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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 model for data containing continuous and categorical variables both.The categorical data are summarized by the contingency table and their marginal distribution, 182by the mult
The number of employees absent from work at a large electronics manufacturing plant over aperiod of 106 days is given in the table below. 146 141 139 140 145 141 142 131 142 140
Window variables are the variables measured during the constrained interval of an observation period which is accepted as the proxies for the information over the whole period. Fo
Locally weighted regression is the method of regression analysis in which the polynomials of degree one (linear) or two (quadratic) are used to approximate regression function in
Dr. Stallter has been teaching basic statistics for many years. She knows that 80% of the students will complete the assigned problems. She has also determined that among those who
what is operational gaining
The graphical method for studying the behavior of the seasonal time series. In such a plot, the January values of seasonal component are graphed for the upcoming years, then the
Partial least squares is an alternative to the multiple regressions which, in spite of using the original q explanatory variables directly, constructs the new set of k regressor v
It is the multivariate normal random vector which satisfies certain conditional independence suppositions. This can be viewed as a model framework which contains a wide range of st
The graphic representation of the alternatives in a decision making problem which summarizes all the possibilities foreseen by the decision maker. For instance, suppose we are give
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