Consider a discrete model given by
Investigate the linear stability about the positive steady state N by setting . Show that nt satisfies the equation
Hence show that is a bifurcation value and that as
the steady state bifurcates to a periodic solution of period 6.
The positive steady state (N^*) is obtained by setting (N_{t+1}=N_t=N_{t-1}=N^*) in [ N_{t+1}=\frac{rN_t}{1+bN_{t-1}^2}, ] which gives [ N^*=\frac{rN^*}{1+b(N^*)^2}. ] Since (N^*>0), this reduces to [ 1+b(N^*)^2=r \quad \Rightarrow \quad b(N^*)^2=r-1. ] To investigate linear stability, write [ _____ ____ ___ _________ ______ _____ __________ ____ ____ ______ ________.
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Return distributions of the two securities are given below:
| Return | Probabilities | |
| X | Y |
|
| 0.20 | 0.15 | 0.30 |
| 0.15 | 0.08 | 0.25 |
| 0.10 | 0.05 | 0.15 |
| 0.11 | 0.09 | 0.25 |
The population consisting of all married couples is collected. The data showing the age of 12 married couples is as follows:
| Husband’s age (years) | Wife’s age (years) | Husband’s age (years) | Wife’s age (years) |
| 32 | 27 | 51 | 50 |
| 25 | 30 | 48 | 46 |
| 36 | 34 | 37 | 36 |
| 72 | 65 | 50 | 42 |
| 37 | 37 | 51 | 46 |
| 36 | 38 | 36 | 35 |
Consider a discrete model given by
Investigate the linear stability about the positive steady state N by setting . Show that nt satisfies the equation
Hence show that is a bifurcation value and that as
the steady state bifurcates to a periodic solution of period 6.
Companies considering the purchase of a computer must first assess their future needs in order to determine the proper equipment. A computer scientist collected data from seven similar company sites so that computer hardware requirements for inventory management could be developed. The data collected is as follows:
| Customer Orders (in thousands) | Add-delete items (in thousands) | CPU time (in hours) |
| 123.5 | 2.108 | 141.5 |
| 146.1 | 9.213 | 168.9 |
| 133.9 | 1.905 | 154.8 |
| 128.5 | 0.815 | 146.5 |
| 151.5 | 1.061 | 172.8 |
| 136.2 | 8.603 | 160.1 |
| 92.0 | 1.125 | 108.5 |
A library wants to improve its service facilities in terms of the waiting time of its borrowers. The library has two counters at present and borrowers arrive according to Poisson distribution with arrival rate 2 every 10 minutes and service time follows exponential distribution with a mean of 15 minutes. The library has relaxed its membership rules and a substantial increase in the number of borrowers is expected. Find the number of additional counters to be provided if the arrival rate is expected to be twice the present value and the average waiting time of the borrower must be limited to half the present value.
Maximize , subject to the constraints
and
and are integers.
Let be a portfolio of two securities X and Y. Find the values of w1 and w2 in the following situations:
i) and P is risk free.
ii) and variance P is minimum.
iii) Variance P is minimum and and
.