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Mat 540 Assignment 1

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we consider that a_{u}=0, this make the search paths more flexible and will take the form: x_{iu}(t)=[t/2]R_{u}cos t; #5 x_{ju}(t)=[t/2]R_{u}sin t; #6 x_{ku}(t)=q_{u}sin(_{u}t), #7 see figure 2. Assuming that the searchers motion is independent of the target motion and is not specifically directed toward escape. Moreover, it is to be noted that S_{u}'s path intersects the target's path at maximum number of points, if the speed of each searcher is much larger than the target's speed and E(|X₀₁X₀₂...X_{0N}|) (the N^{th} order moment of the target's initial position) is finite. Also, let the first meeting time be a random variable valued in ℝ⁺ and it will happen at the position min(B₁(t),B₂(t),...,B_{N}(t)) through the …show more content…

After time t, in the first revolution S_{u} will arrive to the point ([t/2]R_{u}cos t,[t/2]R_{u}sin t,q_{u}sin(_{u}t)]). To calculate the volume within _{1u} of the searching path we take the volume element r_{u}dtdr_{u}dx_{ku} (see, figure 3) and integrate it over the domain included by _{1u} as in Theorem 1. Theorem 1 For the slinky-turn-spiral searching paths of S_{u},u=1,2,...,N, the total volume between the spiral revolutions _{u} and _{(-1)u},=1,2,... is ((²(6-1)q_{u}R_{u}²)/2). By using the polar coordinates we get the volume between _{u} and _{(-1)u} as, V_{iu}=∫_{-q_{u}}^{q_{u}}∫₂₍₋₁₎²∫_{((R_{u}(t+(-1)))/2)}^{((R_{u}(t+))/2)}r_{u}dtdr_{u}dx_{ku} =((q_{u}R_{u}²)/4)∫₂₍₋₁₎²[(2+1)+2t]dt =((²(6-1)q_{u}R_{u}²)/2). It is clear that the volume between _{u} and _{(-1)u} is depending on q_{u},R_{u} and (the number of the revolution). Also, we notice that for the complete revolution _{u}, there exists a relationship between the random distance L_{u} (the distance between the target position at time t and the origin (searcher's starting point)) and the revolution number m_{u}. [Figure] [Figure] (a) (b) Fig. 3: (a) S_{u}′s position at time t and t+2; and (b) The projection of the volume element on …show more content…

The leaping in the first revolution _{1u} is given by ((tR_{u})/2)-0 and the leaping in _{2u} is given by (((t+2)R_{u})/2)-((tR_{u})/2). Furthermore, the leaping in _{um_{u}} is given by ((R_{u}(t+m_{u}))/2)-((R_{u}(t+(m_{u}-1)))/2). Then, the distance L_{u} is given by: L_{u} =((tR_{u})/2)-0+(((t+2)R_{u})/2)-((tR_{u})/2)+...+((R_{u}(t+m_{u}))/2)-((R_{u}(t+(m_{u}-1)))/2)=((R_{u}(t+m_{u}))/2). Let the search plan that S_{u},u=1,2,...,N follow be the combination of continuous functions (t)=(₁(t),₂(t),...,_{N}(t)) with speed vector [v₁,v₂,...,v_{N}] and given by (t):ℝ⁺→ℝ such that |_{u}(t)| 0 the curvature peaks generally decrease by increasing the value of

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