An Elimination Theory for Differential Algebra by A Seidenberg

By A Seidenberg

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14. Consider a lake where some chemical product is being poured in. 17). 14. We assume that the volume V of the lake remains constant. A simple model for the pollution of a lake was given in [AGU 99]. 17. Schematic diagram of a lake balance, we obtain: dM = Fin Cin − Fout Cout − kCV, dt where M , k, Fin , Fout , Cin and Cout represent the mass of the contaminant, the rate of reaction of the chemical pollutant, the input flow rate, the output flow rate, the concentration of the contaminant at the input and the concentration of the contaminant at the output, respectively.

38) for k = 1. Let us look for solutions of the form6: Ik = α exp (λk) + β exp (−λk) . 39) We have to calculate the parameters α, β and λ. 38), we obtain: (2z + z1 ) [α exp (λk) + β exp (−λk)] − zα exp (λ (k + 1)) −zβ exp (−λ (k + 1)) − z [α exp (λ (k − 1)) + β exp (−λ (k − 1))] = 0, [α exp (λk) + β exp (−λk)] [2z + z1 − z exp λ − z exp (−λ)] = 0. It follows readily that: 2z + z1 − z exp λ − z exp (−λ) = 0. 40) Recall the expressions for the hyperbolic cosine and sine functions, cosh x = exp x + exp (−x) 2 and sinh x = exp x − exp (−x) .

Therefore, the inductor behaves like a closed switch. For high frequencies, Lω → ∞, and the inductor behaves like an open switch, or, in other words, the impedance becomes infinite. 32 constitutes a high-pass filter. 33, and following the comments made above, we derive the result that for low frequencies, the inductor acts as a closed switch, and for high frequency the inductor behaves like an open switch (infinite impedance). Consequently, this system represents a low-pass filter. Observe that this kind of reasoning and this kind of the behavior of an electronic circuit is commonly used by electronic engineers.

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