Shapes and geometries: metrics, analysis, differential calculus, and optimization
Gespeichert in:
| Beteiligte Personen: | , |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Philadelphia
Society for Industrial and Applied Mathematics
2011
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Advances in design and control
22 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020845893&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Beschreibung: | Includes bibliographical references and index |
| Umfang: | XXIII, 622 S. Ill. |
| ISBN: | 9780898719369 |
Internformat
MARC
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| 100 | 1 | |a Delfour, Michel C. |d 1943- |e Verfasser |0 (DE-588)17252931X |4 aut | |
| 245 | 1 | 0 | |a Shapes and geometries |b metrics, analysis, differential calculus, and optimization |c M. C. Delfour ; J.-P. Zolésio |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Philadelphia |b Society for Industrial and Applied Mathematics |c 2011 | |
| 300 | |a XXIII, 622 S. |b Ill. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Advances in design and control |v 22 | |
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Shape theory (Topology) | |
| 650 | 0 | 7 | |a Form |0 (DE-588)4154987-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Strukturoptimierung |0 (DE-588)4183811-7 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Formgebung |0 (DE-588)4113598-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Gestaltoptimierung |0 (DE-588)4329076-0 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Geometrie |0 (DE-588)4020236-7 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Gestaltoptimierung |0 (DE-588)4329076-0 |D s |
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| 700 | 1 | |a Zolésio, Jean P. |d 1947- |e Verfasser |0 (DE-588)121134121 |4 aut | |
| 830 | 0 | |a Advances in design and control |v 22 |w (DE-604)BV021715022 |9 22 | |
| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020845893&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
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Datensatz im Suchindex
| _version_ | 1819361778544410624 |
|---|---|
| adam_text | Contents
List of Figures
xvii
Preface
xix
1
Objectives
and Scope of the Book
.................... xix
2
Overview of the Second Edition
...................... xx
3
Intended Audience
............................ xxii
4
Acknowledgments
............................. xxiii
1
Introduction: Examples, Background, and Perspectives
1
1
Orientation
................................ 1
1.1
Geometry as a Variable
..................... 1
1.2
Outline of the Introductory Chapter
.............. 3
2
A Simple One-Dimensional Example
.................. 3
3
Buckling of Columns
........................... 4
4
Eigenvalue Problems
........................... 6
5
Optimal Triangular Meshing
....................... 7
6
Modeling Free Boundary Problems
................... 10
6.1
Free Interface between Two Materials
............. 11
6.2
Minimal Surfaces
......................... 12
7
Design of a Thermal
Diffuser
...................... 13
7.1
Description of the Physical Problem
.............. 13
7.2
Statement of the Problem
.................... 14
7.3
Reformulation of the Problem
.................. 16
7.4
Scaling of the Problem
...................... 16
7.5
Design Problem
.......................... 17
8
Design of a Thermal Radiator
...................... 18
8.1
Statement of the Problem
.................... 18
8.2
Scaling of the Problem
...................... 20
9
A Glimpse into Segmentation of Images
................ 21
9.1
Automatic Image Processing
.................. 21
9.2
Image Smoothing/Filtering by Convolution and Edge Detectors
22
9.2.1
Construction of the Convolution of
ƒ ........ 23
9.2.2
Space-Frequency Uncertainty Relationship
..... 23
9.2.3
Laplacian Detector
................... 25
vii
viii Contents
9.3
Objective
Functions Defined on the Whole Edge
....... 26
9.3.1
Eulerian Shape
Semiderivative
............ 26
9.3.2
From Local to Global Conditions on the Edge
... 27
9.4
Snakes, Geodesic Active Contours, and Level Sets
...... 28
9.4.1
Objective Functions Defined on the Contours
.... 28
9.4.2
Snakes and Geodesic Active Contours
........ 28
9.4.3
Level Set Method
................... 29
9.4.4
Velocity Carried by the Normal
........... 30
9.4.5
Extension of the Level Set Equations
........ 31
9.5
Objective Function Defined on the Whole Image
....... 32
9.5.1
Tikhonov Regularization/Smoothing
......... 32
9.5.2
Objective Function of Mumford and Shah
...... 32
9.5.3
Relaxation of the (N
-
1)-Hausdorff Measure
.... 33
9.5.4
Relaxation to BV-, Hs-, and SBV-Functions
.... 33
9.5.5
Cracked Sets and Density Perimeter
......... 35
10
Shapes and Geometries: Background and Perspectives
........ 36
10.1
Parametrize Geometries by Functions or Functions by
Geometries?
............................ 36
10.2
Shape Analysis in Mechanics and Mathematics
........ 39
10.3
Characteristic Functions: Surface Measure and Geometric
Measure Theory
......................... 41
10.4
Distance Functions: Smoothness, Normal, and Curvatures
. . 41
10.5
Shape Optimization: Compliance Analysis and Sensitivity
Analysis
.............................. 43
10.6
Shape Derivatives
........................ 44
10.7
Shape Calculus and Tangential Differential Calculus
..... 46
10.8
Shape Analysis in This Book
.................. 46
11
Shapes and Geometries: Second Edition
................ 47
11.1
Geometries Parametrized by Functions
............. 48
11.2
Functions Parametrized by Geometries
............. 50
11.3
Shape Continuity and Optimization
.............. 52
11.4
Derivatives, Shape and Tangential Differential Calculuses, and
Derivatives under State Constraints
.............. 53
2
Classiceli
Descriptions of Geometries and Their Properties
55
1
Introduction
................................ 55
2
Notation and Definitions
......................... 56
2.1
Basic Notation
.......................... 56
2.2
Abelian Group Structures on Subsets of a Fixed Holdall
D
. 56
2.2.1
First Abelian Group Structure on
{V{D),
Δ)
.... 57
2.2.2
Second Abelian Group Structure on (V{D), V)
. . . 58
2.3
Connected Space, Path-Connected Space, and Geodesic
Distance
.............................. 58
2.4
Bouligand s Contingent Cone, Dual Cone, and Normal Cone
59
2.5
Sobolev Spaces
.......................... 60
2.5.1
Definitions
....................... 60
Contents ¡x
2.5.2
The Space
W™ *
(п)
.................. 61
2.5.3
Embedding of
H ¿(íl)
into
#¿(D)
.......... 62
2.5.4
Projection Operator
.................. 63
2.6
Spaces of Continuous and Differentiable Functions
...... 63
2.6.1
Continuous and Ck Functions
............ 63
2.6.2
Holder (C0-*) and Lipschitz (C0·1) Continuous
Functions
........................ 65
2.6.3
Embedding Theorem
................. 65
2.6.4
Identity
Ск г(ЈГ)
=
Wk+1 °°{íl):
From Convex to
Path-Connected Domains via the Geodesic Distance
66
3
Sets Locally Described by an Homeomorphism or a Diffeomorphism
67
3.1
Sets of Classes Ck and Ck e
................... 67
3.2
Boundary Integral, Canonical Density, and Hausdorff Measures
70
3.2.1
Boundary Integral for Sets of Class
С
l
....... 70
3.2.2
Integral on Submanifolds
............... 71
3.2.3
Hausdorff Measures
.................. 72
3.3
Fundamental Forms and Principal Curvatures
......... 73
4
Sets Globally Described by the Level Sets of a Function
....... 75
5
Sets Locally Described by the Epigraph of a Function
........ 78
5.1
Local C° Epigraphs, C° Epigraphs, and Equi-C0 Epigraphs
and the Space
Ή
of Dominating Functions
........... 79
5.2
Local C^-Epigraphs and
Hölderian/Lipschitzian
Sets
.... 87
5.3
Local C^-Epigraphs and Sets of Class
СкЛ
.......... 89
5.4
Locally Lipschitzian Sets: Some Examples and Properties
. . 92
5.4.1
Examples and Continuous Linear Extensions
.... 92
5.4.2
Convex Sets
...................... 93
5.4.3
Boundary Measure and Integral for Lipschitzian Sets
94
5.4.4
Geodesic Distance in a Domain and in Its Boundary
97
5.4.5
Nonhomogeneous Neumann and Dirichlet Problems
100
6
Sets Locally Described by a Geometric Property
........... 101
6.1
Definitions and Main Results
.................. 102
6.2
Equivalence of Geometric Segment and C° Epigraph
Properties
............................. 104
6.3
Equivalence of the Uniform Fat Segment and the Equi-C0
Epigraph Properties
....................... 109
6.4
Uniform Cone/Cusp Properties and
Hölderian/Lipschitzian
Sets
................................ 113
6.4.1
Uniform Cone Property and Lipschitzian Sets
. . . 114
6.4.2
Uniform Cusp Property and
Hölderian
Sets
..... 115
6.5
Hausdorff Measure and Dimension of the Boundary
..... 116
3
Courant
Metrics on Images of a Set
123
1
Introduction
................................ 123
2
Generic Constructions of Micheletti
................... 124
2.1
Space T{Q) of Transformations of RN
............. 124
2.2 Diffeomorphismsforß(RN,RN)
and C£°(RN,RN)
...... 136
Contents
2.3
Closed Subgroups
Ç
....................... 138
2.4
Courant
Metric on the
Quotient Group
T(ßi)IQ ....... 140
2.5
Assumptions for
ßfe(RN,RN),
C^R^R*), and Cofc(RN,RN)
143
2.5.1
Checking the Assumptions
.............. 143
2.5.2
Perturbations of the Identity and Tangent Space
. . 147
2.6
Assumptions for Cfe 1
(RÑ Rn)
and C^1(RN.RN)
...... 149
2.6.1
Checking the Assumptions
.............. 149
2.6.2
Perturbations of the Identity and Tangent Space
. . 151
3
Generalization to All Homeomorphisms and C^-Diffeomorphisms
. . 153
Transformations Generated by Velocities
159
1
Introduction
................................ 159
2
Metrics on Transformations Generated by Velocities
......... 161
2.1
Subgroup Ge of Transformations Generated by Velocities
. . 161
2.2
Complete Metrics on Gq and Geodesies
............ 166
2.3
Constructions of Azencott and
Trouvé
............. 169
3 Semiderivatives
via Transformations Generated by Velocities
.... 170
3.1
Shape Function
.......................... 170
3.2
Gateaux and
Hadamard
Semiderivatives............ 170
3.3
Examples of Families of Transformations of Domains
..... 173
3.3.1
C^-Domains
...................... 173
3.3.2
C^-Domains
...................... 175
3.3.3
Cartesian Graphs
................... 176
3.3.4
Polar Coordinates and Star-Shaped Domains
.... 177
3.3.5
Level Sets
........................ 178
4
Unconstrained Families of Domains
................... 180
4.1
Equivalence between Velocities and Transformations
..... 180
4.2
Perturbations of the Identity
.................. 183
4.3
Equivalence for Special Families of Velocities
......... 185
5
Constrained Families of Domains
.................... 193
5.1
Equivalence between Velocities and Transformations
..... 193
5.2
Transformation of Condition
(V2ß)
into a Linear
Constraint
............................. 200
6
Continuity of Shape Functions along Velocity Flows
......... 202
Metrics via Characteristic Functions
209
1
Introduction
................................ 209
2
Abelian Group Structure on Measurable Characteristic Functions
. . 210
2.1
Group Structure on XM(RN)
.................. 210
2.2
Measure Spaces
.......................... 211
2.3
Complete Metric for Characteristic Functions in
¿^-Topologies
........................... 212
3
Lebesgue Measurable Characteristic Functions
............ 214
3.1
Strong Topologies and C^-Approximations
.......... 214
3.2
Weak Topologies and
Microstructures
............. 215
3.3
Nice or Measure Theoretic Representative
........... 220
Contents
x¡
3.4
The Family of Convex Sets
...................223
3.5
Sobolev Spaces for Measurable Domains
............224
4
Some Compliance Problems with Two Materials
...........228
4.1
Transmission Problem and Compliance
............ 228
4.2
The Original Problem of
Céa
and Malanowski
......... 235
4.3
Relaxation and Homogenization
................ 239
5
Buckling of Columns
........................... 240
6
Caccioppoli or Finite Perimeter Sets
.................. 244
6.1
Finite Perimeter Sets
.......................245
6.2
Decomposition of the Integral along Level Sets
........251
6.3
Domains of Class W^P(D),
0 <
ε < λ/ρ, ρ
> 1,
and a Cascade
of Complete Metric Spaces
...................252
6.4
Compactness and Uniform Cone Property
...........254
7
Existence for the Bernoulli Free Boundary Problem
..........258
7.1
An Example: Elementary Modeling of the Water Wave
. . . 258
7.2
Existence for a Class of Free Boundary Problems
.......260
7.3
Weak Solutions of Some Generic Free Boundary Problems
. . 262
7.3.1
Problem without Constraint
.............262
7.3.2
Constraint on the Measure of the Domain
Ω
.... 264
7.4
Weak Existence with Surface Tension
.............265
6
Metrics via Distance Functions
267
1
Introduction
................................ 267
2
Uniform Metric Topologies
....................... 268
2.1
Family of Distance Functions Cd(D)
.............. 268
2.2
Pompéiu-Hausdorŕf
Metric on Cd(D)
.............. 269
2.3
Uniform Complementary Metric Topology and Cd(D)
.... 275
2.4
Families Ccd(E; D) and Ccd]oc{E;D)
............... 278
3
Projection, Skeleton, Crack, and Differentiability
........... 279
4
W 1 p-Metric Topology and Characteristic Functions
......... 292
4.1
Motivations and Main Properties
................ 292
4.2
Weak W^-P-Topology
....................... 296
5
Sets of Bounded and Locally Bounded Curvature
........... 299
5.1
Examples
............................. 301
6
Reach and Federer s Sets of Positive Reach
.............. 303
6.1
Definitions and Main Properties
................ 303
6.2
C^-Submanifolds
......................... 310
6.3
A Compact Family of Sets with Uniform Positive Reach
. . . 315
7
Approximation by Dilated Sets/Tubular Neighborhoods and Critical
Points
................................... 316
8
Characterization of Convex Sets
.................... 318
8.1
Convex Sets and Properties of
гід
................ 318
8.2
Semiconvexity and BV Character of (¿a
............ 320
8.3
Closed Convex Hull of A and
Fenchel
Transform of dA
. . . . 322
8.4
Families of Convex Sets Cd{D), Ccd{D), Ccd{E D), and
ChocWD)
............................ 323
XII
Contents
9
Compactness Theorems for Sets of Bounded Curvature
........ 324
9.1
Global Conditions in
D
..................... 325
9.2
Local Conditions in Tubular Neighborhoods
.......... 327
Metrics via Oriented Distance Functions
335
1
Introduction
................................ 335
2
Uniform Metric Topology
........................ 337
2.1
The Family of Oriented Distance Functions Cb(D)
...... 337
2.2
Uniform Metric Topology
.................... 339
3
Projection, Skeleton, Crack, and Differentiability
........... 344
4
^1 P(JD)-Metric Topology and the Family C^(D)
........... 349
4.1
Motivations and Main Properties
................ 349
4.2
Weak H^ P-Topology
....................... 352
5
Boundary of Bounded and Locally Bounded Curvature
........ 354
5.1
Examples and Limit of Tubular Norms as
h
Goes to Zero
. . 355
6
Approximation by Dilated Sets/Tubular Neighborhoods
....... 358
7
Federer s Sets of Positive Reach
..................... 361
7.1
Approximation by Dilated Sets/Tubular Neighborhoods
. . . 361
7.2
Boundaries with Positive Reach
................. 363
8
Boundary Smoothness and Smoothness of
Ьа
............. 365
9
Sobolev or Wm>p Domains
........................ 373
10
Characterization of Convex and Semiconvex Sets
........... 375
10.1
Convex Sets and Convexity of bj
................ 375
10.2
Families of Convex Sets Cb(D), Cb(E; D), and
Cb,lûc(E;D)
............................ 379
10.3
BV
Character of
Ьа
and Semiconvex Sets
........... 380
11
Compactness and Sets of Bounded Curvature
............. 381
11.1
Global Conditions on
D
..................... 382
11.2
Local Conditions in Tubular Neighborhoods
.......... 382
12
Finite Density Perimeter and Compactness
.............. 385
13
Compactness and Uniform Fat Segment Property
........... 387
13.1
Main Theorem
.......................... 387
13.2
Equivalent Conditions on the Local Graph Functions
..... 391
14
Compactness under the Uniform Fat Segment Property and a Bound
on a Perimeter
.............................. 393
14.1
De Giorgi
Perimeter of Caccioppoli Sets
............ 393
14.2
Finite Density Perimeter
..................... 394
15
The Families of Cracked Sets
...................... 394
16
A Variation of the Image Segmentation Problem of Mumford
and Shah
................................. 400
16.1
Problem Formulation
...................... 400
16.2
Cracked Sets without the Perimeter
.............. 401
16.2.1
Technical Lemmas
................... 401
16.2.2
Another Compactness Theorem
........... 402
16.2.3
Proof of Theorem
16.1................. 402
16.3
Existence of a Cracked Set with Minimum Density Perimeter
405
Contents xiii
16.4 Uniform
Bound or Penalization Term in the Objective
Function on the Density Perimeter
............... 407
8
Shape Continuity and Optimization
409
1
Introduction and Generic Examples
................... 409
1.1
First Generic Example
...................... 411
1.2
Second Generic Example
..................... 411
1.3
Third Generic Example
..................... 411
1.4
Fourth Generic Example
..................... 412
2
Upper Semicontinuity and Maximization of the First Eigenvalue
. . 412
3
Continuity of the Transmission Problem
................ 417
4
Continuity of the Homogeneous Dirichlet Boundary Value Problem
. 418
4.1
Classical, Relaxed, and Overrelaxed Problems
......... 418
4.2
Classical Dirichlet Boundary Value Problem
.......... 421
4.3
Overrelaxed Dirichlet Boundary Value Problem
........ 423
4.3.1
Approximation by Transmission Problems
...... 423
4.3.2
Continuity with Respect to X(D) in the
Lp(£>)-Topology
.................... 424
4.4
Relaxed Dirichlet Boundary Value Problem
.......... 425
5
Continuity of the Homogeneous Neumann Boundary Value Problem
426
6
Elements of Capacity Theory
...................... 429
6.1
Definition and Basic Properties
................. 429
6.2
Quasi-continuous Representative and
7/
^Functions
...... 431
6.3
Transport of Sets of Zero Capacity
............... 432
7
Crack-Free Sets and Some Applications
................ 434
7.1
Definitions and Properties
.................... 434
7.2
Continuity and Optimization over L(D, r.O,
λ)
........ 437
7.2.1
Continuity of the Classical Homogeneous Dirichlet
Boundary Condition
.................. 437
7.2.2
Minimization/Maximization of the First
Eigenvalue
....................... 438
8
Continuity under Capacity Constraints
................. 440
9
Compact Families OCtr{D) and Lc,r(O,D)
............... 447
9.1
Compact Family
Oc,r(D)
.................... 447
9.2
Compact Family LC,T(C D) and Thick Set Property
..... 450
9.3
Maximizing the Eigenvalue
λ4(Ω)
............... 452
9.4
State Constrained Minimization Problems
........... 453
9.5
Examples with a Constraint on the Gradient
......... 454
9
Shape and Tangential Differential Calculuses
457
1
Introduction
................................ 457
2
Review of Differentiation in Topological Vector Spaces
........ 458
2.1
Definitions of Semiderivat
i ves
and Derivatives
......... 458
2.2
Derivatives in Normed Vector Spaces
.............. 461
2.3
Locally Lipschitz Functions
................... 465
2.4
Chain Rule for
Semiderivatives................. 465
xiv
Contents
2.5 Semiderivatives
of Convex Functions
.............. 467
2.6
Hadamard
Semiderivative
and Velocity Method
........ 469
3
First-Order Shape
Semiderivatives
and Derivatives
.......... 471
3.1
Eulerian and
Hadamard
Semiderivatives............ 471
3.2
Hadamard Semidifferentiability
and
Courant
Metric
Continuity
............................ 476
3.3
Perturbations of the Identity and Gateaux and
Fréchet
Derivatives
............................ 476
3.4
Shape Gradient and Structure Theorem
............ 479
4
Elements of Shape Calculus
....................... 482
4.1
Basic Formula for Domain Integrals
.............. 482
4.2
Basic Formula for Boundary Integrals
............. 484
4.3
Examples of Shape Derivatives
................. 486
4.3.1
Volume of
Ω
and Surface Area of
Γ
......... 486
4.3.2
tf^j-Norm
...................... 487
4.3.3
Normal Derivative
................... 488
5
Elements of Tangential Calculus
.................... 491
5.1
Intrinsic Definition of the Tangential Gradient
........ 492
5.2
First-Order Derivatives
..................... 495
5.3
Second-Order Derivatives
.................... 496
5.4
A Few Useful Formulae and the Chain Rule
.......... 497
5.5
The Stokes and Green Formulae
................ 498
5.6
Relation between Tangential and Covariant Derivatives
. . . 498
5.7
Back to the Example of Section
4.3.3.............. 501
6
Second-Order
Semiderivative
and Shape Hessian
........... 501
6.1
Second-Order Derivative of the Domain Integral
....... 502
6.2
Basic Formula for Domain Integrals
.............. 504
6.3
Nonautonomous Case
...................... 505
6.4
Autonomous Case
........................ 510
6.5
Decomposition of d2J{Q]V(0),W(0))
............. 515
10
Shape Gradients under a State Equation Constraint
519
1
Introduction
................................ 519
2
Min
Formulation
............................. 521
2.1
An Illustrative Example and a Shape Variational Principle
. 521
2.2
Function Space Parametrization
................ 522
2.3
Differentiability of a Minimum with Respect to a
Parameter
............................. 523
2.4
Application of the Theorem
................... 526
2.5
Domain and Boundary Integral Expressions of the Shape
Gradient
.............................. 530
3
Buckling of Columns
........................... 532
4
Eigenvalue Problems
........................... 535
4.1
Transport of
Н$(п)
by ^^-Transformations of RN
.... 536
4.2
Laplacian and Bi-Laplacian
................... 537
4.3
Linear Elasticity
......................... 546
Contents xv
5
Saddle
Point
Formulation and Function Space Parametrization
... 551
5.1
An Illustrative Example
..................... 551
5.2
Saddle Point Formulation
.................... 552
5.3
Function Space Parametrization
................ 553
5.4
Differentiability of a Saddle Point with Respect to a
Parameter
............................. 555
5.5
Application of the Theorem
................... 559
5.6
Domain and Boundary Expressions for the Shape Gradient
. 561
6
Multipliers and Function Space Embedding
.............. 562
6.1
The Nonhomogeneous Dirichlet Problem
............ 562
6.2
A Saddle Point Formulation of the State Equation
...... 563
6.3
Saddle Point Expression of the Objective Function
...... 564
6.4
Verification of the Assumptions of Theorem
5.1........ 566
Elements of Bibliography
571
Index of Notation
615
Index
619
|
| any_adam_object | 1 |
| author | Delfour, Michel C. 1943- Zolésio, Jean P. 1947- |
| author_GND | (DE-588)17252931X (DE-588)121134121 |
| author_facet | Delfour, Michel C. 1943- Zolésio, Jean P. 1947- |
| author_role | aut aut |
| author_sort | Delfour, Michel C. 1943- |
| author_variant | m c d mc mcd j p z jp jpz |
| building | Verbundindex |
| bvnumber | BV036930837 |
| callnumber-first | Q - Science |
| callnumber-label | QA612 |
| callnumber-raw | QA612.7 |
| callnumber-search | QA612.7 |
| callnumber-sort | QA 3612.7 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 870 |
| ctrlnum | (OCoLC)706929721 (DE-599)BVBBV036930837 |
| dewey-full | 514/.24 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 514 - Topology |
| dewey-raw | 514/.24 |
| dewey-search | 514/.24 |
| dewey-sort | 3514 224 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV036930837 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T14:44:52Z |
| institution | BVB |
| isbn | 9780898719369 |
| language | English |
| lccn | 2010028846 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020845893 |
| oclc_num | 706929721 |
| open_access_boolean | |
| owner | DE-703 DE-824 DE-706 DE-11 DE-19 DE-BY-UBM DE-20 DE-355 DE-BY-UBR |
| owner_facet | DE-703 DE-824 DE-706 DE-11 DE-19 DE-BY-UBM DE-20 DE-355 DE-BY-UBR |
| physical | XXIII, 622 S. Ill. |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | marc |
| series | Advances in design and control |
| series2 | Advances in design and control |
| spellingShingle | Delfour, Michel C. 1943- Zolésio, Jean P. 1947- Shapes and geometries metrics, analysis, differential calculus, and optimization Advances in design and control Shape theory (Topology) Form (DE-588)4154987-9 gnd Strukturoptimierung (DE-588)4183811-7 gnd Formgebung (DE-588)4113598-2 gnd Gestaltoptimierung (DE-588)4329076-0 gnd Geometrie (DE-588)4020236-7 gnd |
| subject_GND | (DE-588)4154987-9 (DE-588)4183811-7 (DE-588)4113598-2 (DE-588)4329076-0 (DE-588)4020236-7 |
| title | Shapes and geometries metrics, analysis, differential calculus, and optimization |
| title_auth | Shapes and geometries metrics, analysis, differential calculus, and optimization |
| title_exact_search | Shapes and geometries metrics, analysis, differential calculus, and optimization |
| title_full | Shapes and geometries metrics, analysis, differential calculus, and optimization M. C. Delfour ; J.-P. Zolésio |
| title_fullStr | Shapes and geometries metrics, analysis, differential calculus, and optimization M. C. Delfour ; J.-P. Zolésio |
| title_full_unstemmed | Shapes and geometries metrics, analysis, differential calculus, and optimization M. C. Delfour ; J.-P. Zolésio |
| title_short | Shapes and geometries |
| title_sort | shapes and geometries metrics analysis differential calculus and optimization |
| title_sub | metrics, analysis, differential calculus, and optimization |
| topic | Shape theory (Topology) Form (DE-588)4154987-9 gnd Strukturoptimierung (DE-588)4183811-7 gnd Formgebung (DE-588)4113598-2 gnd Gestaltoptimierung (DE-588)4329076-0 gnd Geometrie (DE-588)4020236-7 gnd |
| topic_facet | Shape theory (Topology) Form Strukturoptimierung Formgebung Gestaltoptimierung Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020845893&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV021715022 |
| work_keys_str_mv | AT delfourmichelc shapesandgeometriesmetricsanalysisdifferentialcalculusandoptimization AT zolesiojeanp shapesandgeometriesmetricsanalysisdifferentialcalculusandoptimization |