Semi-infinite programming

From formulasearchengine
Revision as of 17:24, 28 August 2013 by en>Pdecalculus (External links: spelling in a reference and correct link)
Jump to navigation Jump to search

Discrete Morse theory is a combinatorial adaptation of Morse theory developed by Robin Forman. The theory has various practical applications in diverse fields of applied mathematics and computer science, such as configuration spaces,[1] homology computation[2] and mesh compression.[3]

Notation regarding CW complexes

Let 𝒳 be a CW complex. Define the incidence function κ:𝒳×𝒳 in the following way: given two cells σ and τ in 𝒳, let κ(σ,τ) be the degree of the attaching map from the boundary of σ to τ. The boundary operator on 𝒳 is defined by

(σ)=τ𝒳κ(σ,τ)τ

It is a defining property of boundary operators that 0. In more axiomatic definitions[4] one can find the requirement that σ,τ𝒳

τ𝒳κ(σ,τ)κ(τ,τ)=0

which is a corollary of the above definition of the boundary operator and the requirement that 0.

Discrete Morse functions

A real-valued function μ:𝒳 is a discrete Morse function if it satisfies the following two properties:

  1. For any cell σ𝒳, the number of cells τ𝒳 in the boundary of σ which satisfy μ(σ)μ(τ) is at most one.
  2. For any cell σ𝒳, the number of cells τ𝒳 containing σ in their boundary which satisfy μ(σ)μ(τ) is at most one.

It can be shown[5] that the cardinalities in the two conditions cannot both be one simultaneously for a fixed cell σ, provided that 𝒳 is a regular CW complex. In this case, each cell σ𝒳 can be paired with at most one exceptional cell τ𝒳: either a boundary cell with larger μ value, or a co-boundary cell with smaller μ value. The cells which have no pairs, i.e., their function values are strictly higher than their boundary cells and strictly lower than their co-boundary cells are called critical cells. Thus, a discrete Morse function partitions the CW complex into three distinct cell collections: 𝒳=𝒜𝒦𝒬, where:

  1. 𝒜 denotes the critical cells which are unpaired,
  2. 𝒦 denotes cells which are paired with boundary cells, and
  3. 𝒬 denotes cells which are paired with co-boundary cells.

By construction, there is a bijection of sets between k-dimensional cells in 𝒦 and the (k1)-dimensional cells in 𝒬, which can be denoted by pk:𝒦k𝒬k1 for each natural number k. It is an additional technical requirement that for each K𝒦k, the degree of the attaching map from the boundary of K to its paired cell pk(K)𝒬 is a unit in the underlying ring of 𝒳. For instance, over the integers , the only allowed values are ±1. This technical requirement is guaranteed when one assumes that 𝒳 is a regular CW complex over .

The fundamental result of discrete Morse theory establishes that the CW complex 𝒳 is isomorphic on the level of homology to a new complex 𝒜 consisting of only the critical cells. The paired cells in 𝒦 and 𝒬 describe gradient paths between adjacent critical cells which can be used to obtain the boundary operator on 𝒜. Some details of this construction are provided in the next section.

The Morse complex

A gradient path is a sequence of paired cells

ρ=(Q1,K1,Q2,K2,,QM,KM)

satisfying Qm=p(Km) and κ(Km,Qm+1)0. The index of this gradient path is defined to be the integer

ν(ρ)=m=1M1κ(Km,Qm+1)m=1Mκ(Km,Qm).

The division here makes sense because the incidence between paired cells must be ±1. Note that by construction, the values of the discrete Morse function μ must decrease across ρ. The path ρ is said to connect two critical cells A,A𝒜 if κ(A,Q1)0κ(KM,A). This relationship may be expressed as AρA. The multiplicity of this connection is defined to be the integer m(ρ)=κ(A,Q1)ν(ρ)κ(KM,A). Finally, the Morse boundary operator on the critical cells 𝒜 is defined by

Δ(A)=κ(A,A)+AρAm(ρ)A

where the sum is taken over all gradient path connections from A to A.

Basic Results

Many of the familiar results from continuous Morse theory apply in the discrete setting.

The Morse Inequalities

Let 𝒜 be a Morse complex associated to the CW complex 𝒳. The number mq=|𝒜q| of q-cells in 𝒜 is called the qth Morse number. Let βq denote the qth Betti number of 𝒳. Then, for any N>0, the following inequalities[6] hold

mNβN, and
mNmN1+±m0βNβN1+±β0

Moreover, the Euler characteristic χ(𝒳) of 𝒳 satisfies

χ(𝒳)=m0m1+±mdim𝒳

Discrete Morse Homology and Homotopy Type

Let 𝒳 be a regular CW complex with boundary operator and a discrete Morse function μ:𝒳. Let 𝒜 be the associated Morse complex with Morse boundary operator Δ. Then, there is an isomorphism[7] of Homology groups as well as homotopy groups.

H*(𝒳,)H*(𝒜,Δ)

See also

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

  1. F. Mori and M. Salvetti: (Discrete) Morse theory for Configuration spaces
  2. Perseus: Software for computing persistent homology groups.
  3. T Lewiner, H Lopez and G Tavares: Applications of Forman's discrete Morse theory to topological visualization and mesh compression
  4. Template:Cite web
  5. Forman, Robin: Morse Theory for Cell Complexes, Lemma 2.5
  6. Forman, Robin: Morse Theory for Cell Complexes, Corollaries 3.5 and 3.6
  7. Forman, Robin: Morse Theory for Cell Complexes, Theorem 7.3