AMS Monograph Series Sample

Author One

Author Two

Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803

Current address: Department of Mathematics and Statistics, Case Western Reserve University, Cleveland, Ohio 43403

E-mail address: xyz@math.university.edu

Mathematical Research Section, School of Mathematical Sciences, Australian National University, Canberra ACT 2601, Australia

E-mail address: two@maths.univ.edu.au

2010 Mathematics Subject Classification. Primary 54C40, 14E20; Secondary 46E25, 20C20

Key words and phrases. amsbook, AMS-L^{A}TEX

The first author was supported in part by NSF Grant #000000.

Abstract. This document is a sample prepared to illustrate the use of the American Mathematical Society’s L^{A}TEX document class amsbook and publicationspecific variants of that class.

Contents

Preface vii

Part 1. This is a Part Title Sample 1

Chapter 1. AMS Monograph Series Sample 3 This is an unnumbered first-level section head 3 This is a Special Section Head 3 1.1. This is a numbered first-level section head 3 Exercises 4 1.2. Some more list types 5

Bibliography 7

Index 9

Preface

This is an example of an unnumbered chapter which can be used for a Preface or Foreword.

The purpose of this paper is to establish a relationship between an infinitedimensional Grassmannian and arbitrary algebraic vector bundles of any rank defined over an arbitrary complete irreducible algebraic curve, which generalizes the known connection between the Grassmannian and line bundles on algebraic curves.

Author Name

Part 1

This is a Part Title Sample

CHAPTER 1 AMS Monograph Series Sample

This is an unnumbered first-level section head

This is an example of an unnumbered first-level heading.

This is a Special Section Head

This is an example of a special section head^{1}.

1.1. This is a numbered first-level section head

This is an example of a numbered first-level heading.

1.1.1. This is a numbered second-level section head. This is an example of a numbered second-level heading.

This is an unnumbered second-level section head. This is an example of an unnumbered second-level heading.

1.1.1.1. This is a numbered third-level section head. This is an example of a numbered third-level heading.

This is an unnumbered third-level section head. This is an example of an unnumbered third-level heading.

Lemma 1.1. Let f, g ∈ A(X) and let E, F be cozero sets in X.

(1) If f is E-regular and F ⊆ E, then f is F -regular.

(2) If f is E-regular and F -regular, then f is E ∪ F -regular.

(3) If f(x) ≥ c > 0 for all x ∈ E, then f is E-regular.

The following is an example of a proof.

Proof. Set j(ν) = max(I\a(ν)) − 1. Then we have

∑_{i∉a(ν)} t_{i} ∼ t_{j(ν)+1} = ∏_{j=0}^{j(ν)}(t_{j+1}/t_{j}).

1Here is an example of a footnote. Notice that this footnote text is running on so that it can stand as an example of how a footnote with separate paragraphs should be written.

And here is the beginning of the second paragraph.

Hence we have

\begin{aligned}∏_{ν}( ∑_{i∉a(ν)} t_{i})^{|a(ν−1)|−|a(ν)|} & ∼ ∏_{ν} ∏_{j=0}^{j(ν)}(t_{j+1}/t_{j})^{|a(ν−1)|−|a(ν)|} \\ & = ∏_{j≥0}(t_{j+1}/t_{j})^{∑_{j(ν)≥j}(|a(ν−1)|−|a(ν)|)}. \end{aligned} ; (1.1)

By definition, we have a(ν(j)) ⊃ c(j). Hence, |c(j)| = n − j implies (5.4). If c(j) ∉ a, a(ν(j))c(j) and hence we have (5.5). □

This is an example of an ‘extract’. The magnetization M_{0} of the Ising model is related to the local state probability P (a) : M_{0} = P (1) − P (−1). The equivalences are shown in Table 1.

Table 1 \begin{aligned}& −∞ & +∞ \\ f_{+}(x, k) & e^{\sqrt{−1}kx} + s_{12}(k)e^{−\sqrt{−1}kx} & s_{11}(k)e^{\sqrt{−1}kx} \\ f_{−}(x, k) & s_{22}(k)e^{−\sqrt{−1}kx} & e^{−\sqrt{−1}kx} + s_{21}(k)e^{\sqrt{−1}kx} \end{aligned}

Definition 1.2. This is an example of a ‘definition’ element. For f ∈ A(X), we define

(1.2)  Z(f) = {E ∈ Z[X] : f is E^{c}-regular}.

Remark 1.3. This is an example of a ‘remark’ element. For f ∈ A(X), we define

(1.3)  Z(f) = {E ∈ Z[X] : f is E^{c}-regular}.

Example 1.4. This is an example of an ‘example’ element. For f ∈ A(X), we define

(1.4)  Z(f) = {E ∈ Z[X] : f is E^{c}-regular}.

Exercise 1.5. This is an example of the xca environment. This environment is used for exercises which occur within a section.

Some extra text before the xcb head. The xcb environment is used for exercises that occur at the end of a chapter. Here it contains an example of a numbered list.

Exercises

(1) First item. In the case where in G there is a sequence of subgroups

G = G_{0}, G_{1}, G_{2}, …, G_{k} = e

such that each is an invariant subgroup of G_{i}.

(2) Second item. Its action on an arbitrary element X = λ^{α}X_{α} has the form

(1.5)  [e^{α}X_{α}, X] = e^{α}λ^{β}[X_{α}X_{β}] = e^{α}c_{αβ}^{γ}λ^{β}X_{γ},

(a) First subitem.

−2ψ_{2}(e) = c_{αγ}^{δ}c_{βδ}^{γ}e^{α}e^{β}.

Figure 1. This is an example of a figure caption with text.

Figure 2

(b) Second subitem.

(i) First subsubitem. In the case where in G there is a sequence of subgroups

G = G_{0}, G_{1}, G_{2}, …, G_{k} = e

such that each subgroup G_{i+1} is an invariant subgroup of G_{i} and each quotient group G_{i+1}/G_{i} is abelian, the group G is called solvable.

(ii) Second subsubitem.

(c) Third subitem.

(3) Third item.

Here is an example of a cite. See [A].

Theorem 1.6. This is an example of a theorem.

Theorem 1.7 (Marcus Theorem). This is an example of a theorem with a parenthetical note in the heading.

1.2. Some more list types

This is an example of a bulleted list.

• J_{g} of dimension 3g − 3;

• E_{g}^{2} = {Pryms of double covers of C = with normalization of C hyperelliptic of genus g − 1} of dimension 2g;

• E_{1,g−1}^{2} = {Pryms of double covers of C = _{P^{1}}^{H} with H hyperelliptic of genus g − 2} of dimension 2g − 1;

• P_{t,g−t}^{2} for 2 ≤ t ≤ g/2 = {Pryms of double covers of C = _{C′′}^{C′} with

g(C′) = t − 1 and g(C′′) = g − t − 1} of dimension 3g − 4. This is an example of a ‘description’ list.

Zero case: ρ(Φ) = {0}.

Rational case: ρ(Φ) ≠ {0} and ρ(Φ) is contained in a line through 0 with

rational slope. Irrational case: ρ(Φ) ≠ {0} and ρ(Φ) is contained in a line through 0 with

irrational slope.

Bibliography

[A] T. Aoki, Calcul exponentiel des opérateurs microdifferentiels d’ordre infini. I, Ann. Inst. Fourier (Grenoble) 33 (1983), 227–250.

[B] R. Brown, On a conjecture of Dirichlet, Amer. Math. Soc., Providence, RI, 1993.

[D] R. A. DeVore, Approximation of functions, Proc. Sympos. Appl. Math., vol. 36, Amer. Math. Soc., Providence, RI, 1986, pp. 34–56.

Index

Absorbing barrier, 4 Adjoint partial differential operator, 20 A-harmonic function, 16, 182 A^{∗}-harmonic function, 182

Boundary condition, 20, 22 Dirichlet, 15 Neumann, 16

Boundary value problem the first, 16 the second, 16 the third, 16

Bounded set, 19

Diffusion coefficient, 1 equation, 3, 23

Dirichlet boundary condition, 15 boundary value problem, 16

Elliptic boundary value problem, 14, 158 partial differential equation, 14 partial differential operator, 19

Fick’s law, 1 Flux, 1 Formally adjoint partial differential

operator, 20 Fundamental solution

conceptional explanation, 12 general definition, 23 temporally homogeneous case, 64, 112

Genuine solution, 196 Green function, 156 Green’s formula, 21

Harnack theorems first theorem, 185 inequality, 186 lemma, 186 second theorem, 187 third theorem, 187 Helmhotz decomposition, 214 Hilbert-Schmidt expansion theorem, 120

Initial-boundary value problem, 22 Initial condition, 22 Invariant measure (for the fundamental

solution), 167

Maximum principle for A-harmonic functions, 183 for parabolic differential equations, 65 strong, 83

Neumann boundary condition, 16 boundary value problem, 16 function, 179

One-parameter semigroup, 113

Parabolic initial-boundary value problem,

22 Partial differential equation

of elliptic type, 14

of parabolic type, 22 Positive definite kernel, 121

Reflecting barrier, 4 Regular (set), 19 Removable isolated singularity, 191 Robin problem, 16

Semigroup property (of fundamental

solution), 64, 113 Separation of variables, 131 Solenoidal (vector field), 209 Strong maximum principle, 83 Symmetry (of fundamental solution), 64,

112

Temporally homogeneous, 111

Vector field with potential, 209

Weak solution of elliptic equations, 195 of parabolic equation, 196

10 INDEX

associated with a boundary condition,