Download PDF by Mary Gray: A Radical Approach to Algebra (Addison-Wesley Series in

By Mary Gray

ISBN-10: 020102568X

ISBN-13: 9780201025682

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Therefore there exist n functions xi = φi (˜ x, u, . . , u(γ) ) for 1 ≤i≤n which define a local diffeomorphism φ parametrized by u, . . 1) becomes 24 2 Modeling ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x ˜˙ 1 = x ˜2 x ˜˙ 2 = x ˜4 .. (s ) x), u, . . , u(γ) ) x ˜˙ s1 = h1 1 (φ(˜ ˙x ˜s1 +1 = x ˜s1 +2 .. (s ) x ˜˙ s1 +s2 = h2 2 (φ(˜ x), u, . . , u(γ) ) .. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x ˜˙ s1 +···+sp ⎪ ⎪ ⎪ ˙ ⎪ x ˜ ⎪ s1 +···+sp +i ⎪ ⎪ ⎪ y1 ⎪ ⎪ ⎪ ⎪ y ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ yp = = = = .. 5) (s ) hp p (φ(˜ x), u, .

Infected CD4 cells act as host cells and they produce new HIV virions. An elementary model may be derived. Let T denote the population of healthy CD4 cells. Let T ∗ denote the population of infected CD4 cells. Let v denote the population of HIV virions. As any living specie, the CD4 cells have some finite lifetime 1/δ. The evolution of some independent population is then approximated by the linear first-order system: T˙ = −δT 42 2 Modeling The body is assumed to produce new CD4 cells at some constant rate s; thus, the evolution of T in a noninfected body will be described by T˙ = s − δT and the population T stabilizes at some equilibrium T0 = s/δ.

Dξr } Since A is invariant under time differentiation, in particular, ξ˙1 = f1 (ξ1 , · · · , ξr ) .. 7 Controllability Indices ξ˙1 = f1 (ξ1 , · · · , ξr ) .. ξ˙r = fr (ξ1 , · · · , ξr ) ξ˙r+1 = fr+1 (ξ1 , · · · , ξn ) + gr+1 (ξ1 , · · · , ξn )u .. 1, the ξ1 , . . , ξr are autonomous elements that satisfy a differential equation of order less than or equal to r. 7 Controllability Indices Define, now, hi = dimHi − dimHi+1 , for i ≥ 1 Moreover, hk∗ is nonzero and hk = 0, for any k > k ∗ . ∗ The set of controllability indices {k1∗ , .

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A Radical Approach to Algebra (Addison-Wesley Series in Mathematics) by Mary Gray


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