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 $F$ is a simple, infinite, semidistributive lattice. Its existence was recently proved by R. Freese and J.\,B. Nation. Let $\mathsf{B}_n$ denote the Boolean lattice with $n$ atoms. For a lattice $K$, let $K^+$ denote $K$ with a new unit adjoined. We prove that the finite distributive lattices: $\mathsf{B}_0^+, \mathsf{B}_1^+,\mathsf{B}_2^+, \dots$ can be represented as congruence lattices of infinite semidistributive lattices. The case $n = 0$ is the Freese-Nation result, which is utilized in the proof. We also prove some related representation theorems