1,106 research outputs found
Additive triples of bijections, or the toroidal semiqueens problem
We prove an asymptotic for the number of additive triples of bijections
, that is, the number of pairs of
bijections such that
the pointwise sum is also a bijection. This problem is equivalent
to counting the number of orthomorphisms or complete mappings of
, to counting the number of arrangements of
mutually nonattacking semiqueens on an toroidal chessboard, and to
counting the number of transversals in a cyclic Latin square. The method of
proof is a version of the Hardy--Littlewood circle method from analytic number
theory, adapted to the group .Comment: 22 page
Kronecker products and the RSK correspondence
The starting point for this work is an identity that relates the number of
minimal matrices with prescribed 1-marginals and coefficient sequence to a
linear combination of Kronecker coefficients. In this paper we provide a
bijection that realizes combinatorially this identity. As a consequence we
obtain an algorithm that to each minimal matrix associates a minimal component,
with respect to the dominance order, in a Kronecker product, and a
combinatorial description of the corresponding Kronecker coefficient in terms
of minimal matrices and tableau insertion. Our bijection follows from a
generalization of the dual RSK correspondence to 3-dimensional binary matrices,
which we state and prove. With the same tools we also obtain a generalization
of the RSK correspondence to 3-dimensional integer matrices
Additive monotones for resource theories of parallel-combinable processes with discarding
A partitioned process theory, as defined by Coecke, Fritz, and Spekkens, is a
symmetric monoidal category together with an all-object-including symmetric
monoidal subcategory. We think of the morphisms of this category as processes,
and the morphisms of the subcategory as those processes that are freely
executable. Via a construction we refer to as parallel-combinable processes
with discarding, we obtain from this data a partially ordered monoid on the set
of processes, with f > g if one can use the free processes to construct g from
f. The structure of this partial order can then be probed using additive
monotones: order-preserving monoid homomorphisms with values in the real
numbers under addition. We first characterise these additive monotones in terms
of the corresponding partitioned process theory.
Given enough monotones, we might hope to be able to reconstruct the order on
the monoid. If so, we say that we have a complete family of monotones. In
general, however, when we require our monotones to be additive monotones, such
families do not exist or are hard to compute. We show the existence of complete
families of additive monotones for various partitioned process theories based
on the category of finite sets, in order to shed light on the way such families
can be constructed.Comment: In Proceedings QPL 2015, arXiv:1511.0118
Recollements of Module Categories
We establish a correspondence between recollements of abelian categories up
to equivalence and certain TTF-triples. For a module category we show,
moreover, a correspondence with idempotent ideals, recovering a theorem of
Jans. Furthermore, we show that a recollement whose terms are module categories
is equivalent to one induced by an idempotent element, thus answering a
question by Kuhn.Comment: Comments are welcom
Jordan weak amenability and orthogonal forms on JB*-algebras
We prove the existence of a linear isometric correspondence between the
Banach space of all symmetric orthogonal forms on a JB-algebra
and the Banach space of all purely Jordan generalized derivations
from into . We also establish the existence of a
similar linear isometric correspondence between the Banach spaces of all
anti-symmetric orthogonal forms on , and of all Lie Jordan
derivations from into
Equivariant Kasparov theory and generalized homomorphisms
Let G be a locally compact group. We describe elements of KK^G (A,B) by
equivariant homomorphisms, following Cuntz's treatment in the non-equivariant
case. This yields another proof for the universal property of KK^G: It is the
universal split exact stable homotopy functor.
To describe a Kasparov triple (E, phi, F) by an equivariant homomorphism, we
have to arrange for the Fredholm operator F to be equivariant. This can be done
if A is of the form K(L^2G) otimes A' and more generally if the group action on
A is proper in the sense of Rieffel and Exel.Comment: 22 pages, final version, will appear in K-Theory added references and
a few additional explanations to the tex
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