Introductory Combinatorics
5th Edition
ISBN: 9780134689616
Author: Brualdi, Richard A.
Publisher: Pearson,
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Chapter 11, Problem 55E
To determine
The trees which has a Hamilton path.
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Calculs Insights
πT
| cos x |³ dx
59
2
2. Consider the ODE
u' = ƒ (u) = u² + r
where r is a parameter that can take the values r = −1, −0.5, -0.1, 0.1. For each value of r:
(a) Sketch ƒ(u) = u² + r and determine the equilibrium points.
(b) Draw the phase line.
(d) Determine the stability of the equilibrium points.
(d) Plot the direction field and some sample solutions,i.e., u(t)
(e) Describe how location of the equilibrium points and their stability change as you increase the
parameter r.
(f) Using the matlab program phaseline.m generate a solution for each value of r and the initial
condition u(0) = 0.9. Print and turn in your result for r = −1. Do not forget to add a figure caption.
(g) In the matlab program phaseline.m set the initial condition to u(0) = 1.1 and simulate the ode
over the time interval t = [0, 10] for different values of r. What happens? Why? You do not need to
turn in a plot for (g), just describe what happens.
The following are suggested designs for group sequential studies. Using PROCSEQDESIGN, provide the following for the design O’Brien Fleming and Pocock.• The critical boundary values for each analysis of the data• The expected sample sizes at each interim analysisAssume the standardized Z score method for calculating boundaries.Investigators are evaluating the success rate of a novel drug for treating a certain type ofbacterial wound infection. Since no existing treatment exists, they have planned a one-armstudy. They wish to test whether the success rate of the drug is better than 50%, whichthey have defined as the null success rate. Preliminary testing has estimated the successrate of the drug at 55%. The investigators are eager to get the drug into production andwould like to plan for 9 interim analyses (10 analyzes in total) of the data. Assume thesignificance level is 5% and power is 90%.Besides, draw a combined boundary plot (OBF, POC, and HP)
Chapter 11 Solutions
Introductory Combinatorics
Ch. 11 - Prob. 1ECh. 11 -
Determine each of the 11 nonisomorphic graphs of...Ch. 11 - Does there exist a graph of order 5 whose degree...Ch. 11 - Does there exist a graph of order 5 whose degree...Ch. 11 -
Use the pigeonhole principle to prove that f1...Ch. 11 - Let be a sequence of n nonnegative integers whose...Ch. 11 - Let G be a graph with degree sequence (d1, d2,...Ch. 11 - Draw a connected graph whose degree sequence...Ch. 11 - Prove that any two connected graphs of order n...Ch. 11 - Determine which pairs of the general graphs in...
Ch. 11 - Determine which pairs of the graphs in Figure...Ch. 11 - Prove that, if two vertices of a general graph are...Ch. 11 - Let x and y be vertices of a general graph, and...Ch. 11 - Let x and y be vertices of a general graph, and...Ch. 11 - Let G be a connected graph of order 6 with degree...Ch. 11 - Let γ be a trail joining vertices x and y in a...Ch. 11 - Let G be a general graph and let G' be the graph...Ch. 11 - Prove that a graph of order n with at least
edges...Ch. 11 - Prob. 21ECh. 11 - Prob. 26ECh. 11 - Prob. 27ECh. 11 - Determine if the multigraphs in Figure 11.41 have...Ch. 11 - Which complete graphs Kn have closed Eulerian...Ch. 11 - Determine all nonisomorphic graphs of order at...Ch. 11 - Solve the Chinese postman problem for the complete...Ch. 11 - Call a graph cubic if each vertex has degree equal...Ch. 11 - * Let G be a graph of order n having at...Ch. 11 - Let be an integer. Let Gn be the graph whose...Ch. 11 - Prove Theorem 11.3.4.
Ch. 11 - Which complete bipartite graphs Km, n have...Ch. 11 - Prove that Km,n is isomorphic to Kn,m.
Ch. 11 - Is GraphBuster a bipartite graph? If so, find a...Ch. 11 - Prob. 50ECh. 11 - Prob. 51ECh. 11 - Prob. 53ECh. 11 - Which trees have an Eulerian path?
Ch. 11 - Prob. 55ECh. 11 - Prob. 56ECh. 11 - Prob. 58ECh. 11 - Prove that the removal of an edge from a tree...Ch. 11 - Prob. 60ECh. 11 - Prob. 62ECh. 11 - Prob. 63ECh. 11 - Prob. 64ECh. 11 - How many cycles does a connected graph of order n...Ch. 11 - Prob. 68E
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