User:MRFS/Pending

Here is a z̄. Unfortunately the overbar is barely visible!

where P is a permutation matrix and each B_i is a square matrix that is either irreducible or zero. Now if A is non-negative then so too is each block of PAP^-1, moreover the spectrum of A is just the union of the spectra of the B_i.

[ The inverse of PAP^-1 (if it exists) must have diagonal blocks of the form B_i^-1 so if any B_i isn't invertible then neither is PAP^-1 or A. Conversely let D be the "diagonal" component of PAP^-1, in other words PAP^-1 with the asterisks zeroised. If each B_i is invertible then so is D and D^-1(PAP^-1) is equal to the identity plus a nilpotent matrix. But such a matrix is always invertible (if N^k = 0 the inverse of 1 - N is 1 + N + N² + ... + N^k-1) so PAP^-1 and A are both invertible. ]

Therefore many of the spectral properties of A may be deduced by applying the theorem to the irreducible B_i. For example the Perron root is the maximum of the spectral radii of the B_i. Whilst there will still be eigenvectors with non-negative components it is quite possible that none of these will be positive.

[ To see this let D be the "diagonal" component of PAP^-1, in other words PAP^-1 with the asterisks zeroised. Then D is invertible iff each B_i is invertible and D^-1(PAP^-1) is equal to the identity plus a nilpotent matrix. But such a matrix is always invertible (if N^k = 0 the inverse of 1 - N is 1 + N + N² + ... + N^k-1) so PAP^-1 and A are both invertible. XXXX ]

Formula

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Uninteresting centers

Assume a,b,c are real variables and let α,β,γ be any three real constants.

$f(a,b,c)={\begin{cases}\alpha &{\text{if }}a<b{\text{ and }}a<c\quad {\text{ (equivalently the first variable is the smallest)}},\\\gamma &{\text{if }}a>b{\text{ and }}a>c\quad {\text{ (equivalently the first variable is the largest)}},\\\beta &{\text{otherwise}}\quad \qquad \qquad {\text{ (equivalently the first variable is in the middle)}}.\end{cases}}$

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Let Δ be the area of the triangle with sides a, b, c. Whenever no angle of ΔABC exceeds 120° the sum L of the distances from the Fermat point to the vertices is given by the formula

L = √ ((a² + b² + c² + 4Δ√3)/2

This formula also occurs in the following two seemingly unrelated situations.

• a, b, c are the distances from an internal point to the vertices of an equilateral triangle of side L.

• L is the height of the largest equilateral triangle that can be circumscribed about a triangle with sides a, b, c.

Despite the square roots there are infinitely many solutions (a,b,c,L) in integers. Some like (3,5,7,8) and (7,8,13,15) are simple in the sense that the triangle has an angle of exactly 120° but there are plenty of more interesting ones such as (57,65,73,112) and (73,88,95,147).

Napoleon's Theorem is said to be one of the most-often rediscovered results in mathematics. C&G reference - whilst the theorem has been attributed to Napoleon the possibility of his knowing enough geometry for this feat is as questionable as the possibility of his knowing enough English to compose the famous palindrome : able was i ere i saw elba.

see the web page on the square root symbol in html and css - ugh! best way is probably to resize and drop the superscripts slightly.

Σ√;

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$L={\sqrt {{\mbox{if}}~a\ {\mbox{is even}}^{2}+b^{2}+c^{2}+4\Delta \surd 3)/2}}$ .

A(√3,-30°) and B(√3,30°).

The best possible local realist imitation (red) for the quantum correlation of two spins in the singlet state (blue), insisting on perfect anti-correlation at 0°, perfect correlation at 180°. Many other possibilities exist for the classical correlation subject to these side conditions, but all are characterized by sharp peaks (and valleys) at 0°, 180°, and 360°, and none has more extreme values (±0.5) at 45°, 135°, 225°, and 315°. These values are marked by stars in the graph, and are the values measured in a standard Bell-CHSH type experiment: QM allows ±1/√2 = ±0.7071..., local realism predicts ±0.5 or less.

Here is the original diagram from Bell's Theorem that was removed on 11/3/22.

The image downloaded as a png file but the text is separate. If it is to be used in LaTeX then the png will need converted to pdf.