Showing papers on "Generalization published in 1977"

An implicit technology of generalization.

Abstract: Traditionally, discrimination has been understood as an active process, and a technology of its procedures has been developed and practiced extensively. Generalization, by contrast, has been considered the natural result of failing to practice a discrimination technology adequately, and thus has remained a passive concept almost devoid of a technology. But, generalization is equally deserving of an active conceptualization and technology. This review summarizes the structure of the generalization literature and its implicit embryonic technology, categorizing studies designed to assess or program generalization according to nine general headings: Train and Hope; Sequential Modification; Introduce to Natural Maintaining Contingencies; Train Sufficient Exemplars; Train Loosely; Use Indiscriminable Contingencies; Program Common Stimuli; Mediate Generalization; and Train “To Generalize”.

Generalization of the classical theory of capillarity

Abstract: Gibbs’ concept of dividing surfaces is supplemented explicitly by the concepts of dividing lines and dividing points. The general forms of the fundamental equations for dividing surfaces and lines are established by considering the proper extensive geometric properties, in addition to area and length. The detailed description of the fluid part of a capillary system by these fundamental equations is used to obtain the general conditions of equilibrium. Proper generalizations of the Laplace equation, the Neumann relation, and the Young equation are derived.

The High Order Maximal Principle and Its Application to Singular Extremals

Abstract: The high order maximal principle (HMP) which was announced in [11] is a generalization of the familiar Pontryagin maximal principle. By using the higher derivatives of a large class of control vari...

Some properties of the order of an autoregressive model selected by a generalization of Akaike∘s EPF criterion

Abstract: SUMMARY The asymptotic distribution of the order of an autoregression selected by a generalization of Akaike's FPE criterion is given. Some of the properties of the distribution are investigated. The use of this criterion is illustrated by a simulation study.

A Cyclic Reduction Algorithm for Solving Block Tridiagonal Systems of Arbitrary Dimension

Abstract: A generalization of the Buneman variant of cyclic odd–even reduction algorithm for solving finite difference approximations to Poisson’s equations is presented. This generalization places no restriction on the block size, n, of the system and computes the solution in $O(n^2 \log _2 n)$ operations.

A simple proof of the generalization of Israel's theorem

Abstract: A simple proof of the generalization of the theorem of Israel concerning the uniqueness of the Schwarzschild black hole is presented.

Categories in systems and control theory

Abstract: Beginning with the well-known concept of a system at the third level of generalization, where it is associated with empirical engineering and science, several new system categories are proposed. These are related to deterministic and stochastic distinctions and show clearly whether system disturbances are internally or externally generated. They are therefore valuable in both input/output and structural approaches for developing conceptual system models and choosing mathematical formalisms. By combining these categories with the concepts of autonomy and non-autonomy, new and more precise definitions for stationarity and non-stationary are formulated. In this way it is shown how the steady-state properties of systems may be inferred from knowledge of their internal properties and from the steady-state levels of inputs or parameters. Several examples are presented to demonstrate how the proposed definitions usefully relate general system concepts to specific physical system properties.

Geometric structure of observers for linear feedback control laws

Abstract: This paper is concerned with constructing observers for linear feedback control laws. Two types of observers (Kalman-type and Luenberger-type) are considered concurrently. Geometric theory of dynamic covers is developed for evaluating the minimal orders of observers. New lower and upper bounds are obtained for the minimal order of function observers possessing an arbitrarily prescribed set of poles. They are expressed simply in terms of observability indices of an augmented system and give a new light on the structural properties of observers. They also suggest the possibility of significant order reduction compared with observers estimating the whole state. A new geometric concept of generator, a natural generalization of cyclic generator, plays a key role in their derivation. A frequency domain characterization of observers is derived which reveals an interesting algebraic property of observers. It is used for devising a design algorithm in the frequency domain; in which the problem is reduced to pole assignment by dynamic compensator of a restricted type. Another design algorithm is presented in the time domain. Some illustrative examples are shown.

Inductive learning of relational productions

Abstract: Relational productions provide a mathematically tractable formal model for operators in discrete systems. Two paradigms for the inductive learning of such operators are considered: before-and-after situation pairs and situation sequences. An operational theory for the generalization of productions for these paradigms is then developed, based on previous work in concept induction. This theory has been computer implemented. In examples three "blocks world" operators are learned from six before-and-after pairs and also from a sequence of fifteen blocks world situations. A transformational grammar learning example of Hayes-Roth is repeated with an improvement in speed of two orders of magnitude.

A Comparison of Power Indices and a Nonsymmetric Generalization

Abstract: The first half of this paper describes and contrasts two well known measures of power in voting systems. The second half develops an explanatory model for a generalization of the Shapley-Shubik measure.

A structural generalization of the Ramsey theorem

Abstract: A generalization of the Ramsey theorem is stated. This solves a problem of P. Erdös and others. The result has recent applications in the theory of ultrafilters and model theory. The Ramsey theorem [3] states: For all positive integers k,m,p there exists an n such that for every coloring c: [n] —• k, there exists a homogeneous m set, I Ç n , \\M\\ = m, with \\c([M]P)\\= 1. This can be generalized to set systems of a given type and to set systems without forbidden subsystems. The purpose of this note is to announce this result. A family A = (Ô,.; i E I), 8, > 1, is called a type. (X, M) = (X, (M,; i E ƒ)) is a set system of type A if MzÇ [X] * and X is a finite ordered set. ƒ: (X, M) + (Y9 N) = (Y, (fy; / E ƒ)) is called an embedding if f: X * Y is a monotone 1-1 mapping and f(M) E Ut *=* M E M,for every i E I. (X, M) is a subsystem of (Y, M) if the inclusion X ÇY is an embedding. Denote by Emb (A, B) the set of all embeddings A —•> B and by Set (A) the category of all set systems of type A and all embeddings. The following holds: THEOREM. Let a type A be fixed. Let k be a positive integer and A E Set(A). Then for every B E Set(A) there exists C E Set(A) such that the following holds: for every coloring c: Emb (A, C) —• k there exists a subsystem B' of C which is isomorphic to B such that \\c(Emb(A9B'))\\ = 1. Moreover, if B does not contain a fundamental set system D, then C may be chosen with the same property. Here D = (X, M) is fundamental if for every i EI either M/ = 0 or Mt=[X} . This generalizes the Ramsey theorem and has the following consequences: COROLLARY 1. For every graph G = (V, E) without a complete graph with k-vertices, there exists a graph H = (W, F) without a complete subgraph AMS (MOS) subject classifications (1970). Primary 05A99; Secondary 04A20, 02H05.

On Some Integral Inequalities Related to Hardy's

Abstract: We obtain mainly by using Jensen's inequality for convex functions an integral inequality, which contains as a special case Shun's generalization of Hardy's inequality.

On Wentzel’s proof of the quantization condition for a single‐well potential

Abstract: Wentzel’s elegant method for deriving the quantal generalization of the Sommerfeld quantization condition, which has been criticized by other authors, is justified and put on an irrefutable basis. Clarifying comments on some questions related to the discussion in the present paper are also made.

Multivariate Generalization of Kendall's Tau with Application to Data Reduction

Abstract: The usual bivariate Kendall's tau estimates the probability of concordance of point pairs minus the probability of discordance. This paper presents a multivariate generalization of this idea. The multivariate versions can be used to investigate dimensions (in the multicollinearity spirit) in the data. Examples are given.