1 The Rayleigh distribution is f(r) = (2λ) r exp( - λ r 2)

2 Rayleigh^2(x) = Limit x®y { Rayleigh(x) x Rayleigh(y)}

3 Correlation function between functions f(x) and g(x).
f * g = Σ f(x) g(x)
= IFT [FFT{f}. Conj (FFT{g}) ]
Where g(x) = f(x+h), ‘h’ the ‘delay’ in this case the mean level spacing.

4 The Rice pdf is RICE(R) = R/σ2. exp ( - [(R2 + A2 )/2σ2 ]) I0 (AR/σ2) ; where R = Noise + Signal, where Signal = A sinφ and A2/2σ2 is the mean signal-to-noise ratio (SNR is 10log10(A2/2σ2) in decibels). σ is an rms value. I0 is the Modified Bessel function of the 1st kind (Zeroth order). Note if A=0, we have the Rayleigh (Noise-limited) form. For example, use A = Ö2, s = 1 to give SNR = 1

5 As opposed to 5 log N gain for a Power Square Law detector, whose SNR = N/√N =√N, where signal is coherently summed (N) and noise is incoherently summed √N (rms).

6 Take Expectation operator over the whole Hermitian matrix infinite ensemble {}. In the beamformer correlator E< {(ai1 + ai2 +...)(aj1 + aj2 +...) + (bi1 + bi2 +...)(bj1 + bj2 +...)}> = E< (ai12 + ai22 +...) + (bi12 + bi22 +...) >, The only surviving residual terms are with the same indices i.e.            E< aij akm > ≠ 0 (non-zero) if i=k & j=m in the ensemble, but zero otherwise. All different terms will vary equally in sign and therefore cancel out in the (infinite) ensemble. The a’s and b’s are real and imaginary terms respectively (multiplied together with its conjugate i.e.|z|2 = z*conj(z) for complex z to give magnitude component where z = a + bi):

E<a mi a mk> = 0 if i¹k   mutually independent because the noise terms are independently sampled on different bit streams in the arguments

                    = Non-zero if i=k

E<b mi b mk> = 0 if i¹k  ditto by mutual independence

                    = Non-zero if i=k

E<a mi b mk> = 0 " i, k by the double angle formula sin2q º 2 sinq cosq, so mutually uncorrelated (covariance is zero; they are 90 degree out of phase). (If sampled on the same data stream then there is mutual dependence).

7 How many ways to make N? e.g. N = 5, there are seven ways = 5+0 = 4+1 = 3+2 = 3+1+1 = 2 + 2 + 1 = 2 +1+1+1 = 1+1+1+1+1

8 Note in QM, the Heisenberg Uncertainty Principle (HUP) is actually a direct consequence of this general principle (combined with the de Broglie/ Einstein formula p = h/λ). Namely that a small pulse length (a ‘chirp’) will support high freq components, whereas a long pulse length can only be supported by low frequency components (and therefore low Bandwidth). B. T ≈ 1 by Fourier Transforming the CW pulse

9 For the Rayleigh distribution, its mean E (Rayleigh) = μ = λ-1/2 Π(3/2) = λ-1/2 √π /2 = σ √π/2 ≈ 0.8862 σ, where σ is the RMS or √{variance} of the Normal real or imaginary parts (see next below).

10 In the Critical Strip then, the Root Sum Square RSS (of Log Zeta) is analytic because if any complex function f(z) is analytic then 

|f(z). conj {f(z)} |½ is analytic and its series representation extends down to Re(z) =½ within its own domain of convergence. This is not really surprising. Intuitively, the mean value is ‘smoother’ than the raw data.

11 The magnitude R=√(X2 + Y2) are Rayleigh distributed where X and Y are identical, mutually independent Normal distributions each with mean zero (N (0,1)); variance is normalised to one.

12 Equivalently read as "in the neighbourhood of"

13 Z(t) = exp( iψ(t) ) ζ (½ + it) is from the Riemann-Siegel equation, which is real and Zeta ζ(t) is evaluated on the Critical Line. Z(t) therefore has the same zeros as the zeta function on the Critical line (CL).

14 The analogy of the omega number is that of an infinite "coin tossing number". Randomly toss a coin with p= ½ for a head (or tail) forever. If a result gives a head, write ‘1’, otherwise for tail, write a ‘0’. You don’t now what the exact number instance will be in advance, only that for a very large number, (approximately) half the bits will be heads and half will be tails.

15 However for Zeta there are infinite number (Nops =∞) of these possibilities of δ ’s over an infinite range so that limit Nops*δ = ∞. It is this omega number that gives rise to an infinite variety of physical interpretations or degrees of freedom when the function is (forced to be) zero (at a root) as interpreted within the computation, the Infinitesimally small δ’s are like the "Ghost Variables" (~virtual particles) encountered in Quantum Mechanics.

16 This frequency power spectrum interpretation is exactly the same as that used in Sonar. A mathematical solution Z(t) exp(iω t) is regarded as signal with an associated phase ω, Z(t) is real. This gives a pure tone if we (Discrete) Fourier Transform over the correct frequency cell. Now add some (Gaussian) noise, and the effect of the phase ω is gradually reduced. Eventually, in the noise limited case (signal-to-noise=0), the phase has disappeared. This is seen over a continuous family of distribution curves of signal+noise called Rician distributions (named after S.O.Rice), which I recall are Modified Bessel functions of Zero Order which eventually converge to a noise limited case (the Rayleigh form).

17 Omega contaminates the character/bit sentences of any mathematical theorem that may prove RH, (or, for that matter, any algorithm that disproves it), which any such proof must be finite in order to be able to read it all and transfer it from person to person for programs of essential minimal size (e.g. such as in a MS WORD file/document format).

18 "Infinity" can be (finitely) coded as "Take a number, and add one repeatedly." For example in the L27 system of numeration (Ref Rudy Rucker, Mind Tools - The Mathematics of Information, Penguin Books). An algorithm is finite may have an infinite computation. For instance “one divide by three” is a finite sentence, but its output is = 0.3333333333333333 … (as a recurring decimal), which is not particularly interesting number (lacking any variety), although it is an infinitely long string of numbers.

19 The number N can be reduced to an 'unaliased' representation if it contains non-trivial integer factors p and q, such that N = p x q,  where the maximum bit length [entropy] is compressed to smaller numbers p and q with the multiplication operator "x" may itself be represented as L27 char.

20 The Bohr condition for the quantisation of the angular momentum (mvr) is

mvr = n.ħ       n=1,2,3,4 ....    where ħ = h/2p

By de Broglie  (r/l) = n/2p    e.g. for n=1; (r/l) = 1/2p » 0.1591549

21 The Balmer Series for the 4 visible lines of the Hydrogen spectrum

1/l = R (1/22 - 1/n2)    where R is Ryberg's Constant

(1/22 - 1/42) = 0.1875;  (1/22 - 1/32) ~ 0.1389