9 research outputs found
Checking Admissibility Using Natural Dualities
This paper presents a new method for obtaining small algebras to check the
admissibility-equivalently, validity in free algebras-of quasi-identities in a
finitely generated quasivariety. Unlike a previous algebraic approach of
Metcalfe and Rothlisberger that is feasible only when the relevant free algebra
is not too large, this method exploits natural dualities for quasivarieties to
work with structures of smaller cardinality and surjective rather than
injective morphisms. A number of case studies are described here that could not
be be solved using the algebraic approach, including (quasi)varieties of
MS-algebras, double Stone algebras, and involutive Stone algebras
Anisotropic intrinsic lattice thermal conductivity of phosphorene from first principles
Phosphorene, the single layer counterpart of black phosphorus, is a novel
two-dimensional semiconductor with high carrier mobility and a large
fundamental direct band gap, which has attracted tremendous interest recently.
Its potential applications in nano-electronics and thermoelectrics call for a
fundamental study of the phonon transport. Here, we calculate the intrinsic
lattice thermal conductivity of phosphorene by solving the phonon Boltzmann
transport equation (BTE) based on first-principles calculations. The thermal
conductivity of phosphorene at is
(zigzag) and
(armchair), showing an obvious anisotropy along different directions. The
calculated thermal conductivity fits perfectly to the inverse relation with
temperature when the temperature is higher than Debye temperature (). In comparison to graphene, the minor contribution around
of the ZA mode is responsible for the low thermal conductivity of
phosphorene. In addition, the representative mean free path (MFP), a critical
size for phonon transport, is also obtained.Comment: 5 pages and 6 figures, Supplemental Material available as
http://www.rsc.org/suppdata/cp/c4/c4cp04858j/c4cp04858j1.pd
Canonical extensions of bounded lattices and natural duality for default bilattices
This thesis presents results concerning canonical extensions of bounded lattices and natural dualities for quasivarieties of default bilattices. Part I is dedicated to canonical extensions, while Part II focuses on natural duality for default bilattices.
A canonical extension of a lattice-based algebra consists of a completion of the underlying lattice and extensions of the additional operations to the completion. Canonical extensions find rich application in providing an algebraic method for obtaining relational semantics for non-classical logics.
Part I gives a new construction of the canonical extension of a bounded lattice. The construction is done via successive applications of functors and thus provides an elegant exposition of the fact that the canonical extension is functorial.
Many existing constructions are described via representation and duality theorems. We demonstrate precisely how our new formulation relates to existing constructions as well as proving new results about complete lattices constructed from graphs. Part I ends with an analysis of the untopologised structures used in two methods of construction of canonical extensions of bounded lattices: the untopologised graphs used in our new construction, and the so-called `intermediate structure'. We provide sufficient conditions for the intermediate structure to be a lattice and, for the case of finite lattices, identify when the dual graph is not a minimal representation of the lattice.
Part II applies techniques from natural duality theory to obtain dualities for quasivarieties of bilattices used in default logic. Bilattices are doubly-ordered algebraic structures which find application in reasoning about inconsistent and incomplete information. This account is the first attempt to provide dualities or representations when there is little interaction required between the two orders. Our investigations begin by using computer programs to calculate dualities for specific examples, before using purely theoretical techniques to obtain dualities for more general cases. The results obtained are extremely revealing, demonstrating how one of the lattice orders from the original algebra is encoded in the dual structure.
We conclude Part II by describing a new class of default bilattices. These provide an alternative way of interpreting contradictory information. We obtain dualities for two newly-described quasivarieties and provide insights into how these dual structures relate to previously described classes of dual structures for bilattices.This thesis is not currently available in OR
A Multipurpose Backtracking Algorithm
A backtracking algorithm with element order selection is presented, and its efficiency discussed in relation both to standard examples and to examples concerning relation-preserving maps which the algorithm was derived to solve. 1 Introduction Backtracking has long been used as a strategy for solving combinatorial problems and has been extensively studied (Gerhart & Yelowitz (1976), Roever (1978), Walker (1960), Wells (1971)). In worst case situations it may be highly inefficient, and a systematic analysis of efficiency is very difficult. Thus backtracking has sometimes been regarded as a method of last resort. Nevertheless, backtracking algorithms are widely used, especially on NP-complete problems. In order to make these algorithms computationally feasible on a range of large problems, they are usually tailored to particular applications (see, for example, Butler and Lam's approach to isomorphism-testing in Butler & Lam (1985) and Knuth and Szwarcfiter's approach to topological sort..