Arithmetic Groups and Their Generalizations What, Why, and by Lizhen Ji

By Lizhen Ji

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The dynamics of these changes are defined by the following equation (Hellman, 1985): (   ∂u ( x, t ) = −  w ( x, t ) − w ( x, t ) u ( x, t ) dx  u ( x, t ) , x ∈ X, t ∈ 0, T  . 1) Assume that the density function w is already defined. 1 is −1  t    t   u ( x, t ) = u ( x ) exp  − w ( x, τ ) dτ  ×  u ( x ) exp  − w ( x, τ ) dτ  dx  . 2)  t  The value uˆ ( x, t ) = u ( x ) exp  − w ( x, τ ) dτ  is a joint probability that the target is located in the  0  point x and was not detected up to the time t.

Probably, the first results in this direction formulated in the terms of trajectories were reported in 1952 by Wilkinson (1952) who investigated a possibility of random search in the birds’ wandering. Following the Wilkinson results and based on the work published in 1951 by Skellam (1951), Patlak (1953) in 1953 suggested detailed mathematical techniques for modeling of animals’ migration in the terms of Brownian random walks. These models formed a basis for recent considerations of foraging following the methodology different from the optimization techniques, which are used in classical foraging theory.

According to the group-testing approach, assume that the search is conducted by the search system, which observes the observed areas a ⊂ X and obtains observation results z(a) ∈ {0, 1}. If z(a) = 1 continues search in a; otherwise, it continues with the points, which are not in a. Since the searcher seeks for a single target, the search terminates when the observed area a includes a single point, that is, a = {x}, and for this area, the searcher obtains observation result z ( a ) = z {x} = 1. ( ) As earlier, the search planning starts with the target’s location probabilities u ( xi ) , i = 1, 2,…, n, n u ( xi ) = 1, and results in a set A = {a1, a2 ,…, am } of search areas a j ⊂ X, j = 1, 2,…, m, and a ∑ i =1 sequence d = d ( 0 ) , d (1) , d ( 2 ) ,…, d ( T ) of decision rules d ( t | z ) , which specify the observed areas a(t) given the previous observed areas a(t−1) and observation results z a ( t −1) ; the rule d ( 0 ) is ( ) interpreted as a general decision to start the search in the set X.

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