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with probability 1. Using this, the theorems of 3 can easily be extended to a general stationary ergodic process. We can de ne a typical set in the same way as we did for the i.i.d. case in 3. By the same arguments, we can show that the typical set has a probability close to 1 and that there are about 2nH (X ) typical sequences of length n, each with probability about 2 nH (X ) . We can therefore represent the typical sequences of length n using approximately nH (X) bits. This shows the signi cance of the entropy rate as the average description length for a stationary ergodic process. The entropy rate is well de ned for all stationary processes. The entropy rate is particularly easy to calculate for Markov chains. Markov Chains. For a stationary Markov chain, the entropy rate is given by H (X) = H (X) = lim H (Xn |Xn 1 , . . . , X1 ) = lim H (Xn |Xn 1 ) = H (X2 |X1 ), (4.25)

MST edges, edge e remains the longest edge of C0 , but all the other edges of C0 are now also in MST. This is a contradiction, since MST has no cycles. Therefore each edge AB from MST belongs to both LMST(A) and LMST(B), and therefore to LMST. A 10.5.3 Relative Neighborhood Graph

(physically clean); when all compounds used in cleaning and sanitizing and all the deposits (i.e. scale) are removed by rinsing (chemically clean); and when the number of present microorganisms has been reduced to acceptable levels (microbiologically clean). Monitoring steps in the cleaning program should include: (1) post-cleaning visual inspection to ensure that any visible food residue has been removed; (2) temperature checks for water and chemical solutions to ensure they are within the ranges speci ed in the procedure; (3) concentration checks for detergents and sanitizers to ensure they are used according to directions. Veri cation of the effectiveness of the program should be scheduled and may range from simple procedures that must be performed as a preoperative action on a daily schedule to more complex testing that may need to be outsourced: Processors often rely on sensory inspection (looking, touching, and smelling) to determine the effectiveness of their sanitation program: a ashlight (torch) can be used to inspect the inside of semi-enclosed equipment. Surfaces such as stainless steel should be smooth and nongreasy, and there should be no unpleasant odors. ATP (adenosinetriphosphate) uorescence testing can be used to detect the presence of both microorganisms and food residues. The ATP test is a rapid test; the results are available immediately and give a pass or fail for the sanitizing procedure. Microbiological testing of equipment and surfaces is required to verify if the level of microorganisms present post-sanitizing is within acceptable limits. Each plant should determine the points to sample and the frequency of sampling based on results over time.

5

(a) 2

BD-I B'

Disjoint Participation See Mutually Exclusive Participation Entity Relationship Diagram (ERD) A diagram showing all or part of a CDM written in the ERM ML Entity Relationship Model (ERM) A conceptual modeling language (CML) developed by Dr Peter Pin-Shan Chen in 1976 It captures concepts using ETs, RTs, and ATs We also use this term to refer to the extended or enhanced ERM, which includes ITs for inheritance Entity Type (ET) A CDM construct used to represent anything for which we want to capture details, relationships, and identity Grouping AT A CDM construct that represents an AT that contains other ATs This is analogous to a structure or user de ned data type in a programming language Identifying AT A CDM construct that represents an AT that is used to help uniquely identify an ET instance.

Identifying RT A CDM construct used to connect a Weak ET to the one or more Strong ETs that partially identi es it Inheritance Type (IT) A CDM construct that represents a generalization/specialization between two ETs Intentional Model A model that represents the planned structure or shape rather than the actual contents all the DMs discussed in this chapter are intentional models Logical Data Model (LDM) A middle-level DM that captures details pertinent to the logical design of the data storage for a DB This model will also usually capture the semantic concepts and their interrelationships/interdependencies in a way that is dependent on the data storage design (but not necessarily dependent on the data storage implementation) Modeling Language (ML) A vocabulary and set of rules used to de ne a model.