ALLEN R. MAJEWSKI
A QUALIFYING EXAM PAPER PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
I know Vi Hart discussed it, but she may have not been the first to entertain the idea. Not sure. The question is do they digits of pi contain copies of hamlet?
Yes, yes they do. Not just copies of Hamlet, but actually copies of every book imaginable are contained within the digits of pi an infinite number of times.
Using software package pi and simple shell commands,
I found that any N digit sequence probably appears within the first 10^N digits of pi ; moreover, the sequence would appear about 10 times in the first 10^(N+1) digits of pi, 100 times in the first 10^(N+2) digits, etc.
For example, lets say our sequence is the last 4 digits of my phone number – 1345
That’s 4 digits, so if I scan the first 104 digits of pi, sure enough, my phone number occurs 1 time. If I scan 105 digits, it occurs 12 times. For 109 digits, it occures like 98 times. Its clockwork. See below.
altoidnerd@HADRON:~$ pi 10000|grep -o 1345|grep -c 1345
altoidnerd@HADRON:~$ pi 100000|grep -o 1345|grep -c 1345
altoidnerd@HADRON:~$ pi 1000000|grep -o 1345|grep -c 1345
altoidnerd@HADRON:~$ pi 10000000|grep -o 1345|grep -c 1345
Anyway, so the question is how many digits of pi would we need to search through to likely find about 1 or 2 copies of hamlet in the sequence?
So I got hamlet in plaintext from 2 sources. I found that hamlet when compressed from plain text to a tar.gz archive, the average size was 69 KiB. Therefore, (I think?) that means to represent hamlet as just a binary integer, it would have 69 x 8 x 1024 digits = 565248 digits.
To get the number of digits in base 10, we multiply by log_10 (2) +~ .69 so the base ten hamlet would be like a sequence of about ~390,000 digitsm (0-9).
Ok so back to pi. How many digits of pi must we scan to probably find a string of length 390,000? That would be 10^390000 digits. That’s a lot of pi, but pi has got enough digits to spare. We should see approximately 1 copy of hamlet in the first 10^390000 digits of pi.
Even cooler is that if we just increase the power by 1, we should see 10 hamlets; increase the power by 2, and we should get 100 copies of hamlet. So quickly, we end up with infinite hamlets in the digits of pi. And not only hamlet … this argument should would for any book.
 This is not too surprising since pi is believed to be a normal number, though this is unproven.
This blog post is a glorified (with some images and minor changes) copy pasta of my post on /r/cryptomarkets.
LTC – how low can it go? It might be time to take a shot with LTC for a potential bitcoin profit.
I have done this with LTC a few times – the best of which was last November. Priced in BTC, it is now near parity with DRK and below .01 … which isn’t far from the price I think I paid for a bunch of LTC right before the boom in November, when I started writing about trading in /r/cryptomarkets with “the golden rules for alt trading.”
One thing that makes LTC different from other scrypt alts is its very slow ditribution – it’s just as slow as bitcoin. To me, that makes it still attractive as a swing trade, because big investors are probably thinking about it. What do y’all think?
BTW, just looked. I decided to buy lots off LTC last year at .008, and it’s not far from this mark. I wrote the golden rules to this sub from a 5-star hotel (I don’t stay in those) thanks to that trade. Good times…
If you write “fail” as 0x0FA11, not fail becomes 0xF05EE..or “foresee!”
It’s fun to try and write words in hex, like “deadbeef” of “cafebabe.” If we allow ourselves certain 1337 notations for letters, we can write even more words as Hexadecimal integers like “5ca1ab1e.” Pretty cool!
What is not scalable? Well that’s computable … not 0x5ca1ab1e is 0xa35e54e1. This isn’t a word.
But as I demonstrated above, much fun can be had by taking binary operations on these integers. Not fail is foresee. Can you find any others?
This is a full description of a situation often encountered by scientists in the process of fabrication of the NMR probe. The analysis requires some tedious complex algebra, a bit of circuit theory, and enforces a matching condition. I tried to write this so that one may infer the cirucit theory from context. If there is a problem, just ask.
We will examine the impedance of this reactive L network
Goals of this challenege:
Characterize the parameter space of the variables Cm, Ct, ω, L, r and produce some useful set of tables for lab, in which the relationship between Cm and Ct is known for a given ω L and r. Furthermore to ponder the level of greed allowed. Which parameters limit others? Compare this with what the laboratory reality is.
One must always strive for impedance matching conditions to be satisfied, which for us means 50 ohms real. So we must
Im_Z = 0
Re_Z = R := 50 ohms
These requirements are nasty if you allow the impedance of the coil to have a small (but very physical and influential) real part r
Z_coil = j ω L + r
So the total impedance is
Z_tot = -j / (Cm ω) + Z_coil || Z_Ct = Re_Z + j Im_Z = R + j 0 = 50
* Z_m is the impedance of the matching cap only
* Z_t is the impedance of the tuning cap only
* the notation A || B means “A parallel B” and A || B = ( 1/A + 1/B)^(-1)
Since Z_coil has a real and an imaginary part, the expression for total impedance is a headache.
So I did it by hand, and with mathematica, and iteratively found what I consider decently short code with reasonably concise expressions. Here we go.
Clone the mathematica stuff here
git clone https://github.com/Altoidnerd/NMR-Tank-Circuits
In which there is a file where I do in fact show the real and imaginary parts of Z_tot are:
real part (which we denote Re_Z…please note the sloppyness. Here w is ω)
Re_Z = r/((r^2 +
L^2 w^2) (r^2/(r^2 +
L^2 w^2)^2 + (T w - (L w)/(r^2 + L^2 w^2))^2))
and imaginary part (Im_Z)
ImZ = (-(1/(M w)) - (T w)/(r^2/(r^2 +
L^2 w^2)^2 + (T w - (L w)/(r^2 +
L^2 w^2))^2) + (L w)/((r^2 +
L^2 w^2) (r^2/(r^2 +
L^2 w^2)^2 + (T w - (L w)/(r^2 + L^2 w^2))^2)))
where we eliminated the need for subscripts but denoting Cm := M and Ct := T.
How do we make useful data from these equations? To answer this, we must first assess what the experimenter can really control.
* coils are hard to wind and have prescibed results. In general, the parameter r is less than 1 ohm, but its actual value is not constant through frequency sadly. It must be treated as such.
* A typical coil inductance L satisfied 1.0 uH < L 30 uH. Intermediate values such as 8 uH tend to be the most difficult to fabricate. A coil inductance of 8uH I find would be useful for lower frequencies, below 3MHz, which are currently causing me problems. It is here the equations become extremely sensitive.
* The capacitance T and M can within reason, be expected to continuously vary between 0 < T,M < 1 nF and even more reasonably if the upper boundary is around 300 pF.
* the frequency is going to satisfy 1 MHz < f < 30 MHz; so ω = 6.28 f so we can say about, that
1 e7 < ω < 3e8
I have made many charts. Got any brilliant ideas?
A typical annoying situation in lab would be:
Drat. To reach the target frequency, we must either replace the capacitors with larger ones,
or exchange the coil with one of larger inductance. Which will take me less time?
I usually do not know in fact. I either make a intuitive guess, prepare some primitive tests, or try a bit of each.
The code in the github repo above will give you some parameter sliders. You can try plotting M, and T vs ω as L and that little tiny r are varied…I still must get to the bottom of these matters, such as, the qualitative effect of increasing r at fixed ω and L etc. How to encapsulate all such desirable relations in a single concise set of diagrams is what I truly seek, from the kind theorists of who may read this.
I have studied this problem up down left right…I wrote some interesting special cases down here, but I believe there is more to be known about these equations that could be of service to the designer.
Redditor /u/Mimshot gave the following example:
If an observer is near the path of a small, moving charged particle (unless there’s some special quantum effect I’d love you to tell me about if it exists) the observer will see the E field increase and then decrease and will see the B field ramp from baseline, then reverse direction, which is certainly wave-like. I’m not saying it radiates photons, but I’m wondering if “no, it must be accelerating” is a complete answer.
Is there some quantum effect I’m missing?
I “know” immediately there is no radiation in this case, because the theory of relativity tells us we can use a frame of reference in which the particle is stationary. Hence, as a rule, only accelerating particles radiate and thus give rise to traveling waves. Nevertheless, this question did get me to think about what the fields would be like in such a situation. A passing electron would seem to have some time dependent magnetic fields because the “ramp” explanation above, but it cannot be the case since we should know, just “because”, only accelerating charges radiate.
After some thought I came up with the following proof that the magnetic field is static in this case.
J(r,t) = ρ(r,t)v(r,t) = e δ(r – r’,t)v(r – r’)
v has no time dependence.
The current I is ∫ J d2 x’
I = ∫ d2 x’ e δ(r – r’,t)v(r – r’) = e v = a constant
To find B we use ampere’s law for some closed loop
∫ B dx = μ I = constant
If you’re concerned about the ∂E/∂t term lets look at the full maxwell equation
∇ x B = μ J + μ ε ∂E/∂t
Applying the operation ∫ d2 x to both sides gives
∫ B dx = μ I + μ ε ∂/∂t ( ∫ d2 x E )
The RHS of the above equation is simpified using gauss law, the integral gives the charge enclosed by a surface
∫ d2 x E = q/ε
∫ B dx = μ I + μ ε ∂/∂t ( q/ε )
but ∂/∂t ( q ) = 0
so that term doesn’t change things.
An “organized” source is being built here –
A picture says a thousand words…so here is the tarball of code fragments:
Locating pure NQR spectra precisely would in many cases clarify NMR studies. Furthermore NQR is indicative of internal field geometry in solids and is thus useful in the identification of quantum phase transitions.
The pursuit of pure NQR is difficult however because the resonant frequency is sample specific and is often unknown. Unlike in the case of NMR, the frequency cannot be controlled in the laboratory, but is rather a property of a material that is a fingerprint of the local environment of the nucleus in question.
In general, operating a pulsed spectrometer at various frequencies requires the corresponding adjustment of the two capacitors shown below. Reducing the parameter space to a single value would make sweeping much more efficient. Any shortcuts and tricks to allow easy sweeping could greatly accelerate understanding of NQR in yet unstudied samples.
This general probe topology is common in the practice of nuclear resonance.
The inductive load is tuned and matched to the characteristic impedance of a transmission line Z0 (usually 50 ohms) by the two variable capacitors C1 and C2.
Postulate: If the series losses in the coil are set to
R = Z0 / 4,
C_1 =~ C_2
regardless of the value of Z0, and for any reasonable L and f where f is frequency of operation and f > 500 KHz. For f < 500 KHz the approximation begins to break down for feasible values of L.
Suppose we can utilize transmission line transformers to reduce the effective Zo from 50 ohms to something lower, allowing a higher Q.
If for example the effective characteristic impedance of the tank Z0 = 20 ohms, then one could set r = 5 ohms externally. This results in nice agreement for the caps with L = 30 uH down to around 1 MHz. This would be excellent for sweeping and snooping for unknown quadrupolar resonances in this band, as 14N NQR often appears below 5 MHz.
Source tree on Github
Raw copy pasta for mathematica