Mathematical analysis and numerical methods for science and technology: 2 Functional and variational methods
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| Beschreibung: | XV, 589 S. graph. Darst. |
| ISBN: | 3540660984 |
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| LEADER | 00000nam a22000001cc4500 | ||
|---|---|---|---|
| 001 | BV013031481 | ||
| 003 | DE-604 | ||
| 005 | 20101027 | ||
| 007 | t | ||
| 008 | 000208s2000 gw d||| |||| 00||| eng d | ||
| 016 | 7 | |a 958275319 |2 DE-101 | |
| 020 | |a 3540660984 |9 3-540-66098-4 | ||
| 035 | |a (OCoLC)44551964 | ||
| 035 | |a (DE-599)BVBBV013031481 | ||
| 040 | |a DE-604 |b ger |e rakddb | ||
| 041 | 1 | |a eng |h fre | |
| 044 | |a gw |c DE | ||
| 049 | |a DE-703 |a DE-355 |a DE-20 | ||
| 082 | 0 | |a 519.4 | |
| 100 | 1 | |a Dautray, Robert |d 1928- |0 (DE-588)133309347 |4 aut | |
| 240 | 1 | 0 | |a Analyse mathématique et calcul numérique pour les sciences et les techniques |
| 245 | 1 | 0 | |a Mathematical analysis and numerical methods for science and technology |n 2 |p Functional and variational methods |c Robert Dautray ; Jacques-Louis Lions |
| 250 | |a [Nachdr.] | ||
| 264 | 1 | |a Berlin ; Heidelberg ; New York ; Paris ; Barcelona ; Hong Kong ; London |b Springer-Verlag |c 2000 | |
| 300 | |a XV, 589 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 7 | |a Analyse fonctionnelle |2 ram | |
| 650 | 7 | |a Analyse mathématique |2 ram | |
| 650 | 7 | |a Principes variationnels |2 ram | |
| 700 | 1 | |a Lions, Jacques-Louis |d 1928-2001 |0 (DE-588)124055397 |4 aut | |
| 773 | 0 | 8 | |w (DE-604)BV013031479 |g 2 |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008878226&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-008878226 | ||
Datensatz im Suchindex
| _version_ | 1804127728402694144 |
|---|---|
| adam_text | Table
of Contents
Chapter III. Functional Transformations
Introduction
......................... 1
Part A. Some Transformations Useful in Applications
....... 4
§ 1.
Fourier Series and Dirichlet s Problem
............ 4
1.
Fourier Series
..................... 4
1.1.
Convergence in
Û
(ТГ)
............... 5
1.2.
Pointwise Convergence on
Τ
............. 5
2.
Distributions on
Τ
and Periodic Distributions
....... 7
2.1.
Comparison of
$> (¥)
with the Distributions on
R
... 7
2.2.
Principal Properties of 3> (T)
............. 9
3.
Fourier Series of Distributions
.............. 10
4.
Fourier Series and Fourier Transforms
........... 14
5.
Convergence in the Sense of
Césaro
............ 15
6.
Solution of Dirichlet s Problem with the Help of Fourier Series
17
6.1.
Dirichlet s Problem in a Disk
............. 17
6.2.
Dirichlet s Problem in a Rectangle
........... 20
§2.
The Mellin Transform
................... 24
1.
Generalities
...................... 24
2.
Definition of the Mellin Transform
............ 26
3.
Properties of the Mellin Transform
............ 28
4.
Inverse Mellin Transform
................. 30
5.
Applications of the Mellin Transform
........... 32
6.
Table of Some Mellin Transforms
............. 40
§3.
The Hankel Transform
................... 40
1.
Generalities
...................... 40
2.
Introduction to Bessel Functions
............. 42
3.
Definition of the Hankel Transform
............ 47
4.
The Inversion Formula
................. 48
5.
Properties of the Hankel Transform
............ 50
6.
Application of the Hankel Transform to
Partia!
Differential
Equations
....................... 53
VIH
Table of Contents
6.1.
Dirichlet s Problem for Laplace s Equation in R+.
The Case of Axial Symmetry
............. 53
6.2.
Boundary Value Problem for the Biharmonic Equation
in
R
+ .
with Axial Symmetry
............. 55
7.
Table of Some Hanlcel Transforms
............ 57
Review of Chapter III A
.................... 57
Part B. Discrete Fourier Transforms and Fast Fourier Transforms
. . 59
§1.
Introduction
....................... 59
§2.
Acceleration of the Product of a Matrix by a Vector
...... 62
§3.
The Fast Fourier Transform of Cooley and Tukey
....... 64
§4.
The Fast Fourier Transform of Good-Winograd
........ 66
§5.
Reduction of the Number of Multiplications
......... 69
1.
Relation Between the Discrete Fourier Transform
and the Problem of Cyclic Convolution
.......... 69
2.
Complexity of the Product of Two Polynomials
....... 71
3.
Application to the Cyclic Convolution of Order
2...... 72
4.
Application to the Cyclic Convolution of Order
3...... 73
5.
Application to the Cyclic Convolution of Order
6...... 75
§6.
Fast Fourier Transform in Two Dimensions
.......... 77
§7.
Some Applications of the Fast Fourier Transform
....... 78
1.
Solution of Boundary Value Problems
........... 78
2.
Régularisation
and Smoothing of Functions
........ 81
3.
Practical Calculation of the Fourier Transform of a Signal
. . 83
4.
Determination of the Spectrum of Certain Finite Difference
Operators and Fast Solvers for the Laplacian
........ 84
Review of Chapter III
В
.................... 91
Chapter IV. Sobolev Spaces
Introduction
......................... 92
§1.
Spaces
#!(ΩΧ
ff
(Ω)...................
92
§2.
The Space
ŕľ(W).....................
96
1.
Definition and First Properties
.............. 96
2.
The Topological Dual of
H (WLn)
............. 98
3.
The Equation (-A+k2)u
=ƒ
in R , fceR {0}
....... 100
§3.
Sobolev s Embedding Theorem
............... 100
§4.
Density and Trace Theorems for the Spaces
#m(ß),
NN {0})
.....................
102
Table
of Contents IX
1.
A Density Theorem
................... 102
2.
A Trace Theorem for
# (»+).............. 107
3.
Traces of the Spaces tfm(R +) and Hm(Q)
......... 113
4.
Properties of m-Extension
................ 114
§5.
The Spaces H~m(Q) for all
m
εΝ..............
120
§6.
Compactness
....................... 123
§7.
Some Inequalities in Sobolev Spaces
............. 125
1.
Poincaré s
Inequality for
Η (Ω)
(resp.
Щ{и))
........ 125
2.
Poincaré s
Inequality for
Η1 (Ω)
.............. 127
3.
Convexity Inequalities for
Hm(ß)............. 133
§8.
Supplementary Remarks
.................. 138
1.
Sobolev Spaces Wm-P(Q)
................. 138
1.1.
Definitions
.................... 138
1.2.
Sobolev Injections
................. 139
1.3.
Trace Theorems for the Spaces Wm-P{Q)
........ 140
2.
Sobolev Spaces with Weights
............... 141
2.1.
Unbounded Open Sets
............... 141
2.2.
Polygonal Open Sets
................ 141
Review of Chapter IV
..................... 142
Appendix: The Spaces
НЅ(Г)
with
Γ
the Regular Boundary
of an Open Set
Ω
in R
.................... 143
Chapter V. Linear Differential Operators
Introduction
......................... 148
§1.
Generalities on Linear Differential Operators
......... 149
1.
Characterisation of Linear Differential Operators
...... 149
2.
Various Definitions
................... 152
2.1.
Leibniz s Formula
................. 152
2.2.
Transpose of a Linear Differential Operator
...... 153
2.3.
Order of a Linear Differential Operator
........ 154
3.
Linear Differential Operator on a Manifold
........ 155
4.
Characteristics
..................... 157
4.1.
Concept of Characteristics
.............. 157
4.2.
Bicharacteristics
.................. 159
5.
Operators with Analytic Coefficients.
Theorems of Cauchy-Kowalewsky and of Holmgren
..... 163
§2.
Linear Differential Operators with Constant Coefficients
.... 170
1.
Study of a l.d.o. with Constant Coefficients
by the Fourier Transform
................ 171
X Table of Contents
1.1.
Existence of a Solution of
Pu
=ƒ
in the Space
of Tempered Distributions
.............. 171
1.2.
Example
1 :
The Laplacian
.............. 173
1.3.
Elliptic and Strongly Elliptic Operators
........ 174
1.4.
Hypo-Elliptic and Semi-Elliptic Operators
....... 177
1.5.
Examples
..................... 179
1.6.
Reduction of Operators of Order
2
in a Homogeneous,
Isotropie
Medium
................ 181
2.
Elementary Solutions of a l.d.o. with Constant Coefficients
. . 182
2.1.
Introduction
.................... 182
2.2.
Elementary Solutions in
У
Examples
......... 184
2.3.
Elementary Solution with Support in a Salient Closed
Convex Cone: Hyperbolic Operator
.......... 189
3.
Characterisation of Hyperbolic Operators
......... 191
3.1.
Characteristics of a l.d.o. with Constant Coefficients
. . . 191
3.2.
Algebraic Characterisation of Hyperbolic Operators
... 194
3.3.
Hyperbolic Operators of Order
2........... 198
4.
Parabolic Operators
................... 202
§3.
Cauchy Problem for Differential Operators
with Constant Coefficients
................. 204
1.
Cauchy Problem and the Elementary Solution in
f(l xRł)
205
2.
Propagation in Hyperbolic Cauchy Problems
........ 209
3.
Choice of a Functional Space: Well-Posed Cauchy Problem
. . 214
4.
Well-Posed Cauchy Problem in <f
............ 217
5.
Parabolic and Weakly Parabolic Operators
......... 221
6.
Study of the Particular Case
Ρ
=
õ/õt
+
Po
......... 223
6.1.
Analysis of One-Dimensional Case
.......... 223
6.2.
Case in which Po is Strongly Elliptic
......... 224
6.3. Schrödinger
Operator
................ 225
7.
Well-Posed Cauchy Problem in a1 : Hyperbolic Operators
. . . 226
§4.
Local Regularity of Solutions*
................ 230
1.
Characterisation of Hypo-Ellipticity
............ 230
1.1.
Necessary Condition for Hypo-EHipticity
....... 230
1.2.
Algebraic Transformation of the Necessary Condition
for Hypo-Ellipticity
................. 232
1.3.
The Principal Result
................ 233
2.
Analyticky
of Solutions
................. 234
2.1.
Statement of Results
................ 234
2.2.
Estimates of Analyticity
............... 237
2.3.
Generalisation: Gevrey Classes
............ 240
3.
Comparison of Operators
................ 241
4.
Local Regularity for Operators with Variable Coefficients
and of Constant Force
................. 245
5.
Construction of an Elementary Solution
.......... 247
Table
of
Contents
XI
§5.
The Maximum Principle*
................. 250
1.
Prerequisites
...................... 250
2.
Parabolic Maximum Principle and Dissipativity
....... 252
3.
Characterisation of Operators
Ρ
Satisfying
Maximum Principles
.................. 259
3.1.
The Weak Maximum Principle
............ 259
3.2.
The Comparison Principle
.............. 261
3.3.
The Strong Maximum Principle
............ 263
3.4.
The Principle of the Strong Parabolic Maximum
.... 265
Review of Chapter V
..................... 268
Chapter VI. Operators in Banach Spaces and in Hubert Spaces
Introduction
......................... 269
§1.
Review of Functional Analysis: Banach and
Hubert
Spaces
. . . 270
1.
Locally Convex Topological Vector Spaces. Normed Spaces
and Banach Spaces
................... 270
2.
Linear Operators
.................... 274
3.
Duality
........................ 281
4.
The Hahn-Banach Theorem and its Applications
...... 282
4.1.
Problems of Approximation
............. 282
4.2.
Problems of Existence
................ 283
4.3.
Problems of Separation of Convex Sets
........ 285
5.
Bidual,
Reflexivity,
Weak Convergence, Weak Compactness
. 285
5.1.
Bidual
...................... 285
5.2.
Reflexivity
..................... 286
5.3.
Weak Convergence
................. 287
5.4.
Weak Compactness
................. 289
5.5.
Weak-Star Convergence
............... 290
6.
Hubert Spaces
..................... 291
6.1.
Definitions
.................... 291
6.2.
Projection on a Closed Convex Set
.......... 295
6.3.
Orthonormal
Bases
................. 299
6.4.
The Riesz Representation Theorem.
Reflexivity
..... 302
7.
Ideas About Functions of a Real or Complex Variable
with Values in a Banach Space
.............. 304
7.1.
Weak Topology
.................. 304
7.2.
Weak Differentiability
................ 304
7.3.
Weak Holomorphy
................. 305
§2.
Linear Operators in Banach Spaces
............. 305
1.
Generalities on Linear Operators
............. 305
1.1.
Domain, Kernel and Image of a Linear Operator
.... 305
XII Table of
Contents
1.2.
Nullity and Deficiency
índices............
307
1.3.
Basic
Properties
of Linear
Operators
......... 307
2.
Spaces of Bounded Operators
.............. 310
2.1.
Introduction
.................... 310
2.2.
Various Concepts of Convergence of Operators
..... 313
2.3.
Composition and Inverse of Bounded Operators
.... 316
2.4.
Transpose of a Bounded Operator
.......... 322
2.5.
Some Classes of Bounded Operators
......... 325
2.6.
Some Ideas on Functions of a Real or Complex Variable
with Operator Values; Families of Operators
...... 332
3.
Closed Operators
.................... 334
3.1.
Definition and Examples
.............. 334
3.2.
Basic Properties
.................. 336
3.3.
The Set .¥(X, Y) of Closed Operators from X into
Y
. . 339
3.4.
Transpose of a Closed Operator
........... 342
3.5.
Operators with Closed Image
............. 346
§3.
Linear Operators in Hubert Spaces
............. 348
1.
Bounded Operators in Hubert Spaces
........... 351
1.1.
Adjoint Sesquilinear Form
.............. 353
1.2.
Hermitian Operators
................ 353
1.3.
Orthogonal Projectors
................ 354
1.4.
Isometries and Unitary Operators
........... 355
1.5.
Hilbert-Schmidt Operators
.............. 357
2.
Unbounded Operators in Hubert Spaces
.......... 361
2.1.
Adjoint of an Unbounded Operator
.......... 361
2.2.
Symmetric Operators
................ 361
2.3.
The Cayley Transform
................ 362
2.4.
Normal Operators
................. 366
2.5.
Sesquilinear Forms and Unbounded Operators
..... 367
Review of Chapter VI
..................... 374
Chapter
VII.
Linear Variational Problems, Regularity
Introduction
......................... 375
§1.
Elliptic Variational Theory
................. 375
1.
The Lax-Milgram Theorem
............... 376
2.
First Examples
..................... 378
2.1.
Example
1.
Dirichlet Problem
............ 379
2.2.
Example
2.
Neumann Problem
............ 380
3.
Extensions in the Case in which V and
Я
are Spaces
of Distributions or of Functions
............. 383
4.
Sesquilinear Forms Associated with Elliptic Operators
of Order Two
..................... 384
Table
of Contents
XIII
5.
Sesquilinear Forms Associated with Elliptic Operators
of Order 2m
...................... 387
6.
Miscellaneous Remarks
................. 389
7.
Application to the Solution of General Elliptic Problems
(of Dirichlet Type)
................... 391
§2.
Examples of Second Order Elliptic Problems
......... 393
1.
Generalities
...................... 393
2.
Examples of Variational Problems
............. 394
2.1.
Mixed Problem
................... 394
2.2.
Non-Local Boundary Conditions
........... 397
3.
Problems Relative to Integro-Differential Forms on
Ω + Γ .
. 398
3.1.
Problem of the Oblique Derivative
.......... 398
3.2.
Robin s Problem
.................. 400
4.
Transmission Problem
.................. 400
5.
Miscellaneous Remarks
................. 405
6.
Application: Stationary Multigroup Equation for the Diffusion
of Neutrons
...................... 407
7.
Application: Statical Problems of Elasticity
......... 411
7.1.
Introduction
.................... 411
7.2.
Variational Formulation
............... 412
7.3.
Korn s Inequality
.................. 414
7.4.
Application to Problem
(2.39)............ 418
7.5.
Inhomogeneous Problem
.............. 418
8.
Statical Problems of the Flexure of Plates
......... 420
§3.
Regularity of the Solutions of Variational Problems
...... 425
1.
Introduction
...................... 425
2.
Interior Regularity
................... 426
3.
Global Regularity of the Solutions of Dirichlet
and Neumann Problems for Elliptic Operators of Order
2 . . 433
4.
Miscellaneous Results on Global Regularity
........ 437
5.
Green s Functions
................... 441
5.1.
Case of the Laplacian in a Bounded Open Set
Ω
with Dirichlet Condition
............... 441
5.2.
Some Other Particular Examples
........... 445
5.3.
Green s Functions in a More General Setting
...... 451
Review of Chapter
VII
.................... 456
Appendix. Distributions
§1.
Definition and Basic Properties of Distributions
........ 457
1.
The Space
®(Ω)
.................... 457
1.1.
Definition
..................... 457
XIV Table of
Contents
1.2.
Elementary Properties of the Convolution Product
of Two Functions
.................. 458
1.3.
A Procedure for the Construction of Functions of
¿¿(Ω)
. 459
1.4.
The Notion of Convergence in
SU
(Ω).........
460
1.5.
Some Inclusion and Density Properties
........ 461
2.
The Space
& (Ω)
of Distributions on
Ω
.......... 463
2.1.
Definition of Distributions and the Concept of Convergence
in
& (Ω)
...................... 463
2.2.
First Examples of Distributions: Measures on
Ω
.... 464
2.3.
Differentiation of Distributions. Examples
....... 467
2.4.
Support of a Distribution. Distributions with
Compact Support
.................. 474
3.
Some Elementary Operations on Distributions
....... 476
3.1.
Product by a Function of
Classi00
.......... 476
3.2.
Primitives of a Distribution on a Interval of
R
..... 477
3.3.
Tensor Product of Two Distributions
......... 480
3.4.
Direct Image and Inverse Image of a Function and
of a Distribution by a Function of Class g
...... 481
4.
Some Examples
.................... 482
4.1.
Primitives of the Dirac Measure
........... 482
4.2.
A Division Problem (Case
я
= 1)........... 483
4.3.
Derivative of a Function of R Discontinuous
on a Surface
.................... 484
4.4.
Distributions Defined by Inverse Image from Distributions
on the Real Line
.................. 485
§2.
Convolution of Distributions
................ 492
1.
Convolution of a Distribution on R and a Function
of^CR )
....................... 492
2.
Convolution of Two Distributions of Which One (at Least)
is with Compact Support
................ 494
3.
Distributions with Convolutive Supports
.......... 496
4.
Convolution Algebras
.................. 497
§3.
Fourier Transforms
.................... 500
1.
Fourier Transform of ¿ -Functions
............ 500
2.
The Space .^(R )
.................... 502
3.
Fourier Transform in L2
................. 506
4.
Fourier Transforms of Tempered Distributions
....... 506
5.
Fourier Transform of Distributions with Compact Support
. . 509
6.
Examples of the Calculation of Fourier Transforms
..... 510
7.
Partial Fourier Transform
................ 513
8.
Fourier Transform and Automorphisms of R :
Homogeneous Distributions
............... 518
8.1.
Fourier Transform and Automorphisms of R
..... 518
8.2.
Homogeneous Distributions
............. 519
Table
of
Contents
XV
9.
Fourier Transform and Convolution. Spaces GM (R )
and ^(R )
...................... 520
9.1.
TheSpaceďM(R I)(=af)..............
520
9.2.
The Space &c
................... 521
10.
Fourier Transform of Tempered Measures
......... 523
11.
Distribution of Positive Type. Bochner s Theorem
...... 525
11.1.
Functions of Positive Type
............. 525
11.2.
Distributions of Positive Type
............ 525
12.
Schwartz s Theorem of Kernels
.............. 527
13.
Some Distributions and Their Fourier Transforms
...... 532
Bibliography
......................... 533
Table of Notations
...................... 538
Index
............................ 551
Contents of Volumes
1, 3-6.................. 585
|
| any_adam_object | 1 |
| author | Dautray, Robert 1928- Lions, Jacques-Louis 1928-2001 |
| author_GND | (DE-588)133309347 (DE-588)124055397 |
| author_facet | Dautray, Robert 1928- Lions, Jacques-Louis 1928-2001 |
| author_role | aut aut |
| author_sort | Dautray, Robert 1928- |
| author_variant | r d rd j l l jll |
| building | Verbundindex |
| bvnumber | BV013031481 |
| ctrlnum | (OCoLC)44551964 (DE-599)BVBBV013031481 |
| dewey-full | 519.4 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.4 |
| dewey-search | 519.4 |
| dewey-sort | 3519.4 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | [Nachdr.] |
| format | Book |
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| id | DE-604.BV013031481 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:37:58Z |
| institution | BVB |
| isbn | 3540660984 |
| language | English French |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008878226 |
| oclc_num | 44551964 |
| open_access_boolean | |
| owner | DE-703 DE-355 DE-BY-UBR DE-20 |
| owner_facet | DE-703 DE-355 DE-BY-UBR DE-20 |
| physical | XV, 589 S. graph. Darst. |
| publishDate | 2000 |
| publishDateSearch | 2000 |
| publishDateSort | 2000 |
| publisher | Springer-Verlag |
| record_format | marc |
| spelling | Dautray, Robert 1928- (DE-588)133309347 aut Analyse mathématique et calcul numérique pour les sciences et les techniques Mathematical analysis and numerical methods for science and technology 2 Functional and variational methods Robert Dautray ; Jacques-Louis Lions [Nachdr.] Berlin ; Heidelberg ; New York ; Paris ; Barcelona ; Hong Kong ; London Springer-Verlag 2000 XV, 589 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Analyse fonctionnelle ram Analyse mathématique ram Principes variationnels ram Lions, Jacques-Louis 1928-2001 (DE-588)124055397 aut (DE-604)BV013031479 2 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008878226&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dautray, Robert 1928- Lions, Jacques-Louis 1928-2001 Mathematical analysis and numerical methods for science and technology Analyse fonctionnelle ram Analyse mathématique ram Principes variationnels ram |
| title | Mathematical analysis and numerical methods for science and technology |
| title_alt | Analyse mathématique et calcul numérique pour les sciences et les techniques |
| title_auth | Mathematical analysis and numerical methods for science and technology |
| title_exact_search | Mathematical analysis and numerical methods for science and technology |
| title_full | Mathematical analysis and numerical methods for science and technology 2 Functional and variational methods Robert Dautray ; Jacques-Louis Lions |
| title_fullStr | Mathematical analysis and numerical methods for science and technology 2 Functional and variational methods Robert Dautray ; Jacques-Louis Lions |
| title_full_unstemmed | Mathematical analysis and numerical methods for science and technology 2 Functional and variational methods Robert Dautray ; Jacques-Louis Lions |
| title_short | Mathematical analysis and numerical methods for science and technology |
| title_sort | mathematical analysis and numerical methods for science and technology functional and variational methods |
| topic | Analyse fonctionnelle ram Analyse mathématique ram Principes variationnels ram |
| topic_facet | Analyse fonctionnelle Analyse mathématique Principes variationnels |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008878226&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV013031479 |
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