Multidimensional Representation of a Dynamical System : Global Environment

[Abstract] Earth as a whole can be considered as a multi-dimensional dynamical system if not completely an isolated system. Global environment problem, for example, should be described in multi-dimensional phase space consisting of all observables including space and time. Multi-dimensional representation of one variable, e.g., temperature at one position, changing with time is first described, and is generalized to multi-dimensional representation of several variables, temperatures at several positions, and then is generalized to that of many variables, all the characteristic quantities of the system at various space-time positions. The multi-dimensional representation should be adopted to the dynamical system analysis described by a system of differential equations as well as the analysis of the observational data system of any dynamical system

Morphology of Distant Galaxies

[Abstract] Binarized CCD images obtained from the original monocolor print in Tyson-Seitzer (Astrophys.J.335,552,1988) are image processed with personal image analyzing system (PIAS). Items of image processing for each galaxy are contour line length, area, lengths of major and minor axes of the approximating ellipse, the maximum diameter length, circularity, two factors concerning roundness, the ratio of contour line length to area, the ratio of major axis to minor axis, slant angle of major axis of the approximating ellipse, slant angle of maximum diameter, slant angle of major axis as rectangle enveloping the galaxy, the ratio of occupation area to the rectangle. These 14 factors are calculated for each galaxy at one threshold intensity in each binarized image. The average value of each factor in each binarized image is compared with that of different threshold intensity to get the statistical information of a distant galaxy. CCD images of distant galaxies show the tendency of alignment of maximum diameters towards east-west direction, which may be spurious due to some asymmetric device in the CCD observation or in printing process of the journal. The principal component analysis made with these 14 factors reveals that the complexity of shape is the first principal component correlating with the distance. And the average of circularity of low threshold image is larger than that of high threshold image. This means that a galaxy is the more distant, the shape of galaxy is the more complicated

Theoretical and Empirical Construction of a Dynamical System

[Abstract] A set of simultaneous first-order ordinary differential equations with time is the standard formalism describing a dynamical system. Such standard formulation of a dynamical system can be derived in a straightforward way from the normal form of the general variational principle describing the conservation of mass, momentum, and energy. On the other hand, the principal component analysis can be employed to derive the standard form of a dynamical system empirically from the observed data set, if sufficiently large number of observational quantities are measured in sufficiently large number of times to construct multi-dimensional phase spaces both from the measured quantities and from their tune derivatives. Examples are given for that are constructed the theoretical and the empirical derivations of the standard dynamical system formulation. Improved principal component analysis is emphasized for use in the global environment

Paradox as Basis of Science

[Abstract] Olbers\u27 paradox is known to have profound relation with the cosmological model, viz., the expanding universe and the cosmological large-scale fractal structure. A well-known mathematical paradox that the length of two sides of a triangle is equal to that of the remaining side is considered to be an good introduction of the fractal geometry. A line segment of an infinite length can be constructed in the same way as the triangle paradox is constructed

Basic Method of Dynamic Modeling of the Global Change

[Abstract] The method of dynamic modeling is explored and applied to the global change. The repetitive use of principal component analyses (PCA) in multidimensional phase space describing the dynamical state of the global environment is the main tactics of the method. For various quantities monitored at sites distributed over the entire globe, amplitudes of variation in shorter time scales as well as the relevant time averages are taken as additional variables in order for revealing the nonlinear couplings among different time scales. Spatial and temporal derivatives of all those variables are then taken also as additional variables for describing the evolution of the dynamical state of the system. The first PCA is intended to obtain the description of the system in terms of the principal components, and the second PCA is to describe the dynamical properties of the system. A method of maximum-entropy adjustment of the data is also explored. Thus, large amount of fragmentary data scattered in wide range of accuracy can be used in the PCA constructively, and the applicability of the method of dynamic modeling is greatly extended in the study of the global change as well as in other branches of the natural history

Topological Structure of Mind

[Abstract] Topological representation is undertaken for the function of mind. Klein\u27s bottle so modified as to have the point cross section of infinite area at the intersection of different parts of its own surface is considered as the simplest model of elementary topological structure of mind. Brain may be the manifestation of the multiple combination of such singular elementary structures. Mind is considered to be the excitation of one combination and change of mind is then the transfer of excitation to other combination of those singular elementary structures