Showing papers on "Completeness (order theory) published in 1980"

Aristotle and logical theory

Abstract: Preface 1. Syllogistic consequence 2. Completeness and compactness 3. Hypothetical syllogisms 4. Invalid inference 5. Invalid proofs Appendix: a note on ignorance 6. Proof by refutation Bibliography Indices.

Boundary methods: A criterion for completeness.

Abstract: The application of methods which constitute an alternative to boundary integral equations to specific problems depends on development of complete systems of solutions, convergence of approximating procedures, and formulation of variational principles. This paper establishes a criterion for completeness. In this manner, greater flexibility of the theory is achieved; for example, systems of functions which are complete for different types of boundary conditions are developed.

Convergence and Cauchy structures on lattice ordered groups

Abstract: This paper employs the machinery of convergence and Cauchy struc- tures in the task of obtaining completion results for lattice ordered groups. §§ 1 and 2 concern /-convergence and /-Cauchy structures in general. §4 takes up the order convergence structure; the resulting completion is shown to be the Dedekind- MacNeille completion. §5 concerns the polar convergence structure; the corre- sponding completion has the property of lateral completeness, among others. A simple theory of subset types routinizes the adjoining of suprema in §3. This procedure, nevertheless, is shown to be sufficiently general to prove the existence and uniqueness of both the Dedekind-MacNeille completion in §4 and the lateral completion in §5. A proof of the existence and uniqueness of a proper class of similar completions comes free. The principal new hull obtained by the techniques of adjoining suprema is the type •?) hull, strictly larger than the lateral completion in general. As is nearly always true in the field of lattice ordered groups, this research follows a path first trod by Conrad (10) and Holland (14). The contributions of Papangelou ((19), (20)) and Kenny (15) have also been important. But the novelty of the present approach lies in the systematic application of convergence and Cauchy structure techniques, significantly more powerful than topological and uniformity techniques. These new techniques have been developed recently by Kent and others ((16), (22)). The author owes a more personal debt of gratitude to professors Gary Davis, G. Otis Kenny, and Darrell Kent for many stimulating conversations on these topics. A collection x) all x G X}. LG(X) and UG(X) are often written more simply L(X) and U(X). The order closure of X, written ocl(X), is defined inductively. X0 = X,

A study of the completeness properties of resonant states

Abstract: The completeness properties of the discrete set of bound state, virtual states, and resonant states of a Hamiltonian H is investigated, where H describes a system in which a single nonrelativistic‐spinless particle moves in a central cutoff potential. A limited form of completeness is obtained. It is shown that the convergence of the resulting ’’completeness series’’ is very sensitive to the detailed mathematical structure of the potential.

Some Kinds of Modal Completeness

Abstract: In the modal literature various notions of “completeness” have been studied for normal modal logics. Four of these are defined here, viz. (plain) completeness, first-order completeness, canonicity and possession of the finite model property — and their connections are studied. Up to one important exception, all possible inclusion relations are either proved or disproved. Hopefully, this helps to establish some order in the jungle of concepts concerning modal logics. In the course of the exposition, the interesting properties of first-order definability and preservation under ultrafilter extensions are introduced and studied as well.

Random Walks on Lie Groups

Abstract: We present in this paper a limited selection of topics intended to introduce the reader to the subject of random walks on Lie groups. The selection has been highly individual and makes no pretense of completeness. For a more comprehensive view of the area the reader should consult [1,8–10].

Asymptotic completeness for classes of two, three, and four particle Schrödinger operators

Abstract: Formulas for the resolvent (z — H)~x are derived, where H = H0 + 2» 3. The allowed potentials belong to a space of dilation analytic multiplication operators which fall off more rapidly than r~2~' at oo. In particular, Yukawa potentials, generalized Yukawa potentials, and potentials of the form (1 + r)~2~' are allowed.

A note on completeness

Abstract: Completeness relationships for eigenfunctions of second order differential equations are presented in a form which employs a contour integration rather than the usual integration and summation over eigenvalues. This technique which is particularly applicable for scattering problems simplifies the usual procedures and the proper weight functions are easily obtained. Some examples are given.

A Complete, Nonredundant Algorithm for Reversed Skolemization

Abstract: An algorithm is presented which, for an arbitrary literal containing Skolem functions, outputs a set of closed quantified literals with the following properties. If a and b are formulae we define a ⊃ b iff {sk(a),dsk(b)} is unifiable where sk denotes Skolemization and dsk denotes the dual operation, where the roles of ∀ and ∃ are reversed. If d is an arbitrary literal and X is the output, then: (i) Soundness: if x ∈ X then x ⊃ d (ii) Completeness: if a ⊃ d then ∃x ∈ X such that a ⊃ x (iii) Nonredundancy: if x,y ∈ X then x ⊅ y and y ⊅ x.

Using Sophisticated Models in Resolution Theorem Proving

Abstract: 1 Introduction.- 2 Hereditary lock resolution.- 3 Completeness of HL-resolution.- 4 Models.- 5 Discussion of HLR.

Kant on Mathematical Definition

Abstract: That is to say, the concept, must be presented with all the characteristics (Merkmale) which are sufficient for its distinction and which can be seen to belong to it directly, i.e. without any proof. In other words, the original concept must be presented with completeness and precision (A727 = B755 footnote; Logik §99). According to Kant only a few concepts are definable in the sense of the quoted statement (*) and these few are mathematical concepts.

The completion of a topological group

Abstract: During the 1920's and 30's, two distinct theories of “completions” for topological spaces were being developed: the French school of mathematics was describing the familiar notion of “complete relative to a uniformity”, and the Russian school the less well-known idea of “absolutely closed”. The two agree precisely for compact spaces. The first part of this article describes these two notions of completeness; the remainder is a presentation of the interesting, but apparently unrecorded, fact that the two ideas coincide when put in the context of topological groups.

The quantisation and measurement of momentum observables

Abstract: Mackey's scheme (1963) for the quantisation of classical momenta generating complete vector fields (complete momenta) is introduced, the differential operators corresponding to these momenta are introduced and discussed, and an isomorphism is shown to exist between the subclass of first-order self-adjoint differential operators, whose symmetric restrictions are essentially self-adjoint, and the complete classical momenta. Difficulties in the quantisation of incomplete momenta are discussed, and a critique given. Finally, in an attempt to relate the concept of completeness to measurability, concepts of classical and quantum global measurability are introduced, and shown to require completeness. These results afford strong physical insight into the nature of complete momenta, and lead us to suggest a quantisability condition based upon global measurability.

Theory of Connectivity: A Unified Approach to Boundary Methods.

Abstract: A theory of connectivity recently developed by the author, is described briefly, and a unified formulation of boundary methods is presented. Boundary integral equations and series expansions in terms of a basic set of functions are among approaches included in this unified formulation. The theory of connectivity therefore appears to be a useful tool for discussing questions of completeness of the basic set of functions and convergence of the approximating procedures; in addition, it supplies a systematic formulation of variational principles for this kind of problem.

Complementary Relations and Map Reading

Abstract: When a group has applied the Interpretive stuctural modeling process to develop a map of a theme or Issue, they typically Inspect visually the map they have produced to see whether it is intuitively satisfactory. While this practice is useful and may play a role in map amendment, a more thorough approah is recommended. The working space of the map consists of the ordered element pairs belonging to the Cartesian product S × S, where S is the set of elements modeled through a contextual reation represented by the binary relation R. In a systematic modeling effort it is desirable to establish the completeness of the relation R without requiring mathematical sophistication on the part of the modeling group. (If R Is not complete, only part of the knowledge germane to the working space is embedded in the model.) Moreover, some of the useful knowledge content of the working space may only be implicit in R. if the modeling group is to take full advantage of the knowedge embedded in a map, it is esential that the group understand a systematic procedure for reading maps. Such a procedure is presented, and it is postulated that the group knows how to use it.