Verfügbarkeit: Beyond quasicrystals

Beyond quasicrystals: Les Houches, March 7 - 18, 1994

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Weitere Verfasser:	Axel, Françoise (HerausgeberIn)
Format:	Tagungsbericht Buch
Sprache:	English
Veröffentlicht:	Berlin [u.a.] Springer [u.a.] 1995
Schriftenreihe:	Centre de Physique Les Houches 3
Schlagworte:	Physique mathématique - Congrès Quasicristaux - Congrès Quasikristallen Systèmes, théorie des - Congrès Quasicrystals > Congresses Quasikristall Konferenzschrift > 1994 > Les Houches
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	XVI, 619 S. Ill., graph. Darst.
ISBN:	3540592512 2868832482

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Datensatz im Suchindex

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adam_text	CONTENTS Quasicrystals ... course ι Quasicrystals, diophantine approximation and algebraic numbers by Yves Meyer 1. Introduction .................................................................................... З 2. Almost-periodic functions, Poisson summation formula and algebraic numbers ...... 5 3. Model sets and quasicrystals ................................................................. 9 4. Quasicrystals and diophantine approximation .............................................. 10 5. Poisson summation formula and quasicrystals ............................................ 14 6. Conclusion ..................................................................................... 15 COURSE 2 The pentacrystals by J. Patera 1. Introduction .................................................................................... 17 2. Preliminaries ................................................................................... 18 3. The pentacrystal map .......................................................................... 20 4. Definition of quasicrystals .................................................................... 22 5. Phasons ......................................................................................... 23 6. Quasiaddition.................................................................................. 24 7. Examples ....................................................................................... 25 COURSE 3 Elements of a multimetrical crystallography by A. Janner Abstract ..............................................................................■............. 33 1. Introduction .................................................................................... 33 2. Close-packed structures ...................................................................... 35 2.1 The 2-dimensional case .................................................................. 35 2.2 The З -dimensional case .................................................................. 35 3. Multimetrical symmetry of the 2D hexagonal lattice ....................................... 37 4. Binary integral quadratic forms and quadratic fields ...................................... 39 5. Indefinite ternary integral quadratic forms .................................................. 41 6. Quadratic forms of lattices of 3D close-packed structures ................................ 44 6.1 Reduced metric tensors .................................................................. 44 6.2 Indefinite binary quadratic forms ....................................................... 45 6.3 Indefinite ternary quadratic forms ...................................................... 46 7. Multimetrical point group of the hexagonal close-packed lattice ......................... 47 8. Multimetrical space groups of crystal structures ........................................... 49 VIII 8.1 Hexagonal close-packed structures ........................................................ 50 8.2 The Wurtzite structure ....................................................................... 51 9. Concluding remarks ........................................................................... 52 COURSE 4 Non-commutative models for quasicrystals by P. Kramer and J. Garcia-Escudero 1. Why non-commutative models for quasicrystals ?........................................ 55 2. Free groups and their automorphisms ...................................................... 56 3. Non-commutative crystallography .......................................................... 57 4. Structure and geometry of the group Aut(F2) .............................................. 58 5. Free groups and automorphisms for η > 2................................................ 66 6. Non-commutative models and symmetries for 2D quasiperiodic patterns .............. 66 7. Automata for the triangle and Penrose patterns ............................................ 69 8. Survey of other results ........................................................................ 72 COURSE 5 From quasiperiodic to more complex systems by T. Janssen 1. Structures ....................................................................................... 75 1.1 Introduction ............................................................................... 75 1.2 Classes of quasiperiodic structures ..................................................... 78 1.3 Embedding of quasiperiodic systems .................................................. 81 1.4 Superspace groups ....................................................................... 84 1.5 Action of symmetry groups in 3-dimensional space ................................. 86 1.6 Scale symmetries ......................................................................... 89 1.7 Hierarchy of structures .................................................................. 91 1.8 Physical origin of quasiperiodicity ..................................................... 93 2. Diffraction ...................................................................................... 94 2.1 Structure factor ........................................................................... 94 2.2 Structure factor of quasiperiodic structures ........................................... 97 2.3 Influence of symmetry ................................................................... 99 2.4 Thermal vibrations ....................................................................... 100 2.5 Disorder ................................................................................... 101 3. Phonons ........................................................................................ 104 3.1 Phonons in 1С phases .................................................................... 104 3.2 Spectra ..................................................................................... 110 3.3 Phonons in quasicrystals ................................................................ Ill 3.4 Neutron scattering from quasiperiodic structures ..................................... 116 4. Substitution^ chains .......................................................................... 122 4.1 Introduction ............................................................................... 122 4.2 Atomic surfaces ........................................................................... 124 4.3 Fractal atomic surfaces ................................................................... 127 5. Electrons ........................................................................................ 132 5.1 Models ..................................................................................... 132 5.2 Spectra ..................................................................................... 135 5.3 Wave functions ........................................................................... 136 IX COURSE 6 Matching rules and quasiperiodicity: the octagonal tilings by A. Katz 1. Introduction .................................................................................... 141 2. Quasiperiodic tilings .......................................................................... 142 2.1 Quasiperiodicity .......................................................................... 142 2.2 The atomic surfaces ...................................................................... 143 2.3 The cut algorithm ......................................................................... 144 2.4 Canonical or Penrose like tilings ..................................................... 144 2.4.1 Definition ............................................................................ 144 2.4.2 The oblique tiling ................................................................... 145 2.4.3 Octagonal tilings .................................................................... 146 3. The composition-decomposition method ................................................... 148 3.1 Self-similarity ............................................................................. 148 3.2 Inflation and quasiperiodicity ........................................................... 150 4. The method of forbidden planes ............................................................. 151 4.1 Position of the problem .................................................................. 151 4.2 Non-transversality conditions ........................................................... 153 4.3 The forbidden planes ..................................................................... 154 5. Decoration of the tiles ......................................................................... 156 5.1 A simple case ............................................................................. 156 5.2 The Ammann decoration of vertices .................................................... 159 6. The main theorem ............................................................................. 162 6.1 Position and intersections of the forbidden planes ................................... 162 6.2 Systems of data ........................................................................... 163 6.3 Propagation of order ..................................................................... 164 6.4 Proof of the theorem ..................................................................... 166 6.4.1 The pushing procedure ............................................................ 166 6.4.2 The cone of planes ................................................................. 168 6.5 Quasiperiodic tilings and special tilings ............................................. 170 7. Generalised Ammann tilings ................................................................. 174 7.1 Definitions ................................................................................. 174 7.2 Symmetry considerations ................................................................ 174 7.3 Setting the method ........................................................................ 175 7.3.1 Systems of data in ε, ............................................................... 177 7.4 Reduction to bad prisms .............................................................. 177 7.5 Proof of the theorem ..................................................................... 178 7.6 Order in generalised Ammann tilings of the first kind ............................... 181 7.7 Generalised Ammann tilings of the second kind: an example of weak rules ...... 183 8. Conclusion ..................................................................................... 188 COURSE 7 A mechanism for diffusion in quasicrystals by P.A. Kalugin ............................................................................... 191 COURSE 8 Experimental aspects of the structure analysis of aperiodic materials by W. Steurer 1. Introduction .................................................................................... 203 2. What are aperiodic materials ?............................................................... 204 3. Experimental probes for distinguishing between crystals and aperiodic structures. ... 206 3.1 Diffraction methods ...................................................................... 207 3.2 Imaging techniques ....................................................................... 211 3.3 Spectroscopical methods ................................................................ 212 4. Structure determination methods ............................................................ 212 4.1 The maximum-entropy method (MEM) ................................................ 213 4.2 How many reflections have to be measured ?......................................... 214 4.3 Aperiodic sequences and their Fourier transforms ................................... 216 4.4 Symmetry-minimum function and Patterson Deconvolution ........................ 218 4.5 Quasicrystals versus twinned approximants .......................................... 224 5. Quantitative aperiodic-crystal structure analysis ........................................... 226 COURSE 9 Scattering on aperiodic superlattices by P. Mikulik 1. Introduction .................................................................................... 229 2. Theories of X-ray diffraction ................................................................ 230 2.1 Kinematical theory of X-ray diffraction ............................................... 230 2.1.1 The Fraunhofer approximation ................................................... 230 2.1.2 Stationary phase method ........................................................... 232 2.2 Dynamical theory ......................................................................... 232 2.2.1 Semi-kinematical approximation .................................................. 233 3. Diffraction on multilayers .................................................................... 234 3.1 Periodic lattice ............................................................................ 236 3.2 The Fibonacci lattice ..................................................................... 237 3.2.1 Finite length of the Fibonacci lattice .............................................. 239 3.2.2 Calculations in the semi-kinematical approximation ............................ 240 3.2.3 Maxima of the diffraction curve of Fibonacci SL ............................... 241 3.3 Other aperiodic superlattices ............................................................ 242 4. X-ray reflectivity .............................................................................. 243 5. Gratings ........................................................................................ 244 6. Conclusion ..................................................................................... 246 COURSE 10 Defects in quasicrystals, in systems with deterministic disorder and in amorphous materials by N. Rivier 1. Introduction .................................................................................... 249 2. Defects in condensed matter .................................................................. 252 2.1 Homotopy description of defects ....................................................... 253 2.2 A simple method for calculating homotopy groups .................................. 260 2.3 Defects in crystals: dislocations ......................................................... 261 3. Defects in quasicrystals ....................................................................... 263 3.1 The oblique tiling ......................................................................... 265 3.1.1 Tilings in 2d and their topological defects ....................................... 265 3.1.2 The oblique tiling ................................................................... 267 3.1.3 Defects in quasicrystals; summary ............................................... 269 3.2 Observation of defects ................................................................... 270 3.2.1 Phase contrast and diffraction contrast microscopy ............................ 271 4. Defects in amorphous materials .............................................................. 272 4.1 Symmetry of disorder .................................................................... 273 4.2 Glass as a fibre bundle ................................................................... 273 XI 4.3 Curved space, packing atoms and frustration ......................................... 276 4.4 Iterative decurving ........................................................................ 277 4.4.1 Three-dimensional materials ...................................................... 279 4.4.1.1 The laves phase ........................................................... 280 4.4.1.2 Hierarchic glass ........................................................... 281 4.4.1.3 Spatial disorder ........................................................... 282 4.4.2 The role of entropy ................................................................. 285 5. Conclusions .................................................................................... 286 ...and Beyond COURSE 11 Automata and automatic sequences by J.-P. Allouche and M. Mendès France Part I. Introduction. Substitutions and finite automata ....................................... 293 1. The Fibonacci sequence ...................................................................... 295 2. The Prouhet-Thue-Morse sequence ......................................................... 296 3. The paper folding sequence .................................................................. 297 4. Automatic sequences: definition; a zoo of examples. A warning ........................ 298 5. Where finite automata enter the picture ..................................................... 303 6. How random can an automatic sequence be ?.............................................. 304 7. Miscellanea ..................................................................................... 305 8. Appendix: automata generating the five examples of section 4.......................... 308 Part II. Further properties of paperfolding .................................................... 312 1. Direct reading and reverse reading .......................................................... 312 2. Words and diagrams .......................................................................... 313 3. The folding operators ......................................................................... 314 4. The dimension of a curve ..................................................................... 318 5. Paperfolding and continued fractions ....................................................... 320 Partili. Complements ........................................................................... 324 1. Repetitions in infinite sequences ............................................................. 324 1.1 The beginning of the story .............................................................. 324 1.2 Why study repetitions? .................................................................. 325 1.3 More examples ............................................................................ 325 2. Multidimensional morphisms and (finite) automata ....................................... 326 2.1 An example ................................................................................ 326 2.2 Properties .................................................................................. 327 3. Links with cellular automata ................................................................. 327 Part IV. Fourier Analysis ........................................................................ 331 1. Fourier-Bohr coefficients ..................................................................... 332 2. Besseľs inequality and Parseval s equality ................................................ 333 3. The Wiener spectrum and the spectral measure ............................................ 334 4. Analysing the spectral measure .............................................................. 337 5. Appendix I: the spectral measure of the Thue-Morse sequence .......................... 337 6. Appendix П: the spectral measure of the paperfolding sequence ........................ 343 хп Part V. Complexity of infinite sequences ...................................................... 346 1. Sturmian sequences and generalizations .................................................... 347 2. Complexity of automatic sequences ......................................................... 348 2.1 An upper bound .......................................................................... 348 2.2 Complexity function of some automatic sequences .................................. 348 2.3 The case of non-constant length morphisms .......................................... 349 Part VI. Opacity of an automaton ............................................................... 352 1. Automata revisited ............................................................................. 352 2. Computing the opacity ........................................................................ 355 Part VII. The Ising automaton .................................................................. 358 1. The inhomogeneous Ising chain ............................................................. 358 2. The induced field .............................................................................. 359 3. The Ising automaton .......................................................................... 362 4. An ergodic property ........................................................................... 364 5. Opacity of the Ising automaton .............................................................. 366 COURSE 12 Spectral study of automatic and substitutive sequences by Martine Queffelec Introduction ....................................................................................... 369 1. Measures on Τ ................................................................................. 370 1.1 Basic definitions and notations ......................................................... 370 1.2 Discrete and continuous measures ...................................................... 371 1.3 Singularity and absolute continuity ..................................................... 372 1.4 Constructive examples ................................................................... 373 2. Tools from ergodic theory .................................................................... 376 2.1 For mutual singularity of measures .................................................... 376 2.2 For dichotomy properties ................................................................ 379 3. Correlation measures .......................................................................... 380 3.1 Correlation of sequences ................................................................ 380 3.2 Classification .............................................................................. 381 4. Substitutive sequences - automatic sequences ............................................. 382 4.1 Basic notations ............................................................................ 382 4.2 Basic definitions .......................................................................... 383 4.3 Examples .................................................................................. 383 5. Substitution dynamical systems ............................................................. 385 5.1 Definitions and basic results ............................................................ 385 5.2 Examples .................................................................................. 386 5.3 Spectrum of ζ ............................................................................. 388 5.4 Discrete substitutions .................................................................... 390 6. Substitutions of constant length ............................................................. 391 6.1 Examples .................................................................................. 391 6.2 Discrete substitutions of constant length ............................................... 392 6.3 Spectrum of a substitution of length q ................................................. 394 6.4 Dynamical system as extension of the odometer ...................................... 396 7. Substitutions of non-constant length ........................................................ 398 7.1 Description of eigenvalues .............................................................. 398 7.2 Various situations for A = {a,b} ....................................................... 399 7.3 Miscellaneous ............................................................................. 407 7.4 Conclusion ................................................................................ 410 XIII COURSE 13 Random and automatic walks by F.M. Dekking 1. Introduction .................................................................................... 415 2. Automatic walks ............................................................................... 416 3. Scaling structure and self-similarity ......................................................... 420 4. Quasi lattice walks ............................................................................. 421 5. Dynamical systems, random and automatic ................................................ 424 6. Generalized random walks ................................................................... 425 7. Mean square displacement .................................................................... 428 COURSE 14 Singular words, invertible substitutions and local isomorphisms by Wen Zhi-Ying 1. Introduction .................................................................................... 433 2. Singular words ................................................................................ 434 3. Invertible substitutions ........................................................................ 436 4. Fibonacci-chain as a periodic chain with discommensurations .......................... 438 COURSE 15 Entropy in deterministic and random systems by V. Berthe 1. Introduction .................................................................................... 441 2. Thermodynamical entropy .................................................................... 442 3. Information theory ............................................................................ 442 3.1 Entropy of a single event ................................................................ 443 3.2 Entropy of an experiment ................................................................ 443 3.3 Concavity of the function L ............................................................. 444 3.4 Marginal and conditional entropy ....................................................... 445 3.5 Entropy of a finite curve ................................................................. 446 3.6 The sequence of block entropies ........................................................ 447 4. Topological and measure-theoretic entropies ............................................... 449 4.1 Topological entropy of a sequence ..................................................... 449 4.2 Measure-theoretic entropy of a sequence .............................................. 450 4.3 Measure-theoretic entropy of a partition ............................................... 450 4.4 Topological entropy of an open cover ................................................. 452 4.5 Variational principle ...................................................................... 453 5. Entropy and spectral properties .............................................................. 454 6. Some examples of computation of block entropies ........................................ 455 6.1 Ultimately periodic and random sequences ......................................... 456 6.2 Sturmian sequences ...................................................................... 457 6.3 Block frequencies for some automatic sequences .................................... 459 6.4 Conclusion ................................................................................ 460 XIV COURSE 16 Trace maps by J. Peyrière 1 .Introduction ..................................................................................... 465 2. Some identities for 2 x 2 - matrices ......................................................... 466 3. Reviews and notations for free groups ..................................................... 470 3.1 Free semi-group generated by Я ........................................................ 470 3.2 Free group generated by Я .............................................................. 470 3.3 Representations in SL (2, C) ............................................................ 471 3.4 Endomorphisms .......................................................................... 471 4. Trace maps (two letter alphabet) ............................................................. 472 4.1 Definition of trace maps ................................................................. 472 4.2 First properties of trace maps ........................................................... 474 4.3 Further properties of trace maps ........................................................ 475 5. Trace maps (n letter alphabet) ................................................................ 476 5.1 Three letter alphabet ...................................................................... 476 5.2 и letter alphabet .......................................................................... 477 6. Comments ...................................................................................... 477 COURSE 17 Schrödinger difference equation with deterministic ergodic potentials by A. Sütő 1. Introduction .................................................................................... 483 2. Main examples ................................................................................. 484 3. General results on the Schrödinger difference equation .................................. 488 3.1 Basic observations ....................................................................... 488 3.2 Transfer matrices ......................................................................... 489 3.3 Lyapunov exponent ...................................................................... 492 3.4 Scattering problem: Landauer resistance ............................................... 493 4. Schrödinger equation with periodic potentials ............................................. 495 5. Spectral theory ................................................................................. 498 5.1 Schrödinger operator, /^(Z^space and spectrum ..................................... 498 5.2 Point spectrum ............................................................................ 499 5.3 Cantor sets ................................................................................ 502 5.4 Continuous spectrum .................................................................... 502 5.5 Spectral projections ...................................................................... 503 5.6 Measures .................................................................................. 506 5.7 Cantor function ........................................................................... 509 5.8 Spectral measures and spectral types ................................................... 510 5.9 A spectral measure for tf0 ............................................................... 514 5.10 r(Z) versus ^(N) ...................................................................... 515 5.11 Asymptotic behaviour of generalized eigenfunctions. Subordinacy .............. 515 6. Schrödinger equation with strictly ergodic potentials ..................................... 517 6.1 Strict ergodicity ........................................................................... 517 6.2 The spectrum οΐΗ(ω) = Η0+ V (ω)................................................. 518 6.3 Integrated density of states .............................................................. 520 6.4 IDS and Lyapunov exponent ............................................................ 521 XV 6.5 Results on the set {E :y(E) = 0} ...................................................... 522 6.6 The role of periodic approximants ...................................................... 524 6.7 Gordon-type theorems ................................................................... 524 6.8 Kotani theorem for potentials of finite range .......................................... 525 6.9 Gap labelling .............................................................................. 526 7. Schrödinger equation with Sturmian and substitutional potentials ...................... 527 7.1 Fibonacci potential ....................................................................... 527 7.2 General Sturmian potentials ............................................................. 528 7.3 Period doubling potential ................................................................ 530 7.4 Thue-Morse potential .................................................................... 530 7.5 Systematic study of substitutional potentials .......................................... 531 8. Solutions of the problems .................................................................... 532 COURSE 18 Schrödinger equation in a hierarchical potential by H. Kunz 1. Introduction .................................................................................... 551 2. Hierarchical potentials ........................................................................ 554 3. Spectral properties of the Schrödinger equation ........................................... 557 4. Liapunov exponents ........................................................................... 558 5. Spectral measures ............................................................................. 558 COURSE 19 Introduction to multifractal analysis by J. Peyrière 1. Introduction .................................................................................... 563 2. Short recalls on measures .................................................................... 563 2.1 Borei sets and measures ................................................................. 563 2.2 An example: the trinomial measures .................................................... 564 3. Several notions of dimensions ............................................................... 565 3.1 Definitions ................................................................................. 565 3.1.1 Hausdorff measures and dimension .............................................. 565 3.1.2 The Bouligand-Minkowski dimension (or box dimension ) ................. 566 3.1.3 Packing dimension ................................................................. 566 3.2 Estimating the Hausdorff dimension ................................................... 567 3.2.1 Comparison of these dimensions ................................................. 567 3.2.2 Lower bound for dim .............................................................. 567 3.2.3 Example ............................................................................. 568 4. The multifractal formalism ................................................................... 569 4.1 Definitions ................................................................................. 569 4.1.1 The pointwise Holder exponent .................................................. 569 4.1.2 The τ function ....................................................................... 569 4.1.3 The multifractal spectrum .......................................................... 570 4.1.4 The multifractal formula ........................................................... 570 4.2 A counterexample of the multifractal formalism ...................................... 570 4.3 The trinomial measures .................................................................. 571 XVI 4.4 A few results .............................................................................. 573 4.5 Negative dimensions ..................................................................... 574 4.6 Concluding remarks ...................................................................... 574 Appendix 1. Legendre transforms .............................................................. 575 Appendix 2. Large deviations ................................................................... 575 Annexes Current deterministic sequences ......................................................... 585 Glossary ......................................................................................... 595 Extra references ............................................................................... 613
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genre_facet	Konferenzschrift 1994 Les Houches
id	DE-604.BV010387571
illustrated	Illustrated
indexdate	2024-07-09T17:51:38Z
institution	BVB
institution_GND	(DE-588)2057705-9 (DE-588)5158480-3
isbn	3540592512 2868832482
language	English
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-006916517
oclc_num	32687890
open_access_boolean
owner	DE-355 DE-BY-UBR DE-29T DE-384 DE-12 DE-703 DE-91G DE-BY-TUM DE-83
owner_facet	DE-355 DE-BY-UBR DE-29T DE-384 DE-12 DE-703 DE-91G DE-BY-TUM DE-83
physical	XVI, 619 S. Ill., graph. Darst.
publishDate	1995
publishDateSearch	1995
publishDateSort	1995
publisher	Springer [u.a.]
record_format	marc
series	Centre de Physique Les Houches
series2	Centre de Physique Les Houches
spelling	Beyond quasicrystals Les Houches, March 7 - 18, 1994 [Centre de Physique]. Eds. Françoise Axel ... Berlin [u.a.] Springer [u.a.] 1995 XVI, 619 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Centre de Physique Les Houches 3 Physique mathématique - Congrès ram Quasicristaux - Congrès ram Quasikristallen gtt Systèmes, théorie des - Congrès ram Quasicrystals Congresses Quasikristall (DE-588)4202613-1 gnd rswk-swf (DE-588)1071861417 Konferenzschrift 1994 Les Houches gnd-content Quasikristall (DE-588)4202613-1 s DE-604 Axel, Françoise edt Centre de Physique (Les Houches) Sonstige (DE-588)2057705-9 oth Winter School Beyond Quasicrystals 1994 Les Houches Sonstige (DE-588)5158480-3 oth Erscheint auch als Online-Ausgabe 978-3-662-03130-8 Centre de Physique Les Houches 3 (DE-604)BV011876452 3 Digitalisierung TU Muenchen application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006916517&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Beyond quasicrystals Les Houches, March 7 - 18, 1994 Centre de Physique Les Houches Physique mathématique - Congrès ram Quasicristaux - Congrès ram Quasikristallen gtt Systèmes, théorie des - Congrès ram Quasicrystals Congresses Quasikristall (DE-588)4202613-1 gnd
subject_GND	(DE-588)4202613-1 (DE-588)1071861417
title	Beyond quasicrystals Les Houches, March 7 - 18, 1994
title_auth	Beyond quasicrystals Les Houches, March 7 - 18, 1994
title_exact_search	Beyond quasicrystals Les Houches, March 7 - 18, 1994
title_full	Beyond quasicrystals Les Houches, March 7 - 18, 1994 [Centre de Physique]. Eds. Françoise Axel ...
title_fullStr	Beyond quasicrystals Les Houches, March 7 - 18, 1994 [Centre de Physique]. Eds. Françoise Axel ...
title_full_unstemmed	Beyond quasicrystals Les Houches, March 7 - 18, 1994 [Centre de Physique]. Eds. Françoise Axel ...
title_short	Beyond quasicrystals
title_sort	beyond quasicrystals les houches march 7 18 1994
title_sub	Les Houches, March 7 - 18, 1994
topic	Physique mathématique - Congrès ram Quasicristaux - Congrès ram Quasikristallen gtt Systèmes, théorie des - Congrès ram Quasicrystals Congresses Quasikristall (DE-588)4202613-1 gnd
topic_facet	Physique mathématique - Congrès Quasicristaux - Congrès Quasikristallen Systèmes, théorie des - Congrès Quasicrystals Congresses Quasikristall Konferenzschrift 1994 Les Houches
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006916517&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV011876452
work_keys_str_mv	AT axelfrancoise beyondquasicrystalsleshouchesmarch7181994 AT centredephysiqueleshouches beyondquasicrystalsleshouchesmarch7181994 AT winterschoolbeyondquasicrystalsleshouches beyondquasicrystalsleshouchesmarch7181994

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