law of 0.95*exp(-0.255*r*t) + 0.05, in which r is the diffusion rate. (-0.254644 is the three leading exponents other than 0 of the eigenvalues of the diffusion matrix.) Update: preliminary simulations show a similar decay curve (the decay constant drops quickly) to an end value slightly above 0.05 and with a very fluctuating tail, even with 1000 trajectories. Despite being consistent with most of our speculations, the enormous fluctuation around the 0.05 tail is unaccounted for.

A trivial result

This explains why normal mode analysis always gives doubly-degenerate results at the bottom of the wells. Freshman level proof, but the results are pretty unexpected.

A possible way to investigate the effects of symmetry breaking

With the five-fold symmetry about z-axis intact, the Poincaré surface of section of (φ=0, p_φ>0) will look identical to (φ=2nπ/5, p_φ>0), n=1, 2, 3, 4, ... but won't if the symmetry is broken. What happens if there is a strong localization? One may guess the density on the “right” surface of section is going to be much denser, but again the measure of the section points is always zero; one needs to think about a way to go about that.

Another issue: on generating initial conditions evenly on the surface of identical energy: if we first pick up r then v, the number of v's given r should be proportional to E⁰ in 2d and E^1/2 in 3d.