Below is a list of science problems that I would like someone to solve.
If you decide to pursue any of these ideas, have seen them done already, or would like my help with them, please e-mail me to let me know.
If any of these ideas prove valuable to you (patents, publications, fame, fortune, etc.), appropriate acknowledgment, citation, co-authorship, etc. would be appreciated.
Wikipedia has a brief article about this technique with some useful references.
See Chapters 7 and 8 of Bender and Orszag's "Advanced Mathematical Methods For Scientists and Engineers", 1978, ISBN=0-07-004452-X too.
"Numerical Recipes in C" 2nd Edition, 1992, ISBN=0-521-43108-5
on p.202 says the following about Pade Approximation:
"There is, in general, no way to tell how accurate it is, or how far out in x it can usefully be extended. It is a powerful, but in the end still mysterious, technique."
It seems that many people use Pade Approximation, but is its reliability really known?
How can you tell its accuracy or when its results can be trusted when you have no other methods available for comparison?
What if you base your Pade Approximation on a perturbation series out to 12th order, but a key behavior does not begin until 16th order?
Please examine the equations in [2] and pp.26,165 of [1]
to see if they are a special class with known convergence properties.
Explore what happens when one replaces the discrete frequencies in
[2]
and
[3]
with finite-bandwidth signals:
Finite-bandwidth signals (like Gaussians or Lorentzians vs. frequency) are more realistic than the δ-function frequency distributions used in [2] and [3].
Fig.6 and the 2nd order term in eq.48 of [2] show that the effect of multiple oscillating magnetic fields is not just the sum of the effects of individual oscillating magnetic fields:
Can this nonlinearity let a finite-bandwidth low-amplitude signal be more effective
than a similar-amplitude signal at a single discrete frequency?
How should one normalize the two types of signals to make an appropriate comparison?
At first glance, one could simply replace the sums over discrete frequencies with integrals over a range of frequencies:
Should one also replace the sums over discrete phases with integrals over a range of phases?
Should the phase depend on the frequency?
What would be a good way to represent finite-bandwidth noise signals?
Appendix F of [1] has some discussion useful for this.
The general technique might be called degenerate Wigner perturbation theory.
The non-degenerate version might have been done independently by Lennard-Jones, Brillouin, and Wigner:
Wigner may have published his work on this in a Hungarian journal written in English.
Develop better density functional theory (DFT) methods for obtaining g-tensors, hyperfine couplings A, or exchange interactions J for radical pair systems:
Many papers by Frank Neese's research group are along these lines.
Aβ40 β-amyloid fibrils seem to contain structurally significant mobile water molecules buried or trapped in the fibril core in intersheet cavities or channels.
Would being in a tightly-packed structure with restricted or low-dimensional diffusion help?
Enzymes that use free radicals tend to isolate these free radicals to protect from unwanted side reactions:
Would this isolation extend relaxation times?
Mitochondria, chloroplasts, ribosomes, membranes, bones, etc. all provide special microenvironments:
Would any of these help?
Below are some interesting references along these lines:
This paper describes how in systems with two or more coupled nuclear spins, certain "long-lived states" have much longer nuclear spin relaxation times than others.
If these "long-lived states" are sufficiently populated by an appropriate pulse sequence,
the overall nuclear spin relaxation time can be much longer (e.g. 37 times more) than the usual spin-lattice relaxation time.
Can similar effects be obtained for systems with two or more coupled electron spins?
Study biological free radicals, especially radical pairs, to determine their g-tensors, hyperfine couplings A, exchange interactions J, inter-radical distances and orientations, lifetimes τ, and electron spin relaxation times T1 and T2:
Explore what happens to these parameters at 20-40 degrees C in Earth-strength magnetic fields.
Some enzymes contain very stable, long-lived free radicals. Others contain transient free radicals.
In [4], the most interesting things happen for radicals with lifetimes 50 μsec or longer.
Many proteins contain free radical forms of amino acids. Some examples are glycyl, tryptophanyl, and tyrosyl radicals.
Many enzymes contain transition metals like cobalt, copper, iron, or molybdenum that have unpaired electrons and so behave like free radicals.
Superoxide dismutases (SOD), nitric oxide synthases (NOS), xanthine oxidase (XO), β-amyloid proteins, melanins, cryptochromes, radical SAM enzymes, B12 enzymes, photosynthetic reaction centers, and DNA photolyases are all interesting systems:
Do any exchange-coupled radical pair species lurk in the EPR spectra of Cu-containing β-amyloid complexes?
Find ways to lengthen electron spin relaxation times T1 and T2 in biological radical pair systems at 20-40 degrees C in
Earth-strength magnetic fields:
The free radical photopolymerization of methyl methacrylate (MMA) behaves differently in Earth-strength versus 6400 G magnetic fields.
Having radical pair recombination occur in local sites of roughly nm size seems to be what allows 10-20% different dye yields in Earth-strength versus 1800 G steady magnetic fields in polymeric glasses of polystyrene (PS) or poly(vinyl chloride) (PVC).
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