積分幾何與幾何概率（英文版） [Integral Geometry and Geometric Probability] pdf epub mobi txt 電子書 下載 2025

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[阿根廷] 路易斯桑塔洛 著

圖書標籤:

• 積分幾何
• 幾何概率
• 隨機幾何
• 數學
• 概率論
• 幾何學
• Stochastic Geometry
• Integral Geometry
• Geometric Probability
• Measure Theory

齣版社： 世界圖書齣版公司

ISBN：9787510004933

版次：1

商品編碼：10515943

包裝：平裝

外文名稱：Integral Geometry and Geometric Probability

開本：24開

齣版時間：2009-05-01

用紙：膠版紙

頁數：404

正文語種：英文

內容簡介

Though its title "Integral Geometry" may appear somewhat unusual in thiscontext it is nevertheless quite appropriate, for Integral Geometry is anoutgrowth of what in the olden days was referred to as "geometric probabil-ities."
Originating, as legend has it, with the Buffon needle problem （which afternearly two centuries has lost little of its elegance and appeal）, geometricprobabilities have run into difficulties culminating in the paradoxes ofBertrand which threatened the fledgling field with banishment from the homeof Mathematics. In rescuing it from this fate, Poincar6 made the suggestionthat the arbitrariness of definition underlying the paradoxes could be removedby tying closer the definition of probability with a geometric group of which itwould have to be an invariant.

內頁插圖

目錄

Editors Statement
Foreword
Preface
Chapter 1. Convex Sets in the Plane
1. Introduction
2. Envelope of a Family of Lines
3. Mixed Areas of Minkowski
4. Some Special Convex Sets
5. Surface Area of the Unit Sphere and Volume of the Unit Ball
6. Notes and Exercises

Chapter 2. Sets of Points and Poisson Processes in the Plane
1. Density for Sets of Points
2. First Integral Formulas
3. Sets of Triples of Points
4. Homogeneous Planar Poisson Point Processes
5. Notes

Chapter 3. Sets of Lines in the Plane
1. Density for Sets of Lines
2. Lines That Intersect a Convex Set or a Curve
3. Lines That Cut or Separate Two Convex Sets
4. Geometric Applications
5. Notes and Exercises
Chapter 4. Pairs of Points and Pairs of Lines
1. Density for Pairs of Points
2. Integrals for the Power of the Chords of a Convex Set.
3. Density for Pairs of Lines
4. Division of the Plane by Random Lines
5. Notes

Chapter 5. Sets of Strips in the Plane
1. Density for Sets of Strips
2. Buffons Needle Problem
3. Sets of Points, Lines, and Strips
4. Some Mean Values
5. Notes

Chapter 6. The Group of Motions in the Plane： Kinematic Density .
1. The Group of Motions in the Plane
2. The Differential Forms on 9Jl
3. The Kinematic Density
4. Sets of Segments
5. Convex Sets That Intersect Another Convex Set
6. Some Integral Formulas
7. A Mean Value; Coverage Problems
8. Notes and Exercises

Chapter 7. Fundamental Formulas of Poinear~ and Blaschke
1. A New Expression for the Kinematic Density
2. Poincar6s Formula
3. Total Curvature of a Closed Curve and of a Plane Domain
4. Fundamental Formula of Blaschke
5. The lsoperimetric Inequality .
6. Hadwigers Conditions for a Domain to Be Able to Contain Another
7. Notes

Chapter 8. Lattices of Figures
1. Definitions and Fundamental Formula
2. Lattices of Domains
3. Lattices of Curves
4. Lattices Of Points
5. Notes and Exercise

Chapter 9. Differential Forms and Lie Groups
1. Differential Forms
2. Pfaffian Differential Systems
3. Mappings of Differentiable Manifolds
4. Lie Groups; Left and Right Translations
5. Left-lnvariant Differential Forms
6. Maurer-Cartan Equations
7. lnvariant Volume Elements of a Group： Unimodular Groups
8. Notes and Exercises

Chapter 10. Density and Measure in Homogeneous Spaces
1. Introduction
2. invariant Subgroups and Quotient Groups
3. Other Conditions for the Existence of a Density on Homo-geneous Spaces
4. Examples
5. Lie Transformation Groups
6. Notes and Exercises

Chapter 11. The Affine Groups
1. The Groups of Affine Transformations
2. Densities for Linear Spaces with Respect to Special Homo-geneous Affinities
3. Densities for Linear Subspaces with Respect to the SpecialNonhomogeneous Affine Group
4. Notes and Exercises

Chapter 12. The Group of Motions in E,
1. Introduction
2. Densities for Linear Spaces in E
3. A Differential Formula
4. Density for r-Planes about a Fixed q-Plane
5. Another Form of the Density for r-Planes in
6. Sets of Pairs of Linear Spaces
7. Notes

Chapter 13. Convex Sets in
1. Convex Sets and Quermassintegrale
2. Cauchys Formula
3. Parallel Convex Sets; Steiners Formula
4. Integral Formulas Relating to the Projections of a Convex Set on Linear Subspaces
5. Integrals of Mean Curvature
6. Integrals of Mean Curvature and Quermassintegrale.
7. Integrals of Mean Curvature of a Flattened Convex Body
8. Notes

Chapter 14. Linear Subspaces, Convex Sets, and Compact Manifolds
1. Sets of r-Planes That Intersect a Convex Set
2. Geometric Probabilities
3. Croftons Formulas in En
4. Some Relations between Densities of Linear Subspaces
5. Linear Subspaces That Intersect a Manifold
6. Hypersurfaces and Linear Spaces
7. Notes

Chapter 15. The Kinematic Density in E
1. Formulas on Densities
2. Integral of the Volume
3. A Differential Formula
4. The Kinematic Fundamental Formula
5. Fundamental Formula for Convex Sets
6. Mean Values for the Integrals of Mean Curvature
7. Fundamental Formula for Cylinders
8. Some Mean Values
9. Lattices in En.
10. Notes and Exercise

Chapter 16. Geometric and Statistical Applications; Stereology
1. Size Distribution of Particles Derived from the Size Distribution of Their Sections
2. Intersection with Random Planes
3. Intersection with Random Lines
4. Notes

Chapter 17. Noneuclidean Integral Geometry
1. The n-Dimensional Noneuclidean Space
2. The Gauss-Bonnet Formula for Noneuclidean Spaces
3. Kinematic Density and Density for r-Planes
4. Sets of r-Planes That Meet a Fixed Body
5. Notes

Chapter 18. Croftons Formulas and the Kinematic Fundamental Formula
in Noneuclidean Spaces
1. Croftons Formulas
2. Dual Formulas in Elliptic Space
3. The Kinematic Fundamental Formula in Noneuclidean
Spaces
4. Steiners Formula in Noneuclidean Spaces
5. An Integral Formula for Convex Bodies in Elliptic Space
6. Notes
Chapter 19. Integral Geometry and Foliated Spaces; Trends in Integral Geometry
1. Foliated Spaces
2. Sets of Geodesics in a Riemann Manifold
3. Measure of Two-Dimensional Sets of Geodesics
4. Measure of （2n - 2）-Dimensional Sets of Geodesics
5. Sets of Geodesic Segments
6. Integral Geometry on Complex Spaces
7. Symplectic Integral Geometry
8. The Integral Geometry of Gelfand
9. Notes
Appendix. Differential Forms and Exterior Calculus
1. Differential Forms and Exterior Product
2. Two Applications of the Exterior Product
3. Exterior Differentiation
4. Stokes Formula
5. Comparison with Vector Calculus in Euclidean Three-Dimensional Space
6. Differential Forms over Manifolds
Bibliography and References
Author Index
Subiect Index

前言/序言

　　This monograph is the first in a projected series on Probability Theory.
　　Though its title "Integral Geometry" may appear somewhat unusual in thiscontext it is nevertheless quite appropriate, for Integral Geometry is anoutgrowth of what in the olden days was referred to as "geometric probabil-ities."
　　Originating, as legend has it, with the Buffon needle problem （which afternearly two centuries has lost little of its elegance and appeal）, geometricprobabilities have run into difficulties culminating in the paradoxes ofBertrand which threatened the fledgling field with banishment from the homeof Mathematics. In rescuing it from this fate, Poincar6 made the suggestionthat the arbitrariness of definition underlying the paradoxes could be removedby tying closer the definition of probability with a geometric group of which itwould have to be an invariant.
　　Thus a union of concepts was born that was to become Integral Geometry. It is unfortunate that in the past forty or so years during which ProbabilityTheory experienced its most spectacular rise to mathematical prominence,Integral Geometry has stayed on its fringes. Only quite recently has there beena reawakening of interest among practitioners of Probability Theory in thisbeautiful and fascinating branch of Mathematics, and thus the book byProfessor Santal6, for many years the undisputed leader in the field of IntegralGeometry, comes at a most appropriate time. Complete and scholarly, the book also repeatedly belies the popular beliefthat applicability and elegance are incompatible. Above all the book should remind all of us that Probability Theory ismeasure theory with a "soul" which in this case is provided not by Physics or bygames of chance or by Economics but by the most ancient and noble of allof mathematical disciplines, namely Geometry.

概率的幾何敘事：從測度到宇宙結構 本書深入探討瞭一個迷人且深刻的數學交叉領域——幾何概率。它不僅僅是對經典概率論的簡單延伸，而是一種全新的視角，將概率論的抽象概念植根於我們對空間、形狀和測量的直覺之中。全書的敘事綫索在於如何使用幾何工具，特彆是積分幾何的強大框架，來量化和分析隨機現象在連續空間中的錶現。 我們將從基礎的測度論和拓撲學概念齣發，為後續的幾何概率論奠定堅實的理論基礎。這裏不涉及微積分的繁瑣細節，而是聚焦於“可測集”的本質，以及黎曼積分和勒貝格積分在描述不規則區域上的概率分布時的優勢。讀者將理解為什麼在處理高維空間或復雜形體時，經典的計數方法會失效，而引入積分測度是必然的趨勢。 第一部分：歐氏空間中的隨機點與綫 本部分的核心在於二維和三維歐氏空間中點、綫、平麵以及更一般形體的統計行為。我們首先考察隨機選擇一個點落在給定區域內的概率問題。這看似簡單，實則引齣瞭對“均勻分布”的嚴謹定義。當區域變得復雜，比如一個不規則的凸多邊形，我們如何定義其內部的隨機點分布，以及如何計算其概率密度函數？書中將詳細分析這些基礎構造，並引入幾何概率密度函數（GPDF）的概念，它徹底取代瞭離散概率論中的頻率概念。 隨後，我們將轉嚮更具挑戰性的隨機直綫和隨機平麵的主題。著名的布豐投針問題（Buffon’s Needle Problem）將被作為起點，但我們不會止步於此。更重要的是，我們將探討如何定義空間中“隨機直綫族”的度量。這需要引入運動不變量（kinematic invariant）的概念——即一個度量如何保持不變，無論我們如何平移或鏇轉（剛體運動）我們所考察的係統。通過對鏇轉和平移群的理解，我們將推導齣綫與綫之間相交的概率，以及一條隨機直綫穿過一個給定形狀輪廓的長度期望。這部分內容將為理解物理學中的散射問題和材料科學中的結構分析打下基礎。 第二部分：積分幾何學的核心工具箱 積分幾何是連接分析學、微分幾何和概率論的橋梁。本部分將係統地介紹支撐該理論的兩大核心支柱：外測度（Outer Measure）和運動測度（Measure of Motion）。 在傳統的幾何學中，我們使用長度、麵積、體積來衡量形體。然而，在積分幾何中，我們需要衡量“形體集閤”的集閤。例如，在所有可能的圓形中，隨機選擇一個半徑在 $r$ 到 $r+dr$ 之間，圓心在某個區域 $D$ 內的圓的集閤的“大小”是多少？我們不能用傳統的麵積或體積來度量，因為圓形是二維形體，而圓心集閤是二維點集，它們在四維參數空間（$x, y, r, heta$）中形成一個元素。 運動測度 $mu(K)$ 的引入，提供瞭一種在 $n$ 維空間中，對所有通過剛體運動變換得到的形體 $K$ 的集閤進行積分的方法。書中將詳盡推導二維和三維空間中平麵、圓形、凸體等的運動測度公式。這些公式不僅具有深刻的數學美感，更重要的是，它們是計算幾何概率的“積分因子”。 我們還將深入探討Crofton 公式及其推廣。Crofton 公式揭示瞭形狀的幾何特性（如周長、麵積）與其在隨機直綫族中的積分特性之間的深刻聯係。通過在隨機直綫族上對某個凸體進行積分，我們可以反推齣該凸體的周長。這體現瞭概率測度與確定性幾何量之間的二元對偶關係。 第三部分：隨機幾何與幾何概率的應用前沿 在掌握瞭基礎的積分幾何工具後，本書將目光投嚮這些理論在現代科學中的具體應用，這些應用往往涉及高維隨機過程和非歐幾何的初步探索。 隨機凸體與隨機集閤 我們探討隨機凸集的形成過程。例如，隨機在平麵上投擲許多小的凸體（如圓盤或橢圓），並將它們的凸包取齣，這個最終形成的隨機凸包的期望麵積和期望周長是多少？這需要利用Minkowski泛函數和相關函數來處理凸體之間的相互作用。書中會分析不同生成機製（如：由隨機點構成的凸包，或隨機增長模型）下，隨機凸體幾何屬性的統計規律。 概率在微分幾何中的體現 幾何概率論自然地過渡到微分幾何。在麯麵上（例如球麵或黎曼流形）上，如何定義“隨機”？在這些空間中，平移和鏇轉的意義發生瞭變化，我們需要依賴於測地綫（Geodesics）來代替直綫。書中將分析在麯麵上隨機選取兩條測地綫相交的概率問題，並展示如何將歐氏空間的運動不變量推廣到一般的黎曼流形上。這部分內容為理解廣義相對論中時空隨機幾何的可能性提供瞭理論基礎。 空間結構的統計描述 最後，我們將關注隨機點過程在空間中的應用，特彆是在描述宇宙學、材料微觀結構和空間網絡中的隨機分布。我們采用泊鬆點過程（Poisson Point Process）作為分析框架，研究隨機點集閤的可見性問題——例如，在一個隨機點雲中，從任意一個點能“看到”其他點的概率是多少？這涉及到對隨機視域（random horizon）的分析。通過積分幾何的工具，我們可以計算齣在給定的隨機點密度下，空間中任意兩點之間存在無遮擋連綫的概率。 本書的結構設計旨在引導讀者從直觀的概率概念齣發，逐步攀升到嚴謹的積分幾何理論，最終展示如何利用這些抽象工具來解決具體的、涉及連續空間測量的難題。全書強調的是對“隨機形體”進行積分的藝術，而非僅僅是計算一個離散事件的頻率。它是一部關於如何用幾何的語言來書寫概率的專著。

評分☆☆☆☆☆

這本書的封麵設計真是充滿瞭古典與現代的交融感，深邃的藍色背景上浮動著抽象的幾何圖形，似乎預示著一場思維的探險即將展開。我第一次翻開它時，就被那種嚴謹而又不失優雅的數學語言深深吸引住瞭。雖然我並不是這個領域的專傢，但作者試圖用一種清晰、有條理的方式，將那些看似遙不可及的積分概念與我們日常所見的幾何形態聯係起來，這種努力本身就值得稱道。特彆是開篇關於“測度”和“不變性”的探討，為後續復雜的定理鋪設瞭堅實的基礎。它不是那種一上來就拋齣深奧公式的書籍，而是更像一位耐心的老師，一步步引導你進入這個迷人的世界。對於那些希望係統性理解幾何學在概率論中扮演關鍵角色的讀者來說，這本書提供瞭一個絕佳的起點，它挑戰瞭我們對空間和隨機性的傳統認知，讓人在閱讀過程中不斷産生“原來如此”的頓悟感。

評分☆☆☆☆☆

這本書的排版和裝幀質量確實達到瞭國際一流水準，紙張的質感拿在手裏非常舒適，即便是長時間的閱讀也不會感到眼睛疲勞。我特彆欣賞其中大量精美的插圖和圖示，它們不僅僅是公式的輔助說明，更像是獨立完成的藝術品，將抽象的幾何概念以可視化的方式呈現齣來。比如，書中關於“綫束的運動”和“積分不變量”的章節，如果沒有那些細緻的配圖，我恐怕會深陷於符號的迷宮中無法自拔。這種對細節的關注，體現瞭齣版方對學術內容的尊重。在處理復雜的微積分和微分幾何時，圖文並茂的呈現方式極大地降低瞭初學者的理解門檻，讓人感覺作者是真正站在讀者的角度來組織材料的，而不是僅僅滿足於學術展示的需要。

評分☆☆☆☆☆

閱讀這本書的過程中，我深刻體會到瞭一種跨學科的思維碰撞。它不僅僅是數學工具的堆砌，更像是一部關於“度量世界”哲學的探討錄。作者在闡述某些定理時，總會不經意地引申到物理學、甚至統計學中的實際應用場景，這使得原本枯燥的理論推導變得生動起來。我尤其對其中關於“隨機切割”和“幾何概率悖論”的討論印象深刻，它揭示瞭如何在不規則的設定下建立可靠的概率模型。這種將純粹的數學邏輯應用於解決現實世界中模糊問題的能力，是這本書最寶貴的地方。它教會我，幾何學並非隻是研究形狀的靜態描述，更是描述事物演化和隨機分布的動態語言，這種深刻的洞察力是其他同類教材難以企及的。

評分☆☆☆☆☆

坦率地說，這本書的深度對非專業人士構成瞭不小的挑戰，但這種挑戰本身就是一種價值。它絕不是一本“速成”指南，更像是一部需要反復研讀的經典著作。在深入到高級主題，例如黎曼流形上的積分幾何概念時，我發現自己不得不頻繁地查閱相關的拓撲學和微分幾何預備知識。這提醒我，要真正掌握這些內容，需要一個紮實的數學基礎作為支撐。然而，即便是那些暫時無法完全理解的段落，它們所蘊含的數學美感和邏輯嚴密性也令人摺服。對於那些追求學術深度、渴望在數學研究領域有所建樹的讀者來說，這本書無疑是一塊試金石，它能幫助你搭建起更高維度的數學認知框架。

評分☆☆☆☆☆

這本書在語言的運用上，展現齣一種獨特的英式學術風格——精確、剋製，卻又在關鍵時刻爆發齣思想的光芒。譯文（假設我閱讀的是外文原版或高質量譯本）的流暢性毋庸置疑，它成功地保留瞭原著中那種邏輯推進的韻律感，沒有齣現生硬的術語堆砌或不自然的句式結構。特彆是那些長難句的拆解和重組，都處理得非常到位，使得復雜的數學定義在清晰度和準確性之間取得瞭完美的平衡。它讓你感覺，作者是在與一位學識淵博的同行進行著一場高水平的學術對話，而不是在被動地接收知識灌輸。這種閱讀體驗是極其愉悅且富有啓發性的，它極大地提升瞭對高等數學學習的興趣和耐性。