2,419 research outputs found
Congruence lattices of semilattices
The main result of this paper is that the class of congruence
lattices of semilattices satisfies no nontrivial lattice
identities. It is also shown that the class of subalgebra
lattices of semilattices satisfies no nontrivial lattice identities.
As a consequence it is shown that if V is a semigroup variety
all of whose congruence lattices satisfy some fixed nontrivial
lattice identity, then all the members of V are groups with exponent dividing a fixed finite number
A class of infinite convex geometries
Various characterizations of finite convex geometries are well known. This
note provides similar characterizations for possibly infinite convex geometries
whose lattice of closed sets is strongly coatomic and lower continuous. Some
classes of examples of such convex geometries are given.Comment: 10 page
Ultra-Strong Optomechanics Incorporating the Dynamical Casimir Effect
We propose a superconducting circuit comprising a dc-SQUID with mechanically
compliant arm embedded in a coplanar microwave cavity that realizes an
optomechanical system with a degenerate or non-degenerate parametric
interaction generated via the dynamical Casimir effect. For experimentally
feasible parameters, this setup is capable of reaching the single-photon,
ultra-strong coupling regime, while simultaneously possessing a parametric
coupling strength approaching the renormalized cavity frequency. This opens up
the possibility of observing the interplay between these two fundamental
nonlinearities at the single-photon level.Comment: 7 pages, 1 figure, 1 tabl
Iterative solutions to the steady state density matrix for optomechanical systems
We present a sparse matrix permutation from graph theory that gives stable
incomplete Lower-Upper (LU) preconditioners necessary for iterative solutions
to the steady state density matrix for quantum optomechanical systems. This
reordering is efficient, adding little overhead to the computation, and results
in a marked reduction in both memory and runtime requirements compared to other
solution methods, with performance gains increasing with system size. Either of
these benchmarks can be tuned via the preconditioner accuracy and solution
tolerance. This reordering optimizes the condition number of the approximate
inverse, and is the only method found to be stable at large Hilbert space
dimensions. This allows for steady state solutions to otherwise intractable
quantum optomechanical systems.Comment: 10 pages, 5 figure
The Bases of Association Rules of High Confidence
We develop a new approach for distributed computing of the association rules
of high confidence in a binary table. It is derived from the D-basis algorithm
in K. Adaricheva and J.B. Nation (TCS 2017), which is performed on multiple
sub-tables of a table given by removing several rows at a time. The set of
rules is then aggregated using the same approach as the D-basis is retrieved
from a larger set of implications. This allows to obtain a basis of association
rules of high confidence, which can be used for ranking all attributes of the
table with respect to a given fixed attribute using the relevance parameter
introduced in K. Adaricheva et al. (Proceedings of ICFCA-2015). This paper
focuses on the technical implementation of the new algorithm. Some testing
results are performed on transaction data and medical data.Comment: Presented at DTMN, Sydney, Australia, July 28, 201
Lattices of quasi-equational theories as congruence lattices of semilattices with operators, part I
We show that for every quasivariety K of structures (where
both functions and relations are allowed) there is a semilattice S with
operators such that the lattice of quasi-equational theories of K (the dual
of the lattice of sub-quasivarieties of K) is isomorphic to Con(S;+; 0; F).
As a consequence, new restrictions on the natural quasi-interior operator
on lattices of quasi-equational theories are found
Planar, infinite, semidistributive lattices
An FN lattice is a simple, infinite, semidistributive lattice. Its
existence was recently proved by R. Freese and J.\,B. Nation. Let
denote the Boolean lattice with atoms. For a lattice ,
let denote with a new unit adjoined.
We prove that the finite distributive lattices: can be represented as congruence lattices
of infinite semidistributive lattices. The case is the Freese-Nation
result, which is utilized in the proof.
We also prove some related representation theorems
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