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Parity of Sets of Mutually Orthogonal Latin Squares
Every Latin square has three attributes that can be even or odd, but any two
of these attributes determines the third. Hence the parity of a Latin square
has an information content of 2 bits. We extend the definition of parity from
Latin squares to sets of mutually orthogonal Latin squares (MOLS) and the
corresponding orthogonal arrays (OA). Suppose the parity of an
has an information content of bits. We show that
. For the case corresponding to projective
planes we prove a tighter bound, namely when
is odd and when is even. Using the
existence of MOLS with subMOLS, we prove that if
then for all sufficiently large .
Let the ensemble of an be the set of Latin squares derived by
interpreting any three columns of the OA as a Latin square. We demonstrate many
restrictions on the number of Latin squares of each parity that the ensemble of
an can contain. These restrictions depend on and
give some insight as to why it is harder to build projective planes of order than for . For example, we prove that when it is impossible to build an for which all
Latin squares in the ensemble are isotopic (equivalent to each other up to
permutation of the rows, columns and symbols)
Multi-layer S=1/2 Heisenberg antiferromagnet
The multi-layer square lattice Heisenberg antiferromagnet with
up to 6 layers is studied via various series expansions. For the systems with
an odd number of coupled planes, the ground-state energy, staggered
magnetization, and triplet excitation spectra are calculated via two different
Ising expansions. The systems are found to have long range N\'eel order and
gapless excitations for all ratios of interlayer to intralayer couplings, as
for the single-layer system. For the systems with an even number of coupled
planes, there is a second order transition point separating the gapless Ne\'el
phase and gapped quantum disordered spin liquid phase, and the critical points
are located via expansions in the interlayer exchange coupling. This transition
point is found to vary about inversely as the number of layers. The triplet
excitation spectra are also computed, and at the critical point the normalized
spectra appear to follow a universal function, independent of number of layers.Comment: 13 pages plus 8 figure
Central aspects of skew translation quadrangles, I
Except for the Hermitian buildings , up to a combination
of duality, translation duality or Payne integration, every known finite
building of type satisfies a set of general synthetic
properties, usually put together in the term "skew translation generalized
quadrangle" (STGQ). In this series of papers, we classify finite skew
translation generalized quadrangles. In the first installment of the series, as
corollaries of the machinery we develop in the present paper, (a) we obtain the
surprising result that any skew translation quadrangle of odd order is
a symplectic quadrangle; (b) we determine all skew translation quadrangles with
distinct elation groups (a problem posed by Payne in a less general setting);
(c) we develop a structure theory for root-elations of skew translation
quadrangles which will also be used in further parts, and which essentially
tells us that a very general class of skew translation quadrangles admits the
theoretical maximal number of root-elations for each member, and hence all
members are "central" (the main property needed to control STGQs, as which will
be shown throughout); (d) we solve the Main Parameter Conjecture for a class of
STGQs containing the class of the previous item, and which conjecturally
coincides with the class of all STGQs.Comment: 66 pages; submitted (December 2013
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