472 research outputs found
Multi-latin squares
A multi-latin square of order and index is an array of
multisets, each of cardinality , such that each symbol from a fixed set of
size occurs times in each row and times in each column. A
multi-latin square of index is also referred to as a -latin square. A
-latin square is equivalent to a latin square, so a multi-latin square can
be thought of as a generalization of a latin square.
In this note we show that any partially filled-in -latin square of order
embeds in a -latin square of order , for each , thus
generalizing Evans' Theorem. Exploiting this result, we show that there exist
non-separable -latin squares of order for each . We also show
that for each , there exists some finite value such that for
all , every -latin square of order is separable.
We discuss the connection between -latin squares and related combinatorial
objects such as orthogonal arrays, latin parallelepipeds, semi-latin squares
and -latin trades. We also enumerate and classify -latin squares of small
orders.Comment: Final version as sent to journa
Universal quadratic forms, small norms and traces in families of number fields
We obtain good estimates on the ranks of universal quadratic forms over
Shanks' family of the simplest cubic fields and several other families of
totally real number fields. As the main tool we characterize all the
indecomposable integers in these fields and the elements of the codifferent of
small trace. We also determine the asymptotics of the number of principal
ideals of norm less than the square root of the discriminant.Comment: 20 page
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