Unimodular property

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

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 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 ... 

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. 

It is both conceptually and computationally useful to cast the equation into a form which can be interpreted more easily. This possibility hinges on a simple property of determinants the fact that the value of a determinant is invariant against a unimodular transformation amongst its rows (or columns). The molecular orbitals of which the single-determinant wavefunction is composed may be... 

Such a series has the important property of being unimodular, which means that consecutive terms have... 

This is a set packing formulation and is NP-hard [89]. There are special cases under which the structure of this problem simplifies and allows for polynomial time solutions. Many special cases arise out of constraints that reduce the constraint matrix to be totally unimodular [35]. A common example is the case where adjacent plots of land are being sold and bidders might want multiple plots but they need to be adjacent. However, real world problems will often not satisfy the fairly severe restrictions that provide a totally unimodular constraint matrix. Moreover, if the bidding language is not expressive then this can interact with the incentive properties of an auction because a bidder is not able to express her true valuation, even if that would be her equilibrium strategy. We wait until Section 4 for an extensive discussion of the interaction between computational constraints and incentives. 

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