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1. Verfasser:	Mehta, Madan Lal 1932-2006 (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Amsterdam [u.a.] Elsevier 2004
Ausgabe:	3. ed.
Schriftenreihe:	Pure and applied mathematics 142
Schlagworte:	Matrices aléatoires Mecânica estatística Random matrices Energieniveau Quantenmechanik Wahrscheinlichkeitsrechnung Stochastische Matrix
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	1. Aufl. u.d.T.: Mehta, Madan L.: Random matrices and the statistical theory of energy levels
Beschreibung:	XVIII, 688 S. graph. Darst.
ISBN:	0120884097 9780120884094

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Datensatz im Suchindex

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adam_text	CONTENTS Preface to the Third Edition ......................................... xiii Preface to the Second Edition ........................................ xv Preface to the First Edition ......................................... xvii Chapter 1. Introduction ......................................... 1 1.1. Random Matrices in Nuclear Physics ............................... 1 1.2. Random Matrices in Other Branches of Knowledge ....................... 5 1.3. A Summary of Statistical Facts about Nuclear Energy Levels .................. 8 1.3.1. Level Density ....................................... 8 1.3.2. Distribution of Neutron Widths .............................. 9 1.3.3. Radiation and Fission Widths ............................... 9 1.3.4. Level Spacings ....................................... 10 1.4. Definition of a Suitable Function for the Study of Level Correlations .............. 10 1.5. Wigner Surmise ........................................... 13 1.6. Electromagnetic Properties of Small Metallic Particles ...................... 15 1.7. Analysis of Experimental Nuclear Levels ............................. 16 1.8. The Zeros of The Riemann Zeta Function ............................. 16 1.9. Things Worth Consideration, But Not Treated in This Book ................... 30 Chapter 2. Gaussian Ensembles. The Joint Probability Density Function for the Matrix Elements . 33 2.1. Preliminaries ............................................. 33 2.2. Time-Reversal Invariance ...................................... 34 2.3. Gaussian Orthogonal Ensemble .................................. 36 2.4. Gaussian Symplectic Ensemble .................................. 38 2.5. Gaussian Unitary Ensemble .................................... 42 2.6. Joint Probability Density Function for the Matrix Elements ................... 43 2.7. Gaussian Ensemble of Hermitian Matrices With Unequal Real and Imaginary Parts ...... 48 2.8. Anti-Symmetric Hennitian Matrices ................................ 48 Summary of Chapter 2....................................... 49 vi Contents Chapters. Gaussian Ensembles. The Joint Probability Density Function for the Eigenvalues ... 50 3.1. Orthogonal Ensemble ........................................ 50 3.2. Symplectic Ensemble ........................................ 54 3.3. Unitary Ensemble .......................................... 56 3.4. Ensemble of Anti-Symmetric Hermitian Matrices ........................ 59 3.5. Gaussian Ensemble of Hermitian Matrices With Unequal Real and Imaginary Parts ...... 60 3.6. Random Matrices and Information Theory ............................ 60 Summary of Chapter 3....................................... 62 Chapter 4. Gaussian Ensembles Level Density ............................. 63 4.1. The Partition Function ....................................... 63 4.2. The Asymptotic Formula for the Level Density. Gaussian Ensembles ............. 65 4.3. The Asymptotic Formula for the Level Density. Other Ensembles ............... 67 Summary of Chapter 4....................................... 69 Chapters. Orthogonal, Skew-Orthogonal and Bi-Orthogonal Polynomials ............. 71 5.1. Quaternions, Pfaffians, Determinants ............................... 72 5.2. Average Value of П /Ui f (χ jY· Orthogonal and Skew-Orthogonal Polynomials ........ 77 5.3. Case β =2; Orthogonal Polynomials ............................... 78 5.4. Case β = 4; Skew-Orthogonal Polynomials of Quaternion Type ................ 82 5.5. Case β = I ; Skew-Orthogonal Polynomials of Real Type .................... 84 5.6. Average Value of ГЈ ·_\| ф(хј. y¡): Bi-Orthogonal Polynomials ................ 88 5.7. Correlation Functions ........................................ 89 5.8. Proof of Theorem 5.7.1....................................... 93 5.8.1. Case β = 2......................................... 93 5.8.2. Case β = 4......................................... 94 5.8.3. Case β = 1, Even Number of Variables ......................... 96 5.8.4. Case β = 1, Odd Number of Variables .......................... 99 5.9. Spacing Functions .......................................... 101 5.10. Determinantal Representations ................................... 101 5.11. Integral Representations ...................................... 103 5.12. Properties of the Zeros ....................................... 106 5.13. Orthogonal Polynomials and the Riemann—Hubert Problem ................... 107 5.14. A Remark (Balian) ......................................... 108 Summary of Chapter 5....................................... 108 Chapter 6. Gaussian Unitary Ensemble ................................. 110 6.1. Generalities ............................................. Ill 6.1.1. About Correlation and Cluster Functions ......................... Ill 6.1.2. About Level-Spacings ................................... 113 6.1.3. Spacing Distribution .................................... 118 6.1.4. Correlations and Spacings ................................. 118 6.2. The л -Point Correlation Function ................................. 118 6.3. Level Spacings ........................................... 122 6.4. Several Consecutive Spacings ................................... 127 6.5. Some Remarks ........................................... 134 Summary of Chapter 6....................................... 144 Contents vii Chapter?. Gaussian Orthogonal Ensemble ............................... 146 7.1. Generalities ............................................. 147 7.2. Correlation and Cluster Functions ................................. 148 7.3. Level Spacings. Integration Over Alternate Variables ...................... 154 7.4. Several Consecutive Spacings: n=2r ............................... 157 7.5. Several Consecutive Spacings: η = 2r — 1 ............................ 162 7.5.1. Case/! = 1......................................... 163 7.5.2. Case n = 2r- ...................................... 164 7.6. Bounds for the Distribution Function of the Spacings ...................... 168 Summary of Chapter 7....................................... 172 Chapter 8. Gaussian Symplectic Ensemble ............................... 175 8.1. A Quaternion Determinant ..................................... 175 8.2. Correlation and Cluster Functions ................................. 177 8.3. Level Spacings ........................................... 179 Summary of Chapter 8....................................... 181 Chapter 9. Gaussian Ensembles: Brownian Motion Model ...................... 182 9.1. Stationary Ensembles ........................................ 1S2 9.2. Nonstationary Ensembles ...................................... 183 9.3. Some Ensemble Averages ..................................... 187 Summary of Chapter 9....................................... 189 Chapter 10. Circular Ensembles ..................................... 191 10.1. Orthogonal Ensemble ........................................ 192 10.2. Symplectic Ensemble ........................................ 194 10.3. Unitary Ensemble .......................................... 196 10.4. The Joint Probability Density of the Eigenvalues ......................... 197 Summary of Chapter 10....................................... 201 Chapter 11. Circular Ensembles (Continued) .............................. 203 11.1. Unitary Ensemble. Correlation and Cluster Functions ...................... 203 11.2. Unitary Ensemble. Level Spacings ................................. 205 11.3. Orthogonal Ensemble. Correlation and Cluster Functions .................... 207 11.3.1. The Case Λ = 2w. Even ................................. 209 11.3.2. The Case A = Ъп + 1, Odd ............................... 210 11.3.3. Conditions of Theorem 5.1.4............................... 211 11.3.4. Correlation and Cluster Functions ............................ 212 11.4. Orthogonal Ensemble. Level Spacings ............................... 213 11.5. Symplectic Ensemble. Correlation and Cluster Functions .................... 216 11.6. Relation Between Orthogonal and Symplectic Ensembles .................... 218 11.7. Symplectic Ensemble. Level Spacings ............................... 219 11.8. Brownian Motion Model ...................................... 221 Summary of Chapter 11....................................... 223 viii ___________Contents Chapter 12. Circular Ensembles. Thermodynamics ........................... 224 12.1. The Partition Function ....................................... 224 12.2. Thermodynamic Quantities ..................................... 227 12.3. Statistical Interpretation of U and С ................................ 229 12.4. Continuum Model for the Spacing Distribution .......................... 231 Summary of Chapter 12....................................... 236 Chapter 13. Gaussian Ensemble of Anti-Symmetric Hermitian Matrices ............... 237 13.1. Level Density. Correlation Functions ............................... 237 13.2. Level Spacings ........................................... 240 13.2.1. Central Spacings ...................................... 240 13.2.2. Non-Central Spacings ................................... 242 Summary of Chapter 13....................................... 243 Chapter 14. A Gaussian Ensemble of Hermitian Matrices With Unequal Real and Imaginary Parts . 244 14.1. Summary of Results. Matrix Ensembles From GOE to GUE and Beyond ........... 245 14.2. Matrix Ensembles From GSE to GUE and Beyond ........................ 250 14.3. Joint Probability Density for the Eigenvalues ........................... 254 14.3.1. Matrices From GOE to GUE and Beyond ........................ 256 14.3.2. Matrices From GSE to GUE and Beyond ........................ 260 14.4. Correlation and Cluster Functions ................................. 263 Summary of Chapter 14....................................... 264 Chapter 15. Matrices With Gaussian Element Densities But With No Unitary or Hermitian Conditions Imposed ..................................... 266 15.1. Complex Matrices .......................................... 266 15.2. Quaternion Matrices ........................................ 273 15.3. Real Matrices ............................................ 279 15.4. Determinants: Probability Densities ................................ 281 Summary of Chapter 15....................................... 286 Chapter 16. Statistical Analysis of a Level-Sequence .......................... 287 16.1. Linear Statistic or the Number Variance .............................. 290 16.2. Least Square Statistic ........................................ 294 16.3. Energy Statistic ........................................... 298 1 6.4. Covariance of Two Consecutive Spacings ............................. 301 16.5. The F-Statistic ............................................ 302 16.6. The Л -Statistic ........................................... 303 16.7. Statistics Involving Three and Four Level Correlations ...................... 303 16.8. Other Statistics ........................................... 307 Summary of Chapter 16....................................... 308 Chapter 17. Selberg s Integral and Its Consequences .......................... 309 17.1. Selberg Integral .......................................... 309 17.2. Selberg s Proof of Eq. (17.1.3)................................... 311 Contents ix 17.3. Aomoto s Proof of Eqs. (17.1.4) and (17.1.3)........................... 315 17.4. Other Averages ........................................... 318 17.5. Other Forms of Selberg s Integral ................................. 318 17.6. Some Consequences of Selberg s Integral ............................. 320 17.7. Normalization Constant for the Circular Ensembles ....................... 323 17.8. Averages With Laguerre or Hermite Weights ........................... 323 17.9. Connection With Finite Reflection Groups ............................ 325 17.10. A Second Generalization of the Beta Integral ........................... 327 17.11. Some Related Difficult Integrals .................................. 329 Summary to Chapter 17....................................... 334 Chapter 18. Asymptotic Behaviour of Ε ρ (0. л ) by Inverse Scattering ................. 335 18.1. Asymptotics of kn(t) ........................................ 336 18.2. Asymptotics of Toeplitz Determinants ............................... 339 18.3. Fredholm Determinants and the Inverse Scattering Theory ................... 340 18.4. Application of the Geľfand-Levitan Method ........................... 342 18.5. Application of the Marchenko Method .............................. 347 18.6. Asymptotic Expansions ....................................... 350 Summary of Chapter 18....................................... 353 Chapter 19. Matrix Ensembles and Classical Orthogonal Polynomials ................ 354 19.1. Unitary Ensemble .......................................... 355 19.2. Orthogonal Ensemble ........................................ 357 19.3. Symplectic Ensemble ........................................ 361 19.4. Ensembles With Other Weights .................................. 363 19.5. Conclusion .............................................. 363 Summary of Chapter 19....................................... 364 Chapter 20. Level Spacing Functions Ε β i r. s ) ; Inter-relations and Power Series Expansions .... 365 20.1. Three Sets of Spacing Functions; Their Inter-Relations ..................... 365 20.2. Relation Between Odd and Even Solutions of Eq. (20.1.13)................... 368 20.3. Relation Between /■ ](;. .v) and F±(z.x) .............................. 371 20.4. Relation Between FąIz.s) and t±iz.s) .............................. 375 20.5. Power Series Expansions of Epi r. .v) ............................... 376 Summary of Chapter 20....................................... 381 Chapter 21. Fredholm Determinants and Painlevé Equations ...................... 382 21.1. Introduction ............................................. 382 21.2. Proof of Eqs. (21.1. 11И21. 1.17) ................................. 385 21.3. Differential Equations for the Functions A, S and S ....................... 394 21.4. Asymptotic Expansions lor Large Positive r ........................... 396 21.5. Fifth and Third Painlevé Transcendents .............................. 400 21.6. Solution of Eq. (21.3.6) for Larger ................................ 406 Summary of Chapter 21....................................... 408 Contents Chapter 22. Moments of the Characteristic Polynomial in the Three Ensembles of Random Matrices 409 22.1. Introduction ............................................. 409 22.2. Calculation of Ip(n. m; x) ..................................... 411 22.2.1. Iß (n. m ;x) as a determinant ora Pfaffian of a matrix of size depending on η .... 412 22.2.2. IaUi.m; x) as determinants of size depending on m .................. 415 22.3. Special Case of the Gaussian Weight ............................... 419 22.4. Average Value of Π?1ι detU,·/ - ΛίΠ ^ι d««;// - ЛГ1 .................. 42] Summary of Chapter 22....................................... 424 Chapter 23. Hermitian Matrices Coupled in a Chain .......................... 426 23.1. General Correlation Function .................................... 428 23.2. Proof of Theorem 23.1.1...................................... 430 23.3. Spacing Functions .......................................... 435 23.4. The Generating Function R(z .I : ...-.Zp.Ip) .......................... 437 23.5. The Zeros of the Bi-Orthogonal Polynomials ........................... 441 Summary of Chapter 23....................................... 448 Chapter 24. Gaussian Ensembles. Edge of the Spectrum ........................ 449 24.1. Level Density Near the Inflection Point .............................. 450 24.2. Spacing Functions .......................................... 452 24.3. Differential Equations: Painlevé .................................. 454 Summary to Chapter 24....................................... 458 Chapter 25. Random Permutations. Circular Unitary Ensemble (CUE) and Gaussian Unitary Ensemble (GUE) ....................................... 460 25.1. Longest Increasing Subsequences in Random Permutations ................... 460 25.2. Random Permutations and the Circular Unitary Ensemble .................... 461 25.3. Robinson-Schensted Correspondence ............................... 463 25.4. Random Permutations and GUE .................................. 468 Summary of Chapter 25....................................... 468 Chapter 26. Probability Densities of the Determinants; Gaussian Ensembles ............. 469 26.1. Introduction ............................................. 469 26.2. Gaussian Unitary Ensemble .................................... 473 26.2.1. Mellin Transform of the PDD .............................. 473 26.2.2. Inverse Mellin Transforms ................................ 475 26.3. Gaussian Symplectic Ensemble .................................. 477 26.4. Gaussian Orthogonal Ensemble .................................. 480 26.5. Gaussian Orthogonal Ensemble. Case ;i = 2m + 1 Odd ..................... 482 26.6. Gaussian Orthogonal Ensemble. Case η = 2m Even ....................... 483 Summary of Chapter 26....................................... 486 Chapter 27. Restricted Trace Ensembles ................................. 487 27.1. Fixed Trace Ensemble; Equivalence of Moments ......................... 487 27.2. Probability Density of the Determinant .............................. 490 Contents xi 27.3. Bounded Trace Ensembles..................................... 492 Summary of Chapter 27....................................... 493 Appendices .................................................. 494 A.I. Numerical Evidence in Favor of Conjectures 1.2.1 and 1.2.2 .................. 494 A.2. The Probability Density of the Spacing» Resulting from a Random Superposition of η Unrelated Sequences of Energy Levels .............................. 495 A.3. Some Properties of Hermitian, Unitary, Symmetric or Self-Dual Matrices ........... 498 A. 4. Counting the Dimensions of Tpc and Тп(. ............................ 499 A.5. An Integral Over the Unitary Group ................................ 500 A.6. The Minimum Value of W ..................................... 504 АЛ. Relation Between R„, Т„ and I-An; 26).............................. 506 A.8. Relation Between E(n;s), Fin: si and /;(/;; .v) .......................... 510 A.9. The Limit of Σο ^ ψ){χ) ..................................... 51() АЛО. The Limits of Σο ~ P/(.v)<ŕ/(v) ................................. 511 A.ll. The Fourier Transforms of the Two-Point Cluster Functions ................... 514 A. 12. Some Applications of Gram s Formula .............................. 516 A. 13. Power Series Expansions of Eigenvalues, of Spheroidal Functions and of Various Probabilities 517 A.14. Numerical Tables of Ay(i). /;;(л) and Epitr.s) for fi = 1.2 ¿md4 ............... 521 A.15. Numerical Values of fy (0; л). Φ^(.ν) and ;^(0:.ν) for β = 1.2 and л- < 3.7.......... 524 A.16. Proof of Eqs. (21.1.1 1)-(21. 1.16), (24.3.11), (24.3.15) and (24.3.20) Using a Compact Notation 525 A. 16.1. Preliminaries ........................................ 525 A.16.2. The Sine Kernel: Definitions and Properties ....................... 527 A.16.3. The Airy Kernel ...................................... 531 АЛ7. Use of Pfaffians in Some Multiple Integrals ............................ 536 A. 18. Calculation of Certain Determinants ................................ 53X A.19. Power-Series Expansion of /„,(«). Eq. (7.6.12).......................... 550 A.20. Proof of the Inequalities (7.6.15).................................. 551 A.21. ProofofEqs.(10.1.11)and(10.2.11) ............................... 552 A.22. Proof of the Inequality (12.1.5)................................... 553 A.23. Good s Proof of Eq. (12.1.16)................................... 554 A. 24. Some Recurrence Relations and Integrals Used in Chapter 14.................. 555 A.25. Normalization Integral, Eq. (14.1.11)............................... 560 A.26. Another Normalization Integral, Eq. (14.2.9)........................... 564 A. 27. Joint Probability Density as a Determinant of a Self-Dual Quaternion Matrix. Section 14.4, Eqs. (14.4.2) and (14.4.5)...................................... 565 A.28. Verification of Eq. (14.4.3)..................................... 569 A.29. The Limits of /yU. v) and D (x. y) as Λ- -»■ oc ........................ 572 A.30. Evaluation of the Integral (15.1.9) for Complex Matrices .................... 573 A.31. A Few Remarks About the Eigenvalues of a Quaternion Real Matrix .............. 577 A.32. Evaluation of the Integral Corresponding to (15.2.9)....................... 579 A.33. Another Proof of Eqs. (15.1.10) and (15.2.10) .......................... 582 A.34. Proof of Eq. (15.2.38)........................................ 5X4 A.35. Partial Triangulation of a Matrix .................................. 585 A. 36. Average Number of Real Eigenvalues of a Real Gaussian Random Matrix ........... 587 A. 37. Probability Density of the Eigenvalues of a Real Random Matrix When к of Its Eigenvalues Are Real ............................................... 588 A.38. Variance of the Number Statistic. Section 16.1.......................... 594 A.38.1. Averages of the Powers of и via;; -Level Spaeings .................... 595 A.39. Optimum Linear Statistic. Section 16.1.............................. 601 xii ______ _____ Contents A.40. Mean Value of Δ. Section 16.2................................... 603 A.41. Tables of Functions Βρ(χλ,χ2) and PpUi ,.γτ) for /J = 1 and 2................ 607 A.42. Sums aj^ and a {r¡ for n = 1,2 and 3, Section 20.5....................... 611 A.43. Values of α ^ , ^1. and« ѓД for Low Values of j and я ................... 613 A.44. A Personal Recollection ...................................... 619 A.45. About Painlevé Transcendents ................................... 620 A.45.1. Six Painlevé Equations .................................. 620 A.45.2. Lax Pairs .......................................... 621 A.45.3. Hamiltonian Equations .................................. 624 A.45.4. Confluences of the Hamiltonians and the Painlevé Equations ............. 627 A.46. Inverse Power Series Expansions of <S,¡( г ), Α,Ατ), В„( г ), etc .................. 629 А.47. Table of Values of«,, in Eq. (21.4.6) for Small Values of и ................... 632 A.48. Some Remarks About the Numerical Computations ....................... 633 A.49. Convolution of Two Gaussian Kernels ............................... 634 A.50. Method of the Change of Variables. Wick s Theorem ...................... 636 A.51. Some Remarks About the Integral Hk.n),Eq. (25.2.12)..................... 638 A.52. Meijer G-functions for Small and Large Values of the Variable ................. 640 A.52. 1. Computation of the G-functions Near the Origin .................... 640 A.52. 2. G-functions for Large Values of the Variable ...................... 642 A. 53. About Binary Quadratic Forms ................................... 642 Notes ..................................................... 645 References .................................................. 655 Author Index ................................................. 680 Subject Index ................................................. 684
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id	DE-604.BV019697335
illustrated	Illustrated
indexdate	2024-07-09T20:04:04Z
institution	BVB
isbn	0120884097 9780120884094
language	English
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-013024987
oclc_num	56987512
open_access_boolean
owner	DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-29T DE-355 DE-BY-UBR DE-703 DE-83 DE-11 DE-188
owner_facet	DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-29T DE-355 DE-BY-UBR DE-703 DE-83 DE-11 DE-188
physical	XVIII, 688 S. graph. Darst.
publishDate	2004
publishDateSearch	2004
publishDateSort	2004
publisher	Elsevier
record_format	marc
series	Pure and applied mathematics
series2	Pure and applied mathematics
spelling	Mehta, Madan Lal 1932-2006 Verfasser (DE-588)13927667X aut Random matrices Madan Lal Mehta 3. ed. Amsterdam [u.a.] Elsevier 2004 XVIII, 688 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics 142 1. Aufl. u.d.T.: Mehta, Madan L.: Random matrices and the statistical theory of energy levels Matrices aléatoires Mecânica estatística larpcal Random matrices Energieniveau (DE-588)4152225-4 gnd rswk-swf Quantenmechanik (DE-588)4047989-4 gnd rswk-swf Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf Stochastische Matrix (DE-588)4057624-3 gnd rswk-swf Stochastische Matrix (DE-588)4057624-3 s DE-604 Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s 1\p DE-604 Energieniveau (DE-588)4152225-4 s 2\p DE-604 Quantenmechanik (DE-588)4047989-4 s 3\p DE-604 Pure and applied mathematics 142 (DE-604)BV010177228 142 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013024987&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk
spellingShingle	Mehta, Madan Lal 1932-2006 Random matrices Pure and applied mathematics Matrices aléatoires Mecânica estatística larpcal Random matrices Energieniveau (DE-588)4152225-4 gnd Quantenmechanik (DE-588)4047989-4 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Stochastische Matrix (DE-588)4057624-3 gnd
subject_GND	(DE-588)4152225-4 (DE-588)4047989-4 (DE-588)4064324-4 (DE-588)4057624-3
title	Random matrices
title_auth	Random matrices
title_exact_search	Random matrices
title_full	Random matrices Madan Lal Mehta
title_fullStr	Random matrices Madan Lal Mehta
title_full_unstemmed	Random matrices Madan Lal Mehta
title_short	Random matrices
title_sort	random matrices
topic	Matrices aléatoires Mecânica estatística larpcal Random matrices Energieniveau (DE-588)4152225-4 gnd Quantenmechanik (DE-588)4047989-4 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Stochastische Matrix (DE-588)4057624-3 gnd
topic_facet	Matrices aléatoires Mecânica estatística Random matrices Energieniveau Quantenmechanik Wahrscheinlichkeitsrechnung Stochastische Matrix
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013024987&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV010177228
work_keys_str_mv	AT mehtamadanlal randommatrices

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