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ALT Two vectors v and w are drawn in R^2, in standard position. A line is drawn through one of them (w) and turned into a number line in the direction of w. Both vectors are projected orthogonally onto this line (v projects onto the number -4, and w projects onto its magnitude, which is shown as the number 3). The dot product is the product of these projections: -3*4 = -12.
ALT Two vectors v and w are drawn in R^2, in standard position. A line is drawn through one of them (w) and turned into a number line in the direction of w. Both vectors are projected orthogonally onto this line (v projects onto the number -4, and w projects onto its magnitude, which is shown as the number 3). The dot product is the product of these projections: -3*4 = -12 ("comp" is used to denote the projections).
ALT Context menu with the following items: * Cut * Copy * Copy as Markdown * Paste * Paste without formatting * Paste from Markdown * Delete
ALT A linear program in two dimensions, with one axis for each of the two candidate materials (metal-organic frameworks), measured in kg. Constraints are based on drinking-water delivery and mass. The corners of the feasible region are marked, with the optimal composition being located at a corner in the interior of the first quadrant, and the optimal single-material designs located on the axes.
ALT A linear program in two dimensions, with one axis for each of the two candidate materials (metal-organic frameworks), measured in kg. Constraints are based on drinking-water delivery and mass. The corners of the feasible region are marked, with the optimal composition being located at a corner in the interior of the first quadrant, and the optimal single-material designs located on the axes.
ALT Four American white pelicans on a pond, seen through tree branches.
ALT Screenshot of the following text: Population models employing Gompertz’s insight in the most transparent way take the following form: dN/dt = G(t)N(t), dG/dt= −αG(t), (2.11) with initial data N(0) = N0 > 0, G(0) = G0 > 0. Here, N(t) is the number of individuals in the population and G(t) represents per capita growth rate (birth rate minus death rate at time t) and is loosely related to Gompertz’s notion of the “power to avoid death.” Really, however, this model assumes that for some unspecified reason, the population growth rate decays exponentially over time at rate α, either from increasing death rates, as Gompertz assumed, or decaying reproduction rates or both. We assume, therefore, that α > 0. From the second equation in system (2.11), G(t) = G0e^(−αt), where G0 is some measure of the initial population’s net fecundity. So we can write the model as a nonautonomous ODE: dN/dt= G0e^(−αt)N(t), (2.12)