where we note that the first component is fully fixed by $F_3$. This neatly illustrates the $(1,2)$ block form of transformed magic mixing matrices, which is basically a combination of the diagonal of $1$-circulant eigenvalues and a $2$-circulant codiagonal. For each $R_2$ factor, there is an $F_3$ transform that fixes it. It follows that for a suitable twisted Fourier transform, the whole triple product will remain fixed.
and the identity matrix. But a mixed $XYZ$ product results in the same two factor (tribimaximal type) form, since a scaled identity does not alter the mixing matrix, and the use of $i F_2$ only affects the phases.
This variation does not greatly alter the five large CKM entries, or the entry corresponding to $c$, but it does move the probability matrix well away from current experimental bounds. At small $b$, the phase $2 \beta_s$ (called phi on the graph) goes like $2abc$.
Our views of the physical world are changing rapidly. Humanity's continuing search for coherent structures in physics, biology, and cosmology has frequently led to surprises as well as confusion. Discovering new phenomena is one thing, putting them into context with other pieces of knowledge, and inferring their fundamental consequences is quite something else. There are controversies, differences of opinion, and sometimes even religious feelings which come into play. These should be discussed openly.From the website of the journal Foundations of Physics, which appears to have accepted Carl Brannen's paper on spin path integrals. Congratulations, Carl.
where it is easy to see that the four interesting phases are given by
which one may easily verify on an old calculator, remembering to normalise and take square roots. Once again the $(td)$ term is a little higher than expected. The third variable $c$ only enters as the $(ub)$ correction to tribimaximal type mixing. The astute reader will see that this $(ub)$ term destroys the unitarity of the CKM matrix, although unitarity is restored in the full product form.
For the neutrino mixing choice of $(a,b) = (\sqrt{2}, 1)$ we observe that these nine numbers become the entries of the tribimaximal matrix. In order to generalise to the $27$ possibilities for three variables, we will place these $9$ numbers in a square, 
which is basically the mixing matrix. Interpreting the zero as $(a^2 + 1)(b^2 + 1)$, the three points on the edges all obey the sum rule $P + Q = R$. This rule extends naturally to the cube in three dimensions, where the three parameters $(a,b,c)$ are symmetrically represented.
Kea, I used to think that they win if we leave. Now I think the only way we'll win is if we leave. In droves. Then they'll have to fund studies to find out why we're leaving. Then they'll overhaul the system. It won't happen fast enough to help people like us in general.
We often say that $R_{3}^{3}$ is the identity, but strictly speaking it is $-i \cdot \mathbf{1}$. In quark boson notation, where phases are written as powers of the angle $\pi / 6$, we have 
and so on. That is, the true relation is $R_{3}^{12} = \mathbf{1}$. As always, the Fourier matrix satisfies $F^{4} = \mathbf{1}$. This is analogous to the rule $R_{2}^{8} = \mathbf{1}$ for the $2 \times 2$ operator, which also relies on the complex number $i$. The Fourier matrix encourages the circulant $R_d$ matrices to take on that extra factor of four. So in dimension $6$, which is the best dimension for studying the particle spectrum, we would have an $R_6$ operator obeying $R_{6}^{24} = \mathbf{1}$. If we rephased the matrix $R_3$ by the factor $-i$, then the cubic rule is obeyed. This matrix is in fact precisely a quark boson matrix for $\overline{u}_{L}$.
The CKM elements $| V_{td} |$ and $| V_{ts} |$ cannot be measured from tree level decays of the top quark, so one has to rely on determinations from $B \overline{B}$ oscillations mediated by box diagrams or loop mediated rare K and B decays. Theoretical uncertainties in hadronic effects limit the accuracy of the current determinations.The 2008 review suggests that $| V_{td} / V_{ts} | < 0.216$, and this bound is broken by the parameter fits (which used this data) in recent posts. Although this fit was rough, it yields a value of $0.242$, which is significantly higher than the Fermilab estimate. It is still within the other bounds given by $B$ and $K$ rates. Note, however, that the current best estimates for CKM parameters also puts $| V_{td} / V_{ts} | $ at too high a value, so perhaps one should not take this bound too seriously, since it relies on Standard Model calculations.
which displays the Pythagorean triple $(7,24,25)$. There are many ways to factor these scaled matrices into two or more components.
The email I wrote was an attempt to deal with a difficult and complex organisational problem internal to the conference.Translation: some people at Imperial, perhaps not Valentini personally, like to protect the university's fine reputation. Valentini seems to think this is OK, because that's how things work, right? But he realises that
The internet is an evolving medium, and one can query the suitability of standard constraints in this context.Oh, yes, we can. And we will. Let us not forget the guy with the broken guitar, whose YouTube video resulted in a $10%$ fall in stock value for United Airlines. An Electronic Mob it may be, but revolutionary mobs are not merely noisy crowds. And I would like somebody to explain to me how I'm less human than people who play 20th century politics.
leaving zero sum corrections given by 
where $\epsilon$ can be zero to within experimental precision. Note that the $704/24$ is just $29$ and a third. This is the norm square matrix, so on taking square roots, the $144$ becomes $12$ and so on. Numerology, maybe, but $0.999133$ still matches an $R_2$ factor at $r = 24$.
is the general case for a super magic circulant, because scale factors will always be absorbed into the normalisation. Under an interchange of the real and complex parameters in $M$, the row sums and norm square sums remain unaltered. This means that $(r,1)$ is equivalent to $(r^{-1},1)$, a kind of S duality. For example, in the tribimaximal case it doesn't matter if we use $\sqrt{2}$ or $\sqrt{2}^{-1}$. This is an inversion of parameters with respect to the unit circle, as in 
Thus magic circulants, although very simple, provide an interesting parameter space in which to study mixing matrices. Note that the inverse of an $R_2$ factor is given by sending $i$ to $-i$, so we might think of the (charged lepton) identity mixing matrix as two inverse $R_2$ factors with parameters $a = r$ and $b = -r$. This gives three parameters in total for lepton mixing.
Note that the choice of $b = 24$ appears to be quite optimal, but I was not doing any real optimization.
