Natural Duality Research Articles

Nonlinear gravito-electromagnetism

There is a non-linear and covariant electromagnetic analogy for gravity, in which the full Bianchi identities are Maxwell-type equations for the free gravitational field, encoded in the Weyl tensor. This tensor gravito-electromagnetism is based on a covariant generalization of spatial vector algebra and calculus to spatial tensor fields, and includes all non-linear effects from the gravitational field and matter sources. The non-linear vacuum Bianchi equations are invariant under spatial duality rotation of the gravito-electric and gravito-magnetic tensor fields. The super-energy density and super-Poynting vector of the gravitational field are natural duality invariants, and satisfy a super-energy conservation equation.

Singular Integrals and Potential Theory

In this paper I consider a class of non-standard singular integrals motivated by potential theoretic and probabilistic considerations. The probabilistic applications, which are by far the most interesting part of this circle of ideas, are only outlined in Section 1.5: They give the best approximation of the solution of the classical Dirichlet problem in a Lipschitz domain $$\Omega\subset {\mathbb{R}}^{n+1}$$ by the corresponding solution by finite differences. The potential theoretic estimate needed for this gives rise to a natural duality between the L p functions on the boundary ∂Ω and a class $$D_q\, ( \frac{1}{p} + \frac{1}{q}=1)$$ of functions A on Ω that was first considered by Dahlberg. The actual duality is given by ∫Ω $$\nabla^2$$ S f(x)A(x)dx = (f, A) where S f(x) = ∫∂Ω |x − y|1−n f(y)dy is the Newtonian potential. We can identify the upper half Lipschitz space $$\Omega$$ with $${\mathbb{R}}^{n+1}_+$$ in the obvious way and express $$(f,A) = \int K(x, y)f(x)A(y)dxdy\, (x\, \in {\mathbb{R}}^n, y \in {\mathbb{R}}^{n+1}_+)$$ for an appropriate kernel K. It is the boundedness properties of the above (for $$f \in L_p$$ , $$A \in D_q$$ ) that is the essential part of this work. This relates with more classical (but still “rough”) singular integrals that have been considered by Christ and Journe.

Natural dualities for semilattice-based algebras

While every finite lattice-based algebra is dualisable, the same is not true of semilattice-based algebras. We show that a finite semilattice-based algebra is dualisable if all its operations are compatible with the semilattice operation. We also give examples of infinite semilattice-based algebras that are dualisable. In contrast, we present a general condition that guarantees the inherent non-dualisability of a finite semilattice-based algebra. We combine our results to characterise dualisability amongst the finite algebras in the classes of flat extensions of partial algebras and closure semilattices. Throughout, we emphasise the connection between the dualisability of an algebra and the residual character of the variety it generates.

Not every full duality is strong!

Is every full duality strong? We show that the answer is ‘no’, thereby answering a question that goes back to the earliest foundations of the theory of natural dualities.

Full but not strong dualities at the finite level: Extending the realm

The realm of natural dualities that are known to be full but not strong at the finite level is a very small one, consisting of a single example. This example, based on the three-element bounded distributive lattice, was presented by Davey, Haviar and Willard [8]. In this paper, we extend this realm to the class of all natural dualities based on a finite non-boolean bounded distributive lattice.

Lattice of subalgebras in the finitely generated varieties of MV-algebras

In this paper, we use the theory of natural duality to study subalgebra lattices in the finitely generated varieties of MV-algebras. With this tool, we obtain the dual atomicity of these lattices, and characterize the members of these varieties in which every subalgebra is an intersection of maximal subalgebras. Then, we determine the algebras that have a modular or distributive lattice of subalgebras.

Choiceless, Pointless, but not Useless: Dualities for Preframes

We provide the appropriate common ‘(pre)framework’ for various central results of domain theory and topology, like the Lawson duality of continuous domains, the Hofmann–Lawson duality between continuous frames and locally compact sober spaces, the Hofmann–Mislove theorems about continuous semilattices of compact saturated sets, or the theory of stably continuous frames and their topological manifestations. Suitable objects for the pointfree approach are quasiframes, i.e., up-complete meet-semilattices with top, and preframes, i.e., meet-continuous quasiframes. We introduce the pointfree notion of locally compact well-filtered preframes, show that they are just the continuous preframes (using a slightly modified definition of continuity) and establish several natural dualities for the involved categories. Moreover, we obtain various characterizations of preframes having duality. Our results hold in ZF set theory without any choice principles.

The Semigroup of Marcinkiewicz Modulars with Involution

The set M of all concave Marcinkiewicz modulars on [0, 1] is a semigroup with respect to the usual composition of functions. We show that some properties of modulars (which are of importance in interpolation and in general Banach theory) distinguish subsets of M that form ideals of the semigroup. These ideals turn out to be in a natural duality relation, which is also studied. Bibliography: 8 titles.

Natural duality in minimal constrained self modeling curve resolution

AbstractSelf modeling curve resolution (SMCR) was introduced by Lawton and Sylvestre (LS) [Technometrics 1971; 13: 617–633] to decompose raw spectroscopic data of two component systems into product of two physically interpretable profile matrices provided that both concentrations and absorbances are non‐negative, accepting both as minimal constraints. Later Borgen et al. in 1985–86 [Anal. Chim. Acta 1985; 174: 1–26; Microchim. Acta 1986; 11: 63–73] generalized LS method for three‐component systems with the same minimal constraints. The concepts of Borgen were rather difficult to understand and to implement, that is why several chemometricians turned to developing approximation methods. Very recently, Rajkó and István [J. Chemom. 2005; 19: 448–463] have revisited Borgen's method and they have given clearer interpretation and used computational geometry tools to find inner and outer polygons. In the meantime Henry [Chemom. Intell. Lab. Syst. 2005; 77: 59–63] has introduced the duality relationship, but he has described it only for multivariate receptor modeling of compositional data of airborne pollution. Generalization of his duality principle will be given in this paper for universally using it in SMCR which is based on singular value decomposition (SVD) or principal component analysis (PCA). Copyright © 2007 John Wiley & Sons, Ltd.

Natural and Doubly Natural Dualities

ABSTRACT Representable dualities between categories of modules satisfying natural closure properties are investigated and shown to be determined by homological properties shared by various classical and recently investigated types of dualities.

A Duality between Forward and Adjoint MPI Communication Routines

In this article, we explore a natural duality that exist between MPI communication routines in parallel programs, and show the ease of its adjoint implementation via pointers.

Duality for finite Hilbert algebras

In this work we shall give a characterization of the Hilbert algebras given by the order and we will prove a duality for finite Hilbert algebras by means of finite ordered sets endowed with a distinguished set of subsets. We will also study the case when the finite Hilbert algebras are join-semilattices or meet-semilattices relative to the natural order defined by the implication. Finally we will prove that Hilbert do not admit a natural duality.

On some normability conditions

Various normability conditions of locally convex spaces (including Vogt interpolation classes DN φ and Ωφ as well as quasi- and asymptotic normability) are investigated. In particular, it is shown that on the class of Schwartz spaces the property of asymptotic normability coincides with the property GS , which is a natural generalization of Gelfand–Shilov countable normability (cf. [9, 25], where the metrizable case was treated). It is observed also that there are certain natural duality relationships among some of normability conditions. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Full does not imply strong, does it?

We give a duality for the variety of bounded distributive lattices that is not full (and therefore not strong) although it is full but not strong at the finite level. While this does not give a complete solution to the “Full vs Strong” Problem, which dates back to the beginnings of natural duality theory in 1980, it does solve it at the finite level. One consequence of this result is that although there is a Duality Compactness Theorem, which says that if an alter ego of finite type yields a duality at the finite level then it yields a duality, there cannot be a corresponding Full Duality Compactness Theorem.

Dual categories for endodualisable Heyting algebras: optimization and axiomatization

This paper is both a contribution to natural duality theory as it applies to varieties of Heyting algebras and to the theory of standard topological quasi-varieties (see [3]) in general. We prove that the n-element Heyting chain C n has an alter ego, consisting of n−2 endomorphisms, that yields an optimal duality on the variety generated by C n . In the case n = 4, we give a set of quasi-equations that describe the dual category. We also give a set of quasi-equations that describe the strong dual category that is obtained by adding a partial endomorphism of C4 to the type of the alter ego. A quasi-equational description of the optimal dual category for the variety of Heyting algebras generated by \({\bf 2}^{2} \oplus {\bf 1}\) is also given. En route, we prove that if a finite topological unary algebra is injective amongst the finite algebras in the variety it generates, then it is standard. The results of [3] then allow us to give non-topological proofs of our topological descriptions of the dual categories.

Natural dualities for varieties of BL-algebras

BL-algebras are the Lindenbaum algebras for Hajek's Basic Logic, just as Boolean algebras correspond to the classical propositional calculus. The finite totally ordered BL-algebras are ordinal sums of MV-chains. We develop a natural duality, in the sense of Davey and Werner, for each subvariety generated by a finite BL-chain, and we use it to describe the injective and the weak injective members of these classes.

Finiteness based results in BL-algebras

BL-algebras were introduced by P. Hajek as algebraic structures of Basic Logic. The aim of this paper is to survey known results about the structure of finite BL-algebras and natural dualities for varieties of BL-algebras. Extending the notion of ordinal sum of BL-algebras , we characterize a class of finite BL-algebras, actually BL-comets, which can be seen as a generalization of finite BL-chains. Then, just using BL-comets, we can represent any finite BL-algebra A as a direct product of BL-comets. This result can be seen as a generalization of the representation of finite MV-algebras as a direct product of MV-chains. Then we consider the varieties generated by one finite non-trivial totally ordered BL-algebra. For each of these varieties, we show the existence of a strong duality. As an application of the dualities, the injective and the weak injective members of these classes are described.

On projective and injective polynomial modules

Let k denote a field. For each n ≥ 0 we have a category Pol(n) of finite dimensional polynomial modules and a category Rat(n) of rational modules over k. For r ≥ 0, we write Pol(n, r) for the full subcategory of Pol(n) whose objects are the polynomial modules of degree r. Each V ∈ Pol(n) has a unique decomposition V = ⊕∞ r=0 V (r), where V (r) is polynomial of degree r. Furthermore, the category Pol(n, r) is naturally equivalent to the category of modules for the (finite dimensional, associative) Schur algebra S(n, r). It follows that every polynomial module has a projective cover and an injective hull. We here attempt to describe those polynomial modules which are both projective and injective. We give a complete description in the cases n = 2, 3 and make a conjecture which, if true, gives a complete description in the general case. A finite dimensional polynomial module which is injective and projective must be a tilting module and, thanks to a natural duality on the category of polynomial modules, our problem is equivalent to that of determining which

HOW FINITE IS A THREE-ELEMENT UNARY ALGEBRA?

We show that, within the class of three-element unary algebras, there is a tight connection between a finitely based quasi-equational theory, finite rank, enough algebraic operations (from natural duality theory) and a special injectivity condition.

Duality Between Prefetching and Queued Writing with Parallel Disks

Parallel disks promise to be a cost effective means for achieving high bandwidth in applications involving massive data sets, but algorithms for parallel disks can be difficult to devise. To combat this problem, we define a useful and natural duality between writing to parallel disks and the seemingly more difficult problem of prefetching. We first explore this duality for applications involving read-once accesses using parallel disks. We get a simple linear time algorithm for computing optimal prefetch schedules and analyze the efficiency of the resulting schedules for randomly placed data and for arbitrary interleaved accesses to striped sequences. Duality also provides an optimal schedule for prefetching plus caching, where blocks can be accessed multiple times. Another application of this duality gives us the first parallel disk sorting algorithms that are provably optimal up to lower-order terms. One of these algorithms is a simple and practical variant of multiway mergesort, addressing a question that had been open for some time.