The notion of loops

We have argued in the introductory section that the cluster expansion in powers of j3c breaks down as soon as a chain strongly interacts with a lot of others. To overcome this problem we must resum the cluster expansion to arrive at the so-called loop expansion . We first define the number of loops. 

The number of loops nr of a Feynman diagram is the number of internal momentum integrations which survive after all momentum conserving 6-functions have been exploited. 

For the diagrams of Fig. 5.3 we, for instance, find n (5.3a) = 1. nt(5.3 ) = 2. 71 (5.3c) = 1. We may show that with each independent internal momentum we may associate a dosed curve on the graph, through which that momentum flows. This explains the notion of loop . A connected diagram without loops is called a tree . 

5- Grand Canonical Description of Solutions at Finite Concentration 

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A remarkable property of graphs is used here—the notion of loop (Berge 1962). A loop is a circular path in a graph that encodes identity or equivalence or a neutral operation. Many physical meanings can be attributed to the existence of a loop it is not possible to discuss them in detail for... 

The notion of quantum feedback control naturally suggests a closed-loop process in the laboratory to stabilize or guide a system to a desired state. In addition, feedback is important in the design of molecular controls. These points will be made clear below, starting with considerations of design followed by a discussion of its role in the laboratory and finally leading to feedback concepts for the inversion of laboratory data. 

Often one relates this type of stabilization to the so-called steric factor [48-51], the notion of which was introduced much later than Rehbinder s concept of lyophilic structural-mechanical barrier. Steric factor primarily reflects configurational elasticity of tails and loops of adsorbed macromolecules as well as osmotic effects. Steric factor represents only an entropic ( generally speaking, small) contribution to the elastic resistance of film, and by itself can not account for the strong stabilization. 

Discuss the notion of the feedback loop and describe the hardware elements needed for its implementation. 

While designing a feedback control system (i.e., selecting its components and tuning its controller), we are seriously concerned about its stability characteristics. Therefore, before we proceed with the particular details of designing a feedback control loop, we will study the notion of stability and analyze the stability characteristics of closed-loop systems. 

The process variables do not always fall into such neat categories. For instance, temperature may be manipulated to adjust average molecular weight. In this case temperature is the manipulated variable for an MW control loop, but may at the same time be the controlled variable for a temperature control loop which uses the flow rate of a coolant as a manipulated variable. In this case the value of the manipulated variable for the MW control loop (temperature) is the desired value for the temperature control loop. This, of course, is the notion of cascade control. 

Some other very fundamental notions are equally handled by this tool, such as the notion of commutativity of operators and its link with the notions of graph symmetry and of graph closure or of circularity (paths in a loop). 

The notion of spillover is important with respect to neglected structural modes. Other modelling errors include parametric uncertainties, which are more difficult to model and may have a substantial impact on the stability and performance of the closed-loop system. 

The work in [3] introduced the notion of a Parallelism Opportunity (POP). This is a code fragment or construct that can be executed by processing elements in parallel. This could be a parallel block, parallel iterations of a for loop over a structure or container, parallel evaluations of subprogram calls, and so on. That work also introduced the term tasklet to capture the notion of a single execution trace within a POP, which the programmer can express with special syntax, or the compiler can implicitly create. 

An electronic system is a concurrent parallel system. All circuits in an electronic assembly are continuously operating. In contrast, software systems incorporate the notion of program flow, in which the sequence of instructions executed is explidtly stated through features such as loops and conditional statements. 

Now, from its essential notion, we have the feedback interconnection implies that a portion of the information from a given system returns back into the system. In this chapter, two processes are discussed in context of the feedback interconnection. The former is a typical feedback control systems, and consists in a bioreactor for waste water treatment. The bioreactor is controlled by robust asymptotic approach [33], [34]. The first study case in this chapter is focused in the bioreactor temperature. A heat exchanger is interconnected with the bioreactor in order to lead temperature into the digester around a constant value for avoiding stress in bacteria. The latter process is a fluid mechanics one, and has feedforward control structure. The process was constructed to study kinetics and dynamics of the gas-liquid flow in vertical column. In this second system, the interconnection is related to recycling liquid flow. The experiment comprises several superficial gas velocity. Thus, the control acting on the gas-liquid column can be seen as an open-loop system where the control variable is the velocity of the gas entering into the column. There is no measurements of the gas velocity to compute a fluid dynamics... 

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