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Dynamics of Population
Published in Robert Ehrlich, Harold A. Geller, John R. Cressman, Renewable Energy, 2023
Robert Ehrlich, Harold A. Geller, John R. Cressman
One noticeable trend related to the number of people in the United States from 1790 to 2020 is that the number has always increased. In fact, it has not only always increased, but the rate of change has also been increasing itself. Of course, if you know calculus, you know that there is a difference between the rate of change of a value and the rate of change of the rate of change. The first is the first derivative of the function and the second is the second derivative of the function. Thus, while the rate of change of human population has always been positive, the rate of change of the rate of change has also always been positive. But can this continue forever? In fact, just because the trend is this way, does this mean that is how it will be for years to come? Think about events that can change the trends in human population in the United States.
Lagrange Multipliers
Published in Yogesh Jaluria, Design and Optimization of Thermal Systems, 2019
In order to determine whether the point is a maximum or a minimum, the second derivative is calculated. Because the slope goes from positive to negative, through zero, at the maximum, the second derivative is negative. Similarly, the slope increases at a minimum and, thus, the second derivative is positive. These conditions may be written as (Keisler, 2012) Foramaximum:d2ydx2<0Foraminimum:d2ydx2>0
Prototype optimization
Published in Fuewen Frank Liou, Rapid Prototyping and Engineering Applications, 2019
There are many types of optimization problems such as continuous, discrete, constrained, unconstrained, and sometimes even a binary (Y/N) problems, which can also be used for optimization. Differential calculus is used for simpler kinds of problems like first-order and second-order equations. The derivative is taken, and then, it is decided whether one should go for maximum or minimum. The second derivative determines whether the value attains a maximum or minimum depending upon whether the value of the derivative is positive or negative. Differential calculus is also called classical optimization, to find the optimal solution to a set of equations. For a smooth and continuous function y=U(x)
Treating cancerous cells with viruses: insights from a minimal model for oncolytic virotherapy
Published in Letters in Biomathematics, 2018
Adrianne L. Jenner, Adelle C. F. Coster, Peter S. Kim, Federico Frascoli
The first stationary point listed, , is fixed on the vertical axis . The corresponding value of the characteristic equation at the stationary point is . The second derivative at the stationary point is and therefore will be a maximum for and a minimum for . This is summarized in Figure 4(a) along with the sign of .
Mechanism underlying initiation of migration of film-like residual oil
Published in Journal of Dispersion Science and Technology, 2022
Xu Han, Lihui Wang, Huifen Xia, Peihui Han, Ruibo Cao, Lili Liu
Because the radius of curvature includes a second derivative, it can be positive or negative. When the radius of curvature is negative, the surface is convex. Conversely, when the radius of curvature is positive, the surface is concave. The radius of curvature changes from positive to negative, indicating that the unevenness of the curved surface has changed. The boundary point of the unevenness of the curved surface is the inflection point. Calculations reveal that there are three inflection points over the entire oil film interface. Along the flow direction, when the oil film interface moves from point (19.3543, 10.4332) to point (19.904, 10.6011), the radius of curvature changes from positive to negative, indicating that the oil film interface changes from concave to convex. From point (31.0911, 7.8806) to point (31.1928, 7.316), the radius of curvature changes from negative to positive, indicating that the oil film interface changes from convex to concave. From (22.7676, 0.7759) to (22.2127, 0.6249), the radius of curvature changes from positive to negative, indicating that the oil film changes from concave to convex. As shown in Figure 18, from left to right, the oil film changes from concave to convex. A calculation reveals that point A (19.451, 10.4846) is the inflection point; then, the oil film interface changes from convex to convex. The oil film is concave, point B (31.1103, 7.7738) is the inflection point, and then it changes from concave to convex, and point C (22.7001, 0.7596) is the inflection point. Although they are all inflection points, the direction of the capillary force formed at the oil film interface before and after the inflection point is unclear.
Hybrid self-inertia weight adaptive particle swarm optimisation with local search using C4.5 decision tree classifier for feature selection problems
Published in Connection Science, 2020
Arfan Ali Nagra, Fei Han, Qing Hua Ling, Muhammad Abubaker, Farooq Ahmad, Sumet Mehta, Abeo Timothy Apasiba
Newton’s technique for gradient and for the minimisation of optimisation function the Hessian matrix of the second derivative is used (Sun & Yuan, 2006). In Quasi-Newton methods, the conversion of Hessian had restructured by examining successive gradient vectors. By the changes in gradients, Quasi-Newton approaches hypothesis a model of the desired function, which is well sufficient to create super-linear convergence. Moreover, the SIW-APSO has been integrated with BFGS. The BFGS illustrated in Algorithm (Noel, 2012).