Intervención humana altera la trama trófica del ecosistema costero
A NALIZARD ESARROLLO DE CONTENIDOS
Standing waves and nodal patterns may be formed when u∗, where |u| = s, is transmitted from one source T0 and is retransmitted from another sourceT1 after reaching it. Any point inZ2having both signals fromT0 andT1will have identified its nodal pattern as the ‘difference’ between them as addressed with concerns to the standard two transmitter model.
Nodal patterns depends upon the distance and the time from the initial trans- mitter and transmission respectively. Let us assume that the symbol wi would
have been received by the vertex at coordinate x from the transmitter at co- ordinate xT at time t. In order to refer to the index i for wi the function
N(xT, x, t) = (x, w(t−(x−xT)) (mod|W|)) is defined. This forms the very intuitive notion of waves that may pass at over each other simultaneously and is directly comparable to the notion that is seen in the physical world. However such a rep- resentation is not possible in broadcasting automata model. In this model due to the message passing primitive for both the synchronous and the asynchronous models whereby the automata sends a message and then refuses all other messages for a single round so as not to receive the message it has just sent. This affords broadcasting automata a simple way to send messages such that they are broad- cast away from the transmitter which in turn generates the wave formations. As such the construction of nodal patterns on the plane is examined here with re- spects to the model for broadcasting automata that had been described. It shall be seen later that the nodal patterns, used for the construction of algorithms, are
structurally equivalent regardless of the method of propagation used. This section aims to bring together and show the clear equivalence of both models, by showing propagation equivalence, and dimension equivalence. Both of these goals may be made possible by discussing the distance functions that represent the transmission of Moore and Von Neumann waves by broadcasting automata.
As already addressed in waves for one dimension there is no difference to the resulting propagation when using either a Moore or Von Neumann transmission pattern. However, upon moving the waves to two dimensions, Z2, it is important to define the distance between the points in terms of (square or diamond) wave propagation for correct calculation of the nodal patterns, as it is not the same as in Cartesian geometry. The following definitions correspond to the well known Lp spaces, specifically, L1, representing the Von Neumann neighbourhood and, L∞, representing the Moore Neighbourhood where, L2, is the standard Euclidean metric [69].
Definition 3.6. Square waves adhere to the distance function d for distances
from p to q where d : Zn ×
Zn → Z and is defined by the correspondence
d((p1, . . . , pn),(q1, . . . , qn)) = max1≤i≤n{|pi−qi|},(p1, . . . , pn),(q1, . . . , qn)∈Zn.
Definition 3.7. Diamond waves adhere to the distance function d♦ for distances from p to q where d♦ : Zn × Zn → Z and is defined by the correspondence
d♦((p1, p2, ..., pn),(q1, q2, ..., qn)) =
Pn
i=1|pi−qi|,(p1, p2, ..., pn),(q1, q2, ..., qn)∈Zn.
Informally speaking, the distance functions defined above provide the time taken for the diamond/square wave front to reach an automata at coordinate (p1, p2, ..., pn)
from a transmitter at (q1, q2, ..., qn).
Let us consider two ways of computing nodal patterns for this process in terms of synchronous and asynchronous cases with transmitters T0 and T1. In both cases we aim to get the same distribution of nodal patterns on Z2 without the need to continuously propagate values from the sources. It is also with this method that it will finally become apparent that all of the preceding intuition regarding these calculations, in which the waves are able to pass over each other much like
occurrences in physical media, coincides with the reality of the calculation that happens within the model. The local distribution of these patterns will allow us later to locate points in a non-oriented environment.
In order to illustrate how the two different models, synchronous and asynchronous, can be reconciled with the more intuitive notion of how the waves might move in the plane the introduction of the following definition of a nodal pattern for a point on the plane will prove insightful.
Definition 3.8. An indexiof a nodal pattern Pi, resulting from the transmission
of two waves, in a point ρ∈Z2 is defined as
iρ =|d(T0, ρ)−(d(T0, T1) +d(T1, ρ))| (mod s)
where d is understood as d ord♦ for square or diamond wave respectively.
This definition illustrates all that is needed to know about the way in which these nodal patterns are formed on the plane. It is the distance, which is, ultimately, equatable to time, as each automata takes a constant amount of time to process and pass the message, that is between the time taken for the first message, from the first transmitter, to reach the automata and the time taken for that message to reach the second transmitter, thus activating it, and for that second transmit- ters first message to reach the automata which solidifies the nodal pattern after the receipt of two messages. The following diagram depicts this labelling of the automata which is seen in Figure. 3.3.
Synchronous model with a single message. In case of transmitting a single message, waves here are activation waves, which the automata use to start internal clocks. Transmissions begin fromT0 where the activation wave propagates through the use of a square (or diamond) wave arriving at T1 which is activated when reached by the first wave after a constant delay s. Any point that has received the first signal will start its internal clock (which counts modulo s), and then after receipt of the second signal the clock is stopped and the value of clocks
T
1T
0d(T
0,T
1)
d(T
0,p)
d(T
1,p)
p
Figure 3.3: A depiction of the labelling of automata on the plane. Each of the
distances are computed with the appropriate distance function for the shape of the broadcast, here, square or diamond. As the nodal pattern is decided by the distance between the first transmission to be received byp and the second the distances must be calculated as shown and the absolute difference found in
order to label p.
corresponds to the index of the nodal pattern for this point which is |(d(T0, ρ) (mod s))−((d(T0, T1) (mod s)) + (d(T1, ρ) (mod s)))|.
Asynchronous model with multiple messages. In the case of the asyn- chronous model, the same distribution of nodal patterns can be simulated by sending a wave from T0 where, on the wave front, every point that receives a symbol ui immediately transmits the symbol u(i+1) (mods). The synchronisation of wave propagation is achieved by assuming that every transmission takes the same constant time. Then transmitter T1 operates in the same way once reached by ui, transmitting the next symbol corresponding to ui+1 but using a different alphabet{v1, ..., vs}to avoid problems whereby transmitting in the same alphabet
could have a blocking effect on the wave. Each node should now contain a pair of symbols (ui0, vi00) which is enough to define the patternP|i0−i00|, where i0 =d(T0, ρ)
(mod s) and i00= (d(T0, T1) +d(T1, ρ)) (mod s).
In the next theorem the properties of nodal pattern distribution is shown resulting in a new approach for partitioning Z2 via non-oriented transmissions and is one of the core tools for the geometric algorithms discussed in this thesis.