Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 401In our example–case of symmetrical 3D load–lifting, the velocity andforce [µσ]−Hamiltonian biodynamics equations (3.216–3.217) become⎛ ⎧ ⎡ ⎛⎪⎨dq i ⎜= ⎝p i [⎪ ¯J i ] −1 ⎢i∑+ ⎣ ¯m i⎝⎩j=1¯L j cos q j ⎞⎠dp i = B ij [q i (t), t] dW j (t) + ⎝ ¯Fi (t, q i , p i , {σ} µ ) − g−10−i∑j=i¯L j sin q j p i p j⎡⎛⎣ ¯m i( i∑k=1¯L k cos q k ) 3⎤⎤ 2−1⎫ ⎞⎪⎬⎥ ⎦ + ∂R ⎟⎠ dt,⎪ ⎭ ∂p i⎦−110−i∑j=i¯L j ¯m j sin q j⎞+ ∂R ⎟∂q i ⎠ dt.In this way, the crisp stochastic σ−-Hamiltonian phase–flow φ σt(3.212)extends into fuzzy–stochastic [µσ]–Hamiltonian phase–flow φ [µσ]tφ [µσ]t : G 1 × I ∗ M → I ∗ M : (¯p 0 i , ¯q i 0) ↦→ (p(t), q(t)), (3.219)(φ [µσ]t ◦ φ [µσ]s = φ [µσ]t+s , φ [µσ]0 = identity).[µσ]−Hamiltonian biodynamical system (3.216–3.218), with its phase–flow φ [µσ]t (3.219), i.e., the functor Fuzzy[Stoch[Can]], represents our final,continual–discrete and fuzzy–stochastic model for the biodynamics functormachine F[S] with the nonlinear system behavior F[r].3.13.5 Biodynamical Topology3.13.5.1 (Co)Chain Complexes in BiodynamicsIn this section we present the category of (co)chain complexes, as used inmodern biodynamics. The central concept in cohomology theory is thecategory S • (C) of generalized cochain complexes in an Abelian category C[Dieudonne (1988)]. The objects of the category S • (C) are infinite sequencesA • : · · · −→ A n−1 d n−1 ✲ An d n ✲ A n+1 −→ · · ·where, for each n ∈ Z, A n is an object of C and d n a morphism of C, withthe conditionsd n−1 ◦ d n = 0