Guerzhoy P.'s A p-adic Property of Fourier Coefficients of Modular Forms PDF

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42) the following system of difference equations pi (t + h) = (1 + h fi (p(t))pi (t), t ∈ R, for i = 1, . . , n. If we define a vector function gh : Rn → Rn by gh (p) = (1 + h fi (p))pi , i = 1, . . 42) (with stepsize h). 42)), ˆ = pˆ . ,n Jgh ( p) where ⎧ ⎪ ⎪ ⎨0, δi j = ⎪ ⎪ ⎩1, for i j . 66) In = n × n-unit matrix. Therefore λ ∈ C is an eigenvalue of JF ( pˆ ), if and only if 1+hλ is an eigenvalue ˆ Further we obtain of Jgh ( p). |1 + hλ|2 = (1 + Re(λ)h)2 + (Im(λ)h)2 = 1 + 2Re(λ)h + h2 |λ|2 = 1 + h(2Re(λ) + h|λ|2 ).

6] W. Krabs: Stability in Predator-Prey Models and Discretization of a Modified Volterra-Lotka-Model. Mathematical and Computer Modelling of Dynamical Systems. 12 (2006), 577-588. [7] W. Krabs: Spieltheorie: Dynamische Behandlung von Spielen. G. Teubner: Stuttgart-Leipzig-Wiesbaden, 2005. [8] W. Krabs: and S. W. Pickl: Analysis, Controllability and Optimisation of Time-Discrete Systems and Dynamical Games. Lecture Notes in Economics and Mathematical Systems. Springer-Verlag Berlin-Heidelberg, 2003.

S 6 . Let us consider u1 ∈ Δ with S (u1 ) = S 1 = {1, 3}. Then u1 is of the form u1 = (t, 0, 1 − t, 0, 0) with 0 ≤ t ≤ 1 and we get Au1T = (2 − t, 2(1 − t), t + 1, 2t, 2t)T . , 1 1 1 1 3 u1 = e1 + e2 = , 0, , 0, 0 and u1 Au1T = . 2 2 2 2 2 Now let u = (t, 0, 1 − t, 0, 0) with 0 ≤ t ≤ 1 and t 1 2, then it follows that 1 1 3 u1 AuT = (2 − t) + (t + 1) = and 2 2 2 uAuT = t(2 − t) + (1 − t)(t + 1) = 1 + 2t(1 − t) < 3 2 which shows that u1 is an evolutionarily stable Nash equilibrium. In a similar way one can show that every u ∈ Δ of the form u = 12 ei + 12 ek for 1 ≤ i ≤ 2 and 3 ≤ k ≤ 5 belongs to ESS.

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