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The absurd high dimensionality of random graphs
The space of random networks is so ridiculously large that one needs to take drastic measures to make it manageable.
Achieving desired tolerance of a Taylor polynomial on desired interval
For a given interval around a point, how many terms of a Taylor polynomial expanded around that point must be used to achieve a required tolerance?
Definition: Adjacency matrix definition
The adjacency matrix is a matrix of ones and zeros where a one indicates the presence of the corresponding edge in the network.
An algebra trick for finding critical points
An old algebraic trick that can make finding critical points much easier.
Another differential equation: projectile motion
Calculating an equation for the position of an object acted upon by only gravity.
Approximating functions by quadratic polynomials
Determining how to use information from the second derivative to form a quadratic approximation to a function.
Approximating a nonlinear function by a linear function
The secant line and tangent line are two ways to approximate a nonlinear function by a linear one.
Area and definite integrals
Integrating to find the area under a curve or the area between two curves.
Introduction to autonomous differential equations
Introduction to solving autonomous differential equations, using a linear differential equation as an example.
Averages and weighted averages
Using integrals to calculate averages and weighted averages of a function.
Developing an initial model to describe bacteria growth
By analyzing some data and hypothesizing rules for cell division, we develop a discrete dynamical system for the growth of a population of bacteria.
Bacteria growth model exercise answers
Answers to selected exercises on developing discrete dynamical systems for the growth of a population of bacteria.
Bacteria growth model exercises
Exercises on developing discrete dynamical systems for the growth of a population of bacteria.
Developing a logistic model to describe bacteria growth
Modeling the slowing growth rate caused by environmental carrying capacity in bacteria growth data.
Basic integration formulas
Formulas for integration based on reversing formulas for differentiation.
Introduction to bifurcations of a differential equation
Representing the influence of a parameter on the behavior of a differential equation in terms of a bifurcation diagram of equilibria.
A birth-death process
Calculating the area under a curve using Riemann sums
Approximating the area under the graph of a positive function as sum of the areas of rectangles.
Illustration of Cartesian coordinates in two and three dimensions.
Definition: Cartesian product definition
The Cartesian product between two sets is the set of all possible ordered pairs with first element from the first set and second element from the second set.
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