The Euler characteristic of a (monodimensional) polyhedron as a - PowerPoint PPT Presentation

The Euler-Poincar´ e characteristic Vector lattices The Main Result The Euler characteristic of a (monodimensional) polyhedron as a valuation on a vector lattice Andrea Pedrini andrea.pedrini@unimi.it Universit` a degli Studi di Milano Dipartimento di Informatica e Comunicazione Algebraic Semantics for Uncertainty and Vagueness 18th - 20th May 2011 A. Pedrini Euler characteristic and vector lattices

The Euler-Poincar´ e characteristic Polyhedra Vector lattices The Euler-Poincar´ e characteristic The Main Result Polyhedra Let x 0 , . . . , x n ∈ R m be affinely independent points (i.e. x 1 − x 0 , . . . , x n − x 0 linearly independent) An n -simplex is the set of points � n n � � � σ n = ( x 0 , . . . , x n ) = λ i x i : λ i ∈ R , λ i ≥ 0 , λ i = 1 i =0 i =0 A. Pedrini Euler characteristic and vector lattices

The Euler-Poincar´ e characteristic Polyhedra Vector lattices The Euler-Poincar´ e characteristic The Main Result Polyhedra Let x 0 , . . . , x n ∈ R m be affinely independent points (i.e. x 1 − x 0 , . . . , x n − x 0 linearly independent) An n -simplex is the set of points � n n � � � σ n = ( x 0 , . . . , x n ) = λ i x i : λ i ∈ R , λ i ≥ 0 , λ i = 1 i =0 i =0 A. Pedrini Euler characteristic and vector lattices

The Euler-Poincar´ e characteristic Polyhedra Vector lattices The Euler-Poincar´ e characteristic The Main Result Polyhedra Let x 0 , . . . , x n ∈ R m be affinely independent points (i.e. x 1 − x 0 , . . . , x n − x 0 linearly independent) An n -simplex is the set of points � n n � � � σ n = ( x 0 , . . . , x n ) = λ i x i : λ i ∈ R , λ i ≥ 0 , λ i = 1 i =0 i =0 A face of σ n is any τ p = ( x i 0 , . . . , x i p ), { x i 0 , . . . , x i p } ⊆ { x 0 , . . . , x n } A. Pedrini Euler characteristic and vector lattices

The Euler-Poincar´ e characteristic Polyhedra Vector lattices The Euler-Poincar´ e characteristic The Main Result Polyhedra A simplicial complex K is a finite set of simplices such that ◮ if σ n ∈ K and τ p is a face of σ n , then τ p ∈ K , ◮ if σ n , τ p ∈ K , then σ n ∩ τ p is a common (possibly empty) face of σ n and τ p A. Pedrini Euler characteristic and vector lattices

The Euler-Poincar´ e characteristic Polyhedra Vector lattices The Euler-Poincar´ e characteristic The Main Result Polyhedra A simplicial complex K is a finite set of simplices such that ◮ if σ n ∈ K and τ p is a face of σ n , then τ p ∈ K , ◮ if σ n , τ p ∈ K , then σ n ∩ τ p is a common (possibly empty) face of σ n and τ p A. Pedrini Euler characteristic and vector lattices

The Euler-Poincar´ e characteristic Polyhedra Vector lattices The Euler-Poincar´ e characteristic The Main Result Polyhedra A simplicial complex K is a finite set of simplices such that ◮ if σ n ∈ K and τ p is a face of σ n , then τ p ∈ K , ◮ if σ n , τ p ∈ K , then σ n ∩ τ p is a common (possibly empty) face of σ n and τ p A polyhedron is a set P of points of R m that is the union of the simplices of some simplicial complex K . K is called a triangulation of P . A. Pedrini Euler characteristic and vector lattices

The Euler-Poincar´ e characteristic Polyhedra Vector lattices The Euler-Poincar´ e characteristic The Main Result Polyhedra A simplicial complex K is a finite set of simplices such that ◮ if σ n ∈ K and τ p is a face of σ n , then τ p ∈ K , ◮ if σ n , τ p ∈ K , then σ n ∩ τ p is a common (possibly empty) face of σ n and τ p A polyhedron is a set P of points of R m that is the union of the simplices of some simplicial complex K . K is called a triangulation of P . A. Pedrini Euler characteristic and vector lattices

The Euler-Poincar´ e characteristic Polyhedra Vector lattices The Euler-Poincar´ e characteristic The Main Result The Euler-Poincar´ e characteristic Let K be a triangulation of the polyhedron P , the Euler-Poincar´ e characteristic of P is the number m � ( − 1) n α n χ ( P ) = n =0 where, for all n , α n is the number of n -simplices of K . A. Pedrini Euler characteristic and vector lattices

The Euler-Poincar´ e characteristic Polyhedra Vector lattices The Euler-Poincar´ e characteristic The Main Result The Euler-Poincar´ e characteristic Let K be a triangulation of the polyhedron P , the Euler-Poincar´ e characteristic of P is the number m � ( − 1) n α n χ ( P ) = n =0 where, for all n , α n is the number of n -simplices of K . It is well-defined: two different triangulations of P give the same number χ ( P ): A. Pedrini Euler characteristic and vector lattices

The Euler-Poincar´ e characteristic Polyhedra Vector lattices The Euler-Poincar´ e characteristic The Main Result The Euler-Poincar´ e characteristic Let K be a triangulation of the polyhedron P , the Euler-Poincar´ e characteristic of P is the number m � ( − 1) n α n χ ( P ) = n =0 where, for all n , α n is the number of n -simplices of K . It is well-defined: two different triangulations of P give the same number χ ( P ): A. Pedrini Euler characteristic and vector lattices

The Euler-Poincar´ e characteristic Polyhedra Vector lattices The Euler-Poincar´ e characteristic The Main Result The Euler-Poincar´ e characteristic Let K be a triangulation of the polyhedron P , the Euler-Poincar´ e characteristic of P is the number m � ( − 1) n α n χ ( P ) = n =0 where, for all n , α n is the number of n -simplices of K . It is well-defined: two different triangulations of P give the same number χ ( P ): A. Pedrini Euler characteristic and vector lattices

The Euler-Poincar´ e characteristic Polyhedra Vector lattices The Euler-Poincar´ e characteristic The Main Result The Euler-Poincar´ e characteristic Let K be a triangulation of the polyhedron P , the Euler-Poincar´ e characteristic of P is the number m � ( − 1) n α n χ ( P ) = n =0 where, for all n , α n is the number of n -simplices of K . It is well-defined: two different triangulations of P give the same number χ ( P ): A. Pedrini Euler characteristic and vector lattices

Definition The Euler-Poincar´ e characteristic Representation Vector lattices vl-Schauder hats The Main Result The Euler-Poincar´ e characteristic of a function Vector lattices A (real) vector lattice is an algebra V = ( V , + , ∧ , ∨ , { λ } λ ∈ R , 0) such that ◮ ( V , + , { λ } λ ∈ R , 0) is a vector space, ◮ ( V , ∧ , ∨ ) is a lattice, ◮ for all t , v , w ∈ V , t + ( v ∧ w ) = ( t + v ) ∧ ( t + w ), ◮ for all v , w ∈ V and for all λ ∈ R , if λ ≥ 0 then λ ( v ∧ w ) = λ v ∧ λ w . A. Pedrini Euler characteristic and vector lattices

Definition The Euler-Poincar´ e characteristic Representation Vector lattices vl-Schauder hats The Main Result The Euler-Poincar´ e characteristic of a function Vector lattices A (real) vector lattice is an algebra V = ( V , + , ∧ , ∨ , { λ } λ ∈ R , 0) such that ◮ ( V , + , { λ } λ ∈ R , 0) is a vector space, ◮ ( V , ∧ , ∨ ) is a lattice, ◮ for all t , v , w ∈ V , t + ( v ∧ w ) = ( t + v ) ∧ ( t + w ), ◮ for all v , w ∈ V and for all λ ∈ R , if λ ≥ 0 then λ ( v ∧ w ) = λ v ∧ λ w . The lattice structure induces a partial order (defined as usual): v ≤ w if and only if v ∧ w = v . A. Pedrini Euler characteristic and vector lattices

Definition The Euler-Poincar´ e characteristic Representation Vector lattices vl-Schauder hats The Main Result The Euler-Poincar´ e characteristic of a function Vector lattices A (real) vector lattice is an algebra V = ( V , + , ∧ , ∨ , { λ } λ ∈ R , 0) such that ◮ ( V , + , { λ } λ ∈ R , 0) is a vector space, ◮ ( V , ∧ , ∨ ) is a lattice, ◮ for all t , v , w ∈ V , t + ( v ∧ w ) = ( t + v ) ∧ ( t + w ), ◮ for all v , w ∈ V and for all λ ∈ R , if λ ≥ 0 then λ ( v ∧ w ) = λ v ∧ λ w . The lattice structure induces a partial order (defined as usual): v ≤ w if and only if v ∧ w = v . A strong unit is an element u ∈ V such that for all 0 ≤ v ∈ V there exists a 0 ≤ λ ∈ R such that v ≤ λ u . A. Pedrini Euler characteristic and vector lattices

Definition The Euler-Poincar´ e characteristic Representation Vector lattices vl-Schauder hats The Main Result The Euler-Poincar´ e characteristic of a function Vector lattices A (real) vector lattice is an algebra V = ( V , + , ∧ , ∨ , { λ } λ ∈ R , 0) such that ◮ ( V , + , { λ } λ ∈ R , 0) is a vector space, ◮ ( V , ∧ , ∨ ) is a lattice, ◮ for all t , v , w ∈ V , t + ( v ∧ w ) = ( t + v ) ∧ ( t + w ), ◮ for all v , w ∈ V and for all λ ∈ R , if λ ≥ 0 then λ ( v ∧ w ) = λ v ∧ λ w . The lattice structure induces a partial order (defined as usual): v ≤ w if and only if v ∧ w = v . A strong unit is an element u ∈ V such that for all 0 ≤ v ∈ V there exists a 0 ≤ λ ∈ R such that v ≤ λ u . A unital vector lattice is a pair ( V , u ), where V is a vector lattice and u is a strong unit of V . A. Pedrini Euler characteristic and vector lattices

Definition The Euler-Poincar´ e characteristic Representation Vector lattices vl-Schauder hats The Main Result The Euler-Poincar´ e characteristic of a function Vector lattices A function f : R m → R n is piecewise linear if there are finitely many linear polynomials w 1 , . . . , w s such that ∀ x ∈ R m ∃ i ∈ { 1 , . . . , s } : f ( x ) = w i ( x ) . A. Pedrini Euler characteristic and vector lattices