Logical Gap Research Articles

Computer Experiments with Fractional Gaussian Noises: Part 3, Mathematical Appendix

The present ‘Appendix’ is devoted to mathematical considerations designed to fill in the innumerable logical gaps left in the preceding paper. Even though most proofs will be merely sketched or omitted, the notation remains heavy, and some readers may wish to skip to formulas 1 and 2, which define ‘Type 1 functions.’Most of what follows is closely patterned after Mandelbrot and Van Ness [1968], which contains additional mathematical details and references to the few earlier articles, where fractional Gaussian noises were fleetingly considered as purely mathematical entities.

Anthropomorphism

Anthropomorphism

Some Hilbert spaces of analytic functions. III

This paper continues the study of Hilbert spaces of analytic functions which are involved in the structural analysis of nonself-adjoint transformations in Hilbert space. The theory of nonself-adjoint transformations originates in the quantum mechanical problems of nuclear scattering theory. Although the ordinary use of the Schriidinger equation leads to self-adjoint transformations, it is customary for physicists to subdivide the nuclear reaction in such a way that nonself-adjoint transformations occur. This observation, made by Livgic [l], caused him to found a general theory of nonself-adjoint transformations. Although LivBic’s theory is extensive and powerful, it has received very little recognition because no one, apparently, can follow his arguments. The difficulties are so great that Dolph and Penzlin [2] have attempted an independent derivation of the main results. The trouble is due not so much to logical gaps as it is to insufficient motivation of the main tool, the characteristic operator function. This quantity arises in the description of fundamental solutions of formally self-adjoint differential equations under variations of the boundary conditions. Yet the characteristic operator function is applied to transformations which have no connection with differential equations. The trouble is that LivGc has missed the meaning of the characteristic operator function, which is to be found in the construction of certain Hilbert spaces of analytic functions [3]. To explain how these spaces originate, we must go back to the basic work of Stone [4] and to our previous work with entire functions [5-81. Stone’s book has two different objectives, apart from a general formulation of concepts. The first is the study of self-adjoint transformations. It is important to note how he goes about the study of these transformations. From his point of view the structure theorem is an abstract analogue of the integral representation of functions which are analytic and have a nonnegative real part in the upper half-plane. (See, for example, Nevanlinna and Nieminen [9] for a more detailed reconstruction of the same argument.) It has

Matrix Derivation of Multiple Correlation

IN MANY textbooks on statistics for students in the behavioral sciences, a rather full discussion can be found of the type of problem situation and as sumptions involved in multiple correlation. How ever, their authors usually then present without derivation the and finally dis cuss computational techniques for solving these equations to obtain constants for a regression equa tion (3:406; 4:178; 5:237). Thus, for students of be havioral science a formal derivation of the normal equations expressed in terms of m atr ix algebra will serve to fill this logical gap. Assume that N persons have been measured on m predictor variables and one criterion variable. We can define the following vectors and m atr ices: j^isanNxi column vector of observed standard scores on the criterion, XisanNxm matrix of observed standard scores on the predictors, J3 is an m x 1 column vector of regression coefficients to be determined, and e is an N x 1 column vector of errors of estimate.

"The Nature and Significance of Economic Science" in Recent Discussion

I. Suffocation by analytical definition, 377. — II. Rationality and Individualist Exchange Economy, 387. — III. Scientific Precision and Stationary State, 391. — IV. Pseudo-Positivism and doctrine of the logical gap, 401. — V. The Mecca of twentieth century economics, 408. … One day in looking over his books he discovered a remarkable law. For every item on credit side an equal item appeared somewhere else on debit side. ‘Ha!’ said Bursar, ‘I have discovered one of great laws controlling college. It is a perfect and exact law of real world. Credit must be called plus and debit minus; and so we have law of conservation of £.s.d. This is true way to find out things, and there is no limit to what may ultimately be discovered by this scientific method. I will pay no more heed to superstitions held by some of Fellows as to a beneficent spirit called King or evil spirits called University Commissioners. I have only to go on in this way and I shall succeed in understanding why prices are always going up.’ The Nature of Physical World.