Chaos in dynamic systems:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English Russian |
| Published: |
Chur u.a.
Harwood
1985
|
| Subjects: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000211229&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Item Description: | Aus d. Russ. übers. |
| Physical Description: | XIX, 370 S. |
| ISBN: | 3718602253 |
Staff View
MARC
| LEADER | 00000nam a2200000 c 4500 | ||
|---|---|---|---|
| 001 | BV000339978 | ||
| 003 | DE-604 | ||
| 005 | 00000000000000.0 | ||
| 007 | t| | ||
| 008 | 870612s1985 xx |||| 00||| eng d | ||
| 020 | |a 3718602253 |9 3-7186-0225-3 | ||
| 035 | |a (OCoLC)11316574 | ||
| 035 | |a (DE-599)BVBBV000339978 | ||
| 040 | |a DE-604 |b ger |e rakddb | ||
| 041 | 1 | |a eng |h rus | |
| 049 | |a DE-12 |a DE-91G |a DE-83 | ||
| 050 | 0 | |a QC174.84 | |
| 082 | 0 | |a 530.1/3 |2 19 | |
| 084 | |a SK 820 |0 (DE-625)143258: |2 rvk | ||
| 084 | |a SK 880 |0 (DE-625)143266: |2 rvk | ||
| 084 | |a MAT 587f |2 stub | ||
| 084 | |a PHY 066f |2 stub | ||
| 100 | 1 | |a Zaslavsky, George M. |d 1935-2008 |e Verfasser |0 (DE-588)13657985X |4 aut | |
| 240 | 1 | 0 | |a Stochastičnost' dinamičeskich sistem |
| 245 | 1 | 0 | |a Chaos in dynamic systems |c G. M. Zaslavsky |
| 264 | 1 | |a Chur u.a. |b Harwood |c 1985 | |
| 300 | |a XIX, 370 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Aus d. Russ. übers. | ||
| 650 | 4 | |a Chaos | |
| 650 | 7 | |a Chaos (théorie des systèmes) |2 ram | |
| 650 | 4 | |a Processus stochastiques | |
| 650 | 4 | |a Chaotic behavior in systems | |
| 650 | 4 | |a Nonlinear theories | |
| 650 | 4 | |a Stochastic systems | |
| 650 | 0 | 7 | |a Dynamisches System |0 (DE-588)4013396-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Chaotisches System |0 (DE-588)4316104-2 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Chaotisches System |0 (DE-588)4316104-2 |D s |
| 689 | 0 | 1 | |a Dynamisches System |0 (DE-588)4013396-5 |D s |
| 689 | 0 | |5 DE-604 | |
| 856 | 4 | 2 | |m HBZ Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000211229&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 940 | 1 | |n oe | |
| 940 | 1 | |q TUB-nveb | |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-000211229 | |
Record in the Search Index
| DE-BY-TUM_call_number | 0102 PHY 066f 2001 A 36683 |
|---|---|
| DE-BY-TUM_katkey | 66169 |
| DE-BY-TUM_location | 01 |
| DE-BY-TUM_media_number | 040020030506 |
| _version_ | 1821939719822376960 |
| adam_text | Contents
Preface xv
Notation xix
CHAPTER 1. ELEMENTS OF DYNAMICS AND ERGODIC
THEORY
1.1. Motion in Phase Space
Liouville s theorem. Poincare s theorem on
recurrences. Time of recurrence. Kac s formula,
Liouville equation 1
1.2. Action Angle Variables
What are convenient variables? Adiabatic
invariance. Oscillations of a nonlinear pendulum.
Plasma oscillations. Spectral properties of
nonlinear oscillations 6
1.3. Nonlinear Resonance
Resonance induced by an external force.
Resonance of coupled oscillations 12
1.4. Kolmogorov Arnold Moser Theory
Multidimensional motion. Invariant tori. Action
angle variables. Irreducible contours. Poincare
mappings. Stability theorem. The role of
dimensionality of phase space 22
1.5. Ergodicity and Mixing
Basic definitions. Local instability. K systems.
Krylov s ideas. Mixing as the condition of
finiteness of relaxation time 28
1.6. Entropy
Coarsening of the distribution function.
Kolmogorov dynamic entropy. K systems and
entropy. Isomorphism of K systems 35
1.7. Historical Background
Foundations of statistical physics. Zermelo s and
vii
Loschmidt s paradoxes. The viewpoints of
Boltzmann and Ehrenfest. More on mixing. •
Krylov s principle of complementarity 41
Notes to Chapter 1 45
CHAPTER 2. STOCHASTICITY CRITERION
2.1. Two Models of Mixing !
Continued fractions and stochasticity. Stretching
transformation. Steady state distribution
function 49
2.2. Stochasticity Criterion
Sinai s criterion. An example of two
dimensional stretching. Saw tooth
transformation. Anosov flow 64
2.3. Collisions of Absolutely Rigid Spheres
Krylov s analysis. Role played by negative
curvature. Difficulties in analyzing the model 69
2.4. Scattering Billiards (Sinai s Billiards)
Sinai s analysis of a gas of rigid disks.
Problems with scattering billiards. Isomorphism
of scattering billiards 74
Notes to Chapter 2 76
CHAPTER 3. STOCHASTIC ACCELERATION OF
PARTICLES (FERMI ACCELERATION)
3.1. Mechanism of Stochastic Acceleration
Ulam model. Equations of mappings.
Acceleration criterion. Kinetics of acceleration 79
3.2. Gravitational Engine
How to use a gravitational field for unlimited
acceleration of a body 86
3.3. Mixing of Skipping Electrons
Surface electrons in a magnetic field. Stochastic
acceleration and scattering billiards. Momentum
distribution of surface electrons 89
Notes to Chapter 3 96
viii
CHAPTER 4. STOCHASTIC INSTABILITY OF
OSCILLATIONS
4.1. Universal Transformation (Mapping) of Nonlinear
Oscillations
Derivation of a universal transformation. Why is
the universal transformation universal! Mixing
criterion. Stability islands. Why is there no
rigorous theory? Correlation decay time 97
4.2. Criterion of Overlapping of Resonances (Chirikov
Overlap Criterion)
Interaction of resonances. Overlapping of
resonances. Relation to the condition of local
instability. Importance of the number of
resonances 105
4.3. Sine Transform
Properties of phase trajectories as functions of the
stretching parameter. Why is the sine
transformation singled out? Stationary
distribution function 110
Notes to Chapter 4 113
CHAPTER 5. STOCHASTIC LAYER: THEORY OF
FORMATION
5.1. Stochastic Destruction of a Separatrix
Peculiarities of motion close to a separatrix.
Construction of transformation. Stochasticity
condition. Width of stochastic layer 115
5.2. Stochastic Layer: Specifics of Formation
Qualitative analysis of the region of separatrix
destruction. Destruction of nonlinear resonance.
Destruction of magnetic surfaces 119
5.3. General Pattern of Stochastic Destruction in
Phase Space
Resonance and stochasticity regions. Stochasticity
quantum in phase space. Origin of stability islands.
Destruction of motion integrals. Remark about Pom
care s theorem. Two examples: motion of a particle in
the field of two plane waves and in the field of a wave
packet. 126
ix
5.4. Homoclinic Structure in the Neighborhood of a
Separatrix
Splitting of a separatrix. Homoclinic points. More
about isomorphism of stochasticity. More about
K systems 133
Notes to Chapter 5 136
CHAPTER 6. MIXING AND THE KINETIC EQUATION
6.1. Principles of Kinetic Description
Contraction of description. Two methods of
contraction. The Master equation.
Boltzmann type equation. Bogolyubov s ideas.
Random phase approximation and the principle
of correlation attenuation 139
6.2. Kinetics of the Nonlinear Oscillator
Perturbation theory for the Liouville equation.
Memory of initial conditions. Formal content of
the RPA. Method of coarsening the distribution
function. Effect of mixing. Time scale 145
6.3. Diffusional Motion of a Particle in a Wave Packet
Field
Choice of variables. Equation for the distribution
function. Approximation of wave packets.
Quasilinear equation. Remark on the trapping
time of a particle 159
Notes to Chapter 6 165
CHAPTER 7. NONLINEAR WAVE FIELD
7.1. The Fermi Pasta Ulam Problem
Origin and formulation of the problem.
Overlapping of resonances and stochasticity
threshold 167
7.2. Stochastization of a Wave Field
Hamiltonian of interacting phonons. Decay
spectra. Construction of discrete mapping.
Condition of stability and condition of
stochasticity. Role played by the number of
degrees of freedom. Kolmogorov entropy in
multidimensional systems 172
X
7.3. Kinetic Description of a Wave Field
Method of the Liouville equation. Loss of
memory of initial conditions. Part played by
mixing 186
7.4. Kinetic Equation for Phonons 191
Notes to Chapter 7 192
CHAPTER 8. STOCHASTICITY OF NONLINEAR
WAVE
8.1. Stationary Dynamics of Nonlinear Waves
Phase plane. Singularities and solitons. Nonlinear
dispersion. Overturning of a wave. Critical
velocity. Spectral properties of nonlinear waves 195
8.2. Perturbation of Nonlinear Waves
Equations for Fourier components. Hamiltonian.
Canonical equations of motion. Poisson bracket. . . . 200
8.3. Nonlinear Resonance
Truncated equations. Nonlinear resonance.
Modulation of wave parameters. Effect of spatial
structure of perturbation 204
8.4. Stochastic Instability of Nonlinear Waves
Spacing between resonances. Stochasticity
conditions. Action angle variables. Diffusion
equation of nonlinear waves 209
Notes to Chapter 8 215
CHAPTER 9. STOCHASTICITY OF QUANTUM
SYSTEMS. NONSTATIONARY PROBLEMS (PART I)
9.1. Quantum K Systems
Comparison of classical and quantum systems.
Stability problem. Problem of integrals of motion
and integrability. Peculiarities of quasiclassical
approximation. Quantum K systems and
formulation of the problem 217
9.2. Quantum Mappings
Mapping in the Schrodinger representation.
Mapping of Heisenberg operators. Projecting on
two phase space. Mapping of projections. Non
Markovian mapping of projections 221
xi
9.3. Projecting in the Basis of Coherent States
Coherent states and their properties. Normal
ordering and projecting. Expansion in powers of
h. Non Hamiltonian equations for quantum
mechanical expectation values. Divergence of
expansion in h 226
9.4. Spreading of Wave Packets
Role played by nonlinearity. Time of spreading
of a free wave packet. Condition of validity
of quasiclassical approximation 232
9.5. J Mapping and Stochasticity Condition
(Approximate Analysis)
Derivation of SF mapping. Classical limit.
Quantum corrections and 3) forms. Exponential
divergence of quantum corrections. Quantum
threshold of stochasticity. Quasiclassical region
and conditions of its existence. 235
Notes to Chapter 9 244
CHAPTER 10. STOCHASTICITY OF QUANTUM
SYSTEMS. NONSTATIONARY PROBLEMS (Part II)
10.1. Quantum Mapping of Wave Functions
Mapping formulas. Calculation of expectation
values. Generalizations 245
10.2. Analysis of Quantum Mappings
Phase correlations. Local instability. Quantum
threshold. Diffusion 250
10.3 Interaction of Quantum Resonances
Quantum nonlinear resonance in an external
field. Effective Hamiltonian. Interaction of two
overlapping resonances. Numerical analysis of
stochasticity in the case of overlapping
resonances 256
Notes to Chapter 10 269
CHAPTER 11. KINETIC DESCRIPTION OF
QUANTUM K SYSTEMS
11.1. Equation for the Density Matrix
xii
Expansion in powers of perturbation. Separation
of the P correlator. Role played by off diagonal
elements of the density matrix 272
11.2. Derivation of the Kinetic Equation
Green s function in the quasiclassical
approximation. Coarsening of the density
matrix. Decay of the (p correlator and attenuation
of correlations. Transition to a kinetic description.
Example 276
CHAPTER 12. DESTRUCTION OF INTEGRALS OF
MOTION IN QUANTUM SYSTEMS
12.1. Historical Background
Einstein s quantization rules. Integration on
topological orbits. Hypotheses on the spectra of
complex systems. The Wigner Porter Dyson
theory 285
12.2. Formulation of the Problem
Formulation of the problem. Concept of series of
levels. Regular and irregular spectra. Concept of
complexity of a system. Statistical ensemble of
levels 293
12.3. Universality of K Systems and Periodic Orbits
Universality of K systems. Universality scales.
Homogeneity of K systems. Periodic trajectories
in K systems. An analog of the ergodic theorem
for approximate cycles 2%
12.4. Quantization Rules
Summation over periodic orbits. Quantization
rules for K systems. Relation to approximate
cycles. Relation to intersection theory. 303
12.5. Distribution of Spacings between Neighboring
Levels
Typical trajectories and orbits. Stability of a
random curve. Law of distribution of small
spacings between levels. Distribution of large
spacings between levels. Results of numerical
analysis 305
xiii
12.6. Some General Remarks on Quantum K Systems
On the distribution of eigenvalues. Can the
shape of a drum be heard? Wave functions and
quasitnodes. Concept of mean field 316
12.7. Stochastic Destruction of a Bound State of Atoms
and Radiation Field
Hamiltonian of field plus atoms systems.
Resonance approximation and integrals of
motion. Destruction of integrals of motion 319
12.8. Intramolecular Energy Exchange
Two limiting cases of energy exchange. Excited
molecules. Predissociation and formation of
molecular bonding 326
Notes to Chapter 12 329
Appendix 1. Mixing Billiards 331
Appendix 2. Arnold Diffusion 339
Appendix 3. Stochasticity in Dissipative Dynamic Systems . . 343
A3.1. Stochasticity and Turbulence 343
A3.2. Strange Attractors 344
A3.3. Model of a Strange Attractor 345
A3.4. Distribution Function on a Strange Attractor:
Example of a Fractal 354
A3.5. Dimensionality of a Strange Attractor 356
A3.6. Relation between the Model and the Problem of
the Generation of Turbulence 359
References 363
Subject Index 369
xiv
|
| any_adam_object | 1 |
| author | Zaslavsky, George M. 1935-2008 |
| author_GND | (DE-588)13657985X |
| author_facet | Zaslavsky, George M. 1935-2008 |
| author_role | aut |
| author_sort | Zaslavsky, George M. 1935-2008 |
| author_variant | g m z gm gmz |
| building | Verbundindex |
| bvnumber | BV000339978 |
| callnumber-first | Q - Science |
| callnumber-label | QC174 |
| callnumber-raw | QC174.84 |
| callnumber-search | QC174.84 |
| callnumber-sort | QC 3174.84 |
| callnumber-subject | QC - Physics |
| classification_rvk | SK 820 SK 880 |
| classification_tum | MAT 587f PHY 066f |
| ctrlnum | (OCoLC)11316574 (DE-599)BVBBV000339978 |
| dewey-full | 530.1/3 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 530 - Physics |
| dewey-raw | 530.1/3 |
| dewey-search | 530.1/3 |
| dewey-sort | 3530.1 13 |
| dewey-tens | 530 - Physics |
| discipline | Physik Mathematik |
| format | Book |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01820nam a2200517 c 4500</leader><controlfield tag="001">BV000339978</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">00000000000000.0</controlfield><controlfield tag="007">t|</controlfield><controlfield tag="008">870612s1985 xx |||| 00||| eng d</controlfield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">3718602253</subfield><subfield code="9">3-7186-0225-3</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)11316574</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)BVBBV000339978</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">rakddb</subfield></datafield><datafield tag="041" ind1="1" ind2=" "><subfield code="a">eng</subfield><subfield code="h">rus</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-12</subfield><subfield code="a">DE-91G</subfield><subfield code="a">DE-83</subfield></datafield><datafield tag="050" ind1=" " ind2="0"><subfield code="a">QC174.84</subfield></datafield><datafield tag="082" ind1="0" ind2=" "><subfield code="a">530.1/3</subfield><subfield code="2">19</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 820</subfield><subfield code="0">(DE-625)143258:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 880</subfield><subfield code="0">(DE-625)143266:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">MAT 587f</subfield><subfield code="2">stub</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">PHY 066f</subfield><subfield code="2">stub</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Zaslavsky, George M.</subfield><subfield code="d">1935-2008</subfield><subfield code="e">Verfasser</subfield><subfield code="0">(DE-588)13657985X</subfield><subfield code="4">aut</subfield></datafield><datafield tag="240" ind1="1" ind2="0"><subfield code="a">Stochastičnost' dinamičeskich sistem</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Chaos in dynamic systems</subfield><subfield code="c">G. M. Zaslavsky</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">Chur u.a.</subfield><subfield code="b">Harwood</subfield><subfield code="c">1985</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">XIX, 370 S.</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">Aus d. Russ. übers.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Chaos</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Chaos (théorie des systèmes)</subfield><subfield code="2">ram</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Processus stochastiques</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Chaotic behavior in systems</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Nonlinear theories</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Stochastic systems</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Dynamisches System</subfield><subfield code="0">(DE-588)4013396-5</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Chaotisches System</subfield><subfield code="0">(DE-588)4316104-2</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Chaotisches System</subfield><subfield code="0">(DE-588)4316104-2</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2="1"><subfield code="a">Dynamisches System</subfield><subfield code="0">(DE-588)4013396-5</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">HBZ Datenaustausch</subfield><subfield code="q">application/pdf</subfield><subfield code="u">http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000211229&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield><datafield tag="940" ind1="1" ind2=" "><subfield code="n">oe</subfield></datafield><datafield tag="940" ind1="1" ind2=" "><subfield code="q">TUB-nveb</subfield></datafield><datafield tag="943" ind1="1" ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-000211229</subfield></datafield></record></collection> |
| id | DE-604.BV000339978 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T07:19:27Z |
| institution | BVB |
| isbn | 3718602253 |
| language | English Russian |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-000211229 |
| oclc_num | 11316574 |
| open_access_boolean | |
| owner | DE-12 DE-91G DE-BY-TUM DE-83 |
| owner_facet | DE-12 DE-91G DE-BY-TUM DE-83 |
| physical | XIX, 370 S. |
| psigel | TUB-nveb |
| publishDate | 1985 |
| publishDateSearch | 1985 |
| publishDateSort | 1985 |
| publisher | Harwood |
| record_format | marc |
| spellingShingle | Zaslavsky, George M. 1935-2008 Chaos in dynamic systems Chaos Chaos (théorie des systèmes) ram Processus stochastiques Chaotic behavior in systems Nonlinear theories Stochastic systems Dynamisches System (DE-588)4013396-5 gnd Chaotisches System (DE-588)4316104-2 gnd |
| subject_GND | (DE-588)4013396-5 (DE-588)4316104-2 |
| title | Chaos in dynamic systems |
| title_alt | Stochastičnost' dinamičeskich sistem |
| title_auth | Chaos in dynamic systems |
| title_exact_search | Chaos in dynamic systems |
| title_full | Chaos in dynamic systems G. M. Zaslavsky |
| title_fullStr | Chaos in dynamic systems G. M. Zaslavsky |
| title_full_unstemmed | Chaos in dynamic systems G. M. Zaslavsky |
| title_short | Chaos in dynamic systems |
| title_sort | chaos in dynamic systems |
| topic | Chaos Chaos (théorie des systèmes) ram Processus stochastiques Chaotic behavior in systems Nonlinear theories Stochastic systems Dynamisches System (DE-588)4013396-5 gnd Chaotisches System (DE-588)4316104-2 gnd |
| topic_facet | Chaos Chaos (théorie des systèmes) Processus stochastiques Chaotic behavior in systems Nonlinear theories Stochastic systems Dynamisches System Chaotisches System |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000211229&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT zaslavskygeorgem stochasticnostdinamiceskichsistem AT zaslavskygeorgem chaosindynamicsystems |
Table of contents
Order paper/chapter scan
Order paper/chapter scan
Branch Library Mathematics & Informatics
| Call Number: |
0102 PHY 066f 2001 A 36683
Floor plan |
|---|---|
| Copy 1 | Available for loan On Shelf |