Canonical Extension Research Articles

Self-adjoint extensions and stochastic completeness of the Laplace–Beltrami operator on conic and anticonic surfaces

We study the evolution of the heat and of a free quantum particle (described by the Schrödinger equation) on two-dimensional manifolds endowed with the degenerate Riemannian metric ds2=dx2+|x|−2αdθ2, where x∈R, θ∈T and the parameter α∈R. For α≤−1 this metric describes cone-like manifolds (for α=−1 it is a flat cone). For α=0 it is a cylinder. For α≥1 it is a Grushin-like metric. We show that the Laplace–Beltrami operator Δ is essentially self-adjoint if and only if α∉(−3,1). In this case the only self-adjoint extension is the Friedrichs extension ΔF, that does not allow communication through the singular set {x=0} both for the heat and for a quantum particle. For α∈(−3,−1] we show that for the Schrödinger equation only the average on θ of the wave function can cross the singular set, while the solutions of the only Markovian extension of the heat equation (which indeed is ΔF) cannot. For α∈(−1,1) we prove that there exists a canonical self-adjoint extension ΔB, called bridging extension, which is Markovian and allows the complete communication through the singularity (both of the heat and of a quantum particle). Also, we study the stochastic completeness (i.e., conservation of the L1 norm for the heat equation) of the Markovian extensions ΔF and ΔB, proving that ΔF is stochastically complete at the singularity if and only if α≤−1, while ΔB is always stochastically complete at the singularity.

Extended tensor products and an operator-valued spectral mapping theorem

We introduce the notion of an extended tensor product of Banach spaces and . It is defined as a triple consisting of a Banach space and two full subalgebras and of the algebras and of all bounded linear operators on and respectively. It is assumed that is an extension of the ordinary tensor product , and the functionals on and operators on have a canonical extension from to . Every pseudo- resolvent generates a functional calculus that sends analytic - valued functions in a neighbourhood of the singular set of the pseudo- resolvent to operators on . We prove an analogue of the spectral mapping theorem for such a functional calculus.

Gluing together Proof Environments: Canonical extensions of LF Type Theories featuring Locks

We present two extensions of the LF Constructive Type Theory featuring monadic locks. A lock is a monadic type construct that captures the effect of an external call to an oracle. Such calls are the basic tool for gluing together diverse Type Theories and proof development environments. The oracle can be invoked either to check that a constraint holds or to provide a suitable witness. The systems are presented in the canonical style developed by the CMU School. The first system, CLLFP, is the canonical version of the system LLFP, presented earlier by the authors. The second system, CLLFP?, features the possibility of invoking the oracle to obtain a witness satisfying a given constraint. We discuss encodings of Fitch-Prawitz Set theory, call-by-value lambda-calculi, and systems of Light Linear Logic. Finally, we show how to use Fitch-Prawitz Set Theory to define a type system that types precisely the strongly normalizing terms.

TiRS graphs and TiRS frames: a new setting for duals of canonical extensions

We consider properties of the graphs that arise as duals of bounded lattices in Ploscica’s representation via maximal partial maps into the two-element set. We introduce TiRS graphs, which abstract those duals of bounded lattices. We demonstrate their one-to-one correspondence with so-called TiRS frames, which are a subclass of the class of RS frames introduced by Gehrke to represent perfect lattices. This yields a dual representation of finite lattices via finite TiRS frames, or equivalently finite TiRS graphs, which generalises the well-known Birkhoff dual representation of finite distributive lattices via finite posets. By using both Ploscica’s and Gehrke’s representations in tandem, we present a new construction of the canonical extension of a bounded lattice. We present two open problems that will be of interest to researchers working in this area.

Jónsson-style canonicity for ALBA-inequalities

The theory of canonical extensions typically considers extensions of maps A→B to maps Aδ→Bδ⁠. In the present article, the theory of canonical extensions of maps A→Bδ to maps Aδ→Bδ is developed, and is applied to obtain a new canonicity proof for those inequalities in the language of Distributive Modal Logic (DML) on which the algorithm ALBA [9] is successful.

Finite quotients of Galois pro- $$p$$ p groups and rigid fields

For a prime number $p$, we show that if two certain canonical finite quotients of a finitely generated Bloch-Kato pro-$p$ group $G$ coincide, then $G$ has a very simple structure, i.e., $G$ is a $p$-adic analytic pro-$p$ group. This result has a remarkable Galois-theoretic consequence: if the two corresponding canonical finite extensions $F^{(3)}/F$ and $F^{\{3\}}/F$ of a field $F$ -- with $F$ containing a primitive $p$-th root of unity -- coincide, then $F$ is $p$-rigid. The proof relies only on group-theoretic tools, and on certain properties of Bloch-Kato pro-$p$ groups.

Bivariate Newton-Raphson method and toroidal attraction basins

When the numerical Newton-Raphson method is applied to find the intersections of two algebraic curves (that is, the roots of a pair of bivariate polynomials), some difficulties appear when the value of a denominator of the corresponding bivariate rational functions is zero. In this paper we give a solution to these problems by using adequate homogeneous coordinates and extending the domain of the iteration function. The iteration of a map given by a pair of bivariate rational maps is analyzed by taking a canonical extension, which is defined on the product of two copies of an (real or complex) augmented projective line. This method gives a global description of the basins of attraction of fixed points of an iteration. In particular, the attraction basins of fixed points associated to the intersection points of the two algebraic curves is obtained. As a consequence of our techniques, we are able to plot the attraction basins of a real root either in a torus (considered as a compactification of the real plane) or in an open square, which is homeomorphic to the global real plane containing the two algebraic curves.

Foundations for proper-time relativistic quantum theory

This paper is a progress report on the foundations for the canonical proper-time approach to relativistic quantum theory. We first review the the standard square-root equation of relativistic quantum theory, followed by a review of the Dirac equation, providing new insights into the physical properties of both. We then introduce the canonical proper-time theory. For completeness, we give a brief outline of the canonical proper-time approach to electrodynamics and mechanics, and then introduce the canonical proper-time approach to relativistic quantum theory. This theory leads to three new relativistic wave equations. In each case, the canonical generator of proper-time translations is strictly positive definite, so that it represents a particle. We show that the canonical proper-time extension of the Dirac equation for Hydrogen gives results that are consistently closer to the experimental data, when compared to the Dirac equation. However, these results are not sufficient to account for either the Lamb shift or the anomalous magnetic moment.

Canonical Extension of Submanifolds and Foliations in Noncompact Symmetric Spaces

We propose a method to extend submanifolds, singular Riemannian foliations and isometric actions from a boundary component of a noncompact symmetric space to the whole space. This extension method preserves minimal submanifolds, isoparametric foliations and polar actions, among other properties. One of the several applications yields the first examples of inhomogeneous isoparametric hypersurfaces in noncompact symmetric spaces of rank at least two.

The canonical FEP construction

A class K of algebras has the finite embeddability property (FEP) if every finite partial subalgebra of some member of K can be embedded into some finite member of K⁠. We prove the FEP for varieties of decreasing residuated lattice-ordered algebras using a construction based on the canonical extension. This construction produces a (generally) different finite member of the class from alternative FEP constructions for similar classes of algebras. Additionally, the constructed algebra is internally compact, in contrast to other FEP constructions. We give a description of the σ- and π-extensions of operations that do not rely on the notions of closed and open elements and we use this to obtain a syntactic description of a class of inequalities s≤t that are preserved by the construction.

Reconciliation of approaches to the construction of canonical extensions of bounded lattices

Abstract We provide new insights into the relationship between different constructions of the canonical extension of a bounded lattice. This follows on from the recent construction of the canonical extension using Ploščica’s maximal partial maps into the two-element set by Craig, Haviar and Priestley (2012). We show how this complete lattice of maps is isomorphic to the stable sets of Urquhart’s representation and to the concept lattice of a specific context, and how to translate our construction to the original construction of Gehrke and Harding (2001). In addition, we identify the completely join- and completely meet-irreducible elements of the complete lattice of maximal partial maps.

Noncommutative Quantum Anisotropic cosmology in K-essence

We study a canonical noncommutative extension of the Bianchi type I Minisuperspace, with a barotropic perfect fluid, in the context of a simplified form of k-essence known as the Sáez-Ballester theory. Noncommutativity is implemented in the pure gravitational sector. The corresponding noncommutative Wheeler-DeWitt equation is constructed and solved. Quantum solutions are obtained for any value of the barotropic parameter. It is seen that the effect of this particular noncommutativity is that of modulating the amplitude of the commutative wave function.

Fit States on Girard Algebras

Recently Weber proposed to define ``weakly additive states on a Girard algebra by the additivity only on its sub-$MV$-algebras and characterized such states on the canonical Girard algebra extensions of any finite $MV$-chain. In the present paper, we take another viewpoint: the arguable sub-$MV$-algebras are replaced by suitable substructures coming from author, H\{o}hle and Weber's own previous investigations. We propose a new notion of \emph{fit} states on a Girard algebra by the additivity on the mentioned substructures and consider such states on the ``non-effectible Girard algebra ``$n$-extensions (= canonical extensions when $n=1$) of $MV$-chains restricting ourselves to ones having less than six nontrivial elements. Our fit states appear as solutions of certain inconsistent systems of linear equations. They have extensive enough domains of the additivity-in any comparable case more extensive than Weber's states have.

A novel semi-supervised canonical correlation analysis and extensions for multi-view dimensionality reduction

Canonical correlation analysis (CCA) is an efficient method for dimensionality reduction on two-view data. However, as an unsupervised learning method, CCA cannot utilize partly given label information in multi-view semi-supervised scenarios. In this paper, we propose a novel two-view semi-supervised learning method, called semi-supervised canonical correlation analysis based on label propagation (LPbSCCA). LPbSCCA incorporates a new sparse representation based label propagation algorithm to infer label information for unlabeled data. Specifically, it firstly constructs dictionaries consisting of all labeled samples; and then obtains reconstruction coefficients of unlabeled samples using sparse representation technique; at last, by combining given labels of labeled samples, estimates label information for unlabeled ones. After that, it constructs soft label matrices of all samples and probabilistic within-class scatter matrices in each view. Finally, in order to enhance discriminative power of features, it is formulated to maximize the correlations between samples of the same class from cross views, while minimizing within-class variations in the low-dimensional feature space of each view simultaneously. Furthermore, we also extend a general model called LPbSMCCA to handle data from multiple (more than two) views. Extensive experimental results from several well-known datasets demonstrate that the proposed methods can achieve better recognition performances and robustness than existing related methods.

On dB spaces with nondensely defined multiplication operator and the existence of zero-free functions

In this work we consider de Branges spaces where the multiplication operator by the independent variable is not densely defined. First, we study the canonical selfadjoint extensions of the multiplication operator as a family of rank-one perturbations from the viewpoint of the theory of de Branges spaces. Then, on the basis of the obtained results, we provide new necessary and sufficient conditions for a real, zero-free function to lie in a de Branges space.

Sheaf representations of MV-algebras and lattice-ordered abelian groups via duality

We study representations of MV-algebras — equivalently, unital lattice-ordered abelian groups — through the lens of Stone–Priestley duality, using canonical extensions as an essential tool. Specifically, the theory of canonical extensions implies that the (Stone–Priestley) dual spaces of MV-algebras carry the structure of topological partial commutative ordered semigroups. We use this structure to obtain two different decompositions of such spaces, one indexed over the prime MV-spectrum, the other over the maximal MV-spectrum. These decompositions yield sheaf representations of MV-algebras, using a new and purely duality-theoretic result that relates certain sheaf representations of distributive lattices to decompositions of their dual spaces. Importantly, the proofs of the MV-algebraic representation theorems that we obtain in this way are distinguished from the existing work on this topic by the following features: (1) we use only basic algebraic facts about MV-algebras; (2) we show that the two aforementioned sheaf representations are special cases of a common result, with potential for generalizations; and (3) we show that these results are strongly related to the structure of the Stone–Priestley duals of MV-algebras. In addition, using our analysis of these decompositions, we prove that MV-algebras with isomorphic underlying lattices have homeomorphic maximal MV-spectra. This result is an MV-algebraic generalization of a classical theorem by Kaplansky stating that two compact Hausdorff spaces are homeomorphic if, and only if, the lattices of continuous [0,1]-valued functions on the spaces are isomorphic.

Canonical extensions of posets

We study a construction that produces a variety of completions of a given poset. The denseness and compactness properties of the completions obtained in this way are investigated. Next we focus our attention on three specific completions of a given poset that can be obtained through this construction—two of which have been called ‘the canonical extension’ of the poset in the literature. We investigate extensions of maps to these three completions. Although the extensions of unary operators need not be operators on the completions, we show that the extensions of unary residuated maps are residuated. We also investigate extensions of n-ary maps. In particular, we have a closer look at order-preserving n-ary maps and binary residuated maps.

Natural extensions for piecewise affine maps via Hofbauer towers

We use canonical Markov extensions (Hofbauer towers) to give an explicit construction of the natural extensions of various measure preserving endomorphisms, and present some applications to particular examples.

Topological duality and lattice expansions, I: A topological construction of canonical extensions

The two main objectives of this paper are (a) to prove purely topological duality theorems for semilattices and bounded lattices, and (b) to show that the topological duality from (a) provides a construction of canonical extensions of bounded lattices. In previously known dualities for semilattices and bounded lattices, the dual spaces are compact 0-dimensional spaces with additional algebraic structure. For example, semilattices are dual to 0-dimensional compact semilattices. Here we establish dual categories in which the spaces are characterized purely in topological terms, with no additional algebraic structure. Thus the results can be seen as generalizing Stone’s duality for distributive lattices rather than Priestley’s. The paper is the first of two parts. The main objective of the sequel is to establish a characterization of lattice expansions, i.e., lattices with additional operations, in the topological setting built in this paper.

A Lefschetz Fixed Point Formula for Elliptic Quasicomplexes

In a recent paper, the Lefschetz number for endomorphisms (modulo trace class operators) of sequences of trace class curvature was introduced. We show that this is a well defined, canonical extension of the classical Lefschetz number and establish the homotopy invariance of this number. Moreover, we apply the results to show that the Lefschetz fixed point formula holds for geometric quasiendomorphisms of elliptic quasicomplexes.