Propagation space

In order to transform a nested loop with linear dependencies into an equivalent URE form, the propagation space of each variable should be determined and the localization of the data movements should be performed. Generally, propagation exists when the propagation space of a variable instance contains more than one node of the index space. The propagation in the index space can take two different forms, namely broadcast operations and Jan-in . More specifically, broadcasting occurs when a value of an instance is distributed to many nodes of the space, while fan-in occurs when different values of an instance are concentrated from many nodes to one node. The latter is the main feature of WSACs [20]. 

Formally, the propagation space of a variable X (f (i)) is the linear subspace of the index space defined by the set of equations... 

In this case, the DVs can be seen as integer vectors that express the movement of the value of a variable instance within its propagation space. Consider any two nodes ii — (in,. ., Hiv) and i2 = ( 21, , i2iv) included in the propagation space of a variable. The coordinates of ii and 12 are not independent since equation (1) holds for both nodes. Therefore, the following equations should be satisfied ... 

Scheuerer, R., Haufiler, M., Renk, K.F., Schomburg, E., Koschurinov, Y., Pavelev, D., Maleev, N., Ustinov, V., and Zhukov, A. (2003) Frequency multiplication of microwave radiation by propagating space-charge domains in a semiconductor superlattice. Appl. Phys. Lett., 82 (17), 2825-2828. 

Equation (A 1,6.8). along with the definitions (Al.h.S) and (Al.6.6) constitute the central equation for the propagation of electromagnetic waves in free space. The fonn of section A 1,6.4 admits hamionic solutions of the fonn... 

The last attribute of tire electromagnetic field we need to discuss is wave polarization. The nature of tire transverse field is such tliat tire oscillating field disturbance (which is perjDendicular to tire propagation direction) has a particular orientation in space. The polarization of light is detennined by tire time evolution of tire direction of tire electric field... 

Excitable media are some of tire most commonly observed reaction-diffusion systems in nature. An excitable system possesses a stable fixed point which responds to perturbations in a characteristic way small perturbations return quickly to tire fixed point, while larger perturbations tliat exceed a certain tlireshold value make a long excursion in concentration phase space before tire system returns to tire stable state. In many physical systems tliis behaviour is captured by tire dynamics of two concentration fields, a fast activator variable u witli cubic nullcline and a slow inhibitor variable u witli linear nullcline [31]. The FitzHugh-Nagumo equation [34], derived as a simple model for nerve impulse propagation but which can also apply to a chemical reaction scheme [35], is one of tire best known equations witli such activator-inlribitor kinetics ... 

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