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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.
Post stratification adjustmen t: One of the most often used population weighting adjustments used in the complex surveys, in which weights for the elements in a class are multiplie
This is the branch of mathematics which deals with the theory of contests between two or more players under the specified sets of rules. The subject supposes a statistical aspect w
Collective risk models : The models applied to insurance portfolios which do not create direct reference to the risk characteristics of individual members of the portfolio when des
Band matrix: A matrix which has its non zero elements arranged uniformly near to the diagonal, so that aij = 0 if (i - j)> ml or (j - i)> mu where aij are the elements of matrix a
Independent component analysis (ICA) is the technique for analyzing the complex measured quantities thought to be mixtures of other more fundamental quantities, into their fundamen
Models which make use of the smoothing techniques such as locally weighted regression to identify and represent the possible non-linear relationships between the explanatory and th
how to describe association between quantitative and categorical variables
Machine learning is a term which literally means the ability of a machine to recognize patterns which have occurred repetitively and to improve its performance based on the past
1. You are interested in investigating if being above or below the median income (medloinc) impacts ACT means (act94) for schools. Complete the necessary steps to examine univariat
Categorizing continuous variables : A practice which involves the conversion of the continuous variables into the series of the categories, which is common in the field of medical
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