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1.

図書
図書
1. Sieves in number theory
George Greaves
出版情報: Berlin ; Tokyo : Springer, c2001  xii, 304 p. ; 24 cm
シリーズ名: Ergebnisse der Mathematik und ihrer Grenzgebiete ; 3. Folge, v. 43
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Introduction
The Structure of Sifting Arguments / 1：
The Sieves of Eratosthenes and Legendre / 1.1：
The Contribution of Eratosthenes / 1.1.1：
Legendre's Sieve / 1.1.2：
An Estimate for n(X) / 1.1.3：
The Distribution of Primes / 1.1.4：
Examples of Sifting Situations / 1.2：
Notations / 1.2.1：
The Integers in an Interval (Y - X, Y ) / 1.2.2：
Numbers Given by Polynomial Expressions / 1.2.3：
Arithmetic Progressions / 1.2.4：
Sums of Two Squares / 1.2.5：
Polynomials with Prime Arguments / 1.2.6：
A General Formulation of a Sifting Situation / 1.3：
The Basic Formulation / 1.3.1：
Legendre's Sieve in a General Setting / 1.3.2：
A Generalised Formulation / 1.3.3：
A Further Generalisation / 1.3.4：
Sifting Density / 1.3.5：
The Sifting Limit Β(k) / 1.3.6：
Composition of Sieves / 1.3.7：
Notes on Chapter 1 / 1.4：
Selberg's Upper Bound Method / 2：
The Sifting Apparatus / 2.1：
Selberg's Theorem / 2.1.1：
The Numbers (lambda)(d) / 2.1.2：
A Simple Application / 2.1.3：
General Estimates of G(x) and E(D, P) / 2.2：
An Estimate by Rankin's Device / 2.2.1：
Asymptotic Formulas / 2.2.2：
The Error Term / 2.2.3：
Applications / 2.3：
Prime Twins and Goldbach's Problem / 2.3.1：
Polynomial Sequences / 2.3.3：
Notes on Chapter 2 / 2.4：
Combinatorial Methods / 3：
The Construction of Combinatorial Sieves / 3.1：
Preliminary Discussion of Brun's Ideas / 3.1.1：
Fundamental Inequalities and Identities / 3.1.2：
Buchstab's Identity / 3.1.3：
The Combinatorial Sieve Lemma / 3.1.4：
Brun's Pure Sieve / 3.2：
Inequalities and Identities / 3.2.1：
The "Pure Sieve" Theorem / 3.2.2：
A Corollary / 3.2.3：
Prime Twins / 3.2.4：
A Modern Edition of Brun's Sieve / 3.3：
Rosser's Choice of X / 3.3.1：
A Technical Estimate / 3.3.2：
A Simplifying Approximation / 3.3.3：
A Combinatorial Sieve Theorem / 3.3.4：
Brun's Version of his Method / 3.3.5：
Brun's Choice of x / 3.4.1：
The Estimations / 3.4.2：
The Result / 3.4.3：
Notes on Chapter 3 / 3.5：
Rosser's Sieve / 4：
Approximations by Continuous Functions / 4.1：
The Recurrence Relations / 4.1.1：
Partial Summation / 4.1.2：
The Leading Terms / 4.1.3：
The Functions F and f / 4.2：
The Difference-Differential Equations / 4.2.1：
The Adjoint Equation and the Inner Product / 4.2.2：
Solutions of the Adjoint Equation / 4.2.3：
Particular Values of F(s) and f(s) / 4.2.4：
Asymptotic Analysis as k -> $(infinity$) / 4.2.5：
The Convergence Problem / 4.3：
The Auxiliary Functions / 4.3.1：
Adjoints and Inner Products / 4.3.2：
The Case k
A Sieve Theorem Following Rosser / 4.4：
The Case k >/= 1/2: a First Result / 4.4.1：
Theorem 1 when k
An Improved Version of Proposition 1 / 4.4.3：
A Two-Sided Estimate / 4.4.4：
Extremal Examples / 4.5：
The Linear Case / 4.5.1：
The Case k=1/2 / 4.5.2：
Notes on Chapter 4 / 4.6：
The Sieve with Weights / 5：
Simpler Weighting Devices / 5.1：
Logarithmic Weights / 5.1.1：
Modified Logarithmic Weights / 5.1.2：
Some Applications / 5.1.3：
More Elaborate Weighted Sieves / 5.2：
An Improved Weighting Device / 5.2.1：
Buchstab's Weights / 5.2.2：
A Weighted Sieve Following Rosser / 5.3：
Combining Sieving and Weighting / 5.3.1：
The Reduction Identities / 5.3.2：
An Identity for the Main Term / 5.3.3：
The Estimate for the Main Term / 5.3.4：
Notes on Chapter 5 / 5.4：
The Remainder Term in the Linear Sieve / 6：
The Bilinear Nature of Rosser's Construction / 6.1：
The Factorisation of x.d / 6.1.1：
Discretisations of Rosser's Sieve / 6.1.2：
Specification of Details / 6.1.3：
The Leading Contributions to the Main Term / 6.1.4：
The Remainder Term / 6.1.5：
Sifting Short Intervals / 6.2：
The Smoothed Formulation / 6.2.1：
The Remainder Sums / 6.2.2：
Trigonometrical Sums / 6.2.3：
Notes on Chapter 6 / 6.3：
Lower Bound Sieves when k > 1 / 7：
An Extension of Selberg's Upper Bound / 7.1：
The Integral Equation and the Function $(sigma$) (s) / 7.1.1：
The Estimation of G(s) / 7.1.2：
A Lower Bound Sieve via Buchstab's Identity / 7.2：
Buchstab's Iterations / 7.2.1：
The Buchstab Transform of the $(lambda$)2 Method / 7.2.2：
The Sifting Limit as k -> $(infinity$) / 7.2.3：
Selberg's a2 a" Method / 7.3：
The Improved Sifting Limit for Large k / 7.3.1：
Notes on Chapter 7 / 7.4：
References
Index
Introduction
The Structure of Sifting Arguments / 1：
The Sieves of Eratosthenes and Legendre / 1.1：
2.

図書
図書
2. Nonlinear diffusion equations
Zhuoqun Wu ... [et al.]
出版情報: Singapore : World Scientific, c2001  xvii, 502 p. ; 23 cm
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Preface
Newtonian Filtration Equations / Chapter 1：
Introduction / 1.1：
Physical examples / 1.1.1：
Definitions of generalized solutions / 1.1.2：
Special solutions / 1.1.3：
Existence and Uniqueness of Solutions: One Dimensional Case / 1.2：
Uniqueness of solutions / 1.2.1：
Existence of solutions / 1.2.2：
Comparison theorems / 1.2.3：
Some extensions / 1.2.4：
Existence and Uniqueness of Solutions: Higher Dimensional Case / 1.3：
Comparison theorem and uniqueness of solutions / 1.3.1：
Regularity of Solutions: One Dimensional Case / 1.3.2：
Lemma / 1.4.1：
Regularity of solutions / 1.4.2：
Regularity of Solutions: Higher Dimensional Case / 1.4.3：
Generalized class B[subscript 2] / 1.5.1：
Some lemmas / 1.5.2：
Properties of functions in the generalized class B[subscript 2] / 1.5.3：
Holder continuity of solutions / 1.5.4：
Properties of the Free Boundary: One Dimensional Case / 1.6：
Finite propagation of disturbances / 1.6.1：
Localization and extinction of disturbances / 1.6.2：
Differential equation on the free boundary / 1.6.3：
Continuously differentiability of the free boundary / 1.6.4：
Some further results / 1.6.5：
Properties of the Free Boundary: Higher Dimensional Case / 1.7：
Monotonicity and Holder continuity of the free boundary / 1.7.1：
Lipschitz continuity of the free boundary / 1.7.2：
Initial Trace of Solutions / 1.7.3：
Harnack inequality / 1.8.1：
Main result / 1.8.2：
Extension of existence and uniqueness theorem / 1.8.3：
Other Problems / 1.9：
Equations with strongly nonlinear sources / 1.9.1：
Asymptotic properties of solutions / 1.9.2：
Non-Newtonian Filtration Equations / Chapter 2：
Introduction Preliminary Knowledge / 2.1：
Introduction Physical example / 2.1.1：
Basic spaces and some lemmas / 2.1.2：
Existence of Solutions / 2.1.3：
The case u[subscript 0] [set membership] C[superscript [infinity] subscript 0](R[superscript N]) or u[subscript 0] [set membership] L[superscript 1](R[superscript N]) [intersection of] L[superscript [infinity](R[superscript N]) / 2.2.1：
The case u[subscript 0] [set membership] L[superscript 1 subscript loc](R[superscript N]) / 2.2.2：
Some remarks / 2.2.3：
Harnack Inequality and the Initial Trace of Solutions / 2.3：
Local Harnack inequality / 2.3.1：
Global Harnack inequality / 2.3.2：
Initial trace of solutions / 2.3.3：
Regularity of Solutions / 2.4：
Boundedness of solutions / 2.4.1：
Boundedness of the gradient of solutions / 2.4.2：
Holder continuity of the gradient of solutions / 2.4.3：
Uniqueness of Solutions / 2.5：
Auxiliary propositions / 2.5.1：
Uniqueness theorem and its proof / 2.5.2：
Properties of the Free Boundary / 2.6：
p-Laplacian equation with strongly nonlinear sources / 2.6.1：
General Quasilinear Equations of Second Order / 2.7.2：
Weakly Degenerate Equations in One Dimension / 3.1：
Uniqueness of bounded and measurable solutions / 3.2.1：
Existence of continuous solutions / 3.2.2：
Weakly Degenerate Equations in Higher Dimension / 3.2.3：
Existence of continuous solutions for equations with two points of degeneracy / 3.3.1：
Uniqueness of BV solutions / 3.3.2：
Existence of BV solutions / 3.3.3：
Strongly Degenerate Equations in One Dimension / 3.3.4：
Definitions of solutions with discontinuity / 3.4.1：
Interior discontinuity condition / 3.4.2：
Uniqueness of BV solutions of the Cauchy problem / 3.4.3：
Formulation of the boundary value problem / 3.4.4：
Boundary discontinuity condition / 3.4.5：
Uniqueness of BV solutions of the first boundary value problem / 3.4.6：
Existence of BV solutions of the first boundary value problem / 3.4.7：
Equations with degeneracy at infinity / 3.4.8：
Properties of the curves of discontinuity / 3.4.10：
Degenerate Equations in Higher Dimension without Terms of Lower Order / 3.5：
Uniqueness of bounded and integrable solutions / 3.5.1：
A lemma on weak convergence / 3.5.2：
General Strongly Degenerate Equations in Higher Dimension / 3.5.3：
Appendix Classes BV and BV[subscript x] / 3.6.1：
Nonlinear Diffusion Equations of Higher Order / Chapter 4：
Similarity Solutions of a Fourth Order Equation / 4.1：
Definition of similarity solutions / 4.2.1：
Existence and uniqueness of global solutions of the Cauchy problem / 4.2.2：
Properties of solutions at zero points / 4.2.3：
Properties of unbounded solutions / 4.2.5：
Bounded solutions on the half line / 4.2.6：
Bounded solutions on the whole line / 4.2.7：
Properties of solutions in typical cases k = 1,2,3,4 / 4.2.8：
Behavior of similarity solutions as t [right arrow] 0[superscript +] / 4.2.9：
Equations with Double-Degeneracy / 4.3：
Weighted energy equality of solutions / 4.3.1：
Some auxiliary inequalities / 4.3.4：
Asymptotic behavior of solutions / 4.3.5：
Extinction of solutions at finite time / 4.3.7：
Nonexistence of nonnegative solutions / 4.3.8：
Infinite propagation case / 4.3.9：
Cahn-Hilliard Equation with Constant Mobility / 4.4：
Existence of classical solutions / 4.4.1：
Blowing-up of solutions / 4.4.2：
Global existence of solutions for small initial value / 4.4.3：
Cahn-Hilliard Equations with Positive Concentration Dependent Mobility / 4.5：
A modified Campanato space / 4.5.1：
Holder norm estimates for a linear problem / 4.5.2：
Zero potential case / 4.5.3：
General case / 4.5.4：
Thin Film Equation / 4.6：
Definition of generalized solutions / 4.6.1：
Approximate solutions / 4.6.2：
Nonnegativity of solutions / 4.6.3：
Zeros of nonnegative solutions / 4.6.5：
Monotonicity of the support of solutions / 4.6.6：
Cahn-Hilliard Equation with Degenerate Mobility / 4.7：
Models with degenerate mobility / 4.7.1：
Definition of physical solutions / 4.7.2：
Physical solutions / 4.7.3：
Bibliography
Preface
Newtonian Filtration Equations / Chapter 1：
Introduction / 1.1：
3.

図書
図書
3. Numerical bifurcation analysis for reaction-diffusion equations
Zhen Mei
出版情報: Berlin : Springer, c2000  xiv, 414 p. ; 24 cm
シリーズ名: Springer series in computational mathematics ; 28
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Reaction-Diffusion Equations / 1：
Introduction / 1.1：
Bifurcations and Pattern Formations / 1.2：
Boundary Conditions / 1.3：
Continuation Methods / 2：
Parameterization of Solution Curves / 2.1：
Natural parameterization / 2.1.1：
Parameterization with arclength / 2.1.2：
Parameterization with pseudo-arclength / 2.1.3：
Local Parameterization of Solution Manifolds / 2.2：
Predictor-Corrector Methods / 2.3：
Euler-Newton method / 2.3.1：
A continuation-Lanczos algorithm / 2.3.2：
A continuation-Arnoldi algorithm / 2.3.3：
Computation of Multi-Dimensional Solution Manifolds / 2.4：
Detecting and Computing Bifurcation Points / 3：
Generic Bifurcation Points / 3.1：
One-parameter problems / 3.1.1：
Two-parameter problems / 3.1.2：
Test Functions / 3.2：
Test functions for turning points / 3.2.1：
Test functions for simple bifurcation point / 3.2.2：
Test functions for Hopf bifurcations / 3.2.3：
Minimally extended systems / 3.2.4：
Computing Simple Bifurcation Points / 3.3：
Simple bifurcation points / 3.3.1：
Extended systems / 3.3.2：
Newton-like methods / 3.3.3：
Rank-1 corrections for sparse problems / 3.3.4：
A numerical example / 3.3.5：
Computing Hopf Bifurcation Points / 3.4：
Hopf points / 3.4.1：
Newton method for extended systems / 3.4.2：
Branch Switching at Simple Bifurcation Points / 4：
Structure of Bifurcating Solution Branches / 4.1：
Behavior of the Linearized Operator / 4.2：
Euler-Newton Continuation / 4.3：
Branch Switching via Regularized Systems / 4.4：
Other Branch Switching Techniques / 4.5：
Bifurcation Problems with Symmetry / 5：
Basic Group Concepts / 5.1：
Equivariant Bifurcation Problems / 5.2：
Equivariant Branching Lemma / 5.3：
A Semi-linear Elliptic PDE on the Unite Square / 5.4：
Liapunov-Schmidt Method / 6：
Liapunov-Schmidt Reduction / 6.1：
Equivariance of the Reduced Bifurcation Equations / 6.2：
Derivatives and Taylor Expansion / 6.3：
Equivalence, Determinacy and Stability / 6.4：
Simple Bifurcation Points / 6.5：
Truncated Liapunov-Schmidt Method / 6.6：
Branch Switching at Multiple Bifurcation Points / 6.7：
Branch switching with prescribed tangents / 6.7.1：
Branch switching with scaling techniques / 6.7.2：
Corank-2 Problems with Dm-symmetry / 6.8：
Semilinear elliptic PDEs on a square / 6.8.1：
A semilinear elliptic PDE on a hexagon / 6.8.2：
Center Manifold Theory / 7：
Center Manifolds and Their Properties / 7.1：
Approximation of Center Manifolds / 7.2：
Symmetry and Normal Form / 7.3：
Hopf bifurcations / 7.4.1：
Waves in Reaction-Diffusion Equations / 7.5：
Oscillating waves / 7.5.1：
Long waves / 7.5.2：
Long time and large spatial behavior / 7.5.3：
A Bifurcation Function for Homoclinic Orbits / 8：
A Bifurcation Function / 8.1：
Approximation of Homoclinic Orbits / 8.2：
Solving the Adjoint Variational Problem / 8.3：
Preserving the inner product / 8.3.1：
Systems with continuous symmetries / 8.3.2：
The Approximate Bifurcation Function / 8.4：
Examples / 8.5：
Freire et al.'s circuit / 8.5.1：
Kuramoto-Sivashinsky equation / 8.5.2：
One-Dimensional Reaction-Diffusion Equations / 9：
Linear Stability Analysis / 9.1：
The general system / 9.2.1：
The Brusselator equations / 9.2.2：
Solution Branches at Double Bifurcations / 9.3：
The reflection symmetry and its induced action / 9.3.1：
(k,m) = (odd, odd) or (odd, even) / 9.3.2：
(k,m) = (even, even) / 9.3.3：
Central Difference Approximations / 9.3.4：
General systems / 9.4.1：
Numerical Results for the Brusselator Equations / 9.4.2：
The length <$>\ell = 1<$>, diffusion rates d1 = 1, d2 = 2 / 9.5.1：
The length <$>\ell = 10<$>, diffusion rates d1 = 1, d2 = 2 / 9.5.2：
Reaction-Diffusion Equations on a Square / 10：
D4-Symmetry / 10.1：
Eigenpairs of the Laplacian / 10.2：
Bifurcation Points / 10.3：
Steady state bifurcation points / 10.4.1：
Hopf bifurcation points / 10.4.2：
Mode Interactions / 10.5：
Steady/steady state mode interactions / 10.5.1：
Hopf/steady state mode interactions / 10.5.2：
Hopf/Hopf mode interactions / 10.5.3：
Kernels of Du G0 and <$>(D_u G_0)^{\ast}<$> / 10.6：
Simple and Double Bifurcations / 10.7：
Simple bifurcations / 10.8.1：
Double bifurcations induced by the D4 symmetries / 10.8.2：
Normal Forms for Hopf Bifurcations / 11：
Domain Symmetries and Their Extensions / 11.1：
Actions of D4 on the Center Eigenspace / 11.3：
The Normal Form / 11.4：
Analysis of the Normal Form / 11.5：
Odd parity / 11.5.1：
Even parity / 11.5.2：
Brusselator Equations / 11.6：
Linear stability analysis / 11.6.1：
Bifurcation scenario / 11.6.2：
Nonlinear degeneracy / 11.6.3：
Steady/Steady State Mode Interactions / 12：
Induced Actions / 12.1：
Interaction of Two D4-Modes / 12.2：
Interaction of two even modes / 12.2.1：
Interaction of an even mode with an odd mode / 12.2.2：
Interaction of two odd modes / 12.2.3：
Mode Interactions of Three Modes / 12.3：
Induced actions / 12.3.1：
Interactions of the modes (m,n,k) =(even, odd, odd) / 12.3.2：
Interactions of the modes (m,n,k) =(even, odd, even) / 12.3.3：
Interactions of Four Modes / 12.4：
Interactions of the modes (m, n, k, l) = (even, odd, even, odd) / 12.4.1：
Interactions of the modes (m, n, k, l) = (even, even, even, odd) / 12.4.2：
Reactions with Z2-Symmetry / 12.5：
Hopf/Steady State Mode Interactions / 13：
Normal Forms / 13.1：
Bifurcation Scenario / 13.4：
Calculations of the Normal Form / 13.5：
Homotopy of Boundary Conditions / 14：
Homotopy of boundary conditions / 14.1：
Boundary conditions for different components / 14.1.2：
Mixed boundary conditions along the sides / 14.1.3：
Dynamical boundary conditions / 14.1.4：
A Brief Review of Sturm-Liouville Theory / 14.2：
Laplacian with Robin Boundary Conditions / 14.3：
Variational Form / 14.4：
Continuity of Solutions along the Homotopy / 14.5：
Neumann and Dirichlet Problems / 14.6：
Properties of Eigenvalues / 14.7：
One-dimensional problems / 14.7.1：
Two-dimensional problems / 14.7.2：
Bifurcations along a Homotopy of BCs / 15：
Stability and Symmetries / 15.1：
Variations of Bifurcations along the Homotopy / 15.3：
(κ1, κ2) = (odd, even) or (even, odd) / 15.4.1：
(κ1, κ2) = (odd, odd) / 15.4.2：
(κ1, κ2) = (even, even) / 15.4.3：
A Numerical Example / 15.5：
Discretization with finite difference methods / 15.5.1：
Homotopy of (κ1(μ), κ2(μ)) from (1,2) to (2,3) / 15.5.2：
Homotopy of (κl(μ), κ2(μ)) from (1,3) to (2,4) / 15.5.3：
Homotopy of (κ1(μ), κ2(μ)) from (2,4) to (3,5) / 15.5.4：
Forced Symmetry-Breaking in BCs / 15.6：
Bifurcation points / 15.6.1：
Bifurcation scenarios / 15.6.2：
A Mode Interaction on a Homotopy of BCs / 16：
Symmetries and Normal Forms / 16.1：
Generic Bifurcation Behavior / 16.3：
Solutions with the modes φ1, φ2 / 16.3.1：
Pure φ3-mode solutions / 16.3.2：
Interactions of three modes / 16.3.3：
Scales of Solution Branches / 16.4：
Secondary Bifurcations / 16.5：
Secondary Hopf bifurcations / 16.5.1：
Truncated Bifurcation Equations / 16.6：
Derivatives with respect to homotopy parameter / 16.6.1：
Reduced Stability / 16.7：
Stability of solution branches at (0, λ1(μ),μ) / 16.7.1：
Stability of solution branches at (0, λ2(μ), μ) / 16.7.2：
Stability of solution branches at mode interaction / 16.7.3：
Solution branches along (0; λ1(μ),μ) / 16.8：
Solution branches along (0, λ2(μ),μ) / 16.8.2：
Mode interaction / 16.8.3：
Switching and continuation of solution branches / 16.8.4：
List of Figures
List of Tables
Bibliography
Index
Reaction-Diffusion Equations / 1：
Introduction / 1.1：
Bifurcations and Pattern Formations / 1.2：
4.

図書
図書
4. Wave Motion (: hard ; : pbk.)
J. Billingham, A. C. King
出版情報: Cambridge : Cambridge University Press, 2000  ix, 468 p. ; 23 cm
シリーズ名: Cambridge texts in applied mathematics
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Introduction
Linear Waves / Part 1：
Basic Ideas / 1：
Exercises
Waves on a Stretched String / 2：
Derivation of the Governing Equation / 2.1：
Standing Waves on Strings of Finite Length / 2.2：
D'Alembert's Solution for Strings of Infinite Length / 2.3：
Reflection and Transmission of Waves by Discontinuities in Density / 2.4：
A Single Discontinuity / 2.4.1：
Two Discontinuities: Impedance Matching / 2.4.2：
Sound Waves / 3：
Plane Waves / 3.1：
Acoustic Energy Transmission / 3.3：
Plane Waves In Tubes / 3.4：
Acoustic Waveguides / 3.5：
Reflection of a Plane Acoustic Wave by a Rigid Wall / 3.5.1：
A Planar Waveguide / 3.5.2：
A Circular Waveguide / 3.5.3：
Acoustic Sources / 3.6：
The Acoustic Source / 3.6.1：
Energy Radiated by Sources and Plane Waves / 3.6.2：
Radiation from Sources in a Plane Wall / 3.7：
Linear Water Waves / 4：
Derivation of the Governing Equations / 4.1：
Linear Gravity Waves / 4.2：
Progressive Gravity Waves / 4.2.1：
Standing Gravity Waves / 4.2.2：
The Wavemaker / 4.2.3：
The Extraction of Energy from Water Waves / 4.2.4：
The Effect of Surface Tension: Capillary--Gravity Waves / 4.3：
Edge Waves / 4.4：
Ship Waves / 4.5：
The Solution of Initial Value Problems / 4.6：
Shallow Water Waves: Linear Theory / 4.7：
The Reflection of Sea Swell by a Step / 4.7.1：
Wave Amplification at a Gently Sloping Beach / 4.7.2：
Wave Refraction / 4.8：
The Kinematics of Slowly Varying Waves / 4.8.1：
Wave Refraction at a Gently Sloping Beach / 4.8.2：
The Effect of Viscosity / 4.9：
Waves in Elastic Solids / 5：
Waves in an Infinite Elastic Body / 5.1：
One-Dimensional Dilatation Waves / 5.2.1：
One-Dimensional Rotational Waves / 5.2.2：
Plane Waves with General Orientation / 5.2.3：
Two-Dimensional Waves in Semi-infinite Elastic Bodies / 5.3：
Normally Loaded Surface / 5.3.1：
Stress-Free Surface / 5.3.2：
Waves in Finite Elastic Bodies / 5.4：
Flexural Waves in Plates / 5.4.1：
Waves in Elastic Rods / 5.4.2：
Torsional Waves / 5.4.3：
Longitudinal Waves / 5.4.4：
The Excitation and Propagation of Elastic Wavefronts / 5.5：
Wavefronts Caused by an Internal Line Force in an Unbounded Elastic Body / 5.5.1：
Wavefronts Caused by a Point Force on the Free Surface of a Semi-infinite Elastic Body / 5.5.2：
Electromagnetic Waves / 6：
Electric and Magnetic Forces and Fields / 6.1：
Electrostatics: Gauss's Law / 6.2：
Magnetostatics: Ampere's Law and the Displacement Current / 6.3：
Electromagnetic Induction: Farady's Law / 6.4：
Plane Electromagnetic Waves / 6.5：
Conductors and Insulators / 6.6：
Reflection and Transmission at Interfaces / 6.7：
Boundary Conditions at Interfaces / 6.7.1：
Reflection by a Perfect Conductor / 6.7.2：
Reflection and Refraction by Insulators / 6.7.3：
Waveguides / 6.8：
Metal Waveguides / 6.8.1：
Weakly Guiding Optical Fibres / 6.8.2：
Radiation / 6.9：
Scalar and Vector Potentials / 6.9.1：
The Electric Dipole / 6.9.2：
The Far Field of a Localised Current Distribution / 6.9.3：
The Centre Fed Linear Antenna / 6.9.4：
Nonlinear Waves / Part 2：
The Formation and Propagation of Shock Waves / 7：
Traffic Waves / 7.1：
Small Amplitude Disturbances of a Uniform State / 7.1.1：
The Nonlinear Initial Value Problem / 7.1.3：
The Speed of the Shock / 7.1.4：
Compressible Gas Dynamics / 7.2：
Some Essential Thermodynamics / 7.2.1：
Equations of Motion / 7.2.2：
Construction of the Characteristic Curves / 7.2.3：
The Rankine--Hugoniot Relations / 7.2.4：
Detonations / 7.2.5：
Nonlinear Water Waves / 8：
Nonlinear Shallow Water Waves / 8.1：
The Dam Break Problem / 8.1.1：
A Shallow Water Bore / 8.1.2：
The Effect of Nonlinearity on Deep Water Gravity Waves: Stokes' Expansion / 8.2：
The Korteweg-de Vries Equation for Shallow Water Waves: the Interaction of Nonlinear Steepening and Linear Dispersion / 8.3：
Derivation of the Korteweg-de Vries Equation / 8.3.1：
Travelling Wave Solutions of the KdV Equation / 8.3.2：
Nonlinear Capillary Waves / 8.4：
Chemical and Electrochemical Waves / 9：
The Law of Mass Action / 9.1：
Molecular Diffusion / 9.2：
Reaction-Diffusion Systems / 9.3：
Autocatalytic Chemical Waves with Unequal Diffusion Coefficients* / 9.4：
Existence of Travelling Wave Solutions / 9.4.1：
Asymptotic Solution for [delta] [[ 1 / 9.4.2：
The Transmission of Nerve Impulses: the Fitzhugh-Nagumo Equations / 9.5：
The Fitzhugh-Nagumo Model / 9.5.1：
The Existence of a Threshold / 9.5.2：
Travelling Waves / 9.5.3：
Advanced Topics / Part 3：
Burgers' Equation: Competition between Wave Steepening and Wave Spreading / 10：
Burgers' Equation for Traffic Flow / 10.1：
The Effect of Dissipation on Weak Shock Waves in an Ideal Gas / 10.2：
Simple Solutions of Burgers' Equation / 10.3：
Asymptotic Solutions for v [[ 1 / 10.3.1：
Diffraction and Scattering / 11：
Diffraction of Acoustic Waves by a Semi-infinite Barrier / 11.1：
Preliminary Estimates of the Potential / 11.1.1：
Pre-transform Considerations / 11.1.2：
The Fourier Transform Solution / 11.1.3：
The Diffraction of Waves by an Aperture / 11.2：
Scalar Diffraction: Acoustic Waves / 11.2.1：
Vector Diffraction: Electromagnetic Waves / 11.2.2：
Scattering of Linear, Deep Water Waves by a Surface Piercing Cylinder / 11.3：
Solitons and the Inverse Scattering Transform / 12：
The Korteweg-de Vries Equation / 12.1：
The Scattering Problem / 12.1.1：
The Inverse Scattering Problem / 12.1.2：
Scattering Data for KdV Potentials / 12.1.3：
Examples: Solutions of the KdV Equation / 12.1.4：
The Nonlinear Schrodinger Equation / 12.2：
Derivation of the Nonlinear Schrodinger Equation for Plane Electromagnetic Waves / 12.2.1：
Solitary Wave Solutions of the Nonlinear Schrodinger Equation / 12.2.2：
The Inverse Scattering Transform for the Nonlinear Schrodinger Equation / 12.2.3：
Useful Mathematical Formulas and Physical Data / Appendix 1：
Cartesian Coordinates / A1.1：
Cylindrical Polar Coordinates / A1.2：
Spherical Polar Coordinates / A1.3：
Some Vector Calculus Identities and Useful Results for Smooth Vector Fields / A1.4：
Physical constants / A1.5：
Bibliography
Index
Introduction
Linear Waves / Part 1：
Basic Ideas / 1：
5.

図書
図書
5. 岩波講座基礎数学 (総索引 ; 月報 ; 月報(第3次))
小平邦彦監修 ; 岩堀長慶 [ほか] 編
出版情報: 東京 : 岩波書店, 1976.5-  冊 ; 22cm
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6.

図書
図書
6. Perturbation methods
Ali Hasan Nayfeh
出版情報: New York : Wiley, c1973  xii, 425 p. ; 23 cm
シリーズ名: Pure and applied mathematics
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目次情報: 続きを見る
Introduction / 1.：
Parameter Perturbations / 1.1.：
An Algebraic Equation / 1.1.1.：
The van der Pol Oscillator / 1.1.2.：
Coordinate Perturbations / 1.2.：
The Bessel Equation of Zeroth Order / 1.2.1.：
A Simple Example / 1.2.2.：
Order Symbols and Gauge Functions / 1.3.：
Asymptotic Expansions and Sequences / 1.4.：
Asymptotic Series / 1.4.1.：
Asymptotic Expansions / 1.4.2.：
Uniqueness of Asymptotic Expansions / 1.4.3.：
Convergent versus Asymptotic Series / 1.5.：
Nonuniform Expansions / 1.6.：
Elementary Operations on Asymptotic Expansions / 1.7.：
Exercises
Straightforward Expansions and Sources of Nonuniformity / 2.：
Infinite Domains / 2.1.：
The Duffing Equation / 2.1.1.：
A Model for Weak Nonlinear Instability / 2.1.2.：
Supersonic Flow Past a Thin Airfoil / 2.1.3.：
Small Reynolds Number Flow Past a Sphere / 2.1.4.：
A Small Parameter Multiplying the Highest Derivative / 2.2.：
A Second-Order Example / 2.2.1.：
High Reynolds Number Flow Past a Body / 2.2.2.：
Relaxation Oscillations / 2.2.3.：
Unsymmetrical Bending of Prestressed Annular Plates / 2.2.4.：
Type Change of a Partial Differential Equation / 2.3.：
Long Waves on Liquids Flowing down Incline Planes / 2.3.1.：
The Presence of Singularities / 2.4.：
Shift in Singularity / 2.4.1.：
The Earth-Moon-Spaceship Problem / 2.4.2.：
Thermoelastic Surface Waves / 2.4.3.：
Turning Point Problems / 2.4.4.：
The Role of Coordinate Systems / 2.5.：
The Method of Strained Coordinates / 3.：
The Method of Strained Parameters / 3.1.：
The Lindstedt-Poincare Method / 3.1.1.：
Transition Curves for the Mathieu Equation / 3.1.2.：
Characteristic Exponents for the Mathieu Equation (Whittaker's Method) / 3.1.3.：
The Stability of the Triangular Points in the Elliptic Restricted Problem of Three Bodies / 3.1.4.：
Characteristic Exponents for the Triangular Points in the Elliptic Restricted Problem of Three Bodies / 3.1.5.：
A Simple Linear Eigenvalue Problem / 3.1.6.：
A Quasi-Linear Eigenvalue Problem / 3.1.7.：
The Quasi-Linear Klein-Gordon Equation / 3.1.8.：
Lighthill's Technique / 3.2.：
A First-Order Differential Equation / 3.2.1.：
The One-Dimensional Earth-Moon-Spaceship Problem / 3.2.2.：
A Solid Cylinder Expanding Uniformly in Still Air / 3.2.3.：
Expansions by Using Exact Characteristics--Nonlinear Elastic Waves / 3.2.4.：
Temple's Technique / 3.3.：
Renormalization Technique / 3.4.：
Limitations of the Method of Strained Coordinates / 3.4.1.：
The Methods of Matched and Composite Asymptotic Expansions / 3.5.1.：
The Method of Matched Asymptotic Expansions / 4.1.：
Introduction--Prandtl's Technique / 4.1.1.：
Higher Approximations and Refined Matching Procedures / 4.1.2.：
A Second-Order Equation with Variable Coefficients / 4.1.3.：
Reynolds' Equation for a Slider Bearing / 4.1.4.：
The Method of Composite Expansions / 4.1.5.：
A Second-Order Equation with Constant Coefficients / 4.2.1.：
An Initial Value Problem for the Heat Equation / 4.2.2.：
Limitations of the Method of Composite Expansions / 4.2.4.：
Variation of Parameters and Methods of Averaging / 5.：
Variation of Parameters / 5.1.：
Time-Dependent Solutions of the Schrodinger Equation / 5.1.1.：
A Nonlinear Stability Example / 5.1.2.：
The Method of Averaging / 5.2.：
Van der Pol's Technique / 5.2.1.：
The Krylov-Bogoliubov Technique / 5.2.2.：
The Generalized Method of Averaging / 5.2.3.：
Struble's Technique / 5.3.：
The Krylov-Bogoliubov-Mitropolski Technique / 5.4.：
The Duffiing Equation / 5.4.1.：
The Klein-Gordon Equation / 5.4.2.：
The Method of Averaging by Using Canonical Variables / 5.5.：
The Mathieu Equation / 5.5.1.：
A Swinging Spring / 5.5.3.：
Von Zeipel's Procedure / 5.6.：
Averaging by Using the Lie Series and Transforms / 5.6.1.：
The Lie Series and Transforms / 5.7.1.：
Generalized Algorithms / 5.7.2.：
Simplified General Algorithms / 5.7.3.：
A Procedure Outline / 5.7.4.：
Algorithms for Canonical Systems / 5.7.5.：
Averaging by Using Lagrangians / 5.8.：
A Model for Dispersive Waves / 5.8.1.：
A Model for Wave-Wave Interaction / 5.8.2.：
The Nonlinear Klein-Gordon Equation / 5.8.3.：
The Method of Multiple Scales / 6.：
Description of the Method / 6.1.：
Many-Variable Version (The Derivative-Expansion Procedure) / 6.1.1.：
The Two-Variable Expansion Procedure / 6.1.2.：
Generalized Method--Nonlinear Scales / 6.1.3.：
Applications of the Derivative-Expansion Method / 6.2.：
Forced Oscillations of the van der Pol Equation / 6.2.1.：
Parametric Resonances--The Mathieu Equation / 6.2.4.：
The van der Pol Oscillator with Delayed Amplitude Limiting / 6.2.5.：
Limitations of the Derivative-Expansion Method / 6.2.6.：
Limitations of This Technique / 6.3.：
Generalized Method / 6.4.：
A General Second-Order Equation with Variable Coefficients / 6.4.1.：
A Linear Oscillator with a Slowly Varying Restoring Force / 6.4.3.：
An Example with a Turning Point / 6.4.4.：
The Duffing Equation with Slowly Varying Coefficients / 6.4.5.：
Reentry Dynamics / 6.4.6.：
Advantages and Limitations of the Generalized Method / 6.4.7.：
Asymptotic Solutions of Linear Equations / 7.：
Second-Order Differential Equations / 7.1.：
Expansions Near an Irregular Singularity / 7.1.1.：
An Expansion of the Zeroth-Order Bessel Function for Large Argument / 7.1.2.：
Liouville's Problem / 7.1.3.：
Higher Approximations for Equations Containing a Large Parameter / 7.1.4.：
Homogeneous Problems with Slowly Varying Coefficients / 7.1.5.：
Reentry Missile Dynamics / 7.1.7.：
Inhomogeneous Problems with Slowly Varying Coefficients / 7.1.8.：
Successive Liouville-Green (WKB) Approximations / 7.1.9.：
Systems of First-Order Ordinary Equations / 7.2.：
Expansions Near an Irregular Singular Point / 7.2.1.：
Asymptotic Partitioning of Systems of Equations / 7.2.2.：
Subnormal Solutions / 7.2.3.：
Systems Containing a Parameter / 7.2.4.：
Homogeneous Systems with Slowly Varying Coefficients / 7.2.5.：
The Langer Transformation / 7.3.：
Problems with Two Turning Points / 7.3.3.：
Higher-Order Turning Point Problems / 7.3.4.：
Higher Approximations / 7.3.5.：
An Inhomogeneous Problem with a Simple Turning Point--First Approximation / 7.3.6.：
An Inhomogeneous Problem with a Simple Turning Point--Higher Approximations / 7.3.7.：
An Inhomogeneous Problem with a Second-Order Turning Point / 7.3.8.：
Turning Point Problems about Singularities / 7.3.9.：
Turning Point Problems of Higher Order / 7.3.10.：
Wave Equations / 7.4.：
The Born or Neumann Expansion and The Feynman Diagrams / 7.4.1.：
Renormalization Techniques / 7.4.2.：
Rytov's Method / 7.4.3.：
A Geometrical Optics Approximation / 7.4.4.：
A Uniform Expansion at a Caustic / 7.4.5.：
The Method of Smoothing / 7.4.6.：
References and Author Index
Subject Index
Introduction / 1.：
Parameter Perturbations / 1.1.：
An Algebraic Equation / 1.1.1.：
7.

図書
図書
7. Trigonometric series. 2nd ed (v. 1 ; v. 2)
A. Zygmund
出版情報: Cambridge : Cambridge University Press, 1959  2 v. (ix, 383, 354 p) ; 27 cm
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8.

図書
図書
8. Automorphic forms on SL[2](R)
Armand Borel
出版情報: Cambridge, U.K. ; New York, NY, USA : Cambridge University Press, 1997  x, 192 p. ; 24 cm
シリーズ名: Cambridge tracts in mathematics ; 130
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目次情報: 続きを見る
Basic Material On SL2(R), Discrete Subgroups and the Upper-Half Plane / Part I：
Prerequisites and notation / 1：
Review of SL2(R), differential operators, convolution / 2：
Action of G on X, discrete subgroups of G, fundamental domains / 3：
The unit disc model / 4：
Automorphic Forms and Cusp Forms / Part II：
Growth conditions, automorphic forms / 5：
Poincare series / 6：
Constant term:the fundamental estimate / 7：
Finite dimensionality of the space of automorphic forms of a given type / 8：
Convolution operators on cuspidal functions / 9：
Eisenstein Series / Part III：
Definition and convergence of Eisenstein series / 10：
Analytic continuation of the Eisenstein series / 11：
Eisenstein series and automorphic forms orthogonal to cusp forms / 12：
Spectral Decomposition and Representations / Part IV：
Spectral decomposition of L2(G G)m with respect to C / 13：
Generalities on representations of G / 14：
Representations of SL2(R) / 15：
Spectral decomposition of L2(G SL2(R)):the discrete spectrum / 16：
Spectral decomposition of L2(G SL2(R)): the continuous spectrum / 17：
Concluding remarks / 18：
Basic Material On SL2(R), Discrete Subgroups and the Upper-Half Plane / Part I：
Prerequisites and notation / 1：
Review of SL2(R), differential operators, convolution / 2：
9.

図書
図書
9. Geometrical methods of mathematical physics (: pbk)
Bernard F. Schutz
出版情報: Cambridge ; New York : Cambridge University Press, 1980  xii, 250 p. ; 24 cm
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目次情報: 続きを見る
Preface
Some basic mathematics / 1：
The space R[superscript n] and its topology / 1.1：
Mappings / 1.2：
Real analysis / 1.3：
Group theory / 1.4：
Linear algebra / 1.5：
The algebra of square matrices / 1.6：
Bibliography / 1.7：
Differentiable manifolds and tensors / 2：
Definition of a manifold / 2.1：
The sphere as a manifold / 2.2：
Other examples of manifolds / 2.3：
Global considerations / 2.4：
Curves / 2.5：
Functions on M / 2.6：
Vectors and vector fields / 2.7：
Basis vectors and basis vector fields / 2.8：
Fiber bundles / 2.9：
Examples of fiber bundles / 2.10：
A deeper look at fiber bundles / 2.11：
Vector fields and integral curves / 2.12：
Exponentiation of the operator d/d[lambda] / 2.13：
Lie brackets and noncoordinate bases / 2.14：
When is a basis a coordinate basis? / 2.15：
One-forms / 2.16：
Examples of one-forms / 2.17：
The Dirac delta function / 2.18：
The gradient and the pictorial representation of a one-form / 2.19：
Basis one-forms and components of one-forms / 2.20：
Index notation / 2.21：
Tensors and tensor fields / 2.22：
Examples of tensors / 2.23：
Components of tensors and the outer product / 2.24：
Contraction / 2.25：
Basis transformations / 2.26：
Tensor operations on components / 2.27：
Functions and scalars / 2.28：
The metric tensor on a vector space / 2.29：
The metric tensor field on a manifold / 2.30：
Special relativity / 2.31：
Lie derivatives and Lie groups / 2.32：
Introduction: how a vector field maps a manifold into itself / 3.1：
Lie dragging a function / 3.2：
Lie dragging a vector field / 3.3：
Lie derivatives / 3.4：
Lie derivative of a one-form / 3.5：
Submanifolds / 3.6：
Frobenius' theorem (vector field version) / 3.7：
Proof of Frobenius' theorem / 3.8：
An example: the generators of S[superscript 2] / 3.9：
Invariance / 3.10：
Killing vector fields / 3.11：
Killing vectors and conserved quantities in particle dynamics / 3.12：
Axial symmetry / 3.13：
Abstract Lie groups / 3.14：
Examples of Lie groups / 3.15：
Lie algebras and their groups / 3.16：
Realizations and representations / 3.17：
Spherical symmetry, spherical harmonics and representations of the rotation group / 3.18：
Differential forms / 3.19：
The algebra and integral calculus of forms / A：
Definition of volume -- the geometrical role of differential forms / 4.1：
Notation and definitions for antisy mmetric tensors / 4.2：
Manipulating differential forms / 4.3：
Restriction of forms / 4.5：
Fields of forms / 4.6：
Handedness and orientability / 4.7：
Volumes and integration on oriented manifolds / 4.8：
N-vectors, duals, and the symbol [epsilon][subscript ij...k] / 4.9：
Tensor densities / 4.10：
Generalized Kronecker deltas / 4.11：
Determinants and [epsilon][subscript ij...k] / 4.12：
Metric volume elements / 4.13：
The differential calculus of forms and its applications / B：
The exterior derivative / 4.14：
Notation for derivatives / 4.15：
Familiar examples of exterior differentiation / 4.16：
Integrability conditions for partial differential equations / 4.17：
Exact forms / 4.18：
Proof of the local exactness of closed forms / 4.19：
Lie derivatives of forms / 4.20：
Lie derivatives and exterior derivatives commute / 4.21：
Stokes' theorem / 4.22：
Gauss' theorem and the definition of divergence / 4.23：
A glance at cohomology theory / 4.24：
Differential forms and differential equations / 4.25：
Frobenius' theorem (differential forms version) / 4.26：
Proof of the equivalence of the two versions of Frobenius' theorem / 4.27：
Conservation laws / 4.28：
Vector spherical harmonics / 4.29：
Applications in physics / 4.30：
Thermodynamics
Simple systems / 5.1：
Maxwell and other mathematical identities / 5.2：
Composite thermodynamic systems: Caratheodory's theorem / 5.3：
Hamiltonian mechanics
Hamiltonian vector fields / 5.4：
Canonical transformations / 5.5：
Map between vectors and one-forms provided by [characters not reproducible] / 5.6：
Poisson bracket / 5.7：
Many-particle systems: symplectic forms / 5.8：
Linear dynamical systems: the symplectic inner product and conserved quantities / 5.9：
Fiber bundle structure of the Hamiltonian equations / 5.10：
Electromagnetism / C：
Rewriting Maxwell's equations using differential forms / 5.11：
Charge and topology / 5.12：
The vector potential / 5.13：
Plane waves: a simple example / 5.14：
Dynamics of a perfect fluid / D：
Role of Lie derivatives / 5.15：
The comoving time-derivative / 5.16：
Equation of motion / 5.17：
Conservation of vorticity / 5.18：
Cosmology / E：
The cosmological principle / 5.19：
Lie algebra of maximal symmetry / 5.20：
The metric of a spherically symmetric three-space / 5.21：
Construction of the six Killing vectors / 5.22：
Open, closed, and flat universes / 5.23：
Connections for Riemannian manifolds and gauge theories / 5.24：
Introduction / 6.1：
Parallelism on curved surfaces / 6.2：
The covariant derivative / 6.3：
Components: covariant derivatives of the basis / 6.4：
Torsion / 6.5：
Geodesics / 6.6：
Normal coordinates / 6.7：
Riemann tensor / 6.8：
Geometric interpretation of the Riemann tensor / 6.9：
Flat spaces / 6.10：
Compatibility of the connection with volume-measure or the metric / 6.11：
Metric connections / 6.12：
The affine connection and the equivalence principle / 6.13：
Connections and gauge theories: the example of electromagnetism / 6.14：
Solutions and hints for selected exercises / 6.15：
Notation
Index
Appendix
Preface
Some basic mathematics / 1：
The space R[superscript n] and its topology / 1.1：
10.

図書
図書
10. Algebraic curves over finite fields
Carlos Moreno
出版情報: Cambridge [England] ; New York : Cambridge University Press, 1991  ix, 246 p. ; 24 cm
シリーズ名: Cambridge tracts in mathematics ; 97
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目次情報: 続きを見る
Preface
Algebraic curves and function fields / 1：
Geometric aspects / 1.1：
Introduction / 1.1.1：
Affine varieties / 1.1.2：
Projective varieties / 1.1.3：
Morphisms / 1.1.4：
Rational maps / 1.1.5：
Non-singular varieties / 1.1.6：
Smooth models of algebraic curves / 1.1.7：
Algebraic aspects / 1.2：
Points on the projective line P[superscript 1] / 1.2.1：
Extensions of valuation rings / 1.2.3：
Points on a smooth curve / 1.2.4：
Independence of valuations / 1.2.5：
Exercises
Notes
The Riemann-Roch theorem / 2：
Divisors / 2.1：
The vector space L(D) / 2.2：
Principal divisors and the group of divisor classes / 2.3：
The Riemann theorem / 2.4：
Pre-adeles (repartitions) / 2.5：
Pseudo-differentials (the Riemann-Roch theorem) / 2.6：
Zeta functions / 3：
The zeta functions of curves / 3.1：
The functional equation / 3.3：
Consequences of the functional equation / 3.3.1：
The Riemann hypothesis / 3.4：
The L-functions of curves and their functional equations / 3.5：
Preliminary remarks and notation / 3.5.1：
Exponential sums / 3.5.2：
The zeta function of the projective line / 4.1：
Gauss sums: first example of an L-function for the projective line / 4.2：
Properties of Gauss sums / 4.3：
Cyclotomic extensions: basic facts / 4.3.0：
Elementary properties / 4.3.1：
The Hasse-Davenport relation / 4.3.2：
Stickelberger's theorem / 4.3.3：
Kloosterman sums / 4.4：
Second example of an L-function for the projective line / 4.4.1：
A Hasse-Davenport relation for Kloosterman sums / 4.4.2：
Third example of an L-function for the projective line / 4.5：
Basic arithmetic theory of exponential sums / 4.6：
Part I: L-functions for the projective line / 4.6.1：
Part II: Artin-Schreier coverings / 4.6.2：
The Hurwitz-Zeuthen formula for the covering [pi]: C [right arrow] C / 4.6.3：
Goppa codes and modular curves / 5：
Elementary Goppa codes / 5.1：
The affine and projective lines / 5.2：
Affine line A[superscript 1](k) / 5.2.1：
Projective line P[superscript 1] / 5.2.2：
Goppa codes on the projective line / 5.3：
Algebraic curves / 5.4：
Separable extensions / 5.4.1：
Closed points and their neighborhoods / 5.4.2：
Differentials / 5.4.3：
The theorems of Riemann-Roch, of Hurwitz and of the Residue / 5.4.4：
Linear series / 5.4.6：
Algebraic geometric codes / 5.5：
Algebraic Goppa codes / 5.5.1：
Codes with better rates than the Varshamov-Gilbert bound / 5.5.2：
The theorem of Tsfasman, Vladut and Zink / 5.6：
Modular curves / 5.6.1：
Elliptic curves over C / 5.6.2：
Elliptic curves over the fields F[subscript p], Q / 5.6.3：
Torsion points on elliptic curves / 5.6.4：
Igusa's theorem / 5.6.5：
The modular equation / 5.6.6：
The congruence formula / 5.6.7：
The Eichler-Selberg trace formula / 5.6.8：
Proof of the theorem of Tsfasman, Vladut and Zink / 5.6.9：
Examples of algebraic Goppa codes / 5.7：
The Hamming (7,4) code / 5.7.1：
BCH codes / 5.7.2：
The Fermat cubic (Hermite form) / 5.7.3：
Elliptic codes (according to Driencourt-Michon) / 5.7.4：
The Klein quartic / 5.7.5：
Simplification of the singularities of algebraic curves / Appendix：
Homogeneous coordinates in the plane / A.1：
Basic lemmas / A.2：
Dual curves / A.3：
Plucker formulas / A.3.1：
Quadratic transformations / A.4：
Quadratic transform of a plane curve / A.4.1：
Quadratic transform of a singularity / A.4.2：
Singularities off the exceptional lines / A.4.3：
Reduction of singularities / A.4.4：
Bibliography
Index
Preface
Algebraic curves and function fields / 1：
Geometric aspects / 1.1：
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