Hyperbolic systems of balance laws: lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, July 14 - 21, 2003
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| Sprache: | English |
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Springer
2007
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| Schriftenreihe: | Lecture notes in mathematics
1911 |
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| Beschreibung: | XII, 346 S. graph. Darst. 235 mm x 155 mm |
| ISBN: | 9783540721864 354072186X |
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| 020 | |a 9783540721864 |c Pb. : ca. EUR 53.45 (freier Pr.), ca. sfr 82.00 (freier Pr.) |9 978-3-540-72186-4 | ||
| 020 | |a 354072186X |c Pb. : ca. EUR 53.45 (freier Pr.), ca. sfr 82.00 (freier Pr.) |9 3-540-72186-X | ||
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| 245 | 1 | 0 | |a Hyperbolic systems of balance laws |b lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, July 14 - 21, 2003 |c Alberto Bressan ... Ed.: Pierangelo Marcati |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2007 | |
| 300 | |a XII, 346 S. |b graph. Darst. |c 235 mm x 155 mm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Lecture notes in mathematics |v 1911 | |
| 650 | 7 | |a Equações diferenciais parciais hiperbólicas (congressos) |2 larpcal | |
| 650 | 7 | |a Hyperbolische differentiaalvergelijkingen |2 gtt | |
| 650 | 4 | |a Ondes de choc - Mathématiques - Congrès | |
| 650 | 7 | |a Singularidades (congressos) |2 larpcal | |
| 650 | 4 | |a Équations différentielles hyperboliques - Congrès | |
| 650 | 4 | |a Mathematik | |
| 650 | 4 | |a Differential equations, Hyperbolic |v Congresses | |
| 650 | 4 | |a Shock waves |x Mathematics |v Congresses | |
| 650 | 0 | 7 | |a Hyperbolisches Differentialgleichungssystem |0 (DE-588)4496581-3 |2 gnd |9 rswk-swf |
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| 689 | 0 | 1 | |a Hyperbolisches Differentialgleichungssystem |0 (DE-588)4496581-3 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Bressan, Alberto |d 1956- |e Sonstige |0 (DE-588)172009952 |4 oth | |
| 700 | 1 | |a Marcati, Pier A. |e Sonstige |4 oth | |
| 710 | 2 | |a Centro Internazionale Matematico Estivo |e Sonstige |0 (DE-588)1025933-8 |4 oth | |
| 830 | 0 | |a Lecture notes in mathematics |v 1911 |w (DE-604)BV000676446 |9 1911 | |
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Datensatz im Suchindex
| _version_ | 1804136568095506432 |
|---|---|
| adam_text | Contents
BV
Solutions
to Hyperbolic Systems by Vanishing Viscosity
Alberto
Bressan
.................................................. 1
1
Introduction
................................................ 1
2
Review of Hyperbolic Conservation Laws
....................... 6
2.1
Centered Rarefaction Waves
............................. 7
2.2
Shocks and Admissibility Conditions
...................... 8
2.3
Solution of the Rieraann Problem
......................... 11
2.4 Glimm
and Front Tracking Approximations
................ 12
2.5
A Semigroup of Solutions
................................ 15
2.6
Uniqueness and Characterization of Entropy Weak Solutions
. 17
3
The Vanishing Viscosity Approach
............................. 19
4
Parabolic Estimates
.......................................... 25
5
Decomposition by Traveling Wave Profiles
...................... 34
6
Interaction of Viscous Waves
.................................. 48
7
Stability of Viscous Solutions
.................................. 67
8
The Vanishing Viscosity Limit
................................. 70
References
....................... ............................... 76
Discrete Shock Profiles: Existence and Stability
Denis
Serre
..................................................... 79
Introduction
..................................................... 81
1
Existence Theory Rational Case
............................... 86
1.1
Steady Lax Shocks
..................................... 87
More Complex Situations
............................... 91
Other Rational Values of
77.............................. 92
Explicit Profiles for the Godunov Scheme
................. 92
DSPs for Strong Steady Shocks Under the Lax-Wendroff
Scheme
.......................................... 95
Scalar Shocks Under Monotone Schemes
.................. 97
1.2
Under-Compressive Shocks
.............................. 98
An Example from Reaction-Diffusion
..................... 99
X
Contents
Homoclinic and Chaotic Orbits
..........................101
Exponentially Small Splitting
............................102
1.3
Conclusions
...........................................103
2
Existence Theory the Irrational Case
...........................104
2.1
Obstructions
..........................................105
The Small Divisors Problem
.............................105
The Function
Y
........................................106
Counter-Examples to
(2.9) ..............................108
The Lax-Friedrichs Scheme with an Almost Linear Flux
....
Ill
The Scalar Case
.......................................113
2.2
The Approach by Liu and Yu
............................116
3
Semi-Discrete vs Discrete Traveling Waves
......................117
3.1
Semi-Discrete Profiles
...................................118
3.2
A Strategy Towards Fully Discrete Traveling Waves
........118
3.3
Sketch of Proof of Theorem
3.1 ..........................120
The Richness of Discrete Dynamics
.......................123
4
Stability Analysis: The Evans Function
.........................125
4.1
Spectral Stability
......................................126
4.2
The Essential Spectrum of I
............................127
4.3
Construction of the Evans Function
......................131
The Gap Lemma
.......................................132
The Geometric Separation
...............................133
5
Stability Analysis: Calculations
................................135
5.1
Calculations with Lax Shocks
............................137
The Homotopy from
ζ
= 1
to oo
.........................139
The Large Wave-Length Analysis
........................140
Conclusions
...................... 141
5.2
Calculations with Under-Compressive Shocks
..............144
5.3
Results for the Godunov Scheme
.........................145
The Case of Perfect Gases
...............................150
5.4
The Role of the Functional
Y
in the Nonlinear Stability
.....151
References
........................
25g
Stability of Multidimensional Viscous Shocks
Mark Williams
......................
I59
1
Lecture One: The Small Viscosity Limit : introduction,
...........
Approximate Solution
................ .. 160
1-1
Approximate Solution
.................................162
1-2
Summary
..............
Igg
2
Lecture Two: Full Linearization, Reduction to ODEs, Conjugation
to a Limiting Problem
................. ... 167
2.1
Full Versus Partial Linearization
............ ............167
2-2
The Extra Boundary Condition
..........................169
2-3
Corner Compatible Initial Data and Reduction to a Forward
Problem
.............
17Q
Contents
XI
2.4 Principal Parts,
Exponential Weights.....................
171
2.5
Some Difficulties
.......................................172
2.6
Semiclassical Form
.....................................173
2.7
Frozen Coefficients; ODEs Depending on Frequencies as
Parameters
............................................174
2.8
Three Frequency Regimes
...............................175
2.9
First-Order System
.....................................175
2.10
Conjugation
...........................................176
2.11
Conjugation to HP Form
................................178
3
Lecture Three: Evans Functions, Lopatinski Determinants,
Removing the Translational Degeneracy
........................178
3.1
Evans Functions, Instabilities, the Zumbrun-Serre Result
.... 179
3.2
The Evans Function as a Lopatinski Determinant
..........182
3.3
Doubling
..............................................182
3.4
Slow Modes and Fast Modes
.............................183
3.5
Removing the Translational Degeneracy
...................184
4
Lecture Four: Block Structure, Symmetrizers, Estimates
..........187
4.1
The
MF
Regime
........................................187
4.2
The SF Regime
........................................190
4.3
The Sign Condition
....................................192
4.4
Glancing Blocks and Glancing Modes
.....................193
4.5
Auxiliary Hypothesis for Lecture
5.......................195
4.6
The SF Estimate
.......................................196
4.7
The HF Regime
........................................198
4.8
Summary of Estimates
..................................198
5
Lecture Five: Long Time Stability via Degenerate Symmetrizers
... 200
5.1
Nonlinear Stability
.....................................201
5.2
L1
-
L2 Estimates
.....................................202
5.3
Proof of Proposition
5.1.................................204
5.4
The Dual Problem
.....................................205
5.5
Decomposition of Uh±
..................................206
5.6
Interior Estimates
......................................208
5.7
L°° Estimates
.........................................211
5.8
Nonlinear Stability Results
..............................211
6
Appendix A: The Uniform Stability Determinant
................212
7
Appendix B: Continuity of Decaying
Eigenspaces................213
8
Appendix C: Limits
аѕг-» ±оо
of Slow Modes
at Zero Frequency
...........................................215
9
Appendix D: Evans
=>
Transversality
+
Uniform Stability
........216
10
Appendix E: Proofs in Lecture
3...............................219
10.1
Construction of
R
......................................219
10.2
Propositions
3.4
and
3.5 ................................220
11
Appendix F: The HF Estimate
................................221
11.1
Block Stucture
.........................................223
11.2
Symmetrizer and Estimate
..............................223
XII Contents
12 Appendix
G:
Transition
to PDE Estimates
......................225
References
......................................................226
Planar Stability Criteria for Viscous Shock Waves of Systems
with Real Viscosity
Kevin Zumbrun
..................................................229
1
Introduction: Structure of Physical Equations
...................230
2
Description of Results
........................................242
3
Analytical Preliminaries
......................................251
4
Reduction to Low Frequency
..................................266
5
Low Frequency Analysis/Completion of Proofs
..................284
6
Appendices
.................................................305
6.1
Appendix A: Semigroup Facts
...........................305
6.2
Appendix B: Proof of Proposition
1.21....................315
6.3
Appendix C: Proof of Proposition
5.15....................317
References
......................................................320
Tutorial on the Center Manifold Theorem
.....................327
A.I Review of Linear O.D.E s
.....................................327
A.2 Statement of the Center Manifold Theorem
.....................329
A.3 Proof of the Center Manifold Theorem
.........................331
A.
3.1
Reduction to the Case of a Compact Perturbation
..........331
A.3.2 Characterization of the Global Center Manifold
............332
A.
3.3
Construction of the Center Manifold
......................333
A.
3.4
Proof of the
Invariance
Property (ii)
......................335
A.3.5 Proof of
(iv)...........................................335
A.3.6 Proof of the Tangency Property
(iii)
......................335
A.
3.7
Proof of the Asymptotic Approximation Property (v)
.......336
A.
3.8
Smoothness of the Center Manifold
.......................337
A.4 The Contraction Mapping Theorem
............................342
|
| adam_txt |
Contents
BV
Solutions
to Hyperbolic Systems by Vanishing Viscosity
Alberto
Bressan
. 1
1
Introduction
. 1
2
Review of Hyperbolic Conservation Laws
. 6
2.1
Centered Rarefaction Waves
. 7
2.2
Shocks and Admissibility Conditions
. 8
2.3
Solution of the Rieraann Problem
. 11
2.4 Glimm
and Front Tracking Approximations
. 12
2.5
A Semigroup of Solutions
. 15
2.6
Uniqueness and Characterization of Entropy Weak Solutions
. 17
3
The Vanishing Viscosity Approach
. 19
4
Parabolic Estimates
. 25
5
Decomposition by Traveling Wave Profiles
. 34
6
Interaction of Viscous Waves
. 48
7
Stability of Viscous Solutions
. 67
8
The Vanishing Viscosity Limit
. 70
References
.'. 76
Discrete Shock Profiles: Existence and Stability
Denis
Serre
. 79
Introduction
. 81
1
Existence Theory Rational Case
. 86
1.1
Steady Lax Shocks
. 87
More Complex Situations
. 91
Other Rational Values of
77. 92
Explicit Profiles for the Godunov Scheme
. 92
DSPs for Strong Steady Shocks Under the Lax-Wendroff
Scheme
. 95
Scalar Shocks Under Monotone Schemes
. 97
1.2
Under-Compressive Shocks
. 98
An Example from Reaction-Diffusion
. 99
X
Contents
Homoclinic and Chaotic Orbits
.101
Exponentially Small Splitting
.102
1.3
Conclusions
.103
2
Existence Theory the Irrational Case
.104
2.1
Obstructions
.105
The Small Divisors Problem
.105
The Function
Y
.106
Counter-Examples to
(2.9) .108
The Lax-Friedrichs Scheme with an Almost Linear Flux
.
Ill
The Scalar Case
.113
2.2
The Approach by Liu and Yu
.116
3
Semi-Discrete vs Discrete Traveling Waves
.117
3.1
Semi-Discrete Profiles
.118
3.2
A Strategy Towards Fully Discrete Traveling Waves
.118
3.3
Sketch of Proof of Theorem
3.1 .120
The Richness of Discrete Dynamics
.123
4
Stability Analysis: The Evans Function
.125
4.1
Spectral Stability
.126
4.2
The Essential Spectrum of I
.127
4.3
Construction of the Evans Function
.131
The Gap Lemma
.132
The Geometric Separation
.133
5
Stability Analysis: Calculations
.135
5.1
Calculations with Lax Shocks
.137
The Homotopy from
ζ
= 1
to oo
.139
The Large Wave-Length Analysis
.140
Conclusions
. 141
5.2
Calculations with Under-Compressive Shocks
.144
5.3
Results for the Godunov Scheme
.145
The Case of Perfect Gases
.150
5.4
The Role of the Functional
Y
in the Nonlinear Stability
.151
References
.
25g
Stability of Multidimensional Viscous Shocks
Mark Williams
.
I59
1
Lecture One: The Small Viscosity Limit': 'introduction,
.
Approximate Solution
. . 160
1-1
Approximate Solution
.162
1-2
Summary
.
Igg
2
Lecture Two: Full Linearization, Reduction to ODEs, Conjugation
to a Limiting Problem
. . 167
2.1
Full Versus Partial Linearization
.'.167
2-2
The Extra Boundary Condition
.169
2-3
Corner Compatible Initial Data and Reduction to a Forward
Problem
.
17Q
Contents
XI
2.4 Principal Parts,
Exponential Weights.
171
2.5
Some Difficulties
.172
2.6
Semiclassical Form
.173
2.7
Frozen Coefficients; ODEs Depending on Frequencies as
Parameters
.174
2.8
Three Frequency Regimes
.175
2.9
First-Order System
.175
2.10
Conjugation
.176
2.11
Conjugation to HP Form
.178
3
Lecture Three: Evans Functions, Lopatinski Determinants,
Removing the Translational Degeneracy
.178
3.1
Evans Functions, Instabilities, the Zumbrun-Serre Result
. 179
3.2
The Evans Function as a Lopatinski Determinant
.182
3.3
Doubling
.182
3.4
Slow Modes and Fast Modes
.183
3.5
Removing the Translational Degeneracy
.184
4
Lecture Four: Block Structure, Symmetrizers, Estimates
.187
4.1
The
MF
Regime
.187
4.2
The SF Regime
.190
4.3
The Sign Condition
.192
4.4
Glancing Blocks and Glancing Modes
.193
4.5
Auxiliary Hypothesis for Lecture
5.195
4.6
The SF Estimate
.196
4.7
The HF Regime
.198
4.8
Summary of Estimates
.198
5
Lecture Five: Long Time Stability via Degenerate Symmetrizers
. 200
5.1
Nonlinear Stability
.201
5.2
L1
-
L2 Estimates
.202
5.3
Proof of Proposition
5.1.204
5.4
The Dual Problem
.205
5.5
Decomposition of Uh±
.206
5.6
Interior Estimates
.208
5.7
L°° Estimates
.211
5.8
Nonlinear Stability Results
.211
6
Appendix A: The Uniform Stability Determinant
.212
7
Appendix B: Continuity of Decaying
Eigenspaces.213
8
Appendix C: Limits
аѕг-» ±оо
of Slow Modes
at Zero Frequency
.215
9
Appendix D: Evans
=>
Transversality
+
Uniform Stability
.216
10
Appendix E: Proofs in Lecture
3.219
10.1
Construction of
R
.219
10.2
Propositions
3.4
and
3.5 .220
11
Appendix F: The HF Estimate
.221
11.1
Block Stucture
.223
11.2
Symmetrizer and Estimate
.223
XII Contents
12 Appendix
G:
Transition
to PDE Estimates
.225
References
.226
Planar Stability Criteria for Viscous Shock Waves of Systems
with Real Viscosity
Kevin Zumbrun
.229
1
Introduction: Structure of Physical Equations
.230
2
Description of Results
.242
3
Analytical Preliminaries
.251
4
Reduction to Low Frequency
.266
5
Low Frequency Analysis/Completion of Proofs
.284
6
Appendices
.305
6.1
Appendix A: Semigroup Facts
.305
6.2
Appendix B: Proof of Proposition
1.21.315
6.3
Appendix C: Proof of Proposition
5.15.317
References
.320
Tutorial on the Center Manifold Theorem
.327
A.I Review of Linear O.D.E's
.327
A.2 Statement of the Center Manifold Theorem
.329
A.3 Proof of the Center Manifold Theorem
.331
A.
3.1
Reduction to the Case of a Compact Perturbation
.331
A.3.2 Characterization of the Global Center Manifold
.332
A.
3.3
Construction of the Center Manifold
.333
A.
3.4
Proof of the
Invariance
Property (ii)
.335
A.3.5 Proof of
(iv).335
A.3.6 Proof of the Tangency Property
(iii)
.335
A.
3.7
Proof of the Asymptotic Approximation Property (v)
.336
A.
3.8
Smoothness of the Center Manifold
.337
A.4 The Contraction Mapping Theorem
.342 |
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| classification_tum | MAT 357f |
| ctrlnum | (OCoLC)144615195 (DE-599)DNB983624380 |
| dewey-full | 515/.353 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.353 |
| dewey-search | 515/.353 |
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| genre | (DE-588)1071861417 Konferenzschrift 2003 Cetraro gnd-content |
| genre_facet | Konferenzschrift 2003 Cetraro |
| id | DE-604.BV022478121 |
| illustrated | Illustrated |
| index_date | 2024-07-02T17:47:20Z |
| indexdate | 2024-07-09T20:58:28Z |
| institution | BVB |
| institution_GND | (DE-588)1025933-8 |
| isbn | 9783540721864 354072186X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015685507 |
| oclc_num | 144615195 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-824 DE-11 DE-188 |
| owner_facet | DE-91G DE-BY-TUM DE-824 DE-11 DE-188 |
| physical | XII, 346 S. graph. Darst. 235 mm x 155 mm |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| spelling | Hyperbolic systems of balance laws lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, July 14 - 21, 2003 Alberto Bressan ... Ed.: Pierangelo Marcati Berlin [u.a.] Springer 2007 XII, 346 S. graph. Darst. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 1911 Equações diferenciais parciais hiperbólicas (congressos) larpcal Hyperbolische differentiaalvergelijkingen gtt Ondes de choc - Mathématiques - Congrès Singularidades (congressos) larpcal Équations différentielles hyperboliques - Congrès Mathematik Differential equations, Hyperbolic Congresses Shock waves Mathematics Congresses Hyperbolisches Differentialgleichungssystem (DE-588)4496581-3 gnd rswk-swf Erhaltungssatz (DE-588)4131214-4 gnd rswk-swf (DE-588)1071861417 Konferenzschrift 2003 Cetraro gnd-content Erhaltungssatz (DE-588)4131214-4 s Hyperbolisches Differentialgleichungssystem (DE-588)4496581-3 s DE-604 Bressan, Alberto 1956- Sonstige (DE-588)172009952 oth Marcati, Pier A. Sonstige oth Centro Internazionale Matematico Estivo Sonstige (DE-588)1025933-8 oth Lecture notes in mathematics 1911 (DE-604)BV000676446 1911 Digitalisierung TU Muenchen application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015685507&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hyperbolic systems of balance laws lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, July 14 - 21, 2003 Lecture notes in mathematics Equações diferenciais parciais hiperbólicas (congressos) larpcal Hyperbolische differentiaalvergelijkingen gtt Ondes de choc - Mathématiques - Congrès Singularidades (congressos) larpcal Équations différentielles hyperboliques - Congrès Mathematik Differential equations, Hyperbolic Congresses Shock waves Mathematics Congresses Hyperbolisches Differentialgleichungssystem (DE-588)4496581-3 gnd Erhaltungssatz (DE-588)4131214-4 gnd |
| subject_GND | (DE-588)4496581-3 (DE-588)4131214-4 (DE-588)1071861417 |
| title | Hyperbolic systems of balance laws lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, July 14 - 21, 2003 |
| title_auth | Hyperbolic systems of balance laws lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, July 14 - 21, 2003 |
| title_exact_search | Hyperbolic systems of balance laws lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, July 14 - 21, 2003 |
| title_exact_search_txtP | Hyperbolic systems of balance laws lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, July 14 - 21, 2003 |
| title_full | Hyperbolic systems of balance laws lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, July 14 - 21, 2003 Alberto Bressan ... Ed.: Pierangelo Marcati |
| title_fullStr | Hyperbolic systems of balance laws lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, July 14 - 21, 2003 Alberto Bressan ... Ed.: Pierangelo Marcati |
| title_full_unstemmed | Hyperbolic systems of balance laws lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, July 14 - 21, 2003 Alberto Bressan ... Ed.: Pierangelo Marcati |
| title_short | Hyperbolic systems of balance laws |
| title_sort | hyperbolic systems of balance laws lectures given at the c i m e summer school held in cetraro italy july 14 21 2003 |
| title_sub | lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, July 14 - 21, 2003 |
| topic | Equações diferenciais parciais hiperbólicas (congressos) larpcal Hyperbolische differentiaalvergelijkingen gtt Ondes de choc - Mathématiques - Congrès Singularidades (congressos) larpcal Équations différentielles hyperboliques - Congrès Mathematik Differential equations, Hyperbolic Congresses Shock waves Mathematics Congresses Hyperbolisches Differentialgleichungssystem (DE-588)4496581-3 gnd Erhaltungssatz (DE-588)4131214-4 gnd |
| topic_facet | Equações diferenciais parciais hiperbólicas (congressos) Hyperbolische differentiaalvergelijkingen Ondes de choc - Mathématiques - Congrès Singularidades (congressos) Équations différentielles hyperboliques - Congrès Mathematik Differential equations, Hyperbolic Congresses Shock waves Mathematics Congresses Hyperbolisches Differentialgleichungssystem Erhaltungssatz Konferenzschrift 2003 Cetraro |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015685507&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
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