Separable Ones Research Articles

Symmetric indefinite systems for interior point methods

We present a unified framework for solving linear and convex quadratic programs via interior point methods. At each iteration, this method solves an indefinite system whose matrix is\(\left[ {\begin{array}{*{20}c} { - D^{ - 2} } & {A^T } \\ A & 0 \\ \end{array} } \right]\) instead of reducing to obtain the usualAD 2 A T system. This methodology affords two advantages: (1) it avoids the fill created by explicitly forming the productAD 2 A T whenA has dense columns; and (2) it can easily be used to solve nonseparable quadratic programs since it requires only thatD be symmetric. We also present a procedure for converting nonseparable quadratic programs to separable ones which yields computational savings when the matrix of quadratic coefficients is dense.

A "Thermal" Perceptron Learning Rule

The thermal perceptron is a simple extension to Rosenblatt's perceptron learning rule for training individual linear threshold units. It finds stable weights for nonseparable problems as well as separable ones. Experiments indicate that if a good initial setting for a temperature parameter, T0, has been found, then the thermal perceptron outperforms the Pocket algorithm and methods based on gradient descent. The learning rule stabilizes the weights (learns) over a fixed training period. For separable problems it finds separating weights much more quickly than the usual rules.

On dissolute spaces

Compact subspaces and metrizable subspaces of dissolution locales A d (the locale defined by the frame of congruence relations on the frame of A) are investigated. Separable metrizable ones are countable, and every countable metric space can be so embedded. Compact ones are scattered, but there is a further restriction: the space admits a partial ordering whose principal ideals are open and closed.

Approximate separability and aggregation

This note renews the concern of F. M. Fisher for the possibility of approximate aggregation. A geometric measure is introduced to gauge directly the extent to which a group of variables is separable relative to a given function. Bounds on the tightness of approximation of a function by separable ones are derived in terms of this geometric measure and the moduli of continuity of the function. The main conclusion is that approximate aggregation is unlikely to be possible unless the function is approximately separable. Hence, known stringent conditions for the existence of economic aggregates cannot be easily evaded.

Resolving the dilemmas of interaction, separability, and additivity

Motivated by the basic biological concern of attributing phenotypic variation to genotypic versus environmental causes, the general principles of assigning effects to two causal variables and specifying the operation by which they produce (phenotypic) responses are elaborated. It turns out that the effects of a cause can be defined only if the response to it is consistently distinguishable under the operation of the complementary causes. If consistency is realized for all causes of one variable, the effects of this causal variable are termed separable. Interaction occurs when separability is not realized and thus effects are not definable for at least some causes. As a consequence, separability or interaction may be realized for one but not for the other causal variable, or it may be realized for both variables simultaneously. The concept of unidirectional interaction or separability provides new means of interpretation of data. Additivity of effects, even in its most general sense, is shown to be a particular case of mutual separability (separability for both variables). It is shown that there are situations which can be transformed into additivity, thus implying mutual separability, which would definitely have been classified as interaction according to the classical concepts. Additive responses actually form only a vanishingly small proportion among the mutually separable ones. This is demonstrated with the help of necessary conditions for additivity. Therefore, the additive basis of the analysis of variance is not suitable for detecting interaction. The probabilistic interpretation of the concept of separability is outlined, and the problem of using averages for defining effects is discussed and demonstrated.

Forms of the generic norm of a separable Jordan algebra

The purpose of this note is to show that the type of problem considered by Professor Waterhouse in the preceding paper has its natural setting in the theory of generic norms of finite-dimensional Jordan algebras. Using this approach the results can be obtained on forms of the determinant and considerably more by invoking well-known results on Jordon algebras. The most general class of Jordan algebras that shall be considered is that of the separable ones. The key fact, which is easily established, is that forms of the generic norm of such an algebra are algebra forms of isotopes of the algebra. The algebra forms of the important algebras in our class are well known. Perhaps the most important of these were determined in 1949 by F. D. Jacobson and the author by a method based on special universal envelopes. This circumvented the use of Galois descent that had been used earlier for the analogous problems for Lie algebras by W. Landherr and by the author. Another known Jordan result is that similarity of generic norms of separable Jordan algebras holds if and only if the algebras are isotopic. Thus the determination of the forms and conditions for similarity can be derived from the Jordan theory.

On the closed graph theorem

Our main purpose is to describe those separated locally convex spaces which can serve as domain spaces for a closed graph theorem in which the range space is an arbitrary Banach space of (linear) dimension at most c, the cardinal number of the real line R. These are the δ-barrelled spaces which are considered in §4. Many of the standard elementary Banach spaces, including in particular all separable ones, have dimension at most c. Also it is known that an infinite dimensional Banach space has dimension at least c (see e.g. [8]). Thus if we classify Banach spaces by dimension we are dealing, in a natural sense, with the first class which contains infinite dimensional spaces.

Equivalent Local Potentials from Nonlocal Separable Ones

Explicit expressions for the calculation of equivalent local potentials corresponding to given nonlocal separable potentials have been derived. Using these expressions, equivalent local potentials have been calculated for a test case and a self-consistency check of the formulation has been made. An effective-range formalism for nonlocal separable potentials has also been presented following the coordinate representation.