Transseries and real differential algebra:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2006
|
| Schriftenreihe: | Lecture notes in mathematics
1888 |
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. [235] - 239 |
| Beschreibung: | XII, 255 S. graph. Darst. 235 mm x 155 mm |
| ISBN: | 9783540355908 3540355901 |
Internformat
MARC
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| 084 | |a MAT 346f |2 stub | ||
| 084 | |a 68W30 |2 msc | ||
| 100 | 1 | |a Hoeven, Joris van der |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Transseries and real differential algebra |c J. van der Hoeven |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2006 | |
| 300 | |a XII, 255 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 1888 | |
| 500 | |a Literaturverz. S. [235] - 239 | ||
| 650 | 4 | |a Algèbre différentielle | |
| 650 | 7 | |a Differentiaalrekening |2 gtt | |
| 650 | 4 | |a Séries arithmétiques | |
| 650 | 4 | |a Differential algebra | |
| 650 | 4 | |a Series, Arithmetic | |
| 650 | 0 | 7 | |a Differentialalgebra |0 (DE-588)4134657-9 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Differentialalgebra |0 (DE-588)4134657-9 |D s |
| 689 | 0 | |5 DE-604 | |
| 830 | 0 | |a Lecture notes in mathematics |v 1888 |w (DE-604)BV000676446 |9 1888 | |
| 856 | 4 | 2 | |q text/html |u http://deposit.dnb.de/cgi-bin/dokserv?id=2841853&prov=M&dok_var=1&dok_ext=htm |3 Inhaltstext |
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| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-014965735 | |
Datensatz im Suchindex
| _version_ | 1805088433749622784 |
|---|---|
| adam_text |
Table
of
Contents
Foreword
.
XI
Introduction
. 1
The field with no escape
. 1
Historical perspectives
. 3
Outline of the contents
. 7
Notations
. 10
1
Orderings
. 11
1.1
Quasi-orderings
. 12
1.2
Ordinal numbers
. 15
1.3
Well-quasi-orderings
. 17
1.4
Kruskal's theorem
. 19
1.5
Ordered structures
. 2?,
1.6
Asymptotic relations
., , 25
1.7 Hahn
spaces
.,,.,,,,,,,, 29
1.8
Groups and rings with generalized powers
, , , ,., , 30
2
Grid-based series
. 33
2.1
Grid-based sets
. . . . 34
2.2
Grid-based series
. . . . 36
2.3
Asymptotic relations
. 40
2.3.1
Dominance and neglection
relations
. . 40
2.3.2
Flatness relations
. 42
2.3.3
Truncations
. 42
2.4
Strong linear algebra
. 44
2.4.1
Set-like notations for families
. 44
2.4.2
Infmitary operators
. 45
2.4.3
Strong abelian groups
. 46
2.4.4
Other strong structures
. 47
2.5
Grid-based summation
. 48
2.5.1
Ultra-strong grid-based algebras
. 48
2.5.2
Properties of grid-based summation
. . 49
2.5.3
Extension by strong linearity
. 50
2.6
Asymptotic scales
. 53
The Newton polygon method
. 57
3.1
The method illustrated by examples
. 58
3.1.1
The Newton polygon and its slopes
. 58
3.1.2
Equations with asymptotic constraints and refinements
. 59
3.1.3
Almost double roots
. 62
3.2
The implicit series theorem
. 63
3.3
The Newton polygon method
. 65
3.3.1
Newton polynomials and Newton degree
. 65
3.3.2
Decrease of the Newton degree during refinements
. . 66
3.3.3
Resolution of asymptotic polynomial equations
. 67
3.4
Cartesian representations
. 69
3.4.1
Cartesian representations
. 69
3.4.2
Inserting new infinitesimal monomials
. 71
3.5
Local communities
. 71
3.5.1
Cartesian communities
. 72
3.5.2
Local communities
. 72
3.5.3
Faithful Cartesian representations
. 73
3.5.4
Applications of faithful Cartesian representations
. . 74
3.5.5
The Newton polygon method revisited
. 75
Transseries
. 79
4.1
Totally ordered
exp-
log fields
. 80
4.2
Fields of grid-based
transseries
. 84
4.3
The field of grid-based
transseries
in
ж
. 87
4.3.1
Logarithmic
transseries
in
ж
. 88
4.3.2
Exponential extensions
. 88
4.3.3
Increasing unions
. 89
4.3.4
General
transseries
in
ж
. 89
4.3.5
Upward and downward shifting
. 90
4.4
The incomplete
transbasis
theorem
. 92
4.5
Convergent
transseries
. 94
5
Operations on
transseries .
97
5.1
Differentiation
. . 98
5.2
Integration
. 103
5.3
Functional composition
. . 106
5.4
Functional inversion
.
Ill
5.4.1
Existence of functional inverses
.
Ill
5.4.2
The
Translagrange
theorem
. 112
6
Grid-based operators
. 115
6.1
Multilinear grid-based operators
. 116
6.1.1
Multilinear grid-based operators
. 116
6.1.2
Operator supports
. 117
6.2
Strong tensor products
. 118
6.3
Grid-based operators
. 122
6.3.1
Definition and characterization
. 122
6.3.2
Multivariate grid-based operators and compositions
. 123
6.4
Atomic decompositions
. 124
6.4.1
The space of grid-based operators
. 124
6.4.2
Atomic decompositions
. 125
6.4.3
Combinatorial interpretation of atomic families
. 126
6.5
Implicit function theorems
. 127
6.5.1
The first implicit function theorem
. 128
6.5.2
The second implicit function theorem
. 130
6.5.3
The third implicit function theorem
. 130
6.6
Multilinear types
. 133
7
Linear differential equations
. 135
7.1
Linear differential operators
.,.,.,,. 136
7.1.1
Linear differential operators as series
. . 136
7.1.2
Multiplicative conjugation
. 137
7.1.3
Upward shifting
. 137
7.2
Differential Riccati polynomials
. 139
7.2.1
The differential Riccati polynomial
. 139
7.2.2
Properties of differential Riccati polynomials
. 140
7.3
The trace of a linear differential operator
. 141
7.3.1
The trace relative to plane
transbases
. 141
7.3.2
Dependence of the trace on the
transbasis
. 143
7.3.3
Remarkable properties of the trace
. 144
7.4
Distinguished solutions
. 146
7.4.1
Existence of distinguished right inverses
. 146
7.4.2
On the supports of distinguished solutions
. 148
7.5
The deformed Newton polygon method
. 151
7.5.1
Asymptotic Riccati equations modulo o(l)
. 151
7.5.2
Quasi-linear Riccati equations
. 152
7.5.3
Refinements
. 153
7.5.4
An algorithm for finding all solutions
. 155
7.6
Solving the homogeneous equation
. 156
7.7
Oscillating
transseries
. 158
7.7.1
Complex and oscillating
transseries
. 158
7.7.2
Oscillating solutions to linear differential equations
. 159
7.8
Factorization of differential operators
. 162
7.8.1
Existence of factorizations
. 162
7.8.2
Distinguished factorizations
. 163
8
Algebraic differential equations
. 165
8.1
Decomposing differential polynomials
. 166
8.1.1
Serial decomposition
. 166
8.1.2
Decomposition by degrees
. 167
8.1.3
Decomposition by orders
. 167
8.1.4
Logarithmic decomposition
. 168
8.2
Operations on differential polynomials
. 169
8.2.1
Additive conjugation
. 169
8.2.2
Multiplicative conjugation
. 170
8.2.3
Upward and downward shifting
. 170
8.3
The differential Newton polygon method
. 172
8.3.1
Differential Newton polynomials
. 172
8.3.2
Properties of differential Newton polynomials
. 173
8.3.3
Starting terms
. 174
8.3.4
Refinements
. 175
8.4
Finding the starting monomials
. 177
8.4.1
Algebraic starting monomials
. 177
8.4.2
Differential starting monomials
. 179
8.4.3
On the shape of the differential Newton polygon
. . . 180
8.5
Quasi-linear equations
. 182
8.5.1
Distinguished solutions
. 182
8.5.2
General
solutions
. 183
8.6
Unravelling almost multiple solutions
. 186
8.6.1
Partial unravellings
. 186
8.6.2
Logarithmic slow-down of the unravelling process
. . 188
8.6.3
On the stagnation of the depth
. 189
8.6.4
Bounding the depths of solutions
. 190
8.7
Algorithmic resolution
. 192
8.7.1
Computing starting terms
. 192
8.7.2
Solving the differential equation
. 194
8.8
Structure theorems
. 196
8.8.1
Distinguished unravellers
. 196
8.8.2
Distinguished solutions and their existence
. 197
8.8.3
On the intrusion of new exponentials
. 198
The intermediate value theorem
. 201
9.1
Compactification of total
orderings
. 202
9.1.1
The interval topology on total
orderings
. 202
9.1.2
Dedekind cuts
. 203
9.1.3
The compactness theorem
. 204
9.2
Compactification of totally ordered fields
. 206
9.2.1
Functorial properties of compactification
. 206
9.2.2
Compactification of totallj·-ordered fields
. 207
9.3
Compactification of grid-based
algebran
. 208
9.3.1
Monomial cuts
. 208
9.3.2
Width of a cut
. . . . ,. 209
9.3.3
Initializers
.,. . 210
9.3.4
Serial cuts
.,,,.,,.,.,
П0
9.3.5
Decomposition of
nou
serial
ente
. . , , 211
9.4
Compa.ctification of' the
transline
. 212
9.4.1
Exponentiation in
T
. 213
9.4.2
Classification of
transseries
cuts
. 213
9.4.3
Finite nested expansions
. 214
9.4.4
Infinite nested expansions
. 216
9.5
Integral neighbourhoods of cuts
. 219
9.5.1
Differentiation and integration of cuts
. 219
9.5.2
Integral nested expansions
. 219
9.5.3
Integral neighbourhoods
. 220
9.5.4
On the orientation of integral neighbourhoods
. 222
9.6 Differential
polynomials near cuts
. 223
9.6.1
Differential polynomials near serial cuts
. 223
9.6.2
Differential polynomials near constants
. 224
9.6.3
Differential polynomials near nested cuts
. 225
9.6.4
Differential polynomials near arbitrary cuts
. 226
9.6.5
On the sign of a differential polynomial
. 227
9.7
The intermediate value theorem
. 229
9.7.1
The quasi-linear case
. 229
9.7.2
Preserving sign changes during refinements
. 230
9.7.3
Proof of the intermediate value theorem
. 232
References
. 235
Glossary
. 241
Index
. 247 |
| adam_txt |
Table
of
Contents
Foreword
.
XI
Introduction
. 1
The field with no escape
. 1
Historical perspectives
. 3
Outline of the contents
. 7
Notations
. 10
1
Orderings
. 11
1.1
Quasi-orderings
. 12
1.2
Ordinal numbers
. 15
1.3
Well-quasi-orderings
. 17
1.4
Kruskal's theorem
. 19
1.5
Ordered structures
. 2?,
1.6
Asymptotic relations
., , 25
1.7 Hahn
spaces
.,,.,,,,,,,, 29
1.8
Groups and rings with generalized powers
, , , ,., , 30
2
Grid-based series
. 33
2.1
Grid-based sets
. . . . 34
2.2
Grid-based series
. . . . 36
2.3
Asymptotic relations
. 40
2.3.1
Dominance and neglection
relations
. . 40
2.3.2
Flatness relations
. 42
2.3.3
Truncations
. 42
2.4
Strong linear algebra
. 44
2.4.1
Set-like notations for families
. 44
2.4.2
Infmitary operators
. 45
2.4.3
Strong abelian groups
. 46
2.4.4
Other strong structures
. 47
2.5
Grid-based summation
. 48
2.5.1
Ultra-strong grid-based algebras
. 48
2.5.2
Properties of grid-based summation
. . 49
2.5.3
Extension by strong linearity
. 50
2.6
Asymptotic scales
. 53
The Newton polygon method
. 57
3.1
The method illustrated by examples
. 58
3.1.1
The Newton polygon and its slopes
. 58
3.1.2
Equations with asymptotic constraints and refinements
. 59
3.1.3
Almost double roots
. 62
3.2
The implicit series theorem
. 63
3.3
The Newton polygon method
. 65
3.3.1
Newton polynomials and Newton degree
. 65
3.3.2
Decrease of the Newton degree during refinements
. . 66
3.3.3
Resolution of asymptotic polynomial equations
. 67
3.4
Cartesian representations
. 69
3.4.1
Cartesian representations
. 69
3.4.2
Inserting new infinitesimal monomials
. 71
3.5
Local communities
. 71
3.5.1
Cartesian communities
. 72
3.5.2
Local communities
. 72
3.5.3
Faithful Cartesian representations
. 73
3.5.4
Applications of faithful Cartesian representations
. . 74
3.5.5
The Newton polygon method revisited
. 75
Transseries
. 79
4.1
Totally ordered
exp-
log fields
. 80
4.2
Fields of grid-based
transseries
. 84
4.3
The field of grid-based
transseries
in
ж
. 87
4.3.1
Logarithmic
transseries
in
ж
. 88
4.3.2
Exponential extensions
. 88
4.3.3
Increasing unions
. 89
4.3.4
General
transseries
in
ж
. 89
4.3.5
Upward and downward shifting
. 90
4.4
The incomplete
transbasis
theorem
. 92
4.5
Convergent
transseries
. 94
5
Operations on
transseries .
97
5.1
Differentiation
. . 98
5.2
Integration
. 103
5.3
Functional composition
. . 106
5.4
Functional inversion
.
Ill
5.4.1
Existence of functional inverses
.
Ill
5.4.2
The
Translagrange
theorem
. 112
6
Grid-based operators
. 115
6.1
Multilinear grid-based operators
. 116
6.1.1
Multilinear grid-based operators
. 116
6.1.2
Operator supports
. 117
6.2
Strong tensor products
. 118
6.3
Grid-based operators
. 122
6.3.1
Definition and characterization
. 122
6.3.2
Multivariate grid-based operators and compositions
. 123
6.4
Atomic decompositions
. 124
6.4.1
The space of grid-based operators
. 124
6.4.2
Atomic decompositions
. 125
6.4.3
Combinatorial interpretation of atomic families
. 126
6.5
Implicit function theorems
. 127
6.5.1
The first implicit function theorem
. 128
6.5.2
The second implicit function theorem
. 130
6.5.3
The third implicit function theorem
. 130
6.6
Multilinear types
. 133
7
Linear differential equations
. 135
7.1
Linear differential operators
.,.,.,,. 136
7.1.1
Linear differential operators as series
. . 136
7.1.2
Multiplicative conjugation
. 137
7.1.3
Upward shifting
. 137
7.2
Differential Riccati polynomials
. 139
7.2.1
The differential Riccati polynomial
. 139
7.2.2
Properties of differential Riccati polynomials
. 140
7.3
The trace of a linear differential operator
. 141
7.3.1
The trace relative to plane
transbases
. 141
7.3.2
Dependence of the trace on the
transbasis
. 143
7.3.3
Remarkable properties of the trace
. 144
7.4
Distinguished solutions
. 146
7.4.1
Existence of distinguished right inverses
. 146
7.4.2
On the supports of distinguished solutions
. 148
7.5
The deformed Newton polygon method
. 151
7.5.1
Asymptotic Riccati equations modulo o(l)
. 151
7.5.2
Quasi-linear Riccati equations
. 152
7.5.3
Refinements
. 153
7.5.4
An algorithm for finding all solutions
. 155
7.6
Solving the homogeneous equation
. 156
7.7
Oscillating
transseries
. 158
7.7.1
Complex and oscillating
transseries
. 158
7.7.2
Oscillating solutions to linear differential equations
. 159
7.8
Factorization of differential operators
. 162
7.8.1
Existence of factorizations
. 162
7.8.2
Distinguished factorizations
. 163
8
Algebraic differential equations
. 165
8.1
Decomposing differential polynomials
. 166
8.1.1
Serial decomposition
. 166
8.1.2
Decomposition by degrees
. 167
8.1.3
Decomposition by orders
. 167
8.1.4
Logarithmic decomposition
. 168
8.2
Operations on differential polynomials
. 169
8.2.1
Additive conjugation
. 169
8.2.2
Multiplicative conjugation
. 170
8.2.3
Upward and downward shifting
. 170
8.3
The differential Newton polygon method
. 172
8.3.1
Differential Newton polynomials
. 172
8.3.2
Properties of differential Newton polynomials
. 173
8.3.3
Starting terms
. 174
8.3.4
Refinements
. 175
8.4
Finding the starting monomials
. 177
8.4.1
Algebraic starting monomials
. 177
8.4.2
Differential starting monomials
. 179
8.4.3
On the shape of the differential Newton polygon
. . . 180
8.5
Quasi-linear equations
. 182
8.5.1
Distinguished solutions
. 182
8.5.2
General
solutions
. 183
8.6
Unravelling almost multiple solutions
. 186
8.6.1
Partial unravellings
. 186
8.6.2
Logarithmic slow-down of the unravelling process
. . 188
8.6.3
On the stagnation of the depth
. 189
8.6.4
Bounding the depths of solutions
. 190
8.7
Algorithmic resolution
. 192
8.7.1
Computing starting terms
. 192
8.7.2
Solving the differential equation
. 194
8.8
Structure theorems
. 196
8.8.1
Distinguished unravellers
. 196
8.8.2
Distinguished solutions and their existence
. 197
8.8.3
On the intrusion of new exponentials
. 198
The intermediate value theorem
. 201
9.1
Compactification of total
orderings
. 202
9.1.1
The interval topology on total
orderings
. 202
9.1.2
Dedekind cuts
. 203
9.1.3
The compactness theorem
. 204
9.2
Compactification of totally ordered fields
. 206
9.2.1
Functorial properties of compactification
. 206
9.2.2
Compactification of totallj·-ordered fields
. 207
9.3
Compactification of grid-based
algebran
. 208
9.3.1
Monomial cuts
. 208
9.3.2
Width of a cut
. . . . ,. 209
9.3.3
Initializers
.,. . 210
9.3.4
Serial cuts
.,,,.,,.,.,
П0
9.3.5
Decomposition of
nou
serial
ente
. . , , 211
9.4
Compa.ctification of' the
transline
. 212
9.4.1
Exponentiation in
T
. 213
9.4.2
Classification of
transseries
cuts
. 213
9.4.3
Finite nested expansions
. 214
9.4.4
Infinite nested expansions
. 216
9.5
Integral neighbourhoods of cuts
. 219
9.5.1
Differentiation and integration of cuts
. 219
9.5.2
Integral nested expansions
. 219
9.5.3
Integral neighbourhoods
. 220
9.5.4
On the orientation of integral neighbourhoods
. 222
9.6 Differential
polynomials near cuts
. 223
9.6.1
Differential polynomials near serial cuts
. 223
9.6.2
Differential polynomials near constants
. 224
9.6.3
Differential polynomials near nested cuts
. 225
9.6.4
Differential polynomials near arbitrary cuts
. 226
9.6.5
On the sign of a differential polynomial
. 227
9.7
The intermediate value theorem
. 229
9.7.1
The quasi-linear case
. 229
9.7.2
Preserving sign changes during refinements
. 230
9.7.3
Proof of the intermediate value theorem
. 232
References
. 235
Glossary
. 241
Index
. 247 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Hoeven, Joris van der |
| author_facet | Hoeven, Joris van der |
| author_role | aut |
| author_sort | Hoeven, Joris van der |
| author_variant | j v d h jvd jvdh |
| building | Verbundindex |
| bvnumber | BV021752534 |
| callnumber-first | Q - Science |
| callnumber-label | QA247 |
| callnumber-raw | QA247.4 |
| callnumber-search | QA247.4 |
| callnumber-sort | QA 3247.4 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SI 850 |
| classification_tum | MAT 032f MAT 346f |
| ctrlnum | (OCoLC)71747730 (DE-599)BVBBV021752534 |
| dewey-full | 512.56 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.56 |
| dewey-search | 512.56 |
| dewey-sort | 3512.56 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV021752534 |
| illustrated | Illustrated |
| index_date | 2024-07-02T15:32:33Z |
| indexdate | 2024-07-20T09:07:56Z |
| institution | BVB |
| isbn | 9783540355908 3540355901 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014965735 |
| oclc_num | 71747730 |
| open_access_boolean | |
| owner | DE-824 DE-91G DE-BY-TUM DE-29T DE-384 DE-83 DE-11 DE-739 DE-188 |
| owner_facet | DE-824 DE-91G DE-BY-TUM DE-29T DE-384 DE-83 DE-11 DE-739 DE-188 |
| physical | XII, 255 S. graph. Darst. 235 mm x 155 mm |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| spelling | Hoeven, Joris van der Verfasser aut Transseries and real differential algebra J. van der Hoeven Berlin [u.a.] Springer 2006 XII, 255 S. graph. Darst. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 1888 Literaturverz. S. [235] - 239 Algèbre différentielle Differentiaalrekening gtt Séries arithmétiques Differential algebra Series, Arithmetic Differentialalgebra (DE-588)4134657-9 gnd rswk-swf Differentialalgebra (DE-588)4134657-9 s DE-604 Lecture notes in mathematics 1888 (DE-604)BV000676446 1888 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2841853&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014965735&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hoeven, Joris van der Transseries and real differential algebra Lecture notes in mathematics Algèbre différentielle Differentiaalrekening gtt Séries arithmétiques Differential algebra Series, Arithmetic Differentialalgebra (DE-588)4134657-9 gnd |
| subject_GND | (DE-588)4134657-9 |
| title | Transseries and real differential algebra |
| title_auth | Transseries and real differential algebra |
| title_exact_search | Transseries and real differential algebra |
| title_exact_search_txtP | Transseries and real differential algebra |
| title_full | Transseries and real differential algebra J. van der Hoeven |
| title_fullStr | Transseries and real differential algebra J. van der Hoeven |
| title_full_unstemmed | Transseries and real differential algebra J. van der Hoeven |
| title_short | Transseries and real differential algebra |
| title_sort | transseries and real differential algebra |
| topic | Algèbre différentielle Differentiaalrekening gtt Séries arithmétiques Differential algebra Series, Arithmetic Differentialalgebra (DE-588)4134657-9 gnd |
| topic_facet | Algèbre différentielle Differentiaalrekening Séries arithmétiques Differential algebra Series, Arithmetic Differentialalgebra |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=2841853&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014965735&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
| work_keys_str_mv | AT hoevenjorisvander transseriesandrealdifferentialalgebra |