Showing papers on "Maxima and minima published in 1968"

Fermat's methods of maxima and minima and of tangents. A reconstruction

Abstract: I t is well known tha t I~'ERMAT 1 was the first to use the characteristic behaviour of an algebraic expression near its ext rema as a criterion for the determinat ion of these extrema. 2 When it comes, however, to the delineation of what his me thod really consisted of, we are overlooking a field of scholarship of which the famous saying b y a Greek philosopher (HERACLITOS, we are told) seems a singularly apt description. The confusion s tar ted already in FERMAT'S own lifetime. His essays and letters, circulating only in manuscript , were most ly known in selection. The various conceptions of the Fermat ian method were therefore determined by the part icular par t of the corpus tha t had been studied b y the different persons. Add to this the influence of the rapid development of mathemat ics f rom about t630, leading to each succeeding generation of mathemat ic ians interpret ing his method within their own conceptual frame, and we cannot wonder tha t when the Varia Opera 3 appeared in 1679, it was already too late. Tradi t ion had b y then grown strong and was not to be thwar ted b y mere texts. Since then, historians have taken their cues from t7 th cen tury conceptions of FERMAT'S method. They have either represented it as the expansion of [ (x + h) / (x ) ~ 0 in powers of h and leaving out second and higher order terms, h being ' infini tely l i t t le ' , or they have taken it to be based on the criterion tha t the equat ion / ( x ) y = 0 have a double root. 4 Algorithmically there is no difference between these two approaches. In bo th cases the mechanical procedure would be to expand / (x + h) and to take the coefficient of the first order term set equal to zero as determining the extrema. This was certainly FERMAT'S wa y too. However, most historians went further and declared tha t his method (that is, his justification of this algorithm) was identical with one of the two above. As I shall show, there are considerable, if not insurmountable, difficulties in making such views fit with his own words. In m y opinion, therefore, not a single one

Optimum nuclear reactor control theory

Abstract: Publisher Summary The mathematical model of the system to which the optimum control theory is applicable is based on a finite number of ordinary differential equations. A characteristic of the theory is the use made of the adjoint equations, because the problem is found to be non-self-adjoint in nature. The theory is different from the calculus of variations in its treatment of problems in classical dynamics. Dynamic programming is not thought of in the context of the calculus of variations. There is an analogos situation in the optimization of a model based on linear algebraic equations, where the calculus of maxima and minima, and linear programming could be employed. This chapter discusses the relation of dynamic programming to optimal control theory. It further discusses a generalization of the theory to encompass models of distributed rather than “point” systems, to include the usual models of nuclear reactor statics. This helps to specialize the theory and to obtain some results in reactor statics on the optimum design of systems.

Restricted Minima of Functions of Controls

Abstract: Mathematical control theory considering necessary conditions for minimum for relaxed and ordinary controls

Response curves in the theory of atmospheric oscillations

Abstract: It is shown that response curves are characterized by several maxima and minima. With an upper boundary condition which involves total reflection of tidal energy the maximum magnifications are unbounded. With energy propagating upwards at the upper boundary the maximum magnifications are finite and decrease in value with decreasing successive resonant equivalent depths. Thus the first three maxima occur at equivalent depths equal to 9·94 km, 6·71 km, 3·4 km, with magnifications of 9,220, 10 and 1·7 respectively. A minimum magnification of 0·57 occurs at lunar semidiurnal equivalent depth (7·1 km). This is not adequate to explain the observed lunar tide. The features of the response curve are explained through a two layer model atmosphere. It is shown that the values of magnification at small equivalent depths are critically affected by the particular boundary condition whereas at large equivalent depths magnification is insensitive to variations in the top boundary condition.

Shear-Stress Field of a Simple Dislocation Cell

Abstract: The shear‐stress distribution due to a particular example of the simplest, two‐dimensional dislocation array, representing a cell of material misoriented with respect to its surroundings, was calculated numerically. The results are found to be virtually independent of the size of the cell, provided its shape and the magnitude of its angular misorientation are kept constant. For cells of same size and shape the stresses are proportional to the magnitude of the angular misorientation, i.e., inversely proportional to the dislocation spacing. In the example studied, the long‐range stresses are always less intense than the extrema of the short‐range stresses, meaning that the subboundaries, but not the cell interior, could serve as obstacles against glide dislocation motion. The direction of the long‐range stresses is such as to attract from the surroundings dislocations which would reduce the relative misorientation of the cells, so as to clear the cell interior of dislocations.

Effect of experimental resolution on the measured width of cross-section fluctuations

Abstract: The results of a previous note on the number of maxima and minima in a fluctuating excitation curve have been extended to include the effect of the experimental energy resolution. A close formula is derived for the number of maxima as a function ofΓ by assuming a Gaussian distribution for the cross-sections which gives a good approximation in all practical cases. Numerical results are presented for the cases of rectangular and Gaussian resolution functions.