作者：(罗)伊凡·辛格(Ivan Singer
页数：15,519页
出版社：哈尔滨工业大学出版社
出版日期：2020
ISBN：9787560391557
电子书格式：pdf/epub/txt
内容简介
《抽象凸分析（英文）》主要包括从凸分析到抽象凸分析、一个完整格的元素的抽象凸性、集合子集的抽象凸性、集上函数的抽象凸性、完全晶格之间的对偶性、晶格族之间的对偶、函数集合之间的对偶性、抽象的次微分等内容，也包含了关于当代抽象凸分析非常先进且详尽的考查。
《抽象凸分析（英文）》致力于研究通过在一个有序的空间中取得上确界（或下确界）元素族的操作来表示复杂的对象。
在《抽象凸分析（英文）》中，读者可以找到对抽象凸性的几种方法的介绍和它们之间的比较。
《抽象凸分析（英文）》适合对抽象凸分析感兴趣的数学专业学生及教师参考阅读。
目录
Foreword
Preface
Introduction: From Convex Analysis to Abstract Convex Analysis
0.1 Abstract Convexity of Sets
0.1a Inner Approaches
0.1b Intersectional and Separational Approaches
0.1c Approaches via Convexity Systems and Hull Operators
0.2 Abstract Convexity of Functions
0.3 Abstract Convexity of Elements of Complete Lattices
0.4 Abstract Quasi-Convexity of Functions
0.5 Dualities
0,6 Abstract Conjugations
0.7 Abstract Subdifferentials
0.8 Some Applications of Abstract Convex Analysis to Optimization
Theory
0.Sa Applications to Abstract Lagrangian Duality
0.8b Applications to Abstract Surrogate Duality
Chapter One Abstract Convexity of Elements of a Complete Lattice
1.1 The Main (Supremal) Approach: M-Convexity of Elements of a
Complete Lattice E, Where M c E
1.2 lnfimal and Supremal Generators and M-Convexity
1.3 An Equivalent Approach: Convexity Systems
1.4 Another Equivalent Approach: Convexity with Respect to a Hull
Operator
Chapter Two Abstract Convexity of Subsets of a Set
2.1 M-Convexity of Subsets of a Set X, Where M c 2x
2.2 Some Particular Cases
2.2a Convex Subsets of a Linear Space X
2.2b Closed Convex Subsets of a Locally Convex Space X
2.2c Evenly Convex Subsets of a Locally Convex Space X
2.2d Closed Affine Subsets of a Locally Convex Space X
2.2e Evenly Coaffine Subsets of a Locally Convex Space X
2.2f Spherically Convex Subsets of a Metric Space X
2.2g Closed Subsets of a Topological Space X
2.2h Order Ideals and Order Convex Subsets of a Poset X
2.2i Parametrizations of Families □(数理化公式) Where X
Is a Set
2.3 An Equivalent Approach, via Separation by Functions:
W-Convexity of Subsets of a Set X, Where □(数理化公式)
2.4 A Particular Case: Closed Convex Sets Revisited
2.5 Other Concepts of Convexity of Subsets of a Set X, with Respect
to a Set of Functions □(数理化公式)
2.6 (W, □(数理化公式))-Convexity of Subsets of a Set X, Where W Is a Set and
□(数理化公式)R Is a Coupling Function
Chapter Three Abstract Convexity of Functions on a Set
3.1 W-Convexity of Functions on a Set X, Where □(数理化公式)
3.2 Some Particular Cases
3.2a C(X最 + R), Where X Is a Locally Convex Space
3.2b C(X最), Where X Is a Locally Convex Space
3.2c The Case Where X = {0, 1}n and W□(数理化公式)
3.2d The Case Where X = {0, 1}n and W □(数理化公式)
3.2e ot-Ho1der Continuous Functions with Constant N,
Where0 □(数理化公式)
3.2f Suprema of Ho1der Continuous Functions, Where
□(数理化公式)
3.2g The Case Where □(数理化公式)
3.3 (W, →o)-ConvexityofFunctions on a Set X, Where W Is a Set and
□(数理化公式)R Is a Coupling Function
Chapter Four Abstract Quasi-Convexity of Functions on a Set
4.1 M-Quasi-Convexity of Functions on a Set X, Where □(数理化公式)
4.2 Some Particular Cases
4.2a Quasi-Convex Functions on a Linear Space X
4.2b Lower Semicontinuous Quasi-Convex Functions on a
Locally Convex Space X
4.2c Evenly Quasi-Convex Functions on a Locally Convex
Space X
4.2d Evenly Quasi-Coaffine Functions on a Locally Convex
Space X
4.2e Lower Semicontinuous Functions on a Topological
Space X
4.2f Nondecreasing Functions on a Poset X
4.3 An Equivalent Approach: W-Quasi-Convexity of Functions on a
□(数理化公式)
4.4 Relations Between W-Convexity and W-Quasi-Convexity of
Functions on a Set X, Where W □(数理化公式)
4.5 Some Particular Cases
4.5a Lower Semicontinuous Quasi-Convex Functions
Revisited
4.5b Evenly Quasi-Convex Functions Revisited
4.5c Evenly Quasi-Coaffine Functions Revisited
4.6 (W, →0)-Quasi-Convexity of Functions on a Set X, Where W Is a
Set and □(数理化公式) : X x W → R Is a Coupling Function
4.7 Other Equivalent Approaches: Quasi-Convexity of Functions on
a Set X, with Respect to Convexity Systems/3 c 2x and Hull
Operators u : 2x → 2x
4.8 Some Characterizations of Quasi-Convex Hull Operators
among Hull Operators on □(数理化公式)
Chapter Five Dualities Between Complete Lattices
5.1 Dualities and lnfimal Generators
5.2 Duals of Dualities
5.3 Relations Between Dualities and M-Convex Hulls
5.4 Partial Order and Lattice Operations for Dualities
Chapter Six Dualities Between Families of Subsets
6.1 DualitiesA :2x → 2w, Where X and W Are Two Sets
6.2 Some Particular Cases
6.2a Some Minkowski-Type Dualities
6.2b Some Dualities Obtained from the Minkowski-Type
Dualities AM, by Parametrizing the Family M
6.3 Representations of Dualities A : 2x → 2w with the Aid of
Subsets □(数理化公式) of X → W and Coupling Functions □(数理化公式) : X → W →
6.4 Some Particular Cases
6.4a Representations with the Aid of Subsets f2 of X X W
6.4b Representations with the Aid of Coupling Functions
□(数理化公式)
Chapter Seven Dualities Between Sets of Functions
7.1 Dualities A □(数理化公式) Where X and W Are Two Sets
7.2 Representations of Dualities A : Ax → F, Where X Is a Set
and □(数理化公式) and F Are Complete Lattices
7.3 Dualities A : Ax → Bw, Where X Is a Set and (A,
Preface
Introduction: From Convex Analysis to Abstract Convex Analysis
0.1 Abstract Convexity of Sets
0.1a Inner Approaches
0.1b Intersectional and Separational Approaches
0.1c Approaches via Convexity Systems and Hull Operators
0.2 Abstract Convexity of Functions
0.3 Abstract Convexity of Elements of Complete Lattices
0.4 Abstract Quasi-Convexity of Functions
0.5 Dualities
0,6 Abstract Conjugations
0.7 Abstract Subdifferentials
0.8 Some Applications of Abstract Convex Analysis to Optimization
Theory
0.Sa Applications to Abstract Lagrangian Duality
0.8b Applications to Abstract Surrogate Duality
Chapter One Abstract Convexity of Elements of a Complete Lattice
1.1 The Main (Supremal) Approach: M-Convexity of Elements of a
Complete Lattice E, Where M c E
1.2 lnfimal and Supremal Generators and M-Convexity
1.3 An Equivalent Approach: Convexity Systems
1.4 Another Equivalent Approach: Convexity with Respect to a Hull
Operator
Chapter Two Abstract Convexity of Subsets of a Set
2.1 M-Convexity of Subsets of a Set X, Where M c 2x
2.2 Some Particular Cases
2.2a Convex Subsets of a Linear Space X
2.2b Closed Convex Subsets of a Locally Convex Space X
2.2c Evenly Convex Subsets of a Locally Convex Space X
2.2d Closed Affine Subsets of a Locally Convex Space X
2.2e Evenly Coaffine Subsets of a Locally Convex Space X
2.2f Spherically Convex Subsets of a Metric Space X
2.2g Closed Subsets of a Topological Space X
2.2h Order Ideals and Order Convex Subsets of a Poset X
2.2i Parametrizations of Families □(数理化公式) Where X
Is a Set
2.3 An Equivalent Approach, via Separation by Functions:
W-Convexity of Subsets of a Set X, Where □(数理化公式)
2.4 A Particular Case: Closed Convex Sets Revisited
2.5 Other Concepts of Convexity of Subsets of a Set X, with Respect
to a Set of Functions □(数理化公式)
2.6 (W, □(数理化公式))-Convexity of Subsets of a Set X, Where W Is a Set and
□(数理化公式)R Is a Coupling Function
Chapter Three Abstract Convexity of Functions on a Set
3.1 W-Convexity of Functions on a Set X, Where □(数理化公式)
3.2 Some Particular Cases
3.2a C(X最 + R), Where X Is a Locally Convex Space
3.2b C(X最), Where X Is a Locally Convex Space
3.2c The Case Where X = {0, 1}n and W□(数理化公式)
3.2d The Case Where X = {0, 1}n and W □(数理化公式)
3.2e ot-Ho1der Continuous Functions with Constant N,
Where0 □(数理化公式)
3.2f Suprema of Ho1der Continuous Functions, Where
□(数理化公式)
3.2g The Case Where □(数理化公式)
3.3 (W, →o)-ConvexityofFunctions on a Set X, Where W Is a Set and
□(数理化公式)R Is a Coupling Function
Chapter Four Abstract Quasi-Convexity of Functions on a Set
4.1 M-Quasi-Convexity of Functions on a Set X, Where □(数理化公式)
4.2 Some Particular Cases
4.2a Quasi-Convex Functions on a Linear Space X
4.2b Lower Semicontinuous Quasi-Convex Functions on a
Locally Convex Space X
4.2c Evenly Quasi-Convex Functions on a Locally Convex
Space X
4.2d Evenly Quasi-Coaffine Functions on a Locally Convex
Space X
4.2e Lower Semicontinuous Functions on a Topological
Space X
4.2f Nondecreasing Functions on a Poset X
4.3 An Equivalent Approach: W-Quasi-Convexity of Functions on a
□(数理化公式)
4.4 Relations Between W-Convexity and W-Quasi-Convexity of
Functions on a Set X, Where W □(数理化公式)
4.5 Some Particular Cases
4.5a Lower Semicontinuous Quasi-Convex Functions
Revisited
4.5b Evenly Quasi-Convex Functions Revisited
4.5c Evenly Quasi-Coaffine Functions Revisited
4.6 (W, →0)-Quasi-Convexity of Functions on a Set X, Where W Is a
Set and □(数理化公式) : X x W → R Is a Coupling Function
4.7 Other Equivalent Approaches: Quasi-Convexity of Functions on
a Set X, with Respect to Convexity Systems/3 c 2x and Hull
Operators u : 2x → 2x
4.8 Some Characterizations of Quasi-Convex Hull Operators
among Hull Operators on □(数理化公式)
Chapter Five Dualities Between Complete Lattices
5.1 Dualities and lnfimal Generators
5.2 Duals of Dualities
5.3 Relations Between Dualities and M-Convex Hulls
5.4 Partial Order and Lattice Operations for Dualities
Chapter Six Dualities Between Families of Subsets
6.1 DualitiesA :2x → 2w, Where X and W Are Two Sets
6.2 Some Particular Cases
6.2a Some Minkowski-Type Dualities
6.2b Some Dualities Obtained from the Minkowski-Type
Dualities AM, by Parametrizing the Family M
6.3 Representations of Dualities A : 2x → 2w with the Aid of
Subsets □(数理化公式) of X → W and Coupling Functions □(数理化公式) : X → W →
6.4 Some Particular Cases
6.4a Representations with the Aid of Subsets f2 of X X W
6.4b Representations with the Aid of Coupling Functions
□(数理化公式)
Chapter Seven Dualities Between Sets of Functions
7.1 Dualities A □(数理化公式) Where X and W Are Two Sets
7.2 Representations of Dualities A : Ax → F, Where X Is a Set
and □(数理化公式) and F Are Complete Lattices
7.3 Dualities A : Ax → Bw, Where X Is a Set and (A,
