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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.
Pascal's triangle is an arrangement of numbers described by Pascal in his Traité du Triangle Arithmétique published in the year 1665 as 'The number in each cell is equal to in the
Markers of disease progression : Quantities which form a general monotonic series throughout the course of the disease and assist with its modelling. In uasual such quantities are
Link functions: The link function relates the linear predictor ηi to the expected value of the data. In classical linear models the mean and the linear predictor are identical
Method of moments is the procedure for estimating the parameters in a model by equating sample moments to the population values. A famous early instance of the use of the proced
The generalization of the normal distribution used for the characterization of functions. It is known as a Gaussian process because it has Gaussian distributed finite dimensional m
Resentful demoralization is the possible phenomenon in the clinical trials and intervention studies in which comparison groups not attaining a perceived desirable treatment become
Primary Model Below is a regression analysis without 17 outliers that have been removed Regression Analysis: wfood versus totexp, income, age, nk The regression equat
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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
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