Valuation (geometry)

In geometry, a valuation is a finitely additive function from a collection of subsets of a set X to an abelian semigroup. For example, Lebesgue measure is a valuation on finite unions of convex bodies of ℝn. Other examples of valuations on finite unions of convex bodies of ℝn are surface area, mean width, and Euler characteristic.

In geometry, continuity (or smoothness) conditions are often imposed on valuations, but there are also purely discrete facets of the theory. In fact, the concept of valuation has its origin in the dissection theory of polytopes and in particular Hilbert's third problem, which has grown into a rich theory reliant on tools from abstract algebra.

## Contents

* 1 Definition
* 1.1 Examples
* 2 Valuations on convex bodies
* 2.1 Homogeneous valuations
* 2.2 Embedding theorems
* 2.3 Irreducibility Theorem
* 2.4 Smooth valuations
* 3 Operations on translation-invariant valuations
* 3.1 Exterior product
* 3.2 Product
* 3.3 Alesker-Poincaré duality
* 3.4 Convolution
* 3.5 Fourier transform
* 3.6 Pullback and pushforward
* 4 Valuations on manifolds
* 4.1 Examples
* 4.2 Filtration
* 4.3 Product
* 4.4 Pullback and pushforward
* 5 Applications in Integral Geometry
* 6 See also
* 7 References
* 8 Bibliography

## Definition

Let X be a set, and let 𝒮 be a collection of subsets of X. A function ϕ on 𝒮 with values in an abelian semigroup R is called a valuation if it satisfies ϕ(A∪B)+ϕ(A∩B)=ϕ(A)+ϕ(B) whenever A, B, A∪B, and A∩B are elements of 𝒮. If ∅∈𝒮, then one always assumes ϕ(∅)=0.

### Examples

Some common examples of 𝒮 are

* the convex bodies in ℝn
* compact convex polytopes in ℝn
* convex cones
* smooth compact polyhedra in a smooth manifold X

Let 𝒦(ℝn) be the set of convex bodies in ℝn. Then some valuations on 𝒦(ℝn) are

* the Euler characteristic χ:K(ℝn)->ℤ
* Lebesgue measure restricted to 𝒦(ℝn)
* intrinsic volume (and, more generally, mixed volume)
* the map A↦hA, where hA is the support function of A

Some other valuations are

* the lattice point enumerator P↦|ℤn∩P|, where P is a lattice polytope
* cardinality, on the family of finite sets

## Valuations on convex bodies

From here on, let V=ℝn, let 𝒦(V) be the set of convex bodies in V, and let ϕ be a valuation on 𝒦(V).

We say ϕ is translation invariant if, for all K∈𝒦(V) and x∈V, we have ϕ(K+x)=ϕ(K).

Let (K,L)∈𝒦(V)2. The Hausdorff distance dH(K,L) is defined as dH(K,L)=inf⁡{ε>0:K⊂Lε and L⊂Kε}, where Kε is the ε-neighborhood of K under some Euclidean inner product. Equipped with this metric, 𝒦(V) is a locally compact space.

The space of continuous, translation-invariant valuations from 𝒦(V) to ℂ is denoted by Val⁡(V).

The topology on Val⁡(V) is the topology of uniform convergence on compact subsets of 𝒦(V). Equipped with the norm ‖ϕ‖=max⁡{|ϕ(K)|:K⊂B}, where B⊂V is a bounded subset with nonempty interior, Val⁡(V) is a Banach space.

### Homogeneous valuations

A translation-invariant continuous valuation ϕ∈Val⁡(V) is said to be i-homogeneous if ϕ(λK)=λiϕ(K) for all λ>0 and K∈𝒦(V). The subset Vali(V) of i-homogeneous valuations is a vector subspace of Val⁡(V). McMullen's decomposition theorem[1] states that

Val⁡(V)=⨁i=0nVali(V),n=dim⁡V.

In particular, the degree of a homogeneous valuation is always an integer between 0 and n=dim⁡V.

Valuations are not only graded by the degree of homogeneity, but also by the parity with respect to the reflection through the origin, namely Vali=Vali+⊕Vali−, where ϕ∈Valiϵ with ϵ∈{+,−} if and only if ϕ(−K)=ϵϕ(K) for all convex bodies K. The elements of Vali+ and Vali− are said to be even and odd, respectively.

It is a simple fact that Val0(V) is 1-dimensional and spanned by the Euler characteristic χ, that is, consists of the constant valuations on 𝒦(V).

In 1957 Hadwiger[2] proved that Valn(V) (where n=dim⁡V) coincides with the 1-dimensional space of Lebesgue measures on V.

A valuation ϕ∈Val⁡(ℝn) is simple if ϕ(K)=0 for all convex bodies with dim⁡K<n. Schneider[3] in 1996 described all simple valuations on ℝn: they are given by ϕ(K)=cvol⁡(K)+∫Sn−1f(θ)dσK(θ), where c∈ℂ, f∈C(Sn−1) is an arbitrary odd function on the unit sphere Sn−1⊂ℝn, and σK is the surface area measure of K. In particular, any simple valuation is the sum of an n\- and an (n−1)-homogeneous valuation. This in turn implies that an i-homogeneous valuation is uniquely determined by its restrictions to all (i+1)-dimensional subspaces.

### Embedding theorems

The Klain embedding is a linear injection of Vali+(V), the space of even i-homogeneous valuations, into the space of continuous sections of a canonical complex line bundle over the Grassmannian Gri(V) of i-dimensional linear subspaces of V. Its construction is based on Hadwiger's characterization[2] of n-homogeneous valuations. If ϕ∈Vali(V) and E∈Gri(V), then the restriction ϕ|E is an element Vali(E), and by Hadwiger's theorem it is a Lebesgue measure. Hence Klϕ(E)=ϕ|E defines a continuous section of the line bundle Dens over Gri(V) with fiber over E equal to the 1-dimensional space Dens⁡(E) of densities (Lebesgue measures) on E.

Theorem (Klain[4]). The linear map Kl⁡:Vali+(V)->C(Gri(V),Dens⁡) is injective.

A different injection, known as the Schneider embedding, exists for odd valuations. It is based on Schneider's description of simple valuations.[3] It is a linear injection of Vali−(V), the space of odd i-homogeneous valuations, into a certain quotient of the space of continuous sections of a line bundle over the partial flag manifold of cooriented pairs (Fi⊂Ei+1). Its definition is reminiscent of the Klain embedding, but more involved. Details can be found in.[5]

The Goodey-Weil embedding is a linear injection of Vali into the space of distributions on the i-fold product of the (n−1)-dimensional sphere. It is nothing but the Schwartz kernel of a natural polarization that any ϕ∈Valk(V) admits, namely as a functional on the k-fold product of C2(Sn−1), the latter space of functions having the geometric meaning of differences of support functions of smooth convex bodies. For details, see.[5]

### Irreducibility Theorem

The classical theorems of Hadwiger, Schneider and McMullen give fairly explicit descriptions of valuations that are homogeneous of degree 1, n−1, and n=dim⁡V. But for degrees 1<i<n−1 very little was known before the turn of the 21st century. McMullen's conjecture is the statement that the valuations ϕA(K)=voln(K+A),A∈𝒦(V), span a dense subspace of Val⁡(V). McMullen's conjecture was confirmed by Alesker in a much stronger form, which became known as the Irreducibility Theorem:

Theorem (Alesker[6]). For every 0≤i≤n, the natural action of GL(V) on the spaces Vali+(V) and Vali−(V) is irreducible.

Here the action of the general linear group GL(V) on Val⁡(V) is given by (g⋅ϕ)(K)=ϕ(g−1K). The proof of the Irreducibility Theorem is based on the embedding theorems of the previous section and Beilinson-Bernstein localization.

### Smooth valuations

A valuation ϕ∈Val⁡(V) is called smooth if the map g↦g⋅ϕ from GL(V) to Val⁡(V) is smooth. In other words, ϕ is smooth if and only if ϕ is a smooth vector of the natural representation of GL(V) on Val⁡(V). The space of smooth valuations Val∞(V) is dense in Val⁡(V); it comes equipped with a natural Fréchet-space topology, which is finer than the one induced from Val⁡(V).

For every (complex-valued) smooth function f on Gri(ℝn), ϕ(K)=∫Gri(ℝn)voli(PEK)f(E)dE, where PE:ℝn->E denotes the orthogonal projection and dE is the Haar measure, defines a smooth even valuation of degree i. It follows from the Irreducibility Theorem, in combination with the Casselman-Wallach theorem, that any smooth even valuation can be represented in this way. Such a representation is sometimes called a Crofton formula.

For any (complex-valued) smooth differential form ω∈Ωn−1(ℝn×Sn−1) that is invariant under all the translations (x,u)↦(x+t,u) and every number c∈ℂ, integration over the normal cycle defines a smooth valuation:

ϕ(K)=cvoln(K)+∫N(K)ω,K∈𝒦(ℝn).

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(1)

As a set, the normal cycle N(K) consists of the outward unit normals to K. The Irreducibility Theorem implies that every smooth valuation is of this form.

## Operations on translation-invariant valuations

There are several natural operations defined on the subspace of smooth valuations Val∞(V)⊂Val⁡(V). The most important one is the product of two smooth valuations. Together with pullback and pushforward, this operation extends to valuations on manifolds.

### Exterior product

Let V,W be finite-dimensional real vector spaces. There exists a bilinear map, called the exterior product, ⊠:Val∞(V)×Val∞(W)->Val⁡(V×W) which is uniquely characterized by the following two properties:

* it is continuous with respect to the usual topologies on Val and Val∞.
* if ϕ=volV(∙+A) and ψ=volW(∙+B) where A∈𝒦(V) and B∈𝒦(W) are convex bodies with smooth boundary and strictly positive Gauss curvature, and volV and volW are densities on V and W, then

ϕ⊠ψ=(volV⊠volW)(∙+A×B).

### Product

The product of two smooth valuations ϕ,ψ∈Val∞(V) is defined by (ϕ⋅ψ)(K)=(ϕ⊠ψ)(Δ(K)), where Δ:V->V×V is the diagonal embedding. The product is a continuous map Val∞(V)×Val∞(V)->Val∞(V). Equipped with this product, Val∞(V) becomes a commutative associative graded algebra with the Euler characteristic as the multiplicative identity.

### Alesker-Poincaré duality

By a theorem of Alesker, the restriction of the product Valk∞(V)×Valn−k∞(V)->Valn∞(V)=Dens⁡(V) is a non-degenerate pairing. This motivates the definition of the k-homogeneous generalized valuation, denoted Valk−∞(V), as Valn−k∞(V)*⊗Dens⁡(V), topologized with the weak topology. By the Alesker-Poincaré duality, there is a natural dense inclusion Valk∞(V)↪Valk−∞(V)/

### Convolution

Convolution is a natural product on Val∞(V)⊗Dens⁡(V*). For simplicity, we fix a density vol on V to trivialize the second factor. Define for fixed A,B∈𝒦(V) with smooth boundary and strictly positive Gauss curvature vol⁡(∙+A)∗vol⁡(∙+B)=vol⁡(∙+A+B). There is then a unique extension by continuity to a map Val∞(V)×Val∞(V)->Val∞(V), called the convolution. Unlike the product, convolution respects the co-grading, namely if ϕ∈Valn−i∞(V), ψ∈Valn−j∞(V), then ϕ∗ψ∈Valn−i−j∞(V).

For instance, let V(K1,…,Kn) denote the mixed volume of the convex bodies K1,…,Kn⊂ℝn. If convex bodies A1,…,An−i in ℝn with a smooth boundary and strictly positive Gauss curvature are fixed, then ϕ(K)=V(K[i],A1,…,An−i) defines a smooth valuation of degree i. The convolution two such valuations is V(∙[i],A1,…,An−i)∗V(∙[j],B1,…,Bn−j)=ci,jV(∙[n−j−i],A1,…,An−i,B1,…,Bn−j), where ci,j is a constant depending only on i,j,n.

### Fourier transform

The Alesker-Fourier transform is a natural, GL(V)-equivariant isomorphism of complex-valued valuations 𝔽:Val∞(V)->Val∞(V*)⊗Dens⁡(V), discovered by Alesker and enjoying many properties resembling the classical Fourier transform, which explains its name.

It reverses the grading, namely 𝔽:Valk∞(V)->Valn−k∞(V*)⊗Dens⁡(V), and intertwines the product and the convolution: 𝔽(ϕ⋅ψ)=𝔽ϕ∗𝔽ψ.

Fixing for simplicity a Euclidean structure to identify V=V*, Dens⁡(V)=ℂ, we have the identity 𝔽2ϕ(K)=ϕ(−K). On even valuations, there is a simple description of the Fourier transform in terms of the Klain embedding: Kl𝔽ϕ(E)=Klϕ(E⊥). In particular, even real-valued valuations remain real-valued after the Fourier transform.

For odd valuations, the description of the Fourier transform is substantially more involved. Unlike the even case, it is no longer of purely geometric nature. For instance, the space of real-valued odd valuations is not preserved.

### Pullback and pushforward

Given a linear map f:U->V, there are induced operations of pullback f*:Val⁡(V)->Val⁡(U) and pushforward f*:Val⁡(U)⊗Dens⁡(U)*->Val⁡(V)⊗Dens⁡(V)*. The pullback is the simpler of the two, given by f*ϕ(K)=ϕ(f(K)). It evidently preserves the parity and degree of homogeneity of a valuation. Note that the pullback does not preserve smoothness when f is not injective.

The pushforward is harder to define formally. For simplicity, fix Lebesgue measures on U and V. The pushforward can be uniquely characterized by describing its action on valuations of the form vol⁡(∙+A), for all A∈𝒦(U), and then extended by continuity to all valuations using the Irreducibility Theorem. For a surjective map f, f*vol⁡(∙+A)=vol⁡(∙+f(A)). For an inclusion f:U↪V, choose a splitting V=U⊕W. Then f*vol⁡(∙+A)(K)=∫Wvol⁡(K∩(U+w)+A)dw. Informally, the pushforward is dual to the pullback with respect to the Alesker-Poincaré pairing: for ϕ∈Val⁡(V) and ψ∈Val⁡(U)⊗Dens⁡(U)*, ⟨f*ϕ,ψ⟩=⟨ϕ,f*ψ⟩. However, this identity has to be carefully interpreted since the pairing is only well-defined for smooth valuations. For further details, see.[7]

## Valuations on manifolds

In a series of papers beginning in 2006, Alesker laid down the foundations for a theory of valuations on manifolds that extends the theory of valuations on convex bodies. The key observation leading to this extension is that via integration over the normal cycle (1), a smooth translation-invariant valuation may be evaluated on sets much more general than convex ones. Also (1) suggests to define smooth valuations in general by dropping the requirement that the form ω be translation-invariant and by replacing the translation-invariant Lebesgue measure with an arbitrary smooth measure.

Let X be an n-dimensional smooth manifold and let ℙX=ℙ+(T*X) be the co-sphere bundle of X, that is, the oriented projectivization of the cotangent bundle. Let 𝒫(X) denote the collection of compact differentiable polyhedra in X. The normal cycle N(A)⊂ℙX of A∈𝒫(X), which consists of the outward co-normals to A, is naturally a Lipschitz submanifold of dimension n−1.

For ease of presentation we henceforth assume that X is oriented, even though the concept of smooth valuations in fact does not depend on orientability. The space of smooth valuations 𝒱∞(X) on X consists of functions ϕ:𝒫(X)->ℂ of the form ϕ(A)=∫Aμ+∫N(A)ω,A∈𝒫(X), where μ∈Ωn(X) and ω∈Ωn−1(ℙX) can be arbitrary. It was shown by Alesker that the smooth valuations on open subsets of X form a soft sheaf over X.

### Examples

The following are examples of smooth valuations on a smooth manifold X:

* Smooth measures on X.
* The Euler characteristic; this follows from the work of Chern[8] on the Gauss-Bonnet theorem, where such μ and ω were constructed to represent the Euler characteristic. In particular, μ is then the Chern-Gauss-Bonnet integrand, which is the Pfaffian of the Riemannian curvature tensor.
* If X is Riemannian, then the Lipschitz-Killing valuations or intrinsic volumes V0X=χ,V1X,…,VnX=volX are smooth valuations. If f:X->ℝm is any isometric immersion into a Euclidean space, then ViX=f*Viℝm, where Viℝm denotes the usual intrinsic volumes on ℝm (see below for the definition of the pullback). The existence of these valuations is the essence of Weyl's tube formula.[9]
* Let ℂPn be the complex projective space, and let Grkℂ denote the Grassmannian of all complex projective subspaces of fixed dimension k. The function

ϕ(A)=∫Grkℂχ(A∩E)dE,A∈𝒫(ℂPn), where the integration is with respect to the Haar probability measure on Grkℂ, is a smooth valuation. This follows from the work of Fu.[10]

### Filtration

The space 𝒱∞(X) admits no natural grading in general, however it carries a canonical filtration 𝒱∞(X)=W0⊃W1⊃⋯⊃Wn. Here Wn consists of the smooth measures on X, and Wj is given by forms ω in the ideal generated by π*Ωj(X), where π:ℙX->X is the canonical projection.

The associated graded vector space ⨁i=0nWi/Wi+1 is canonically isomorphic to the space of smooth sections ⨁i=0nC∞(X,Vali∞(TX)), where Vali∞(TX) denotes the vector bundle over X such that the fiber over a point x∈X is Vali∞(TxX), the space of i-homogeneous smooth translation-invariant valuations on the tangent space TxX.

### Product

The space 𝒱∞(X) admits a natural product. This product is continuous, commutative, associative, compatible with the filtration: Wi⋅Wj⊂Wi+j, and has the Euler characteristic as the identity element. It also commutes with the restriction to embedded submanifolds, and the diffeomorphism group of X acts on 𝒱∞(X) by algebra automorphisms.

For example, if X is Riemannian, the Lipschitz-Killing valuations satisfy ViX⋅VjX=Vi+jX.

The Alesker-Poincaré duality still holds. For compact X it says that the pairing 𝒱∞(X)×𝒱∞(X)->ℂ, (ϕ,ψ)↦(ϕ⋅ψ)(X) is non-degenerate. As in the translation-invariant case, this duality can be used to define generalized valuations. Unlike the translation-invariant case, no good definition of continuous valuations exists for valuations on manifolds.

The product of valuations closely reflects the geometric operation of intersection of subsets. Informally, consider the generalized valuation χA=χ(A∩∙). The product is given by χA⋅χB=χA∩B. Now one can obtain smooth valuations by averaging generalized valuations of the form χA, more precisely ϕ(X)=∫Sχs(A)ds is a smooth valuation if S is a sufficiently large measured family of diffeomorphisms. Then one has ∫Sχs(A)ds⋅∫S′χs′(B)ds′=∫S×S′χs(A)∩s′(B)dsds′, see.[11]

### Pullback and pushforward

Every smooth immersion f:X->Y of smooth manifolds induces a pullback map f*:𝒱∞(Y)->𝒱∞(X). If f is an embedding, then (f*ϕ)(A)=ϕ(f(A)),A∈𝒫(X). The pullback is a morphism of filtered algebras. Every smooth proper submersion f:X->Y defines a pushforward map f*:𝒱∞(X)->𝒱∞(Y) by (f*ϕ)(A)=ϕ(f−1(A)),A∈𝒫(Y). The pushforward is compatible with the filtration as well: f*:Wi(X)->Wi−(dim⁡X−dim⁡Y)(Y). For general smooth maps, one can define pullback and pushforward for generalized valuations under some restrictions.

## Applications in Integral Geometry

Let

be a Riemannian manifold and let

be a Lie group of isometries of

acting transitively on the sphere bundle

SM.

Under these assumptions the space

𝒱∞(M)G

-invariant smooth valuations on

is finite-dimensional; let

ϕ1,…,ϕm

be a basis. Let

A,B∈𝒫(M)

be differentiable polyhedra in

Then integrals of the form

∫Gϕi(A∩gB)dg

are expressible as linear combinations of

ϕk(A)ϕl(B)

with coefficients

cikl

independent of

and

∫Gϕi(A∩gB)dg=∑k,l=1mciklϕk(A)ϕl(B),A,B∈𝒫(M).

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(2)

Formulas of this type are called kinematic formulas. Their existence in this generality was proved by Fu.[10] For the three simply connected real space forms, that is, the sphere, Euclidean space, and hyperbolic space, they go back to Blaschke, Santaló, Chern, and Federer.

Describing the kinematic formulas explicitly is typically a difficult problem. In fact already in the step from real to complex space forms, considerable difficulties arise and these have only recently been resolved by Bernig, Fu, and Solanes.[12] [13] The key insight responsible for this progress is that the kinematic formulas contain the same information as the algebra of invariant valuations 𝒱∞(M)G. For a precise statement, let kG:𝒱∞(M)G->𝒱∞(M)G⊗𝒱∞(M)G be the kinematic operator, that is, the map determined by the kinematic formulas (2). Let pd⁡:𝒱∞(M)G->𝒱∞(M)G* denote the Alesker-Poincaré duality, which is a linear isomorphism. Finally let mG* be the adjoint of the product map mG:𝒱∞(M)G⊗𝒱∞(M)G->𝒱∞(M)G. The Fundamental theorem of algebraic integral geometry relating operations on valuations to integral geometry, states that if the Poincaré duality is used to identify 𝒱∞(M)G with 𝒱∞(M)G*, then kG=mG*:

## See also

* Hadwiger's theorem
* Integral geometry - Concept in mathematics
* Mixed volume
* Set function - Function from sets to numbers
* Valuation (measure theory)

## References

1. ↑ "Continuous translation-invariant valuations on the space of compact convex sets", Archiv der Mathematik 34 (4): 377–384, 1980, doi:10.1007/BF01224974
2. ↑ 2.0 2.1 Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Die Grundlehren der Mathematischen Wissenschaften, 93, Berlin-Göttingen-Heidelberg: Springer-Verlag, 1957, doi:10.1007/978-3-642-94702-5, ISBN 978-3-642-94703-2
3. ↑ 3.0 3.1 Schneider, Rolf (1996), "Simple valuations on convex bodies", Mathematika 43 (1): 32–39, doi:10.1112/S0025579300011578
4. ↑ Klain, Daniel A. (1995), "A short proof of Hadwiger's characterization theorem", Mathematika 42 (2): 329–339, doi:10.1112/S0025579300014625
5. ↑ 5.0 5.1 Alesker, Semyon (2018), Introduction to the theory of valuations, CBMS Regional Conference Series in Mathematics, 126, Providence, RI: American Mathematical Society
6. ↑ Alesker, Semyon (2001), "Description of translation invariant valuations on convex sets with solution of P. McMullen's conjecture", Geometric and Functional Analysis 11 (2): 244–272, doi:10.1007/PL00001675
7. ↑ Alesker, Semyon (2011), "A Fourier-type transform on translation-invariant valuations on convex sets", Israel Journal of Mathematics 181: 189–294, doi:10.1007/s11856-011-0008-6
8. ↑ "On the curvatura integra in a Riemannian manifold", Annals of Mathematics, Second Series 46 (4): 674–684, 1945, doi:10.2307/1969203
9. ↑ Weyl, Hermann (1939), "On the Volume of Tubes", American Journal of Mathematics 61 (2): 461–472, doi:10.2307/2371513
10. ↑ 10.0 10.1 Fu, Joseph H. G. (1990), "Kinematic formulas in integral geometry", Indiana University Mathematics Journal 39 (4): 1115–1154, doi:10.1512/iumj.1990.39.39052
11. ↑ Fu, Joseph H. G. (2016), "Intersection theory and the Alesker product", Indiana University Mathematics Journal 65 (4): 1347–1371, doi:10.1512/iumj.2016.65.5846
12. ↑ Bernig, Andreas; Fu, Joseph H. G.; Solanes, Gil (2014), "Integral geometry of complex space forms", Geometric and Functional Analysis 24 (2): 403–49, doi:10.1007/s00039-014-0251-1 2
13. ↑ Bernig, Andreas; Fu, Joseph H. G. (2011), "Hermitian integral geometry", Annals of Mathematics, Second Series 173 (2): 907–945, doi:10.4007/annals.2011.173.2.7

## Bibliography

* S. Alesker (2018). Introduction to the theory of valuations. CBMS Regional Conference Series in Mathematics, 126. American Mathematical Society, Providence, RI. ISBN 978-1-4704-4359-7.
* S. Alesker; J. H. G. Fu (2014). Integral geometry and valuations. Advanced Courses in Mathematics. CRM Barcelona.. Birkhäuser/Springer, Basel. ISBN 978-1-4704-4359-7.
* D. A. Klain; G.-C. Rota (1997). Introduction to geometric probability. Lezioni Lincee. [Lincei Lectures]. Cambridge University Press. ISBN 0-521-59362-X.
* R. Schneider (2014). Convex bodies: the Brunn-Minkowski theory.. Encyclopedia of Mathematics and its Applications, 151.. Cambridge University Press, Cambridge, RI. ISBN 978-1-107-60101-7.

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