Nichtlineare Tschebyscheff Approximation mit Nebenbedingungen: Anwendungen

Summary In this paper we give some applications resulting from the theory of nonlinear approximation under side-conditions developed in a preceding paper [3]. Particularly a necessary condition for best approximation in terms of a generalized alternant is discussed, the approximating functions having restricted ranges. As special cases of this kind we deduce theorems for one-sided approximation and for approximations by positive functions. We conclude with a result in the theory of nonlinear programming.

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Zweiter Teil einer gekürzten Fassung der Dissertation des Verfassers [3].     

A survey of multicriteria optimization or the vector maximum problem,part I: 1776–1960

In this survey, the history of the subject from 1776 until 1960 is presented. A brief biographical sketch of Vilfredo Pareto is given first. Then, the more or less simultaneous development of the concepts of utility, preference, and welfare theory follows, with results which go back to Hausdorff and Cantor. A brief discussion of the work of Borel and von Neumann as initiators of game theory is included. Each of these areas has developed enough to warrant its own survey; hence, they are reviewed here only insofar as they provide necessary foundations. Thereafter, the concepts of efficiency, vector maximum problem, and Pareto optimality are reviewed in connection with production theory, programming, and economics. The survey is presented within a unified mathematical framework, and the emphasis is on mathematical results, rather than psychological or socio-economic discussion. To enable the reader to draw conclusions without having to obtain each article himself, the results have been presented in somewhat more detail than usual.     

Instability of optimal equilibria in the minimum mass design of uniform shallow arches

Necessary and sufficient conditions for the minimum mass design of arbitrarily loaded uniform shallow arches are derived. The problem is posed as an optimal control problem with mass as the criterion, initial curvature and axial load as design variables, and with the differential equations of axial and transverse equilibrium of the arch as side conditions. Thus, an optimal equilibrium is associated with each optimal design, and the stability of these equilibria becomes an integral part of the problem solution. As an example, the design process is carried out for the sinusoidally loaded hinged-hinged arch with a fixed span. It turns out that, depending on the given load amplitude, the optimal equilibrium can be unstable, stable after snap-through, and nonunique with one equilibrium unstable and the other stable after snap-through, at the design load of the arch. In addition, a necessary condition for a local minimum is the same as the usual critical point condition in stability analysis, thus assuring the instability of the arch at the optimum. A brief survey of earlier work on the optimal design of arches and curved beams is also included.     

Configuration interaction calculations for the ground and excited states of ozone and its positive ion: Energy locations and transition probabilities

Large-scale configuration interaction calculations (including energy extrapolation) are reported for the various states of ozone and its positive ion. The first four dipole-forbidden electronic transitions in the O₃ spectrum are calculated to occur at 1.20, 1.44, 1.59, and 1.72 eV, respectively, while the corresponding low-energy-allowed species known as the Chappuis, Huggins, and Hartley bands are predicted to possess vertical excitation energies of 1.95, 3.60, and 4.97 eV, respectively. These results all appear to fit in quite well with the observed location of the pertinent spectral features, with respect to both energy and intensity. The 5- to 8-eV region of the ozone spectrum is found to be characterized by a series of double-excitation transitions out of the highest three occupied orbitals to the lowest unoccupied 2b₁(π^*) species. The strong features observed at 9.3 and 10.2 eV are thereupon calculated to result primarily from transitions into the 7a₁ (σ^*) MO (calculated 9.29 and 10.05 eV) and in the former case also from the 3s members of the various O₃ Rydberg series (calculated 9.21 and 9.38 eV). Finally the order of the first three ip's is found to be 6a₁, 4b₂, and 1a₂, while the feature in the neighborhood of 16 eV is attributed to a shake-up state of ²B₁ symmetry.     

A theorem on the uniform convergence of Fourier-type integrals

A Dirichlet-type⁷ theorem for the uniform convergence of integrals of the form ∝^∞₀cos(tf(α)) cos(xα) dα is given. The theorem is illustrated by an application in the solution of a simple fourth order partial differential equation.     

Zur Koecherschen Reduktionstheorie in Positivitätsbereichen. I

Ohne ZusammenfassungGefördert durch National Science Foundation grant GP-3750.