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considering a small system in thermal contact with a large reservoir of temperature T , reads: Helmholtz Potential Minimum Principle. The equilibrium value of any unconstrained internal parameter in a system in d i a t h e m l contact with a heat reservoir minimizes the Helmholtz potential over the manifold of states for which T = T . [p. 1551 Similar results may be derived for the other thermodynamic potentials. There exists an alternative to thermodynamic potentials to condense all the information about a thermodynamic system into a single function that is useful in the context of developing beyond-equilibrium thermodynamics: Whereas the most common functions definable in terms of Legendre transformations are those mentioned . .. [above], another set can be defined by performing the Legendre transformation on the entropy rather than on the energy. That is, the fundamental relation in the form S = S(U,V,N1 ,N z , . . .) can be taken as the relation on which the transformation is performed. Such Legendre transforms of the entropy were invented by Massieu in 1869 and actually predated the transforms of the energy introduced by Gibbs in 1875. We refer to the transforms of the entropy as Massieubncfions,as distinguished from the thermodynamic potentials transformed from the energy. The Massieu funtions will turn out to be particularly useful in the theory of irreversible thermodynamics, and they also arise naturally in statistical mechanics and in the theory of thermal fluctuations. [p. 1511



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This algorithm has a complexity of O(log N) for each scheduling decision Simulation results show that this algorithm performs close to the analytical lower bounds [37] In [12], a low-overhead, bucket-based scheduling algorithm based on the square root rule was also provided In this strategy, the database is partitioned into several buckets, which are kept as cyclical queues The algorithm chooses to broadcast the first item in the bucket for which the expression [T R(Im)]2qm/lm evaluates to the largest value In the expression, T is the current time, R(i) is the time at which an instance of item i was most recently transmitted, Im is the first item in bucket m, and qm and lm are average values of qi s and li s for the items in bucket m Note that the expression [T R(Im)]2qm/lm is similar to equation (113).





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where j = j (rj ) [WT CT e]j and [e]j means j-th element of the vector e. On the basis of these formulas, we can formulate the supervised learning rules. The unsupervised learning algorithms can be alternatively used for updating the output matrices C, D as described in the 11. It is interesting to note that in the special case of binary signals, we can estimate the desired signals by taking d(k) = sign(y(k)), as illustrated for the simpli ed network shown in Fig. 12.6. In this case, instead of back propagating the true error, we can propagate the blind error. Instead of propagating the error e(k), we can back propagate mutual information e(k) = f (y(k)) y(k) [1323, 1324]. The algorithms discussed in this chapter can be considered as extensions and generalizations of some existing algorithms proposed for the linear multichannel blind separation and/or deconvolution problems. Computer simulation experiments con rmed the validity and their good performance of the developed algorithms and ability to recover the source signals even if the mixing dynamical model is nonlinear and its structure is unknown.

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The bucket-based scheduling algorithm is similar to the Bdisk approach, but in contrast to the Bdisk approach, which has a fixed broadcast schedule, the bucketbased algorithm schedules the items online As a result, they differ in the following aspects First, a broadcast program generated using the Bdisk approach is periodic, whereas the bucket-based algorithm cannot guarantee that Second, in the bucket-based algorithm, every broadcast instance is filled up with some data based on the scheduling decision, whereas the Bdisk approach may create holes in its broadcast program Finally, the broadcast frequency for each disk is chosen manually in the Bdisk approach, whereas the broadcast frequency for each item is obtained analytically to achieve the optimal overall system performance in the bucket-based algorithm Regrettably, no study has been carried out to compare their performance In a separate study [33], the broadcast system was formulated as a deterministic Markov decision process (MDP).

Su and Tassiulas [33] proposed a class of algorithms called priority index policies with length (PIPWL- ), which broadcast the item with the largest (pi/li) [T R(i)], where the parameters are defined as above In the simulation experiments, PIPWL-05 showed a better performance than the other settings did 1122 On-Demand Data Scheduling As can be seen, push-based wireless data broadcasts are not tailored to a particular user s needs but rather satisfy the needs of the majority Further, push-based broadcasts are not scalable to a large database size and react slowly to workload changes To alleviate these problems, many recent research studies on wireless data dissemination have proposed using on-demand data broadcast (eg, [5, 6, 13, 34]) A wireless on-demand broadcast system supports both broadcast and on-demand services through a broadcast channel and a low-bandwidth uplink channel.

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service, TL the time it costs to serve all the class Ic customers arrival of the class r customer (note the summation TL the time it costs to serve all the class k customers

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