Statistical mechanics of lattice systems: 2 Exact, series and renormalization group methods
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
1999
|
| Series: | Texts and monographs in physics
|
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 429 S. graph. Darst. |
| ISBN: | 3540644369 |
Staff View
MARC
| LEADER | 00000nam a2200000 cc4500 | ||
|---|---|---|---|
| 001 | BV012380490 | ||
| 003 | DE-604 | ||
| 005 | 19990408 | ||
| 007 | t | ||
| 008 | 990201s1999 d||| |||| 00||| eng d | ||
| 020 | |a 3540644369 |9 3-540-64436-9 | ||
| 035 | |a (OCoLC)174422780 | ||
| 035 | |a (DE-599)BVBBV012380490 | ||
| 040 | |a DE-604 |b ger |e rakddb | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-19 |a DE-703 |a DE-20 |a DE-384 |a DE-29T |a DE-355 |a DE-634 |a DE-11 |a DE-188 | ||
| 084 | |a UG 3100 |0 (DE-625)145625: |2 rvk | ||
| 084 | |a UG 3800 |0 (DE-625)145628: |2 rvk | ||
| 100 | 1 | |a Lavis, David A. |d 1939- |e Verfasser |0 (DE-588)120714310 |4 aut | |
| 245 | 1 | 0 | |a Statistical mechanics of lattice systems |n 2 |p Exact, series and renormalization group methods |c David A. Lavis ; George M. Bell |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1999 | |
| 300 | |a XII, 429 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Texts and monographs in physics | |
| 700 | 1 | |a Bell, George M. |d 1925- |e Verfasser |0 (DE-588)120714329 |4 aut | |
| 773 | 0 | 8 | |w (DE-604)BV012380475 |g 2 |
| 856 | 4 | 2 | |m GBV Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008397658&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-008397658 | ||
Record in the Search Index
| _version_ | 1804127012594384896 |
|---|---|
| adam_text | DAVID A. LAVIS GEORGE M. BELL STATISTICAL MECHANICS OF LATTICE SYSTEMS
VOLUME 2: EXACT, SERIES AND RENORMALIZATION GROUP METHODS WITH 47
FIGURES AND 7 TABLES SPRINGER TABLE OF CONTENTS 1. THERMODYNAMICS AND
STATISTICAL MECHANICS 1 1.1 INTRODUCTION 1 1.2 THERMODYNAMIC FORMULAE
AND VARIABLES 2 1.3 STATISTICAL MECHANICAL FORMULAE 3 1.4 THE
FIELD-DENSITY AND COUPLING-DENSITY REPRESENTATIONS .... 4 1.4.1
THERMODYNAMIC FORMALISM 5 1.4.2 LATTICE SYSTEMS 8 1.5 CORRELATION
FUNCTIONS AND SYMMETRY PROPERTIES 10 1.5.1 CORRELATION FUNCTIONS 10
1.5.2 SYMMETRY PROPERTIES 12 EXAMPLES 14 2. PHASE TRANSITIONS AND
SCALING THEORY 15 2.1 INTRODUCTION 15 2.2 THE GEOMETRY OF PHASE
TRANSITIONS 17 2.2.1 A TWO-DIMENSIONAL PHASE SPACE 18 2.2.2 A
THREE-DIMENSIONAL PHASE SPACE 20 2.3 UNIVERSALITY, FLUCTUATIONS AND
SCALING 21 2.3.1 UNIVERSALITY 22 2.3.2 KADANOFF S SCALING METHOD FOR THE
ISING MODEL 24 2.4 GENERAL SCALING FORMULATION . I 27 2.4.1 THE KADANOFF
SCALING HYPOTHESIS 27 2.4.2 APPROACHES TO THE TRANSITION REGION 30 2.4.3
FIRST-ORDER TRANSITIONS 31 2.4.4 EFFECTIVE EXPONENTS 32 2.5 LOGARITHMIC
SINGULARITIES * 33 2.5.1 THE NIGHTINGALE- T HOOFT SCALING HYPOTHESIS 33
2.5.2 CONSTRAINTS ON SCALING 34 2.5.3 APPROACHES TO THE TRANSITION
REGION 35 2.6 CORRELATION FUNCTIONS 37 2.6.1 SCALING OPERATORS AND
DIMENSIONS 37 2.6.2 VARIABLE SCALING EXPONENTS 40 2.7 DENSITIES AND
RESPONSE FUNCTIONS 41 2.8 CRITICAL POINT AND COEXISTENCE CURVE 42 VIII
TABLE OF CONTENTS 2.8.1 RESPONSE FUNCTIONS 44 2.8.2 CRITICAL EXPONENTS
45 2.8.3 EXPONENT INEQUALITIES 46 2.9 SCALING FOR A CRITICAL POINT 46
2.9.1 SCALING FIELDS FOR THE CRITICAL POINT 47 2.9.2 APPROACHES TO THE
CRITICAL POINT 48 2.9.3 EXPERIMENTAL VARIABLES 50 2.9.4 THE DENSITY AND
RESPONSE FUNCTIONS 51 2.9.5 ASYMPTOTIC FORMS 51 2.9.6 CRITICAL EXPONENTS
AND SCALING LAWS 52 2.9.7 SCALING FOR THE COEXISTENCE CURVE 54 2.10
MEAN-FIELD THEORY FOR THE ISING FERROMAGNET 55 2.11 CORRELATION SCALING
AT A CRITICAL POINT 57 2.12 TRICRITICAL POINT 58 2.13 SCALING FOR A
TRICRITICAL POINT 62 2.13.1 SCALING FIELDS FOR THE TRICRITICAL POINT 62
2.13.2 TRICRITICAL EXPONENTS AND SCALING LAWS 64 2.13.3 CONNECTED
TRANSITION REGIONS 66 2.14 CORRECTIONS TO SCALING 69 2.15 SCALING AND
UNIVERSALITY 70 2.16 FINITE-SIZE SCALING 75 2.16.1 THE FINITE-SIZE
SCALING FIELD 76 2.16.2 THE SHIFT AND ROUNDING EXPONENTS 78 2.16.3
UNIVERSALITY AND FINITE-SIZE SCALING 80 2.17 CONFORMAL INVARIANCE 81
2.17.1 FROM SCALING TO THE CONFORMAL GROUP 82 2.17.2 CORRELATION
FUNCTIONS FOR D 2 83 2.17.3 UNIVERSAL AMPLITUDES FOR D = 2 84 EXAMPLES
86 3. LANDAU AND LANDAU-GINZBURG THEORY 89 3.1 THE FERROMAGNETIC ISING
MODEL 89 3.2 LANDAU THEORY FOR A CRITICAL POINT 91 3.3 LANDAU-GINZBERG
THEORY FOR A CRITICAL POINT 93 3.3.1 THE GAUSSIAN APPROXIMATION 94 3.3.2
GAUSSIAN CRITICAL EXPONENTS 97 3.4 THE EQUILIBRIUM DILUTE ISING MODEL
103 3.5 THE 3-STATE POTTS MODEL 105 3.6 LANDAU THEORY FOR A TRICRITICAL
POINT 108 EXAMPLES ILL TABLE OF CONTENTS IX 4. ALGEBRAIC METHODS IN
STATISTICAL MECHANICS 113 4.1 INTRODUCTION 113 4.2 THE THERMODYNAMIC
LIMIT 116 4.3 LOWER BOUNDS FOR PHASE TRANSITIONS: THE PEIERLS METHOD
.... 117 4.4 LOWER BOUNDS FOR THE SIMPLE LATTICE FLUID 123 4.5 GRAND
PARTITION FUNCTION ZEROS AND PHASE TRANSITIONS 125 4.6 RUELLE S THEOREM
127 4.7 THE YANG-LEE CIRCLE THEOREM 130 4.8 SYSTEMS WITH PAIR
INTERACTIONS 132 4.9 TRANSFER MATRICES 136 4.9.1 THE PARTITION FUNCTION
137 4.9.2 BOUNDARY CONDITIONS 140 4.9.3 THE LIMIT N 2 - OO 140 4.9.4
CORRELATION FUNCTIONS 141 4.9.5 CORRELATION LENGTHS AND PHASE
TRANSITIONS 145 4.10 THE WOOD METHOD 148 4.10.1 EVOLUTION OF PARTITION
FUNCTION ZEROS 149 4.10.2 CONNECTION CURVES AND CROSS-BLOCK CURVES 151
4.10.3 THE SPIN-| SQUARE LATTICE ISING MODEL 154 4.10.4 CRITICAL POINTS
AND EXPONENTS 159 EXAMPLES 163 5. THE EIGHT-VERTEX MODEL 167 5.1
INTRODUCTION 167 5.2 SPIN REPRESENTATIONS 169 5.3 PARAMETER SPACE AND
GROUND STATES 172 5.4 SOME TRANSFORMATIONS 174 5.5 THE WEAK-GRAPH
TRANSFORMATION 176 5.6 TRANSITION SURFACES 177 5.7 THE TRANSFER MATRIX
182 5.8 THE FREE ENERGY AND MAGNETIZATION 186 5.9 CRITICAL BEHAVIOUR *.
187 5.10 THE SPIN REPRESENTATION AND THE ISING MODEL LIMIT 190 5.10.1
THE ISOTROPIC ISING MODEL WITH A FOUR-SPIN COUPLING. 191 5.10.2 THE
JIINGLING SPIN REPRESENTATION 192 5.10.3 THE ISOTROPIC ISING MODEL
WITHOUT A FOUR-SPIN COUPLING 192 5.11 THE SIX-VERTEX MODEL AS A SPECIAL
CASE 195 5.11.1 LOW-TEMPERATURE REGIONS (I) AND (II) 195 5.11.2
LOW-TEMPERATURE REGION (III) 196 5.11.3 HIGH-TEMPERATURE REGIONS (I),
(II) AND (III) 197 5.12 THE EIGHT-VERTEX MODEL AND UNIVERSALITY 198
EXAMPLES 200 X TABLE OF CONTENTS 6. REAL-SPACE RENORMALIZATION GROUP
THEORY 203 6.1 INTRODUCTION 203 6.2 THE BASIC ELEMENTS OF THE
RENORMALIZATION GROUP 204 6.3 RENORMALIZATION TRANSFORMATIONS AND WEIGHT
FUNCTIONS 206 6.4 FIXED POINTS AND THE LINEAR RENORMALIZATION GROUP 211
6.5 FREE ENERGY AND DENSITIES 214 6.6 DECIMATION OF THE ONE-DIMENSIONAL
ISING MODEL 215 6.7 DECIMATION IN TWO DIMENSIONS 222 6.8 LOWER-BOUND AND
UPPER-BOUND APPROXIMATIONS 225 6.8.1 AN UPPER-BOUND METHOD 226 6.8.2 A
LOWER-BOUND METHOD 228 6.9 THE CUMULANT APPROXIMATION 229 6.10 BOND
MOVING APPROXIMATIONS 232 6.11 FINITE-LATTICE APPROXIMATIONS 235 6.12
VARIATIONAL APPROXIMATIONS 239 6.13 PHENOMENOLOGICAL RENORMALIZATION 241
6.13.1 THE SQUARE LATTICE ISING MODEL 244 6.13.2 OTHER MODELS 245 6.13.3
MORE THAN ONE COUPLING 246 6.14 OTHER RENORMALIZATION GROUP METHODS 247
EXAMPLES . 248 7. SERIES METHODS 251 7.1 INTRODUCTION 251 7.2 THE
ANALYSIS OF SERIES 254 7.2.1 THE RATIO METHOD 254 7.2.2 PADE
APPROXIMANTS 256 7.2.3 THE DIFFERENTIAL APPROXIMANT METHOD 258 7.3
LOW-TEMPERATURE SERIES FOR THE SPIN-| ISING MODEL 258 7.4
HIGH-TEMPERATURE SERIES FOR THE SPIN-| ISING MODEL 262 7.4.1 THE FREE
ENERGY . 262 7.4.2 SUSCEPTIBILITY SERIES 264 7.4.3 COEFFICIENT
RELATIONS FOR SUSCEPTIBILITY SERIES 266 7.5 THE LINKED-CLUSTER EXPANSION
270 7.5.1 MULTI-BONDED GRAPHS .. :: 270 7.5.2 CONNECTED GRAPHS AND STARS
273 7.5.3 MOMENTS, CUMULANTS AND FINITE CLUSTERS 275 7.6 APPLICATIONS OF
THE LINKED-CLUSTER EXPANSION 277 7.6.1 THE GENERAL-S ISING MODEL 277
7.6.2 D- VECTOR MODELS 279 7.6.3 THE CLASSICAL HEISENBERG MODEL 280
7.6.4 THE QUANTUM HEISENBERG MODEL 283 7.6.5 CORRELATIONS AND
SUSCEPTIBILITY 286 7.7 FINITE METHODS 290 7.7.1 THE FINITE-CLUSTER
METHOD 291 TABLE OF CONTENTS XI 7.7.2 THE FINITE-LATTICE METHOD 293 7.8
RESULTS AND ANALYSIS 295 7.8.1 HIGH-TEMPERATURE SERIES FOR SPIN-| ISING
AND POTTS MODELS 296 7.8.2 LOW-TEMPERATURE SERIES 299 7.8.3
HIGH-TEMPERATURE K EXPANSIONS 300 EXAMPLES 301 8. DIMER ASSEMBLIES 303
8.1 INTRODUCTION 303 8.2 THE DIMER PARTITION FUNCTION 304 8.2.1 THE
SQUARE LATTICE CASE 306 8.2.2 THE HONEYCOMB LATTICE CASE 310 8.3 THE
MODIFIED KDP MODEL EQUIVALENCE 317 8.4 THE ISING MODEL EQUIVALENCE 320
8.5 K-TYPE AND O-TYPE TRANSITIONS 324 8.6 THE CHAIN CONFORMAL TRANSITION
326 EXAMPLES .332 A. APPENDICES 335 A.I FOURIER TRANSFORMS IN D
DIMENSIONS 335 A.1.1 DISCRETE FINITE LATTICES 335 A.1.2 A CONTINUOUS
FINITE VOLUME 336 A.1.3 A CONTINUOUS INFINITE VOLUME 337 A.1.4 A SPECIAL
CASE 338 A.2 THE CONFORMAL GROUP 340 A.3 GROUP REPRESENTATION THEORY 342
A.3.1 GROUPS.... . 342 A.3.2 REPRESENTATIONS 342 A.3.3 THE BLOCK
DIAGONALIZATION OF TRANSFER MATRICES 347 A.3.4 EQUIVALENCE CLASSES 349
A.3.5 USING EQUIVALENCE CLASSES FOR BLOCK DIAGONALIZATION .. 355 A.3.6
THE TRANSFER MATRIX EIGEN PROBLEM 357 A.4 SOME TRANSFORMATIONS IN THE
COMPLEX PLANE 360 A.5 ALGEBRAIC FUNCTIONS 363 A.6 ELLIPTIC INTEGRALS,
FUNCTIONS AND NOME SERIES 366 A.6.1 ELLIPTIC INTEGRALS 366 A.6.2
ELLIPTIC FUNCTIONS 367 A.6.3 NOME SERIES 368 A.7 LATTICES AND GRAPHS 370
A.7.1 SUBGRAPHS 370 A.7.2 SECTION GRAPHS 373 A.7.3 ZERO-FIELD GRAPHS .
374 A.7.4 MAGNETIC GRAPHS AND COEFFICIENT RELATIONS 374 A.7.5
MULTI-BONDED GRAPHS 379 XII TABLE OF CONTENTS A.7.6 POLYGONS AND
TRIANGULATION 380 A.7.7 ORIENTED GRAPHS 380 A.8 THE WEAK-GRAPH
TRANSFORMATION 381 A.9 THE GENERALIZED MOMENT-CUMULANT RELATIONS 386
A.10 KASTELYN S THEOREM 389 A.10.1 THE CANONICAL FLUX DISTRIBUTION 389
A.10.2 THE DIMER PARTITION FUNCTION 390 A.10.3 SUPERPOSITION POLYNOMIALS
AND PFAFFIANS 391 A.11 DETERMINANTS OF CYCLIC MATRICES 394 A.12 THE T
MATRIX 396 REFERENCES AND AUTHOR INDEX 401 SUBJECT INDEX 423
|
| any_adam_object | 1 |
| author | Lavis, David A. 1939- Bell, George M. 1925- |
| author_GND | (DE-588)120714310 (DE-588)120714329 |
| author_facet | Lavis, David A. 1939- Bell, George M. 1925- |
| author_role | aut aut |
| author_sort | Lavis, David A. 1939- |
| author_variant | d a l da dal g m b gm gmb |
| building | Verbundindex |
| bvnumber | BV012380490 |
| classification_rvk | UG 3100 UG 3800 |
| ctrlnum | (OCoLC)174422780 (DE-599)BVBBV012380490 |
| discipline | Physik |
| format | Book |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01366nam a2200325 cc4500</leader><controlfield tag="001">BV012380490</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">19990408 </controlfield><controlfield tag="007">t</controlfield><controlfield tag="008">990201s1999 d||| |||| 00||| eng d</controlfield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">3540644369</subfield><subfield code="9">3-540-64436-9</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)174422780</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)BVBBV012380490</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="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-19</subfield><subfield code="a">DE-703</subfield><subfield code="a">DE-20</subfield><subfield code="a">DE-384</subfield><subfield code="a">DE-29T</subfield><subfield code="a">DE-355</subfield><subfield code="a">DE-634</subfield><subfield code="a">DE-11</subfield><subfield code="a">DE-188</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">UG 3100</subfield><subfield code="0">(DE-625)145625:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">UG 3800</subfield><subfield code="0">(DE-625)145628:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Lavis, David A.</subfield><subfield code="d">1939-</subfield><subfield code="e">Verfasser</subfield><subfield code="0">(DE-588)120714310</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Statistical mechanics of lattice systems</subfield><subfield code="n">2</subfield><subfield code="p">Exact, series and renormalization group methods</subfield><subfield code="c">David A. Lavis ; George M. Bell</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">Berlin [u.a.]</subfield><subfield code="b">Springer</subfield><subfield code="c">1999</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">XII, 429 S.</subfield><subfield code="b">graph. Darst.</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="490" ind1="0" ind2=" "><subfield code="a">Texts and monographs in physics</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Bell, George M.</subfield><subfield code="d">1925-</subfield><subfield code="e">Verfasser</subfield><subfield code="0">(DE-588)120714329</subfield><subfield code="4">aut</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="w">(DE-604)BV012380475</subfield><subfield code="g">2</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">GBV 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=008397658&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield><datafield tag="999" ind1=" " ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-008397658</subfield></datafield></record></collection> |
| id | DE-604.BV012380490 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:26:35Z |
| institution | BVB |
| isbn | 3540644369 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008397658 |
| oclc_num | 174422780 |
| open_access_boolean | |
| owner | DE-19 DE-BY-UBM DE-703 DE-20 DE-384 DE-29T DE-355 DE-BY-UBR DE-634 DE-11 DE-188 |
| owner_facet | DE-19 DE-BY-UBM DE-703 DE-20 DE-384 DE-29T DE-355 DE-BY-UBR DE-634 DE-11 DE-188 |
| physical | XII, 429 S. graph. Darst. |
| publishDate | 1999 |
| publishDateSearch | 1999 |
| publishDateSort | 1999 |
| publisher | Springer |
| record_format | marc |
| series2 | Texts and monographs in physics |
| spelling | Lavis, David A. 1939- Verfasser (DE-588)120714310 aut Statistical mechanics of lattice systems 2 Exact, series and renormalization group methods David A. Lavis ; George M. Bell Berlin [u.a.] Springer 1999 XII, 429 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Texts and monographs in physics Bell, George M. 1925- Verfasser (DE-588)120714329 aut (DE-604)BV012380475 2 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008397658&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lavis, David A. 1939- Bell, George M. 1925- Statistical mechanics of lattice systems |
| title | Statistical mechanics of lattice systems |
| title_auth | Statistical mechanics of lattice systems |
| title_exact_search | Statistical mechanics of lattice systems |
| title_full | Statistical mechanics of lattice systems 2 Exact, series and renormalization group methods David A. Lavis ; George M. Bell |
| title_fullStr | Statistical mechanics of lattice systems 2 Exact, series and renormalization group methods David A. Lavis ; George M. Bell |
| title_full_unstemmed | Statistical mechanics of lattice systems 2 Exact, series and renormalization group methods David A. Lavis ; George M. Bell |
| title_short | Statistical mechanics of lattice systems |
| title_sort | statistical mechanics of lattice systems exact series and renormalization group methods |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008397658&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV012380475 |
| work_keys_str_mv | AT lavisdavida statisticalmechanicsoflatticesystems2 AT bellgeorgem statisticalmechanicsoflatticesystems2 |