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

▶
the euler characteristic of a monodimensional polyhedron
SMART_READER_LITE
LIVE PREVIEW

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

May 09, 2023 379 likes •840 views

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

slide-1
SLIDE 1

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

slide-2
SLIDE 2

The Euler-Poincar´ e characteristic Vector lattices The Main Result Polyhedra The Euler-Poincar´ e characteristic

Polyhedra

Let x0, . . . , xn ∈ Rm be affinely independent points (i.e. x1 − x0, . . . , xn − x0 linearly independent) An n-simplex is the set of points σn = (x0, . . . , xn) = n

  • i=0

λixi : λi ∈ R, λi ≥ 0,

n

  • i=0

λi = 1

  • A. Pedrini

Euler characteristic and vector lattices

slide-3
SLIDE 3

The Euler-Poincar´ e characteristic Vector lattices The Main Result Polyhedra The Euler-Poincar´ e characteristic

Polyhedra

Let x0, . . . , xn ∈ Rm be affinely independent points (i.e. x1 − x0, . . . , xn − x0 linearly independent) An n-simplex is the set of points σn = (x0, . . . , xn) = n

  • i=0

λixi : λi ∈ R, λi ≥ 0,

n

  • i=0

λi = 1

  • A. Pedrini

Euler characteristic and vector lattices

slide-4
SLIDE 4

The Euler-Poincar´ e characteristic Vector lattices The Main Result Polyhedra The Euler-Poincar´ e characteristic

Polyhedra

Let x0, . . . , xn ∈ Rm be affinely independent points (i.e. x1 − x0, . . . , xn − x0 linearly independent) An n-simplex is the set of points σn = (x0, . . . , xn) = n

  • i=0

λixi : λi ∈ R, λi ≥ 0,

n

  • i=0

λi = 1

  • A face of σn is any τp = (xi0, . . . , xip), {xi0, . . . , xip} ⊆ {x0, . . . , xn}
  • A. Pedrini

Euler characteristic and vector lattices

slide-5
SLIDE 5

The Euler-Poincar´ e characteristic Vector lattices The Main Result Polyhedra The Euler-Poincar´ e characteristic

Polyhedra

A simplicial complex K is a finite set of simplices such that

◮ if σn ∈ K and τp is a face of σn, then τp ∈ K, ◮ if σn, τp ∈ K, then σn ∩ τp is a common (possibly empty) face

  • f σn and τp
  • A. Pedrini

Euler characteristic and vector lattices

slide-6
SLIDE 6

The Euler-Poincar´ e characteristic Vector lattices The Main Result Polyhedra The Euler-Poincar´ e characteristic

Polyhedra

A simplicial complex K is a finite set of simplices such that

◮ if σn ∈ K and τp is a face of σn, then τp ∈ K, ◮ if σn, τp ∈ K, then σn ∩ τp is a common (possibly empty) face

  • f σn and τp
  • A. Pedrini

Euler characteristic and vector lattices

slide-7
SLIDE 7

The Euler-Poincar´ e characteristic Vector lattices The Main Result Polyhedra The Euler-Poincar´ e characteristic

Polyhedra

A simplicial complex K is a finite set of simplices such that

◮ if σn ∈ K and τp is a face of σn, then τp ∈ K, ◮ if σn, τp ∈ K, then σn ∩ τp is a common (possibly empty) face

  • f σn and τp

A polyhedron is a set P of points of Rm 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

slide-8
SLIDE 8

The Euler-Poincar´ e characteristic Vector lattices The Main Result Polyhedra The Euler-Poincar´ e characteristic

Polyhedra

A simplicial complex K is a finite set of simplices such that

◮ if σn ∈ K and τp is a face of σn, then τp ∈ K, ◮ if σn, τp ∈ K, then σn ∩ τp is a common (possibly empty) face

  • f σn and τp

A polyhedron is a set P of points of Rm 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

slide-9
SLIDE 9

The Euler-Poincar´ e characteristic Vector lattices The Main Result Polyhedra The Euler-Poincar´ e characteristic

The Euler-Poincar´ e characteristic

Let K be a triangulation of the polyhedron P, the Euler-Poincar´ e characteristic of P is the number χ(P) =

m

  • n=0

(−1)nαn where, for all n, αn is the number of n-simplices of K.

  • A. Pedrini

Euler characteristic and vector lattices

slide-10
SLIDE 10

The Euler-Poincar´ e characteristic Vector lattices The Main Result Polyhedra The Euler-Poincar´ e characteristic

The Euler-Poincar´ e characteristic

Let K be a triangulation of the polyhedron P, the Euler-Poincar´ e characteristic of P is the number χ(P) =

m

  • n=0

(−1)nαn where, for all n, αn 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

slide-11
SLIDE 11

The Euler-Poincar´ e characteristic Vector lattices The Main Result Polyhedra The Euler-Poincar´ e characteristic

The Euler-Poincar´ e characteristic

Let K be a triangulation of the polyhedron P, the Euler-Poincar´ e characteristic of P is the number χ(P) =

m

  • n=0

(−1)nαn where, for all n, αn 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

slide-12
SLIDE 12

The Euler-Poincar´ e characteristic Vector lattices The Main Result Polyhedra The Euler-Poincar´ e characteristic

The Euler-Poincar´ e characteristic

Let K be a triangulation of the polyhedron P, the Euler-Poincar´ e characteristic of P is the number χ(P) =

m

  • n=0

(−1)nαn where, for all n, αn 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

slide-13
SLIDE 13

The Euler-Poincar´ e characteristic Vector lattices The Main Result Polyhedra The Euler-Poincar´ e characteristic

The Euler-Poincar´ e characteristic

Let K be a triangulation of the polyhedron P, the Euler-Poincar´ e characteristic of P is the number χ(P) =

m

  • n=0

(−1)nαn where, for all n, αn 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

slide-14
SLIDE 14

The Euler-Poincar´ e characteristic Vector lattices The Main Result Definition Representation vl-Schauder hats 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) = λv ∧ λw.

  • A. Pedrini

Euler characteristic and vector lattices

slide-15
SLIDE 15

The Euler-Poincar´ e characteristic Vector lattices The Main Result Definition Representation vl-Schauder hats 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) = λv ∧ λw. 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

slide-16
SLIDE 16

The Euler-Poincar´ e characteristic Vector lattices The Main Result Definition Representation vl-Schauder hats 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) = λv ∧ λw. 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 ≤ λu.

  • A. Pedrini

Euler characteristic and vector lattices

slide-17
SLIDE 17

The Euler-Poincar´ e characteristic Vector lattices The Main Result Definition Representation vl-Schauder hats 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) = λv ∧ λw. 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 ≤ λu. 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

slide-18
SLIDE 18

The Euler-Poincar´ e characteristic Vector lattices The Main Result Definition Representation vl-Schauder hats The Euler-Poincar´ e characteristic of a function

Vector lattices

A function f : Rm → Rn is piecewise linear if there are finitely many linear polynomials w1, . . . , ws such that ∀x ∈ Rm ∃i ∈ {1, . . . , s} : f (x) = wi(x).

  • A. Pedrini

Euler characteristic and vector lattices

slide-19
SLIDE 19

The Euler-Poincar´ e characteristic Vector lattices The Main Result Definition Representation vl-Schauder hats The Euler-Poincar´ e characteristic of a function

Vector lattices

A function f : Rm → Rn is piecewise linear if there are finitely many linear polynomials w1, . . . , ws such that ∀x ∈ Rm ∃i ∈ {1, . . . , s} : f (x) = wi(x). Let P a polyhedron in Rm. F(P) = {f : P → R continuous and piecewise linear} ∇(P) = (F(P), +, min, max, {λ}λ∈R, 0) (∇(P), 1) is a unital vector lattice.

  • A. Pedrini

Euler characteristic and vector lattices

slide-20
SLIDE 20

The Euler-Poincar´ e characteristic Vector lattices The Main Result Definition Representation vl-Schauder hats The Euler-Poincar´ e characteristic of a function

Vector lattices

A function f : Rm → Rn is piecewise linear if there are finitely many linear polynomials w1, . . . , ws such that ∀x ∈ Rm ∃i ∈ {1, . . . , s} : f (x) = wi(x). Let P a polyhedron in Rm. F(P) = {f : P → R continuous and piecewise linear} ∇(P) = (F(P), +, min, max, {λ}λ∈R, 0) (∇(P), 1) is a unital vector lattice. Baker-Beynon duality: each finitely presented (V, u) is isomorphic to (∇(P), 1), for some P in some Euclidean space Rm.

  • A. Pedrini

Euler characteristic and vector lattices

slide-21
SLIDE 21

The Euler-Poincar´ e characteristic Vector lattices The Main Result Definition Representation vl-Schauder hats The Euler-Poincar´ e characteristic of a function

Vector lattices

A triangulation K of the polyhedron P linearizes f ∈ ∇(P) if f is linear on each simplex of K.

  • A. Pedrini

Euler characteristic and vector lattices

slide-22
SLIDE 22

The Euler-Poincar´ e characteristic Vector lattices The Main Result Definition Representation vl-Schauder hats The Euler-Poincar´ e characteristic of a function

Vector lattices

A triangulation K of the polyhedron P linearizes f ∈ ∇(P) if f is linear on each simplex of K. A vl-Schauder hat is an h ∈ ∇(P) such that there is a triangulation Kh of P linearizing h and a 0-simplex ¯ x of Kh such that h(¯ x) = 1 and h(x) = 0 for any other 0-simplices x of Kh.

  • A. Pedrini

Euler characteristic and vector lattices

slide-23
SLIDE 23

The Euler-Poincar´ e characteristic Vector lattices The Main Result Definition Representation vl-Schauder hats The Euler-Poincar´ e characteristic of a function

Vector lattices

A triangulation K of the polyhedron P linearizes f ∈ ∇(P) if f is linear on each simplex of K. A vl-Schauder hat is an h ∈ ∇(P) such that there is a triangulation Kh of P linearizing h and a 0-simplex ¯ x of Kh such that h(¯ x) = 1 and h(x) = 0 for any other 0-simplices x of Kh. The vl-Schauder hats of K is the set of vl-Schauder hats {hi} such that hi(xi) = 1 and hi(xj) = 0, where x0, . . . , xn are the 0-simplices

  • f K.
  • A. Pedrini

Euler characteristic and vector lattices

slide-24
SLIDE 24

The Euler-Poincar´ e characteristic Vector lattices The Main Result Definition Representation vl-Schauder hats The Euler-Poincar´ e characteristic of a function

Vector lattices

A triangulation K of the polyhedron P linearizes f ∈ ∇(P) if f is linear on each simplex of K. A vl-Schauder hat is an h ∈ ∇(P) such that there is a triangulation Kh of P linearizing h and a 0-simplex ¯ x of Kh such that h(¯ x) = 1 and h(x) = 0 for any other 0-simplices x of Kh. The vl-Schauder hats of K is the set of vl-Schauder hats {hi} such that hi(xi) = 1 and hi(xj) = 0, where x0, . . . , xn are the 0-simplices

  • f K.

Each f ∈ ∇(P) can be seen as a sum n

i=0 aihi (where ai ∈ R) of

the vl-Schauder hats h0, . . . , hn of a triangulation Kf linearizing f .

  • A. Pedrini

Euler characteristic and vector lattices

slide-25
SLIDE 25

The Euler-Poincar´ e characteristic Vector lattices The Main Result Definition Representation vl-Schauder hats The Euler-Poincar´ e characteristic of a function

The Euler-Poincar´ e characteristic of a function

Let K a triangulation linearizing |f |; ZK,f = {σ ∈ K : f |σ ≡ 0}.

  • A. Pedrini

Euler characteristic and vector lattices

slide-26
SLIDE 26

The Euler-Poincar´ e characteristic Vector lattices The Main Result Definition Representation vl-Schauder hats The Euler-Poincar´ e characteristic of a function

The Euler-Poincar´ e characteristic of a function

Let K a triangulation linearizing |f |; ZK,f = {σ ∈ K : f |σ ≡ 0}. The supplement SK,f of f in K is an “inner approximation” of the support of f :

  • A. Pedrini

Euler characteristic and vector lattices

slide-27
SLIDE 27

The Euler-Poincar´ e characteristic Vector lattices The Main Result Definition Representation vl-Schauder hats The Euler-Poincar´ e characteristic of a function

The Euler-Poincar´ e characteristic of a function

Let K a triangulation linearizing |f |; ZK,f = {σ ∈ K : f |σ ≡ 0}. The supplement SK,f of f in K is an “inner approximation” of the support of f :

  • A. Pedrini

Euler characteristic and vector lattices

slide-28
SLIDE 28

The Euler-Poincar´ e characteristic Vector lattices The Main Result Definition Representation vl-Schauder hats The Euler-Poincar´ e characteristic of a function

The Euler-Poincar´ e characteristic of a function

Let K a triangulation linearizing |f |; ZK,f = {σ ∈ K : f |σ ≡ 0}. The supplement SK,f of f in K is an “inner approximation” of the support of f :

  • A. Pedrini

Euler characteristic and vector lattices

slide-29
SLIDE 29

The Euler-Poincar´ e characteristic Vector lattices The Main Result Definition Representation vl-Schauder hats The Euler-Poincar´ e characteristic of a function

The Euler-Poincar´ e characteristic of a function

Let K a triangulation linearizing |f |; ZK,f = {σ ∈ K : f |σ ≡ 0}. The supplement SK,f of f in K is an “inner approximation” of the support of f :

  • A. Pedrini

Euler characteristic and vector lattices

slide-30
SLIDE 30

The Euler-Poincar´ e characteristic Vector lattices The Main Result Definition Representation vl-Schauder hats The Euler-Poincar´ e characteristic of a function

The Euler-Poincar´ e characteristic of a function

Let K a triangulation linearizing |f |; ZK,f = {σ ∈ K : f |σ ≡ 0}. The supplement SK,f of f in K is an “inner approximation” of the support of f :

  • A. Pedrini

Euler characteristic and vector lattices

slide-31
SLIDE 31

The Euler-Poincar´ e characteristic Vector lattices The Main Result Definition Representation vl-Schauder hats The Euler-Poincar´ e characteristic of a function

The Euler-Poincar´ e characteristic of a function

Let K a triangulation linearizing |f |; ZK,f = {σ ∈ K : f |σ ≡ 0}. The supplement SK,f of f in K is an “inner approximation” of the support of f : The Euler-Poincar´ e characteristic of f : χ(f ) = χ(supp(f )) = χ(SK,f ) (it does not depend on the choice of K).

  • A. Pedrini

Euler characteristic and vector lattices

slide-32
SLIDE 32

The Euler-Poincar´ e characteristic Vector lattices The Main Result Valuations Characterization Theorem

Valuations

A vl-valuation on (∇(P), 1) is a function ν : ∇(P) → R such that:

◮ ν(0) = 0, ◮ for all f , g ∈ ∇(P), ν(f ∨ g) = ν(f ) + ν(g) − ν(f ∧ g), ◮ for all 0 ≤ f , g ∈ ∇(P), ν(f + g) = ν(f ∨ g), ◮ for all 0 ≤ f , g ∈ ∇(P), if f ∧ g = 0 then

ν(f − g) = ν(f ) − ν(g).

  • A. Pedrini

Euler characteristic and vector lattices

slide-33
SLIDE 33

The Euler-Poincar´ e characteristic Vector lattices The Main Result Valuations Characterization Theorem

Characterization Theorem

Theorem Let P be a polyhedron in Rm, for some integer m ≥ 1, and let (∇(P), 1) be the finitely presented unital vector lattice of real-valued piecewise linear functions on P. Then Euler-Poincar´ e characteristic is the unique vl-valuation χ : ∇(P) → R that assigns the value 1 to each vl-Schauder hat in ∇(P). Moreover, the number χ(1) is the Euler-Poincar´ e characteristic of the polyhedron P.

  • A. Pedrini

Euler characteristic and vector lattices

slide-34
SLIDE 34

The Euler-Poincar´ e characteristic Vector lattices The Main Result Valuations Characterization Theorem

A monodimensional hat

  • A. Pedrini

Euler characteristic and vector lattices

slide-35
SLIDE 35

The Euler-Poincar´ e characteristic Vector lattices The Main Result Valuations Characterization Theorem

A monodimensional hat

  • A. Pedrini

Euler characteristic and vector lattices

slide-36
SLIDE 36

The Euler-Poincar´ e characteristic Vector lattices The Main Result Valuations Characterization Theorem

A monodimensional hat

  • A. Pedrini

Euler characteristic and vector lattices

slide-37
SLIDE 37

The Euler-Poincar´ e characteristic Vector lattices The Main Result Valuations Characterization Theorem

A monodimensional hat

  • A. Pedrini

Euler characteristic and vector lattices

slide-38
SLIDE 38

The Euler-Poincar´ e characteristic Vector lattices The Main Result Valuations Characterization Theorem

A monodimensional hat

  • A. Pedrini

Euler characteristic and vector lattices

slide-39
SLIDE 39

The Euler-Poincar´ e characteristic Vector lattices The Main Result Valuations Characterization Theorem

Characterization Theorem

Theorem Let P be a polyhedron in Rm, for some integer m ≥ 1, and let (∇(P), 1) be the finitely presented unital vector lattice of real-valued piecewise linear functions on P. Then Euler-Poincar´ e characteristic is the unique vl-valuation χ : ∇(P) → R that assigns the value 1 to each vl-Schauder hat in ∇(P). Moreover, the number χ(1) is the Euler-Poincar´ e characteristic of the polyhedron P.

  • A. Pedrini

Euler characteristic and vector lattices

slide-40
SLIDE 40

The Euler-Poincar´ e characteristic Vector lattices The Main Result Valuations Characterization Theorem

χ(1) = χ(P)

  • A. Pedrini

Euler characteristic and vector lattices

slide-41
SLIDE 41

The Euler-Poincar´ e characteristic Vector lattices The Main Result Valuations Characterization Theorem

χ(1) = χ(P)

  • A. Pedrini

Euler characteristic and vector lattices

slide-42
SLIDE 42

The Euler-Poincar´ e characteristic Vector lattices The Main Result Valuations Characterization Theorem

χ(1) = χ(P)

  • A. Pedrini

Euler characteristic and vector lattices

slide-43
SLIDE 43

The Euler-Poincar´ e characteristic Vector lattices The Main Result Valuations Characterization Theorem

Characterization Theorem

Theorem Let P be a polyhedron in Rm, for some integer m ≥ 1, and let (∇(P), 1) be the finitely presented unital vector lattice of real-valued piecewise linear functions on P. Then Euler-Poincar´ e characteristic is the unique vl-valuation χ : ∇(P) → R that assigns the value 1 to each vl-Schauder hat in ∇(P). Moreover, the number χ(1) is the Euler-Poincar´ e characteristic of the polyhedron P.

  • A. Pedrini

Euler characteristic and vector lattices

slide-44
SLIDE 44

The Euler-Poincar´ e characteristic Vector lattices The Main Result Valuations Characterization Theorem

Thank you for your attention.

  • A. Pedrini

Euler characteristic and vector lattices