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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.
Auto correlation : The correlation of the internal observations in the time series, generally expressed as a function of the time lag between the observations. It is also used for
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
Relative risk is the measure of the association between the exposure to a particular factor and the risk or probability of a convinced outcome, calculated as follows therefor
Generalized method of moments (gmm) is the estimation method popular in econometrics which generalizes the method of the moments estimator. Essentially same as what is known as the
You and your team have been hired as strategic consultants by the hugely successful retailer known as “Cutie Pie”. The company sells many products, although one product in particul
Multimodal distribution is the probability distribution or frequency distribution with number of modes. Multimodality is frequently taken as an indication which the observed di
Matching is the method of making a study group and a comparison group comparable with respect to the extraneous factors. Generally used in the retrospective studies when selecting
Mixture experiment is an experiment in which the two or more ingredients are blended together to form an end product. The measurements are taken on the several blends of the ingre
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Median is the value in a set of the ranked observations which divides the data into two parts of equal size. When there are an odd number of observations the median is middle v
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