An optimal narrowing strategy for general canonical systems

The object of this note is to extend the proof, for the general canonical varieties of any dimension k < d on a Vd whose existence was established by Dr Todd in a recent paper, of the property of adjunction given by the relationwhere S is any non-singular variety of dimension d−1 lying on Vd, and Xk[S] denotes the canonical system of dimension k on S. This formula was proved by Todd in the case when Vd is a general non-singular variety, and S is non-singular and belongs to a sufficiently general linear system. Using this result, I show how to remove the latter restriction. (1·1) cannot be taken, without further proof, as defining canonical systems on virtual or isolated varieties, as is done by B. Segre, since the canonical systems on an effective isolated variety are already well defined.

We consider combinations of term rewriting with built-in operations which compute objects of some external domain from ground terms. Combining a canonical term rewriting system with a computation relation does in general not lead to a canonical simplification relation. We show how a canonical simplification system can be constructed for such rewrite systems with built-in operations. Decomposition free simplification systems never need to look at the internal structure of external objects and thus a previous computation never needs to be undone. For some interesting equational specifications the construction of decomposition free canonical simplification systems succeeds. Our construction of canonical simplification systems relies on simulating combined rewriting and computing by pure term rewriting. This allows us to use the wealth of all rewrite techniques and tools to investigate and manipulate these structures.KeywordsExternal ObjectCritical PairGround TermGround InstanceMixed TermThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

It is proved that if an anisotropic medium has an open set of singular directions, then this medium has two slowness surfaces that completely coincide. The coinciding slowness surfaces form one double singular slowness surface. The corresponding anisotropic medium is an elliptical orthorhombic (ORT) medium with equal stiffness coefficients c44=c55=c66 rotated to an arbitrary coordinate system. Based on the representation of the Christoffel matrix as a uniaxial tensor and considering that the elements of the Christoffel matrix are quadratic forms in the components of the slowness vector, a system of homogeneous polynomial equations was derived. Then, the identical equalities between homogeneous polynomials are replaced by the equalities between their coefficients. As a result, a new system of equations is obtained, the solution of which is the values of the reduced (density normalized) stiffness coefficients in a medium with a singular surface. Conditions for the positive definite of the obtained stiffness matrix are studied. For the defined medium, the Christoffel equations and equations of group velocity surfaces are derived. The orthogonal rotation matrix that transforms the medium with a singular surface into an elliptic ORT medium in the canonical coordinate system is determined. In the canonical coordinate system, the slowness surfaces S1 and S2 waves coincide and are given by a sphere with a radius . The slowness surface of qP waves in the canonical coordinate system is an ellipsoid with semi-axes , , . The polarization vectors of S1 and S2 waves can be arbitrarily selected in the plane orthogonal to the polarization vector of the qP wave. However, the qP wave polarization vector can be significantly different from the wave vector. This feature should be taken into account in the joint processing and modelling of S and qP waves. The results are illustrated in one example of an elliptical ORT medium.

ABSTRACT:(Labelled) transition systems are relatively common in theoretical computer science, chiefly as vehicles for operational semantics. The first part of this paper constructs a hierarchy of canonical transition systems and associated maps, aiming to give a strongly extensional theory of transition systems, where any two points with equivalent behaviors are identified. The cornerstone of the development is a notion of convergence in arbitrary transition systems, generalizing the idea of finite (n‐step) approximations to a given point. In particular, our canonical transition systems are also uniform spaces.The resulting hierarchy has very rich combinatorial (and topological) structure, and a lot of the first part of the paper is devoted to its study. We also discuss fixed points in this framework.In the second part of the paper, we show how to obtain a model of set theory with Aczel's Anti‐Foundation Axiom (AFA) from canonical transition systems constructed earlier. We study further the structure of the model thus obtained, and also give a few more abstract results, concerning consistency and independence in the presence of AFA.

The direct spectral problem for the matrix Dirac type systems with locallysummable potentials and the inverse spectral problem for the matrixDirac type systems with locally bounded potentials are solved on theinterval and on the half-line. A direct procedure to recover the potential bythe Weyl–Titchmarsh function is given also. Some important corollarieson the high-energy asymptotics of the Weyl–Titchmarsh functions andon the local uniqueness for the corresponding inverse problem follow.New results are obtained for the general type canonical systems also.

In the theory of de Branges Hilbert spaces of entire functions, so-called ‘functions associated to a space’ play an important role. In the present paper we deal with a generalization of this notion in two directions, namely with functions N-associated \(({N \in\mathbb {Z}})\) to a de Branges Pontryagin space. Let a de Branges Pontryagin space \({\mathcal {P}}\) and \({N \in \mathbb {Z}}\) be given. Our aim is to characterize whether there exists a real and zerofree function N-associated to \({\mathcal {P}}\) in terms of Kreĭn’s Q-function associated with the multiplication operator in \({\mathcal {P}}\) . The conditions which appear in this characterization involve the asymptotic distribution of the poles of the Q-function plus a summability condition. Although this question may seem rather abstract, its answer has a variety of nontrivial consequences. We use it to answer two questions arising in the theory of general (indefinite) canonical systems. Namely, to characterize whether a given generalized Nevanlinna function is the intermediate Weyl-coefficient of some system in terms of its poles and residues, and to characterize whether a given general Hamiltonian ends with a specified number of indivisible intervals in terms of the Weyl-coefficient associated to the system. In addition, we present some applications, e.g., dealing with admissible majorants in de Branges spaces or the continuation problem for hermitian indefinite functions.

39. Transformation of an Arbitrary System of Two Equations into a Canonical System

This paper presents a new study on the adaptive neural network-based control of a class of noncanonical nonlinear systems with large parametric uncertainties. Unlike commonly studied canonical form nonlinear systems whose neural network approximation system models have explicit relative degree structures, which can directly be used to derive parameterized controllers for adaptation, noncanonical form nonlinear systems usually do not have explicit relative degrees, and thus their approximation system models are also in noncanonical forms. It is well-known that the adaptive control of noncanonical form nonlinear systems involves the parameterization of system dynamics. As demonstrated in this paper, it is also the case for noncanonical neural network approximation system models. Effective control of such systems is an open research problem, especially in the presence of uncertain parameters. This paper shows that it is necessary to reparameterize such neural network system models for adaptive control design, and that such reparameterization can be realized using a relative degree formulation, a concept yet to be studied for general neural network system models. This paper then derives the parameterized controllers that guarantee closed-loop stability and asymptotic output tracking for noncanonical form neural network system models. An illustrative example is presented with the simulation results to demonstrate the control design procedure, and to verify the effectiveness of such a new design method.

This paper develops a new adaptive feedback linearization based control scheme for T-S fuzzy systems in general non-canonical forms, which provides an effective control method to control non-canonical form nonlinear systems with large parameter uncertainties. Unlike commonly studied canonical form nonlinear systems whose T-S fuzzy approximation models have explicit relative degree structures which can be directly used to derive parametrized controllers for adaptation, non-canonical form nonlinear systems usually do not have explicit relative degrees as their T-S fuzzy system models are also in non-canonical forms. In order to achieve adaptive output tracking, the system dynamics of non-canonical form T-S fuzzy system models still needs to be parametrized. This paper extends the feedback linearization technique, commonly used for output tracking control of general nonlinear systems, to non-canonical form T-S fuzzy systems, and develops an adaptive feedback linearization based control method which guarantees closed-loop stability and asymptotic output tracking for T-S fuzzy system models. An illustrative example is presented with simulation results to demonstrate the control design procedure, and to verify the effectiveness of such a new design method.

We define a transformation from a left-linear constructor-based conditional rewrite system into an overlapping inductively sequential rewrite system. This transformation is sound and complete for the computations in the source system. Since there exists a sound and complete narrowing strategy for the target system, the combination of these results offers the first procedure for provably sound and complete narrowing computations for the whole class of the leftlinear constructor-based conditional rewrite systems. We address the differences between demand driven and lazy strategies and between narrowing strategies and narrowing calculi. In this context, we analyze the efficiency and practicality of using our transformation for the implementation of functional logic programming languages. The results of this paper complement, extend, and occasionally rectify, previously published results in this area.

Rewriting with non-symmetric relations can be considered as a computational model of many specification languages based on non-symmetric relations. For instance, Logics of Inequalities, Ordered Algebras, Rewriting Logic, Order-Sorted Algebras, Subset Logic, Unified Algebras, taxonomies, subtypes, Refinement Calculus, all them use some kind of non-symmetric relation on expressions. We have developed an operational semantics for these inequality specifications named bi-rewriting systems. In this paper we show the applicability of bi-rewriting systems to Unified Algebras and nondeterministic specifications. In the first case, we give a canonical bi-rewriting system implementing the basic theory of these algebras. In the second case, nondeterministic specifications are viewed as inclusion specifications, thus bi-rewriting is a sound, although not always complete deduction method. We show how a specification has to be completed in order to have both soundness and completeness.

First-order imagery by rays in the neighborhood of some base ray traversing an arbitrary optical system is anamorphotic and characterized by the presence of astigmatism and image rotation. The relationships among these three attributes is discussed; in particular, the effects of requiring that there exist one or more pairs of conjugate points on the base ray. Some inaccuracies in the literature pertaining to the torsion of the image by a general system are cleared up.

We propose an extension of rewriting techniques to derive inclusion relations \(a \subseteq b\) between terms built from monotonic operators. Instead of using only a rewriting relation \(\xrightarrow{ \supseteq }\) and rewriting a to b, we use another rewriting relation \(\xrightarrow{ \supseteq }\) as well and seek a common expression c such that \(a{\xrightarrow{ \subseteq }^*}c\) and \(b{\xrightarrow{ \supseteq }^*}c\). Each component of the bi-rewriting system \(\left\langle {\xrightarrow{ \subseteq },\xrightarrow{ \supseteq }} \right\rangle\) is allowed to be a subset of the corresponding inclusion ⊆ or ⊇. In order to assure the decidability and completeness of the proof procedure we study the commutativity of \(\xrightarrow{ \subseteq }\) and \(\xrightarrow{ \supseteq }\). We also extend the existing techniques of rewriting modulo equalities to bi-rewriting modulo a set of inclusions. We present the canonical bi-rewriting system corresponding to the theory of non-distributive lattices.KeywordsUnify AlgebraConjunctive Normal FormCritical PairCommutativity PropertyRule SchemeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Combining word problems through rewriting in categories with products

Orientability of systems of equalities is the following problem: given a system of equalities s/sub 1/ /spl sime/ t/sub 1/, . . . , s/sub n/ /spl sime/ t/sub n/, does there exist a simplification ordering > which orients the system, that is for every i /spl isin/ {1, ..., n}, either s/sub i/ > t/sub i/ or t/sub i/ > s/sub i/. This problem can be used in rewriting for finding a canonical rewrite system for a system of equalities and in theorem proving for adjusting simplification orderings during completion. We prove that (rather surprisingly) the problem can be solved in polynomial time when we restrict ourselves to the Knuth-Bendix orderings.