## Do copies of Hamlet exist embedded in the digits of pi?

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,

pastebin of shell commands

I found that any N digit sequence probably appears within the first 10^N digits of pi [1]; 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 1 altoidnerd@HADRON:~\$ pi 100000|grep -o 1345|grep -c 1345 12 altoidnerd@HADRON:~\$ pi 1000000|grep -o 1345|grep -c 1345 114 altoidnerd@HADRON:~\$ pi 10000000|grep -o 1345|grep -c 1345 1020```

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.

[1] This is not too surprising since pi is believed to be a normal number, though this is unproven.

## Calculations of the EFG tensor in DTN using the GIPAW method with CASTEP and Quantum Espresso software

One important difference between NQR and NMR is that for NQR, the transition frequency is site specific and cannot be chosen by the experimenter. In NMR studies, the excitation frequency of the nucleus is just the gyromagnetic ratio times the applied field. Thus, the difficulty in NMR is controlling the uniformity of the applied static B. However, the experimenter is free to choose an operating frequency by adjusting B. For NQR, the transition frequency is proportional to the electric field gradient (EFG) at the nuclear site, which is entirely a property of the substance and a function of temperature. While it is theoretically possible to apply a prescribed EFG in the lab, it is unfeasible as the EFG’s in crystal strcutures are often on the order of many kV/m^2. Thus the particular difficulty in NQR spectroscopy is locating the resonance in the first place, which is much like combing the desert if the apparatus runs at a fixed frequency. The home made pulsed superheterodyne spectrometer used in the trial experiments has proven to have excellent resolution. However, it has essentially no ability to perform frequency sweeps as is done in continuous wave methods. The operator must choose a relatively narrow band (< 1 MHz) to target, and design a custom bridge, tank, and receiver configuration just for small band. Band switching is non-trivial with high resolution pulsed techniques.

At one time, it was not possible to calculate the EFG for all but the simplest of structures. However, DFT calculations have become very potent and accessible in the last decade. Beginning in June of 2013, Calculations of the EFG tensor in dichloro-tetrakis-thiourea-nickel NiCl_2 [SC(NH_2)_2]_4 were made using the GIPAW method with closed source CASTEP software (by then graduatre student Dr. Tim Green at Oxford University, with advisor Dr. Jonathan = Yates). Then in October of 2013, Dr. Ari Seitsonen and I repeated the calculation at ETH in Zurich using the open source DFT-GIPAW package Quantum Espresso (QE).

The two calculations show good agreement in many respects and indicate that the 35Cl resonance frequencies in DTN are likely about an order of magnitude lower than initially believed. Because chlorine NQR transitions occurred around 30 MHz in both para-dichlorobenze and NaClO_3, DTN was initally thought to be likely to show 35Cl NQR in the 29-31 MHz band. This was tested first by Robert Baker (REU Student, 2010) and then in 2012-2013 by myself to no avail.

There is no particular reason 35Cl should have to have a resonance in this 30 MHz band. 35Cl NQR measurements have been made in many materials below 18 MHz and above 60 MHz. A high degree of symmetry in a given structure can result in very low NQR frequencies, with correspondingly very poor S/N. Though all nuclei with spin S >= 1 are guaranteed to have at least one non-zero NQR transition frequency, NQR is not always found. This is likely due to poor S/N when the EFG is small and the transition frequency is very low.

Evidence for the validity of the DFT calculations is that the prediction for NaClO3 35Cl NQR matches observation remarkably well. Additionally, since both DFT packages output the full EFG tensor, the NQR transitions for 14N were also predicted by the same calculations. There are sixteen nitrogen nuclei in each unit cell of DTN (right). DTN contains 4 independent thiourea molecules in each unit cell, each having 4 nitrogen centers.

The NQR frequencies calculated by CASTEP and QE for 14N in DTN match very well with those observed in pure thiourea from literature (David H. Smith and R. M. Cotts). The paper by Smith and Cotts quotes NQR in 14N in pure thiourea NQR at 2.6 MHz and 2.0 MHz for inequivalent sites at room temperature. In the absense of any DFT calculation, the 14N resonance in pure thourea would be a best first guess as to the NQR transition for 14N in DTN.

As the NQR frequency at a given nuclear site is directly proportional to Vzz (the electric field gradient, or stretching), crystals with a high degree of symmetry will have low NQR frequencies. Inspection of the crystal structure superficially indicates a widely symmetric nuclear distribution about the chlorine sites, as the thiourea groups are distributed symmetrically and the structure parameters show each Cl is roughly equidistant to both adjacent Ni nuclei.

```The quadrupole coupling constant and NQR transitions
The quadrupole coupling constant Cq is defined as

Cq = e*Vzz*Q/h [1]

where Vzz is the largest absolute eigenvalue, e is the electron charge, Q is the quadrupole moment and h is planck's constant.

We consider a nucleus of spin S and define

A = eVzzQ / (4S(2S-1)) [2]

where S is spin and Q is the quadrupole moment. Then the NQR frequency or (frequencies) are given by

f_nqr = 3|A| / h (2 |m| +1) [3]

where m is the lowest of the two levels m and m+1 over which a transition has occurred. For integral spins there are S unique transitions. For half integral spins there are S – ½ unique transitions.
NQR is observed in nuclei with I=3/2, I=5/2 and I=7/2. For I=1/2., there are no transitions. The I=5/2 and I=7/2 are indeed complicated, but for I=3/2 there is a degeneracy that causes there to be one frequency of half the quadrupole coupling constant, which makes the single transition frequency in the case of I=3/2 equal to one half the quadrupole coupling constant Cq = eVzzQ/h = e^2Qq/h.

NQR Frequencies for half-integral spins
Chlorine has I = 3/2 so there is only 1 transition. Using I=3/2 in the formula above, we obtain

f_nqr = (1/2) eVzzQ/h = (1/2) Cq [4]

for axially symmetric field gradients. For non-axially symmetryic gradients, we define the assymetry parameter η = (Vxx-Vyy)/Vzz. η=0 in the axially symmetric case, where Vzz is the only nonzero component. When η is non-zero, the transitions for I=3/2 are

f_nqr ~ (1/2) eVzzQ/h * (1 + 1/3 η^2)^(1/2) [5]

In general for half integral spin of I = n/2 n 3, 5, 7,... use what is given in Hahn and you should find the coefficients are 3/10, and 3/20 for I=5/2

f_nqr_[5/2 --> 3/2] = (3/10) eVzzQ/h [6]
f_nqr_[3/2->1/2] = (3/20) eVzzQ/h [7]

NQR frequencies for integral spins
For integral spins, we use the formula [2] again. For for I=1 and an axially symmetric field gradient (η=0) , the NQR transition frequency is

f_nqr = (3/4) eVzzQ/h = (3/4) Cq [8]

EFG->C: https://github.com/tfgg/magres-format/blob/master/magres/constants.

CASTEP EFG DATA (DTN – 35Cl)
relaxed
Cl 1 Cq: 8.4298 (MHz) Eta: 0.1272
Cl 2 Cq: 0.1705 (MHz) Eta: 0.5898
Cl 3 Cq: 0.1866 (MHz) Eta: 0.5340
Cl 4 Cq: 8.4513 (MHz) Eta: 0.1264
unrelaxed
Cl 1 Cq: -15.7965 (MHz) Eta: 0.0089
Cl 2 Cq: -16.9778 (MHz) Eta: 0.0048
Cl 3 Cq: -16.9515 (MHz) Eta: 0.0048
Cl 4 Cq: -15.8271 (MHz) Eta: 0.0089

QE EFG DATA (DTN – 35Cl)
relaxed
Cl 1 Cq= 9.4969 MHz Eta=-0.00000
Cl 2 Cq= -0.9388 MHz Eta= 0.00000
Cl 3 Cq= 9.4869 MHz Eta= 0.00000
Cl 4 Cq= -0.9298 MHz Eta= 0.00000

CASTEP EFG DATA (DTN – 14N)
relaxed
N 1 Cq: -3.7067 (MHz) Eta: 0.3400
N 2 Cq: -3.3560 (MHz) Eta: 0.4226
N 3 Cq: -3.6918 (MHz) Eta: 0.3463
N 4 Cq: -3.3621 (MHz) Eta: 0.4256
N 5 Cq: -3.6917 (MHz) Eta: 0.3468
N 6 Cq: -3.3533 (MHz) Eta: 0.4231
N 7 Cq: -3.7068 (MHz) Eta: 0.3410
N 8 Cq: -3.3535 (MHz) Eta: 0.4228
N 9 Cq: -3.6920 (MHz) Eta: 0.3466
N 10 Cq: -3.3611 (MHz) Eta: 0.4251
N 11 Cq: -3.7067 (MHz) Eta: 0.3409
N 12 Cq: -3.3524 (MHz) Eta: 0.4230
N 13 Cq: -3.7073 (MHz) Eta: 0.3399
N 14 Cq: -3.3563 (MHz) Eta: 0.4225
N 15 Cq: -3.6927 (MHz) Eta: 0.3469

unrelaxed
N 1 Cq: 3.0165 (MHz) Eta: 0.8408
N 2 Cq: -2.6850 (MHz) Eta: 0.9198
N 3 Cq: 3.0154 (MHz) Eta: 0.8423
N 4 Cq: -2.6885 (MHz) Eta: 0.9216
N 5 Cq: 3.0164 (MHz) Eta: 0.8419
N 6 Cq: -2.6841 (MHz) Eta: 0.9213
N 7 Cq: 3.0179 (MHz) Eta: 0.8414
N 8 Cq: -2.6854 (MHz) Eta: 0.9186
N 9 Cq: 3.0188 (MHz) Eta: 0.8421
N 10 Cq: -2.6833 (MHz) Eta: 0.9207
N 11 Cq: 3.0214 (MHz) Eta: 0.8411
N 12 Cq: -2.6802 (MHz) Eta: 0.9177
N 13 Cq: 3.0199 (MHz) Eta: 0.8406
N 14 Cq: -2.6798 (MHz) Eta: 0.9189
N 15 Cq: 3.0198 (MHz) Eta: 0.8417
N 16 Cq: -2.6788 (MHz) Eta: 0.9203

QE EFG DATA (DTN – 14N)
relaxed
N 45 Cq= -3.5069 MHz eta= 0.36826
N 46 Cq= -3.5084 MHz eta= 0.36870
N 47 Cq= -3.5084 MHz eta= 0.36870
N 48 Cq= -3.1864 MHz eta= 0.38355
N 49 Cq= -3.1864 MHz eta= 0.38355
N 50 Cq= -3.1880 MHz eta= 0.38408
N 51 Cq= -3.1864 MHz eta= 0.38355
N 52 Cq= -3.1864 MHz eta= 0.38355
N 53 Cq= -3.1880 MHz eta= 0.38408
N 54 Cq= -3.5069 MHz eta= 0.36826
N 55 Cq= -3.5084 MHz eta= 0.36870
N 56 Cq= -3.5084 MHz eta= 0.36870
N 57 Cq= -3.5069 MHz eta= 0.36826
N 58 Cq= -3.1880 MHz eta= 0.38408
N 59 Cq= -3.1880 MHz eta= 0.38408
N 60 Cq= -3.5069 MHz eta= 0.36826
Q= 2.04 1e-30 m^2

For NaClO3, CASTEP arrived at an average NQR frequency (over 4 sites) of 28.9 MHz. For chlorine in DTN, CASTEP calculated NQR at 8.5 MHz (unrelaxed) and 4.5 MHz (relaxed). For 14N in DTN, the calculation yields 2.9 MHz and 1.6 MHz for inequivalent sites in the relaxed EFG calculation. The difference between the relaxed and unrelaxed calculation lies in the structural data preparation. In the unrelaxed case, structure data from direct measurements in literature is fed directly into CASTEP and used. In the relaxed case, the structure parameters itself are adjusted using GIPAW, and then the EFG is calculated.

```

## It may be time to buy some LTC again. Here’s why. The golden rules for altcoin trading.

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…

## Hexadecimal word games. Not fail = foresee!

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?

## A precise analysis of an L-network for impedance matching below 3 MHz. The desire for “an ideal” characterization of the circuit parameter space for utility in fabrication by hand. The reduction to a pure mathematics problem.

### Introduction

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

View post on imgur.com

### 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

### enforce

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

Note:

* 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.

——-

### Final thoughts.

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.

## Sad grep I – V

``` ~\$ sad grep poetry of the internet ```

I
``` # facebook | grep brains```

``` ```

```# ```

II

``` # cat ./telecom| egrep (options|competit.?.?.?.?.?) # # dmesg comcast ```

```# ```

III
``` ~\$ ls```

``` ~\$ ~\$ pwd home/fridge/ ~\$ ls -laR | grep food ```

```~\$ ```

IV
``` # cat ./reddit/r/bitcoin/* | grep criminals```

``` Display all 169,236 possibilities? (y or n) ```

V
``` ~\$ ./github/ | grep working code```

``` OSX ```

```~\$ ```

## Does a single electron moving at constant velocity generate electromagnetic waves?

Nope.

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.

Start here

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

so

∫ B dx = μ I + μ ε ∂/∂t ( q/ε )

but ∂/∂t ( q ) = 0

so that term doesn’t change things.