Holdings: Advanced linear algebra

Advanced linear algebra:

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Main Author:	Loehr, Nicholas A. (Author)
Format:	Book
Language:	English
Published:	Boca Raton [u.a.] CRC Press 2014
Series:	Textbooks in mathematics A Chapman & Hall book
Subjects:	Algebras, Linear Lineare Algebra Electronic books
Online Access:	Inhaltsverzeichnis Klappentext
Item Description:	Includes bibliographical references
Physical Description:	XXII, 609 S. graph. Darst.
ISBN:	9781466559011

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adam_text	Contents й Preface xvii I Background on Algebraic Structures 1 1 Overview of Algebraic Systems 3 1.1 Groups ...................,.................. 3 1.2 Rings and Fields ................................. 4 1.3 Vector Spaces .................................. 6 1.4 Subsystems .................................... 7 1.5 Product Systems ................................. 8 1.6 Quotient Systems ................................ 9 1.7 Homomorphisms ................................. 11 1.8 Spanning, Linear Independence, Basis, and Dimension ............ 13 1.9 Summary ..................................... 15 1.10 Exercises ..................................... 17 2 Permutations 23 2.1 Symmetric Groups ................................ 23 2.2 Representing Functions as Directed Graphs .................. 24 2.3 Cycle Decompositions of Permutations .................... 24 2.4 Composition of Cycles .............................. 26 2.5 Factorizations of Permutations ......................... 27 2.6 Inversions and Sorting .............................. 28 2.7 Signs of Permutations .............................. 29 2.8 Summary ..................................... 30 2.9 Exercises ..................................... 30 3 Polynomials 35 3.1 Intuitive Definition of Polynomials ....................... 35 3.2 Algebraic Operations on Polynomials ..................... 36 3.3 Formal Power Series and Polynomials ..................... 37 3.4 Properties of Degree ............................... 39 3.5 Evaluating Polynomials ............................. 40 3.6 Polynomial Division with Remainder ..................... 41 3.7 Divisibility and Associates ........................... 43 3.8 Greatest Common Divisors of Polynomials .................. 44 3.9 GCDs of Lists of Polynomials ......................... 45 3.10 Matrix Reduction Algorithm for GCDs .................... 46 3.11 Roots of Polynomials .............................. 48 3.12 Irreducible Polynomials ............................. 49 3.13 Factorization of Polynomials into Irreducibles ................. 51 3.14 Prime Factorizations and Divisibility ..................... 52 vii viii Contents 3.15 Irreducible Polynomials in Q[x] ......................... 53 3.16 Irreducibility in Q[x] via Reduction Mod ρ .................. 54 3.17 Eisenstein s Irreducibility Criterion for Q[x] .................. 54 3.18 Kronecker s Algorithm for Factoring in Q[x] ................. 55 3.19 Algebraic Elements and Minimal Polynomials ................ 56 3.20 Multivariable Polynomials ........................... 58 3.21 Summary ..................................... 60 3.22 Exercises ..................................... 62 II Matrices 71 4 Basic Matrix Operations 73 4.1 Formal Definition of Matrices and Vectors .................. 73 4.2 Vector Spaces of Functions ........................... 74 4.3 Matrix Operations via Entries ......................... 75 4.4 Properties of Matrix Multiplication .............·......... 77 4.5 Generalized Associativity ............................ 78 4.6 Invertible Matrices ................................ 79 4.7 Matrix Operations via Columns ........................ 81 4.8 Matrix Operations via Rows .......................... 83 4.9 Elementary Operations and Elementary Matrices .............. 84 4.10 Elementary Matrices and Gaussian Elimination ............... 86 4.11 Elementary Matrices and Invertibility ..................... 87 4.12 Row Rank and Column Rank .......................... 87 4.13 Conditions for Invertibility of a Matrix .................... 89 4.14 Summary ..................................... 90 4.15 Exercises ..................................... 92 5 Determinants via Calculations 101 5.1 Matrices with Entries in a Ring ........................ 101 5.2 Explicit Definition of the Determinant ..................... 102 5.3 Diagonal and Triangular Matrices ....................... 103 5.4 Changing Variables ............................... 103 5.5 Transposes and Determinants ......................... 104 5.6 Multilinearity and the Alternating Property ................. 106 5.7 Elementary Row Operations and Determinants ................ 107 5.8 Determinant Properties Involving Columns .................. 109 5.9 Product Formula via Elementary Matrices .................. 109 5.10 Laplace Expansions ............................... Ill 5.11 Classical Adjoints and Inverses ......................... 113 5.12 Cramer s Rule .................................. 114 5.13 Product Formula for Determinants ....................... 115 5.14 Cauchy-Binet Formula ............................. 116 5.15 Cayley-Hamilton Theorem ........................... 118 5.16 Permanents .................................... 120 5.17 Summary ..................................... 121 5.18 Exercises ..................................... 123 Contents ix 6 Concrete vs. Abstract Linear Algebra 131 6.1 Concrete Column Vectors vs. Abstract Vectors ................ 131 6.2 Examples of Computing Coordinates ..................... 133 6.3 Concrete vs. Abstract Vector Space Operations ................ 135 6.4 Matrices vs. Linear Maps ............................. 136 6.5 Examples of Matrices Associated with Linear Maps ............. 138 6.6 Vector Operations on Matrices and Linear Maps ............... 140 6.7 Matrix Transpose vs. Dual Maps ........................ 141 6.8 Matrix/Vector Multiplication vs. Evaluation of Maps ............ 142 6.9 Matrix Multiplication vs. Composition of Linear Maps ........... 142 6.10 Transition Matrices and Changing Coordinates ................ 143 6.11 Changing Bases ................................. 144 6.12 Algebras of Matrices and Linear Operators .................. 145 6.13 Similarity of Matrices and Linear Maps .................... 147 6.14 Diagonalizability and Triangulability ..................... 147 6.15 Block-Triangular Matrices and Invariant Subspaces ............. 149 6.16 Block-Diagonal Matrices and Reducing Subspaces .............. 150 6.17 Idempotent Matrices and Projections ..................... 151 6.18 Bilinear Maps and Matrices ........................... 152 6.19 Congruence of Matrices ............................. 153 6.20 Real Inner Product Spaces and Orthogonal Matrices ............. 154 6.21 Complex Inner Product Spaces and Unitary Matrices ............ 155 6.22 Summary ..................................... 156 6.23 Exercises ..................................... 161 III Matrices with Special Structure 167 7 Hermitian, Positive Definite, Unitary, and Normal Matrices 169 7.1 Conjugate-Transpose of a Matrix ........................ 169 7.2 Hermitian Matrices ............................... 171 7.3 Hermitian Decomposition of a Matrix ..................... 172 7.4 Positive Definite Matrices ............................ 173 7.5 Unitary Matrices ................................. 174 7.6 Unitary Similarity ................................ 176 7.7 Unitary Triangularization ............................ 177 7.8 Simultaneous Triangularization ......................... 178 7.9 Normal Matrices and Unitary Diagonalization ................ 180 7.10 Polynomials and Commuting Matrices ..................... 181 7.11 Simultaneous Unitary Diagonalization ..................... 182 7.12 Polar Decomposition: Invertible Case ..................... 183 7.13 Polar Decomposition: General Case ...................... 134 7.14 Interlacing Eigenvalues for Hermitian Matrices ................ 185 7.15 Determinant Criterion for Positive Definite Matrices ............ . 187 7.16 Summary ..................................... 188 7.17 Exercises ..................................... 189 8 Jordan Canonical Forms 195 8.1 Examples of Nilpotent Maps ........,................. 196 8.2 Partition Diagrams ............................... 197 8.3 Partition Diagrams and Nilpotent Maps .................... 198 8.4 Computing Images via Partition Diagrams .................. 199 x Contents 8.5 Computing Null Spaces via Partition Diagrams ................ 200 8.6 Classification of Nilpotent Maps (Stage 1) .................. 201 8.7 Classification of Nilpotent Maps (Stage 2) .................. 202 8.8 Classification of Nilpotent Maps (Stage 3) .................. 203 8.9 Fitting s Lemma ................................. 204 8.10 Existence of Jordan Canonical Forms ..................... 205 8.11 Uniqueness of Jordan Canonical Forms .................... 206 8.12 Computing Jordan Canonical Forms ...................... 207 8.13 Application to Differential Equations ..................... 209 8.14 Minimal Polynomials .............................. 210 8.15 Jordan-Chevalley Decomposition of a Linear Operator ........... 211 8.16 Summary ..................................... 212 8.17 Exercises ..................................... 213 9 Matrix Factorizations 219 9.1 Approximation by Orthonormal Vectors .................... 220 9.2 Gram-Schmidt Orthonormalization ...................... 221 9.3 Gram-Schmidt QR Factorization ........................ 222 9.4 Householder Reflections ............................. 224 9.5 Householder QR Factorization ......................... 226 9.6 LU Factorization ................................. 227 9.7 Example of the LU Factorization ........................ 229 9.8 LU Factorizations and Gaussian Elimination ................. 230 9.9 Permuted LU Factorizations .......................... 232 9.10 Cholesky Factorization ............................. 234 9.11 Least Squares Approximation .......................... 235 9.12 Singular Value Decomposition ......................... 236 9.13 Summary ..................................... 237 9.14 Exercises ..................................... 239 10 Iterative Algorithms in Numerical Linear Algebra 245 10.1 Richardson s Algorithm ............................. 245 10.2 Jacobi s Algorithm ................................ 246 10.3 Gauss-Seidel Algorithm ............................. 247 10.4 Vector Norms .................................. 248 10.5 Metric Spaces .................................. 250 10.6 Convergence of Sequences ............................ 250 10.7 Comparable Norms ............................... 251 10.8 Matrix Norms .................................. 252 10.9 Formulas for Matrix Norms ........................... 254 10.10 Matrix Inversion via Geometric Series .................... 255 10.11 Affine Iteration and Richardson s Algorithm ................. 256 10.12 Splitting Matrices and Jacobi s Algorithm .................. 257 10.13 Induced Matrix Norms and the Spectral Radius ............... 257 10.14 Analysis of the Gauss-Seidel Algorithm ................... 259 10.15 Power Method for Finding Eigenvalues .................... 259 10.16 Shifted and Inverse Power Method ...................... 261 10.17 Deflation ..................................... 262 10.18 Summary .................................... 262 10.19 Exercises ..................................... 264 Contents xi IV The Interplay of Geometry and Linear Algebra 271 11 Affine Geometry and Convexity 273 11.1 Linear Subspaces ................................. 273 11.2 Examples of Linear Subspaces ......................... 274 11.3 Characterizations of Linear Subspaces ..................... 275 11.4 Affine Combinations and Affine Sets ...................... 276 11.5 Affine Sets and Linear Subspaces ........................ 277 11.6 Affine Span of a Set ............................... 278 11.7 Affine Independence ............................... 279 11.8 Affine Bases and Barycentric Coordinates ................... 280 11.9 Characterizations of Affine Sets ........................ 281 11.10 Affine Maps ................................... 282 11.11 Convex Sets ................................... 283 11.12 Convex Hulls .................................. 284 11.13 Carathéodory s Theorem ............................ 284 11.14 Hyperplanes and Half-Spaces in Rn ...................... 286 11.15 Closed Convex Sets ............................... 287 11.16 Cones and Convex Cones ............................ 289 11.17 Intersection Lemma for V-Cones ....................... 290 11.18 All Н -Cones Are V-Cones ........................... 291 11.19 Projection Lemma for Н -Cones ........................ 292 11.20 All V-Cones Are Н -Cones ........................... 293 11.21 Finite Intersections of Closed Half-Spaces .................. 294 11.22 Convex Functions ................................ 296 11.23 Derivative Tests for Convex Functions .................... 297 11.24 Summary .................................... 298 11.25 Exercises ..................................... 301 12 Ruler and Compass Constructions 309 12.1 Geometric Constructibility ........................... 310 12.2 Arithmetic Constructibility ........................... 311 12.3 Preliminaries on Field Extensions ....................... 312 12.4 Field-Theoretic Constructibüity ........................ 314 12.5 Proof that GC ç AC .............................. 314 12.6 Proof that AC Q GC .............................. 316 12.7 Algebraic Elements and Minimal Polynomials ................ 320 12.8 Proof that AC = SQC .............................. 322 12.9 Impossibility of Geometric Construction Problems .............. 323 12.10 Constructibility of the 17-Gon ......................... 324 12.11 Overview of Solvability by Radicals ...................... 326 12.12 Summary .................................... 327 12.13 Exercises ..................................... 328 13 Dual Spaces and Bilinear Forms 335 13.1 Vector Spaces of Linear Maps ......................... 335 13.2 Dual Bases .................................... 336 13.3 Zero Sets ..................................... 337 13.4 Annihilators ................................... 338 13.5 Double Dual V** ................................ 339 13.6 Correspondence between Subspaces of V and V* .........·...··... . 341 xii Contents 13.7 Dual Maps .................................... 342 13.8 Nondegenerate Bilinear Forms ......................... 344 13.9 Real Inner Product Spaces ........................... 345 13.10 Complex Inner Product Spaces ........................ 346 13.11 Comments on Infinite-Dimensional Spaces .................. 348 13.12 Affine Algebraic Geometry ........................... 349 13.13 Summary .................................... 350 13.14 Exercises ..................................... 352 14 Metric Spaces and Hubert Spaces 359 14.1 Metric Spaces .................................. 360 14.2 Convergent Sequences .............................. 361 14.3 Closed Sets .................................... 362 14.4 Open Sets .................................... 363 14.5 Continuous Functions .............................. 364 14.6 Compact Sets .................................. 365 14.7 Completeness ................................... 366 14.8 Definition of a Hubert Space .......................... 368 14.9 Examples of Hubert Spaces ........................... 369 14.10 Proof of the Hubert Space Axioms for £2 (X) ................. 371 14.11 Basic Properties of Hubert Spaces ....................... 373 14.12 Closed Convex Sets in Hubert Spaces ..................... 374 14.13 Orthogonal Complements ........................... 375 14.14 Orthonormal Sets ................................ 377 14.15 Maximal Orthonormal Sets .......................... 378 14.16 Isomorphism of H and ¿z(X) ......................... 379 14.17 Continuous Linear Maps ............................ 381 14.18 Dual Space of a Hubert Space ......................... 382 14.19 Adjoints . . . ·.................................. 383 14.20 Summary .................................... 384 14.21 Exercises ..................................... 387 V Modules, Independence, and Classification Theorems 395 15 Finitely Generated Commutative Groups 397 15.1 Commutative Groups .............................. 397 15.2 Generating Sets ................................. 399 15.3 Z-Independence and Z-Bases .......................... 400 15.4 Elementary Operations on Z-Bases ...................... 401 15.5 Coordinates and Z-Linear Maps ........................ 402 15.6 UMP for Free Commutative Groups ...................... 403 15.7 Quotient Groups of Free Commutative Groups ................ 404 15.8 Subgroups of Free Commutative Groups ................... 405 15.9 Z-Linear Maps, and Integer Matrices ...................... 406 15.10 Elementary Operations and Change of Basis ................. 408 15.11 Reduction Theorem for Integer Matrices ................... 411 15.12 Structure of Z-Linear Maps between Free Groups .............. 413 15.13 Structure of Finitely Generated Commutative Groups ........... 414 15.14 Example of the Reduction Algorithm ..................... 415 15.15 Some Special Subgroups ............................ 417 15.16 Uniqueness Proof: Free Case .......................... 418 Contents xiii 15.17 Uniqueness Proof: Prime Power Case ..................... 419 15.18 Uniqueness of Elementary Divisors ...................... 422 15.19 Uniqueness of Invariant Factors ........................ 423 15.20 Uniqueness Proof: General Case ........................ 424 15.21 Summary .................................... 424 15.22 Exercises ..................................... 426 16 Axiomatic Approach to Independence, Bases, and Dimension 437 16.1 Axioms ...................................... 437 16.2 Definitions .................................... 438 16.3 Initial Theorems ................................. 438 16.4 Consequences of the Exchange Axiom ..................... 439 16.5 Main Theorems: Finite-Dimensional Case ................... 440 16.6 Zorn s Lemma .................................. 442 16.7 Main Theorems: General Case ......................... 443 16.8 Bases of Subspaces ............................... 446 16.9 Linear Independence and Linear Bases .................... 446 16.10 Field Extensions ................................ 448 16.11 Algebraic Independence and Transcendence Bases .............. 449 16.12 Independence in Graphs ............................ 452 16.13 Hereditary Systems ............................... 453 16.14 Matroids ..................................... 454 16.15 Equivalence of Matroid Axioms ........................ 455 16.16 Summary .................................... 456 16.17 Exercises ..................................... 457 17 Elements of Module Theory 463 17.1 Module Axioms ................................. 463 17.2 Examples of Modules .............................. 465 17.3 Submodules ................................... 466 17.4 Submodule Generated by a Subset ....................... 467 17.5 Direct Products, Direct Sums, and Hom Modules .............. 468 17.6 Quotient Modules ................................ 470 17.7 Changing the Ring of Scalars .......................... 472 17.8 Fundamental Homomorphism Theorem for Modules ............. 472 17.9 More Module Homomorphism Theorems ................... 474 17.10 Chains of Submodules ............................. 476 17.11 Modules of Finite Length ........................... 478 17.12 Free Modules .................................. 479 17.13 Size of a Basis of a Free Module ........................ 481 17.14 Summary .................................... 483 17.15 Exercises ..................................... 486 18 Principal Ideal Domains, Modules over PIDs, and Canonical Fornas 493 18.1 Principal Ideal Domains ............................. 494 18.2 Divisibility in Commutative Rings ....................... 494 18.3 Divisibility and Ideals .............................. 495 18.4 Prime and Irreducible Elements ........................ 496 18.5 Irreducible Factorizations in РГОѕ ....................... 497 18.6 Free Modules over a PID............................ 498 18.7 Operations on Bases ............................... 499 xiv Contents 18.8 Matrices of Lineax Maps between Free Modules ............... 500 18.9 Reduction Theorem for Matrices over a PID ................. 502 18.10 Structure Theorems for Linear Maps and Modules ............. 504 18.11 Minors and Matrix Invariants ......................... 505 18.12 Uniqueness of Smith Normal Form ...................... 506 18.13 Torsion Submodules .............................. 507 18.14 Uniqueness ofinvariant Factors ........................ 508 18.15 Uniqueness of Elementary Divisors ...................... 509 18.16 .Ffa^-Module Defined by a Linear Operator .................. 510 18.17 Rational Canonical Form of a Linear Map .................. 512 18.18 Jordan Canonical Form of a Linear Map ................... 513 18.19 Canonical Forms of Matrices ........................... 514 18.20 Summary .................................... 516 18.21 Exercises ..................................... 518 VI Universal Mapping Properties and Multilinear Algebra 525 19 Introduction to Universal Mapping Properties 527 19.1 Bases of Free Ä-Modules ............................ 529 19.2 Homomorphisms out of Quotient Modules .................. 529 19.3 Direct Product of Two Modules ........................ 531 19.4 Direct Sum of Two Modules .......................... 532 19.5 Direct Products of Arbitrary Families of Ä-Modules............. 533 19.6 Direct Sums of Arbitrary Families of iř-Modules ............... 534 19.7 Solving Universal Mapping Problems ..................... 537 19.8 Summary ..................................... 539 19.9 Exercises ..................................... 541 20 Universal Mapping Problems in Multilinear Algebra 547 20.1 Multilinear Maps ................................ 547 20.2 Alternating Maps ................................ 548 20.3 Symmetric Maps ................................. 549 20.4 Tensor Product of Modules ........................... 550 20.5 Exterior Powers of a Module .......................... 553 20.6 Symmetric Powers of a Module ......................... 555 20.7 Myths about Tensor Products ......................... 557 20.8 Tensor Product Isomorphisms .......................... 558 20.9 Associativity of Tensor Products ........................ 560 20.10 Tensor Product of Maps ............................ 561 20.11 Bases and Multilinear Maps .......................... 562 20.12 Bases for Tensor Products of Free Ä-Modules ................ 564 20.13 Bases and Alternating Maps .......................... 565 20.14 Bases for Exterior Powers of Free Modules .................. 566 20.15 Bases for Symmetric Powers of Free Modules ................ 567 20.16 Tensor Product of Matrices .......................... 568 20.17 Determinants and Exterior Powers ...................... 569 20.18 From Modules to Algebras ........................... 571 20.19 Summary .................................... 573 20.20 Exercises ..................................... 577 Contents xv Appendix: Basic Definitions 583 Sets........................................... 583 Functions ........................................ 583 Relations........................................ 584 Partially Ordered Sets ................................. 585 Further Reading 587 Bibliography 591 Index 595 TEXTBOOKS in MATHEMATICS Advanced Linear Algebra explores a variety of advanced topics in linear al¬ gebra that highlight the rich interconnections of the subject to geometry, alge¬ bra, analysis, combinatorics, numerical computation, and many other areas of mathematics. The book s 20 chapters are grouped into six main subjects: al¬ gebraic structures, matrices, structured matrices, geometric aspects of linear algebra, modules, and multilinear algebra. The level of abstraction gradually increases as you proceed through the text, moving from matrices to vector spaces to modules. Each chapter consists of a mathematical vignette devoted to the development of one specific topic. Some chapters look at introductory material from a so¬ phisticated or abstract viewpoint while others provide elementary expositions of more theoretical concepts. Several chapters offer unusual perspectives or novel treatments of standard results. Unlike similar advanced mathematical texts, this one minimizes the dependence of each chapter on material found in previous chapters so that you may immediately turn to the relevant chapter without first wading through pages of earlier material to access the necessary algebraic background and theorems. Features • Focuses on theoretical aspects of linear algebra • Gives complete proofs of all results • Supplements the general theory with many specific examples and concrete computations • Requires very little knowledge of abstract algebra beyond fields, vector spaces over a field, subspaces, linear transformations, linear independence, and bases • Includes a summary and exercise sets at the end of each chapter
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owner_facet	DE-703 DE-11 DE-824 DE-83 DE-634
physical	XXII, 609 S. graph. Darst.
publishDate	2014
publishDateSearch	2014
publishDateSort	2014
publisher	CRC Press
record_format	marc
series2	Textbooks in mathematics A Chapman & Hall book
spelling	Loehr, Nicholas A. Verfasser (DE-588)143771701 aut Advanced linear algebra Nicholas Loehr Boca Raton [u.a.] CRC Press 2014 XXII, 609 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Textbooks in mathematics A Chapman & Hall book Includes bibliographical references Algebras, Linear Lineare Algebra (DE-588)4035811-2 gnd rswk-swf Electronic books Lineare Algebra (DE-588)4035811-2 s DE-604 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027544332&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027544332&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext
spellingShingle	Loehr, Nicholas A. Advanced linear algebra Algebras, Linear Lineare Algebra (DE-588)4035811-2 gnd
subject_GND	(DE-588)4035811-2
title	Advanced linear algebra
title_auth	Advanced linear algebra
title_exact_search	Advanced linear algebra
title_full	Advanced linear algebra Nicholas Loehr
title_fullStr	Advanced linear algebra Nicholas Loehr
title_full_unstemmed	Advanced linear algebra Nicholas Loehr
title_short	Advanced linear algebra
title_sort	advanced linear algebra
topic	Algebras, Linear Lineare Algebra (DE-588)4035811-2 gnd
topic_facet	Algebras, Linear Lineare Algebra
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027544332&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027544332&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT loehrnicholasa advancedlinearalgebra

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