981 research outputs found
The set of autotopisms of partial Latin squares
Symmetries of a partial Latin square are determined by its autotopism group.
Analogously to the case of Latin squares, given an isotopism , the
cardinality of the set of partial Latin squares which
are invariant under only depends on the conjugacy class of the latter,
or, equivalently, on its cycle structure. In the current paper, the cycle
structures of the set of autotopisms of partial Latin squares are characterized
and several related properties studied. It is also seen that the cycle
structure of determines the possible sizes of the elements of
and the number of those partial Latin squares of this
set with a given size. Finally, it is generalized the traditional notion of
partial Latin square completable to a Latin square.Comment: 20 pages, 4 table
Absolutely Maximally Entangled states, combinatorial designs and multi-unitary matrices
Absolutely Maximally Entangled (AME) states are those multipartite quantum
states that carry absolute maximum entanglement in all possible partitions. AME
states are known to play a relevant role in multipartite teleportation, in
quantum secret sharing and they provide the basis novel tensor networks related
to holography. We present alternative constructions of AME states and show
their link with combinatorial designs. We also analyze a key property of AME,
namely their relation to tensors that can be understood as unitary
transformations in every of its bi-partitions. We call this property
multi-unitarity.Comment: 18 pages, 2 figures. Comments are very welcom
Algebraic Structures and Variations: From Latin Squares to Lie Quasigroups
In this Master\u27s Thesis we give an overview of the algebraic structure of sets with a single binary operation. Specifically, we are interested in quasigroups and loops and their historical connection with Latin squares; considering them in both finite and continuous variations. We also consider various mappings between such algebraic objects and utilize matrix representations to give a negative conclusion to a question concerning isotopies in the case of quasigroups
Symmetry Regularization
The properties of a representation, such as smoothness, adaptability, generality, equivari- ance/invariance, depend on restrictions imposed during learning. In this paper, we propose using data symmetries, in the sense of equivalences under transformations, as a means for learning symmetry- adapted representations, i.e., representations that are equivariant to transformations in the original space. We provide a sufficient condition to enforce the representation, for example the weights of a neural network layer or the atoms of a dictionary, to have a group structure and specifically the group structure in an unlabeled training set. By reducing the analysis of generic group symmetries to per- mutation symmetries, we devise an analytic expression for a regularization scheme and a permutation invariant metric on the representation space. Our work provides a proof of concept on why and how to learn equivariant representations, without explicit knowledge of the underlying symmetries in the data.This material is based upon work supported by the Center for Brains, Minds and Machines (CBMM), funded by NSF STC award CCF-1231216
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