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Copyright CRC Press 2023 - present CRC Press

Author: Dr Stephen Lynch National Teaching Fellow FIMA SFHEA

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Chapter 6: Biology

Question 6.1: This is a one-dimensional discrete system:

$$c_{n+1}=f(c_n)=(1-a)c_n+bc_n^re^{-sc_n},$$

where $c_n$ is blood cell count.

Fixed points of period one satisfy the equation: $c_{n+1}=c_n=c$, say.

In this case, solve the equation $f(c)-c=0$, to determine the $c^*$ fixed points.

Suppose that there is a fixed point at $c=c^*$. Then, fixed points are stable if:

$$\left| \frac{df(c^*)}{dc} \right|<1.$$

Label the critical points: $O=(0,0)$, $A=(4,0)$ and $B=(\frac{5}{4},\frac{11}{4})$.

The eigenvalues are:

For critical point O: $\lambda_1=4$, $\lambda_2=-1$, and the critical point is unstable, since one eigenvalue is positive.

For critical point A: $\lambda_1=11$, $\lambda_2=-4$, and the critical point is unstable, since one eigenvalue is positive.

For critical point B: $\lambda_1=-2-3.1225i$, $\lambda_2=-2+3.1225i$, and the critical point is stable, as both eigenvalues have negative real part.

Phase portrait of a predator-prey system. The axes would be scaled by large numbers in applications and the species would co-exist with scaled populations of $x=\frac{5}{4}$ and $y=\frac{11}{4}$, assuming of course that neither of the populations are zero.

Chapter 7: Chemistry

Thus, the balanced chemical reaction equation is:

$$3 \mathrm{NaHCO}_3+\mathrm{H}_3\mathrm{C}_6\mathrm{H}_5\mathrm{O}_7 \rightleftharpoons \mathrm{Na}_3\mathrm{C}_6\mathrm{H}_5\mathrm{O}_7+3\mathrm{H}_2\mathrm{O}+3\mathrm{C}\mathrm{O}_2.$$

Chapter 8: Data Science

The optimal solution is, $z=268$, with $\left(x_1,x_2,x_3 \right)=1.8,20.8,1.6$.

Chapter 9: Economics

Chapter 10: Engineering

Exercise 10.4: The Double Pendulum.

Image in a markdown cell

Chapter 11: Fractals and Multifractals

Chapter 12: Image Processing

Important: If you wish to use the plots interactively, use Spyder. In the Console window type:

In[n]: matplotlib qt5

Chapter 13: Numerical Methods for ODEs and PDEs

Exercises 13.1 and 13.2: Use the implicit Euler method (Exercise 13.1) and the two-step Adams-Bashforth method (Exercise 13.2) to solve the initial value problem:

$$\frac{dy}{dx}=(x-3.2)y+8x\exp\left(\frac{(x-3.2)^2}{2} \right)\cos(4x^2),$$

given that $x_0=0, y_0=1, h=0.1$ and $0 \leq x \leq 5$.

Chapter 14: Physics

Chapter 15: Statistics

The F-statistic is 3.295, the $p$-value is $p=0.0188<0.05$, meaning there is statistically significant difference (so, we reject $H_0$, the null hypothesis). There is strong statistical evidence that the treatment has an effect on nitrous oxide levels, there is significant difference in mean concentrations due to drug treatments (we accept $H_1$).

End of Solutions 2