Statistics on special manifolds:
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Springer
2003
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| Schriftenreihe: | Lecture notes in statistics
174 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XXVI, 399 S. |
| ISBN: | 0387001603 |
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Datensatz im Suchindex
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| adam_text | YASUKO CHIKUSE STATISTICS ON SPECIAL MANIFOLDS SPRINGER CONTENTS PREFACE
VII COMMONLY USED NOTATION XXI 1. THE SPECIAL MANIFOLDS AND RELATED
MULTIVARIATE TOPICS 1 1.1. INTRODUCTION, 1 1.2. ANALYTIC MANIFOLDS AND
RELATED TOPICS, 2 1.2.1. SPACES AND GROUPS AND INVARIANT MEASURES, 3
1.2.2. ANALYTIC MANIFOLDS AND EXTERIOR DIFFERENTIAL FORMS, 4 1.3. THE
SPECIAL STIEFEL AND GRASSMANN MANIFOLDS, 8 1.3.1. THE STIEFEL MANIFOLD V
K M , 8 1.3.2. THE GRASSMANN MANIFOLD G K M _ K AND ITS EQUIVALENT
MANIFOLD P K M _ K , 9 1.3.3. EXAMPLES OF ORIENTATION STATISTICS, 12
1.4. THE INVARIANT MEASURES ON THE SPECIAL MANIFOLDS, 14 1.4.1. THE
INVARIANT MEASURE ON THE ORTHOGONAL GROUP O(M) (= V M J, 14 1.4.2. THE
INVARIANT MEASURES ON THE GRASSMANN AND ITS EQUIVALENT MANIFOLDS G K _ K
AND P K M - K , 15 1.4.3. THE INVARIANT MEASURE ON THE STIEFEL MANIFOLD
1.4.4. INTEGRALS OF THE INVARIANT MEASURES, 16 1.5. JACOBIANS AND SOME
RELATED MULTIVARIATE DISTRIBUTIONS, 18 1.5.1. JACOBIANS OF SOME MATRIX
TRANSFORMATIONS, 18 1.5.2. SYMMETRIC MATRIX-VARIATE NORMAL
DISTRIBUTIONS, 22 1.5.3. RECTANGULAR MATRIX-VARIATE NORMAL
DISTRIBUTIONS, 23 1.5.4. (NONCENTRAL) WISHART DISTRIBUTIONS, 25 XIV
CONTENTS 2. DISTRIBUTIONS ON THE SPECIAL MANIFOLDS 27 2.1. INTRODUCTION,
27 2.2. PROPERTIES OF THE UNIFORM DISTRIBUTIONS, 28 X 2.3. NON-UNIFORM
DISTRIBUTIONS, 31 2.3.1. NON-UNIFORM DISTRIBUTIONS ON V KM , 31 2.3.2.
NON-UNIFORM DISTRIBUTIONS ON P K M _ K , 37 2.4. DISTRIBUTIONS OF THE
ORIENTATIONS OF A RANDOM MATRIX, 39 2.4.1. DISTRIBUTIONS OF THE
ORIENTATION OF A RANDOM MATRIX, 39 2.4.2. THE MATRIX ANGULAR CENTRAL
GAUSSIAN DISTRIBUTION, 40 2.4.3. DISTRIBUTIONS OF THE ORIENTATION OF A
LINEAR TRANSFORMATION, 42 2.4.4. THE ORTHOGONAL PROJECTION MATRIX OF A
RANDOM MATRIX, 45 2.5. SIMULATION METHODS FOR GENERATING PSEUDO-RANDOM
MATRICES ON V KM AND P FEITN _ FC) 48 2.5.1. GENERATING UNIFORMLY
DISTRIBUTED MATRICES, 48 2.5.2. GENERATING MATRICES FROM GIVEN
DISTRIBUTIONS, 50 3. DECOMPOSITIONS OF THE SPECIAL MANIFOLDS 53 3.1.
INTRODUCTION, 53 3.2. DECOMPOSITIONS ONTO ORTHOGONALLY COMPLEMENTARY
SUBSPACES OF V K M , 55 3.2.1. DECOMPOSITIONS OF V KM , 55 3.2.2.
DECOMPOSITIONS OF V K M , 65 3.3. OTHER DECOMPOSITIONS OF V K , 69
3.4. ONE-TO-ONE TRANSFORMATIONS OF P K M - K ONTO R M _ K K ORI^-^,72
3.5. ANOTHER DECOMPOSITION OF P K M _ K (OR G K M _ K ), 78 4.
DISTRIBUTIONAL PROBLEMS IN THE DECOMPOSITION THEOREMS AND THE SAMPLING
THEORY 81 4.1. INTRODUCTION, 81 4.2. DISTRIBUTIONS OF THE COMPONENT
MATRIX VARIATES IN THE DECOMPOSITIONS OF THE SPECIAL MANIFOLDS, 82
4.2.1. DECOMPOSITIONS IN THEOREM 3.2.1, 82 4.2.2. DECOMPOSITIONS IN
THEOREM 3.3.1 AND COROLLARY 3.3.2, 85 CONTENTS XV 4.2.3. DECOMPOSITIONS
IN THEOREM 3.3.3, 87 4.2.4. DECOMPOSITIONS IN THEOREMS 3.4.1 AND 3.4.3
AND COROLLARY 3.4.2, 88 4.3. DISTRIBUTIONS OF CANONICAL CORRELATION
COEFFICIENTS OF GENERAL DIMENSION, 89 4.3.1. CANONICAL CORRELATION
COEFFICIENTS OF GENERAL DIMENSION, 89 4.3.2. APPLICATIONS FOR THE
MULTIVARIATE NORMAL DISTRIBUTION, 91 4.3.3. APPLICATIONS FOR THE MATRIX
CONDITIONALLY LANGEVIN DISTRIBUTION, 94 4.4. GENERAL FAMILIES OF
DISTRIBUTIONS ON V KM AND P K M _ K , 95 4.4.1. GENERAL FAMILIES OF
DISTRIBUTIONS ON V KM , 95 4.4.2. A GENERAL FAMILY OF DISTRIBUTIONS ON P
K , M _ K , 98 4.5. SAMPLING THEORY FOR THE MATRIX LANGEVIN
DISTRIBUTIONS, 99 4.5.1. DISTRIBUTIONS OF THE SAMPLE SUM ON V K M , 99
4.5.2. DISTRIBUTIONS OF THE SAMPLE SUM ON P K M _ FC , 106 5. THE
INFERENCE ON THE PARAMETERS OF THE MATRIX LANGEVIN DISTRIBUTIONS 109
5.1. INTRODUCTION, 109 5.2. FISHER SCORING METHODS ON V FEM , 111 5.2.1.
MAXIMUM LIKELIHOOD ESTIMATORS OF THE PARAMETERS F, 0, AND A, 111 5.2.2.
SCORING METHOD FOR THE PARAMETER F, 113 5.2.3. PROFILE LIKELIHOOD
METHOD, 114 5.3. OTHER TOPICS IN THE INFERENCE ON THE ORIENTATION
PARAMETERS ON V K M , 117 5.3.1. BAYES ESTIMATORS OF F AND 0 WITH A
KNOWN, 117 5.3.2. OPTIMALITY PROPERTIES OF THE ESTIMATORS ON V KM , 119
5.3.3. SUFFICIENCY AND ANCILLARITY, 123 5.4. FISHER SCORING METHODS ON P
K - K , 125 5.4.1. MAXIMUM LIKELIHOOD ESTIMATORS OF THE PARAMETERS F AND
A, 125 5.4.2. SCORING METHOD FOR THE PARAMETER B, 127 5.4.3. PROFILE
LIKELIHOOD METHOD, 128 CONTENTS 5.5. OTHER TOPICS IN THE INFERENCE ON
THE ORIENTATION PARAMETER ON P K M _ K , 129 5.5.1. BAYES ESTIMATOR OF F
WITH A KNOWN, 129 5.5.2. OPTIMALITY PROPERTIES OF THE ESTIMATORS ON P K
, M _ K , 129 5.5.3. SUFFICIENCY AND ANCILLARITY, 131 6. LARGE SAMPLE
ASYMPTOTIC THEOREMS IN CONNECTION WITH TESTS FOR UNIFORMITY 133 6.1.
INTRODUCTION, 133 6.2. ASYMPTOTIC EXPANSIONS FOR THE SAMPLE MEAN MATRIX
ON V KIM , 135 6.2.1. THE STANDARDIZED SAMPLE MEAN MATRIX Z, 135 6.2.2.
ASYMPTOTIC DISTRIBUTIONS OF Z AND Z Z, 135 6.2.3. FURTHER RESULTS ON
ASYMPTOTIC DISTRIBUTIONS AND TESTING PROBLEMS, 139 6.3. ASYMPTOTIC
PROPERTIES OF THE PARAMETER ESTIMATION AND THE TESTS FOR UNIFORMITY ON V
K M , 145 6.3.1. A RAYLEIGH-STYLE TEST, 145 6.3.2. THE MAXIMUM
LIKELIHOOD ESTIMATORS AND THE LIKELIHOOD RATIO TEST, 146 6.3.3. THE
SCORE FUNCTIONS AND THE RAO SCORE TEST, 147 6.3.4. THE PROFILE SCORE
FUNCTIONS AND TESTS, 148 6.3.5. THE LOCALLY BEST INVARIANT TEST, 149 .
6.3.6. ASYMPTOTIC EQUIVALENCE OF THE OPTIMAL TESTS FOR UNIFORMITY, 149
6.4. ASYMPTOTIC EXPANSIONS FOR THE SAMPLE MEAN MATRIX ON P FC , M _ FC ,
150 6.4.1. THE STANDARDIZED SAMPLE MEAN MATRIX U, 150 6.4.2. ASYMPTOTIC
DISTRIBUTIONS OF U, 15 1 6.4.3. AN ALTERNATIVE METHOD, 156 6.5.
ASYMPTOTIC PROPERTIES OF THE PARAMETER ESTIMATION AND THE TESTS FOR
UNIFORMITY ON P K M _ FC , 159 6.5.1. A RAYLEIGH-STYLE TEST, 159 6.5.2.
THE MAXIMUM LIKELIHOOD ESTIMATORS AND THE LIKELIHOOD RATIO TEST, 159
6.5.3. THE SCORE FUNCTIONS AND THE RAO SCORE TEST, 160 6.5.4. THE
PROFILE SCORE FUNCTIONS AND TESTS, 161 6.5.5. THE LOCALLY BEST INVARIANT
TEST, 162 6.5.6. ASYMPTOTIC EQUIVALENCE OF THE OPTIMAL TESTS FOR
UNIFORMITY, 163 CONTENTS XVII 7. ASYMPTOTIC THEOREMS FOR CONCENTRATED
MATRIX LANGEVIN DISTRIBUTIONS 165 7.1. INTRODUCTION, 165 7.2. ESTIMATION
OF LARGE CONCENTRATION PARAMETERS, 166 7.2.1. ESTIMATION ON V KM , 166
7.2.2. ESTIMATION ON P K M _ K , 168 7.3. ASYMPTOTIC DISTRIBUTIONS IN
CONNECTION WITH TESTING HYPOTHESES OF THE ORIENTATION PARAMETERS ON V K
M , 170 7.3.1. TESTING HYPOTHESES OF THE ORIENTATION PARAMETERS AND
RELATED STATISTICS, 170 7.3.2. ASYMPTOTIC DISTRIBUTIONS OF THE RELATED
STATISTICS, 171 7.4. ASYMPTOTIC DISTRIBUTIONS IN CONNECTION WITH TESTING
HYPOTHESES OF THE ORIENTATION PARAMETER ON P K M _ K , 176 7.4.1.
TESTING HYPOTHESES OF THE ORIENTATION PARAMETER AND RELATED STATISTICS,
176 7.4.2. ASYMPTOTIC DISTRIBUTIONS OF THE RELATED STATISTICS, 177 7.5.
CLASSIFICATION OF THE MATRIX LANGEVIN DISTRIBUTIONS, 182 7.5.1.
CLASSIFICATION ON V K M , 182 7.5.2. CLASSIFICATION ON P K M _ K , 185
8. HIGH DIMENSIONAL ASYMPTOTIC THEOREMS 187 8.1. INTRODUCTION, 187
8.2. ASYMPTOTIC EXPANSIONS FOR THE MATRIX LANGEVIN DISTRIBUTIONS ON V KM
, 189 8.2.1. THE MATRIX LANGEVIN L(M,K;F) DISTRIBUTION, 189 8.2.2. THE
MATRIX LANGEVIN L(M, FC; M 1 / 2 F) DISTRIBUTION, 194 8.3. ASYMPTOTIC
EXPANSIONS FOR THE MATRIX BINGHAM AND LANGEVIN DISTRIBUTIONS ON V KM AND
P K M _ K , 198 8.3.1. THE MATRIX BINGHAM B(M,K;B) AN D LANGEVIN U P
){M,K;B) DISTRIBUTIONS, 198 8.3.2. THE MATRIX BINGHAM B(M, K; MB) AND
LANGEVIN L - P )(M,K;MB) DISTRIBUTIONS, 203 CONTENTS 8.4. GENERALIZED
STAM S LIMIT THEOREMS, 208 8.4.1. THE FIRST THEOREM, 208 8.4.2. THE
SECOND THEOREM (LIMIT ORTHOGONALITY 8.5. ASYMPTOTIC PROPERTIES OF THE
PARAMETER ESTIMATION AND THE TESTS OF HYPOTHESES, 214 8.5.1. LIMIT
PROPERTIES OF VARIOUS STATISTICS, 214 8.5.2. ASYMPTOTIC PROPERTIES FOR
THE MATRIX LANGEVIN DISTRIBUTIONS ON V K M , 217 8.5.3. ASYMPTOTIC
PROPERTIES FOR THE MATRIX LANGEVIN DISTRIBUTIONS ON P K M _ FC , 222 9.
PROCRUSTES ANALYSIS ON THE SPECIAL MANIFOLDS 231 9.1. INTRODUCTION, 231
9.2. PROCRUSTES REPRESENTATIONS OF THE MANIFOLDS, 232 9.2.1. ORDINARY
PROCRUSTES ANALYSIS, 232 9.2.2. GENERALIZED PROCRUSTES ANALYSIS, 237
9.3. PERTURBATION THEORY, 243 9.3.1. PROCRUSTES STATISTICS WITH ERRORS
IN ONE MATRIX, 243 9.3.2. PROCRUSTES STATISTICS WITH ERRORS IN BOTH
MATRICES, 245 9.4. EMBEDDINGS, 246 10. DENSITY ESTIMATION ON THE SPECIAL
MANIFOLDS 247 10.1. INTRODUCTION, 247 10.2. KERNEL DENSITY ESTIMATION ON
P K M _ K , 249 10.2.1. GENERAL DISCUSSION, 249 10.2.2. APPLICATIONS FOR
A SPECIAL KERNEL FUNCTION, 256 10.3. KERNEL DENSITY ESTIMATION ON V K M
, 258 10.3.1. GENERAL DISCUSSION, 258 10.3.2. APPLICATIONS FOR A SPECIAL
KERNEL FUNCTION, 263 10.4. DENSITY ESTIMATION VIA THE DECOMPOSITIONS (OR
TRANSFORMATIONS) OF P KM _ K AND V FCM , 264 10.4.1. DENSITY ESTIMATION
ON P KM _ K , 264 10.4.2. DENSITY ESTIMATION ON V KM , 265 CONTENTS XIX
10.5. DENSITY ESTIMATION ON THE SPACES M AND # M , P , 266 10.5.1.
DENSITY ESTIMATION ON S M , 266 10.5.2. DENSITY ESTIMATION ON R MP , 271
10.5.3. DENSITY ESTIMATION FOR MULTIPLE RANDOM MATRICES, 276 11.
MEASURES OF ORTHOGONAL ASSOCIATION ON THE SPECIAL MANIFOLDS 283 11.1.
INTRODUCTION, 283 11.2. MEASURES OF ORTHOGONAL ASSOCIATION ON V K M ,
284 11.3. MEASURES OF ORTHOGONAL ASSOCIATION ON P K - K , 286 11.4.
DISTRIBUTIONAL AND SAMPLING PROBLEMS ON V K M , 287 11.4.1. A FAMILY OF
ASSOCIATED DISTRIBUTIONS, 287 11.4.2. DISTRIBUTIONS OF THE ESTIMATED
MEASURE OF ORTHOGONAL ASSOCIATION, 288 11.5. RELATED REGRESSION MODELS
ON V KM , 292 APPENDIX A. INVARIANT POLYNOMIALS WITH MATRIX ARGUMENTS
295 A.I. INTRODUCTION, 295 A.2. ZONAL POLYNOMIALS, 297 A.3. INVARIANT
POLYNOMIALS WITH MULTIPLE MATRIX ARGUMENTS, 299 A.4. BASIC PROPERTIES OF
INVARIANT POLYNOMIALS, 301 A.5. SPECIAL CASES OF INVARIANT POLYNOMIALS,
312 A.6. HYPERGEOMETRIC FUNCTIONS WITH MATRIX ARGUMENTS, 315 A.7. TABLES
OF ZONAL AND INVARIANT POLYNOMIALS, 319 APPENDIX B. GENERALIZED HERMITE
AND LAGUERRE POLYNOMIALS WITH MATRIX ARGUMENTS 321 B.I. INTRODUCTION,
321 B.2. GENERALIZED HERMITE POLYNOMIALS HW. WITH SYMMETRIC MATRIX
ARGUMENTS, 322 B.2.1. SERIES (EDGEWORTH) EXPANSIONS FOR MULTIPLE RANDOM
SYMMETRIC MATRICES, 322 B.2.2. DEFINITION OF THE POLYNOMIALS H^.^, 32 4
B.2.3. VARIOUS PROPERTIES OF THE POLYNOMIALS H^ ;(F) , 325 XX CONTENTS
B.3. GENERALIZED HERMITE POLYNOMIALS H^TFL WITH RECTANGULAR MATRIX
ARGUMENTS, 331 B.3.1. SERIES (EDGEWORTH) EXPANSIONS FOR MULTIPLE RANDOM
RECTANGULAR MATRICES, 331 B.3.2. DEFINITION OF THE POLYNOMIALS FFJ^JFJ,
333 B.3.3. VARIOUS PROPERTIES OF THE POLYNOMIALS #J^JFJ, 333 B.4.
GENERALIZED LAGUERRE POLYNOMIALS IN MULTIPLE MATRICES, 337 B.4.1.
GENERALIZED (CENTRAL) LAGUERRE POLYNOMIALS, 337 B.4.2. GENERALIZED
NONCENTRAL LAGUERRE POLYNOMIALS, 339 B.5. GENERALIZED MULTIVARIATE
MEIXNER CLASSES OF INVARIANT DISTRIBUTIONS OF MULTIPLE RANDOM MATRICES,
344 APPENDIX C. EDGEWORTH AND SADDLE-POINT EXPANSIONS FOR RANDOM
MATRICES 347 C.I. INTRODUCTION, 347 C.2. THE CASE OF RANDOM SYMMETRIC
MATRICES, 348 G.2.1. EDGEWORTH EXPANSIONS, 348 C.2.2. SADDLE-POINT
EXPANSIONS, 354 C.2.3. GENERALIZED EDGEWORTH EXPANSIONS, 356 C.3. THE
CASE OF RANDOM RECTANGULAR MATRICES, 358 C.3.1. EDGEWORTH EXPANSIONS,
358 C.3.2. SADDLE-POINT EXPANSIONS, 365 C.3.3. GENERALIZED EDGEWORTH
EXPANSIONS, 367 C.4. APPLICATIONS, 369 - C.4.1. EXACT SADDLE-POINT
APPROXIMATIONS, 369 C.4.2. APPLICATIONS FOR THE MATRIX LANGEVIN
DISTRIBUTIONS ON V KTM , 370 C.4.3. APPLICATIONS FOR THE MATRIX LANGEVIN
DISTRIBUTIONS ON P FC , M _ FC , 374 BIBLIOGRAPHY 379 INDEX 391
|
| any_adam_object | 1 |
| author | Chikuse, Yasuko |
| author_facet | Chikuse, Yasuko |
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| author_sort | Chikuse, Yasuko |
| author_variant | y c yc |
| building | Verbundindex |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516/.07 21 516/.07 |
| dewey-search | 516/.07 21 516/.07 |
| dewey-sort | 3516 17 221 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| id | DE-604.BV015022671 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T19:08:58Z |
| institution | BVB |
| isbn | 0387001603 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-010091538 |
| oclc_num | 51093121 |
| open_access_boolean | |
| owner | DE-824 DE-739 DE-19 DE-BY-UBM DE-521 DE-83 |
| owner_facet | DE-824 DE-739 DE-19 DE-BY-UBM DE-521 DE-83 |
| physical | XXVI, 399 S. |
| publishDate | 2003 |
| publishDateSearch | 2003 |
| publishDateSort | 2003 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in statistics |
| series2 | Lecture notes in statistics |
| spelling | Chikuse, Yasuko Verfasser aut Statistics on special manifolds Yasuko Chikuse New York [u.a.] Springer 2003 XXVI, 399 S. txt rdacontent n rdamedia nc rdacarrier Lecture notes in statistics 174 Análise multivariada larpcal Estimação de densidades larpcal Manifolds gtt Statistique mathématique Teoria assintótica (inferência estatística) larpcal Variétés (Mathématiques) Manifolds (Mathematics) Mathematical statistics Statistik (DE-588)4056995-0 gnd rswk-swf Graßmann-Mannigfaltigkeit (DE-588)4158085-0 gnd rswk-swf Stiefel-Mannigfaltigkeit (DE-588)4183295-4 gnd rswk-swf Graßmann-Mannigfaltigkeit (DE-588)4158085-0 s Stiefel-Mannigfaltigkeit (DE-588)4183295-4 s Statistik (DE-588)4056995-0 s DE-604 Lecture notes in statistics 174 (DE-604)BV002447846 174 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010091538&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Chikuse, Yasuko Statistics on special manifolds Lecture notes in statistics Análise multivariada larpcal Estimação de densidades larpcal Manifolds gtt Statistique mathématique Teoria assintótica (inferência estatística) larpcal Variétés (Mathématiques) Manifolds (Mathematics) Mathematical statistics Statistik (DE-588)4056995-0 gnd Graßmann-Mannigfaltigkeit (DE-588)4158085-0 gnd Stiefel-Mannigfaltigkeit (DE-588)4183295-4 gnd |
| subject_GND | (DE-588)4056995-0 (DE-588)4158085-0 (DE-588)4183295-4 |
| title | Statistics on special manifolds |
| title_auth | Statistics on special manifolds |
| title_exact_search | Statistics on special manifolds |
| title_full | Statistics on special manifolds Yasuko Chikuse |
| title_fullStr | Statistics on special manifolds Yasuko Chikuse |
| title_full_unstemmed | Statistics on special manifolds Yasuko Chikuse |
| title_short | Statistics on special manifolds |
| title_sort | statistics on special manifolds |
| topic | Análise multivariada larpcal Estimação de densidades larpcal Manifolds gtt Statistique mathématique Teoria assintótica (inferência estatística) larpcal Variétés (Mathématiques) Manifolds (Mathematics) Mathematical statistics Statistik (DE-588)4056995-0 gnd Graßmann-Mannigfaltigkeit (DE-588)4158085-0 gnd Stiefel-Mannigfaltigkeit (DE-588)4183295-4 gnd |
| topic_facet | Análise multivariada Estimação de densidades Manifolds Statistique mathématique Teoria assintótica (inferência estatística) Variétés (Mathématiques) Manifolds (Mathematics) Mathematical statistics Statistik Graßmann-Mannigfaltigkeit Stiefel-Mannigfaltigkeit |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010091538&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV002447846 |
| work_keys_str_mv | AT chikuseyasuko statisticsonspecialmanifolds |