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A directed graph is simple if each ordered pair of vertices is the head and tail of at most one edge; one loop may be present at each vertex. For each n ≥ 1, prove or disprove the following statement. Every simple directed graph with n vertices has two vertices with the same outdegree or two vertices with the same indegree.
(a) How many words of length 2 are there over the alphabet A = { 0; 1; 2; 3}?
(b) Construct the de Bruijn digraph D4,2 and use this digraph to nd an appropriate de Bruijn sequence.
(c) For b = 4 and n = 2, how many de Bruijn sequences can be formed?
Suppose [S, T] and [X, Y ] are two minimal cuts in a network N. Prove that both [S [X U T ∩ Y ] and [S ∩ X, T U Y ] are also minimal cuts in N.
The risk of being able to recognize the respondent's confidential information in the data set. Number of approaches has been proposed to measure the disclosure risk some of which c
Prevented fraction is a measure which can be used to attribute the protection against the disease directly to an intervention. The measure can given by the proportion of disease w
Reliability theory is the theory which attempts to determine the reliability of the complex system from knowledge of the reliabilities of the components. Interest might centre on
the problem that demonstrates inference from two dependent samples uses hypothetical data from TB vaccinations and the number of new cases before and after vaccinations for cases o
Johnson''s Job Sequencing for n jobs and 2 machines
Ask quesoil company is considering whether or not to bid for an offshore drilling contract. If they bid, the value would be $600m with a 65% chance of gaining the contract. The com
Chernoff's faces : A method or technique for representing the multivariate data graphically. Each observation is represented by the computer-created face, the features of which are
Lexis diagram is the diagram for displaying the simultaneous effects of the two time scales (generally age and calendar time) on a rate. For instance, mortality rates from cancer
Hazard regression is the procedure for modeling the hazard function which does not depend on the suppositions made in Cox's proportional hazards model, namely that the log-hazard
Bayes factor : A summary of evidence for the modelM1 against the another modelM0 provided by the set of data D, which can be used in the model selection. Given by the ratio of post
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