Interval logic

P. M. Melhar-Smith, A Graphical Representation of Interval Logic, in CONCURRENCY 88, International Cortference on Concurrency (F. H. Vogt, ed.), pp. 106-120, Springer-Verlag, 1987. 

Prooftest - Initiator 12 months Interval - Logic solver 96 months - Final ehnent 33 months - Initiator 95 % Coverage - Logic solver 95 % - Final element 95 %... 

The principle of applying fuzzy logic to matching of spectra is that, given a sample spectrum and a collection of reference spectra, in a first step the reference spectra are unified and fuzzed, i.e., around each characteristic line at a certain wavenumber k, a certain fuzzy interval [/ o - Ak, + Afe] is laid. The resulting fuzzy set is then intersected with the crisp sample spectrum. A membership function analogous to the one in Figure 9-25 is applied. If a line of the sample spec-... 

For standard deviations, an analogous confidence interval CI(.9jr) can be derived via the F-test. In contrast to Cl(Xmean), ClCij ) is not symmetrical around the most probable value because by definition can only be positive. The concept is as follows an upper limit, on is sought that has the quality of a very precise measurement, that is, its uncertainty must be very small and therefore its number of degrees of freedom / must be very large. The same logic applies to the lower limit. s/ ... 

TIME DELAY SUBROUTINE IS EFFECTED EVERY COMMUNICATION INTERVAL USING THE RESERVED LOGICAL CONTROL VARIABLE... 

It may be useful to point out a few topics that go beyond a first course in control. With certain processes, we cannot take data continuously, but rather in certain selected slow intervals (c.f. titration in freshmen chemistry). These are called sampled-data systems. With computers, the analysis evolves into a new area of its own—discrete-time or digital control systems. Here, differential equations and Laplace transform do not work anymore. The mathematical techniques to handle discrete-time systems are difference equations and z-transform. Furthermore, there are multivariable and state space control, which we will encounter a brief introduction. Beyond the introductory level are optimal control, nonlinear control, adaptive control, stochastic control, and fuzzy logic control. Do not lose the perspective that control is an immense field. Classical control appears insignificant, but we have to start some where and onward we crawl. 

Ferromanganese oxides. An extensive Mo isotope dataset is now available for ferromanganese crusts and nodules, which are enriched in Mo and hence are logical targets for early isotopic investigations (Barling et al. 2001 Siebert et al. 2003). These include nodules from the Paciflc and Atlantic Oceans, and crusts from the Pacific, Atlantic and Indian Oceans. Nodule data are of bulk samples, but Paciflc and Atlantic crust data are time-resolved at l-3 million year intervals from the present back to 60 million years ago. 

A special kind of random noise, pseudo random noise, has the special property of not being really random. After a certain time interval, a sequence, the same pattern is repeated. The most suitable random input function used in CC is the Pseudo Random Binary Sequence (PRBS). The PRBS is a logical function, that has the combined properties of a true binary random signal and those of a reproducible deterministic signal. The PRBS generator is controlled by an internal clock a PRBS is considered with a sequence length N and a clock period t. It is very important to note that the estimation of the ACF, if computed over an integral number of sequences, is exactly equal to the ACF determined over an infinite time. 

What was missing in the previous section was a definition of what is meant by equivalence. Since it is imlikely that two treatments wiU have exactly the same effect we will need to consider how big a difference between the treatments would force us to choose one in preference to the other. In the t)q)hoid example there was a difference in rates of 1.9% and we may well believe that such a small difference would justify us in claiming that the treatment effects were the same. But had the difference been 5% would we still have thought them to be the same Or 10 There will be a difference, say S %, for which we are no longer prepared to accept the equivalence of the treatments. This is the so-called equivalence boimdary. If we want then to have a high degree of confidence that two treatments are equivalent it is logical to require that an appropriately chosen confidence interval (say 95%) for the treatment differences should have its extremes within the boundaries of equivalence. 

It has been suggested that the LOD should be the concentration for which the lower limit of the confidence interval just reaches zero, and while this has a logic to it, as the value is not known until measurements are made, the analyst could be in for a number of speculative experiments before finding the magic concentration that satisfies the condition. 

Coindence Logic pulses at two or more inputs Logic pulse if pulses appear at all inputs within a time interval At... 

As it has been mentioned, apart from point estimates there exist the parameter interval estimates. No matter how well the parameter estimate has been chosen, it is only logical to test the estimate deviation from its correct value, as obtained from the sample. For example, if in numerical analysis one obtains that the solution of an equation is approximately 3.24 and that 0.03 is the maximal possible deviation from the unknown correct solution of the equation, then we are absolutely sure that the range (3.24-0.03=3.21 3.24+0.03=3.27) contains the unknown correct solution of the equation. Therefore the problem of determining the interval estimate is formulated in the following way ... 

Allow a modest proposal T raceability is the ability to demonstrate that measurements are what they are purported to be. Because measurements are always expressed and communicated in the form of numerical values (with associated uncertainties or equivalent intervals between numerical values, at stated levels of confidence) multiplied by measurement units, it then follows by ineluctable logic that the end point of any traceability chain is simply the units... 

The interoccasion variability (IOV) or intraindividual variability [11] arises when a parameter of the model, e.g. CL, varies within a subject between study occasions. The term occasion can be defined arbitrarily, usually logical intervals for an occasion are chosen, e.g. each dosing interval in multiple dose studies or each treatment period of a cross-over study can be defined as an occasion. To assess the IOV of a specific parameter more than one measurement per individual has to be available per occasion. The IOV can be implemented in the random effect model as described in the following ... 

Figure 15.3 refers back to our trial where 42 out of 50 patients showed a successful outcome. Notice that the interval is asymmetrical. The asymmetry arises because possible values are more tightly constrained on one side than the other. The upper limit of the interval could not logically be greater than 100 per cent, so the upper limit cannot be far above the point estimate of 84 per cent. However, the lower limit could be anything down to 0 per cent. Confidence intervals for proportions are always asymmetrical, unless the point estimate happens to be exactly 50 per cent (as in Figure 15.2). 

Most processes operate more efficiently when functions must occur in a desired time sequence or at prescribed intervals of time. In the past, mechanical timers and logic relays were used. Now electronic logic and timing devices are used based on computer software programmable logic controllers. They lend themselves to easy set-up, reprogramming, and provide more accurate control. 

Inference effects relate to systematic and random errors in modelling inducing problems of drawing extrapolations or logic deductions from small statistical samples, from animal data or experimental data onto humans or from large doses to small doses, etc. All of these are usually expressed through statistical confidence intervals ... 

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