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**1 - 8**of**8**### THEME Modeling, Optimization, and Control

"... 3. Scientific Foundations.....................................................................2 3.1. Modeling of complex environment 2 3.2. Analysis of interconnected systems 2 3.3. Stabilization of interconnected systems 3 3.4. Synthesis of reduced complexity controllers 3 ..."

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3. Scientific Foundations.....................................................................2 3.1. Modeling of complex environment 2 3.2. Analysis of interconnected systems 2 3.3. Stabilization of interconnected systems 3 3.4. Synthesis of reduced complexity controllers 3

### and S. Zheng) 1

"... I Differential algebra d(xy) = d(x)y + xd(y) + λd(x)d(y). d(uv) 7→φ d(u)v + ud(v) + λd(u)d(v),∀u, v ∈ R. This leads to normal forms w with no products in d. I Thus free commutative differential algebra (of weight λ) on a set X is of the form k{X}: = k[∆X], ∆X: = {x (n) | x ∈ X,n ≥ 0} with concaten ..."

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I Differential algebra d(xy) = d(x)y + xd(y) + λd(x)d(y). d(uv) 7→φ d(u)v + ud(v) + λd(u)d(v),∀u, v ∈ R. This leads to normal forms w with no products in d. I Thus free commutative differential algebra (of weight λ) on a set X is of the form k{X}: = k[∆X], ∆X: = {x (n) | x ∈ X,n ≥ 0} with concatenation product. Define dX: k{X} → k{X} as follows. Let w = u1 · · · uk,ui ∈ ∆(X), 1 ≤ i ≤ k, be a commutative word from the alphabet set ∆(X). If k = 1, so that w = x (n) ∈ ∆(X), define dX (w) = x (n+1). If k> 1, recursively define dX (w) = dX (u1)u2 · · · uk + u1dX (u2 · · · uk) + λdX (u1)dX (u2 · · · uk). Further define dX (1) = 0. Then (k{X},dX) is the free commutative differential algebra of weight λ on the set X. 2 Free differential algebras

### Composing and Factoring Generalized Green’s Operators and Ordinary Boundary Problems

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### 4 ROTA-BAXTER OPERATORS ON THE POLYNOMIAL ALGEBRAS, INTEGRATION AND AVERAGING OPERATORS

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### MONADS AND DISTRIBUTIVE LAWS FOR ROTA-BAXTER AND DIFFERENTIAL ALGEBRAS

, 2014

"... Monads and distributive laws for Rota-Baxter and differential algebras ..."