Unimodular

Thus, the two wave functions can at most differ by a unimodular complex number e1. It can be shown that the only possibilities occurring in nature are that either the two functions are identical (symmetric wave function, applies to particles called bosons which have inte-... 

It is known as the unitary unimodular group, or the special unitary group denoted by SU(2). Because of the extra condition on the determinant, SU(2) is a three-parameter group. 

Formula (5.29) is the special case (odd k values) of the Casimir operator of the more general so-called unitary unimodular group SU21+1, ie. 

Matrices which represent proper rotations are unimodular, that is they have determinant+ 1 and are unitary (orthogonal, if the space is real, as is ft3). Consider the set of all 2 x 2 unitary matrices with determinant +1. With binary composition chosen to... 

Rotations in a vector space of three orbitals are described by the group SO(3) of orthogonal 3x3 matrices with determinant +1. To embed the octahedral rotation group in this covering group one needs a matrix representation of O which also consists of orthogonal and unimodular 3x3 matrices. Such a matrix representation is sometimes called the fundamental vector representation of the point group. In the case of O the fundamental vector representation is Ti and not T2. Indeed the 7] matrices are unimodular, i.e. have determinant +1, while the determinants of the T2 matrices are equal to the characters of the one-dimensional representation A2. 

A pair of fractions, adjacent to each other in any sequence, has the property of unimodularity, such that for the pair h/k, l/m the quantity hm — kl = 1. 

Each rational fraction, h/k, defines a Ford circle with a radius and y-coordinate of 1/(2k2), positioned at an -coordinate h/k. The Ford circles of any unimodular pair are tangent to each other and to the x-axis. The circles, numbered from 1 to 4 in the construction overleaf, represent the Farey sequence of order 4. This sequence has the remarkable property of one-to-one correspondence with the natural numbers ordered in sets of 2k2 and in the same geometrical relationship as the Ford circles of 4. 

The branch above converges through to r and the lower branch to 0.5802. The numerators of the latter unimodular series ... 

The orbits from Venus to Ceres are represented by the unimodular series 4. In the outer system the Ford circles of only Uranus and Neptune are tangent, but the likeness to Farey sequences in atomic systems is sufficient to support the self-similarity conjecture. 

The principle that governs the periodic properties of atomic matter is the composition of atoms, made up of integral numbers of discrete sub-atomic units - protons, neutrons and electrons. Each nuclide is an atom with a unique ratio of protonsmeutrons, which defines a rational fraction. The numerical function that arranges rational fractions in enumerable order is known as a Farey sequence. A simple unimodular Farey sequence is obtained by arranging the fractions (n/n+1) as a function of n. The set of /c-modular sequences ... 

To explore the periodic structure of the set Sk, and hence of the stable nuclides, it is convenient to represent each fraction h/k by its equivalent Ford circle of radius rp = 1/2k2, centred at coordinates h/k, rp. Any unimodular pair of Ford circles are tangent to each other and to the x-axis. If the x-axis is identified with atomic numbers, touching spheres are interpreted to represent the geometric distribution of electrons in contiguous concentric shells. The predicted shell structure of 2k2 electrons per shell is 2, 8, 8, 18, 18, 32, 32, etc., with sub-shells defined by embedded circles, as 8=2+6,... 

Because MB[Pg.300]

This definition of f together with the unimodular property gives a useful constraint ... 

The unimodular condition requires only a normalizing factor on the matrix. Each term must be divided by the square root of the determinant,... 

A finite sequence is unimodal if it first increases and then decreases, unimodular... 

A square matrix is unimodular if its determinant is 1. unit circle... 

A noteworthy property of matrices found in Eqs. 1.21, 1.22 and 1.24 to 1.26 is their unimodularity - the determinant of every matrix is equal to 1 or -1 for the rotation and inversion (or roto-inversion) operations, respectively, which is shown for the rotation around Z in Eq. 1.27. 

---