text:
# 3v4l.org

run code in 200+ php & hhvm versions
```<?php class dec{ function bin(\$a){ echo rand(); }} \$obj=new dec(); \$obj->bin();```
Output for 7.1.0 - 7.2.0
Fatal error: Uncaught ArgumentCountError: Too few arguments to function dec::bin(), 0 passed in /in/9gmn0 on line 7 and exactly 1 expected in /in/9gmn0:3 Stack trace: #0 /in/9gmn0(7): dec->bin() #1 {main} thrown in /in/9gmn0 on line 3
Process exited with code 255.
Output for 7.0.20
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1923278376
Output for 7.0.14
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 653748171
Output for 7.0.10
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 99375187
Output for 7.0.9
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 75684743
Output for 7.0.8
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 2047842079
Output for 7.0.7
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1485058170
Output for 7.0.6
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 2103318466
Output for 7.0.5
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1276177929
Output for 7.0.4
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1565124778
Output for 7.0.3
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1837295545
Output for 7.0.2
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1538277598
Output for 7.0.1
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1595248616
Output for 7.0.0
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 641072177
Output for 5.6.28
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 171574565
Output for 5.6.25
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1866284565
Output for 5.6.24
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1178996452
Output for 5.6.23
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 876551922
Output for 5.6.22
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 603333155
Output for 5.6.21
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1040280523
Output for 5.6.20
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 375402186
Output for 5.6.19
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1031167308
Output for 5.6.18
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 758533089
Output for 5.6.17
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1015333720
Output for 5.6.16
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 799355094
Output for 5.6.15
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1557234488
Output for 5.6.14
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1252385154
Output for 5.6.13
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1574971425
Output for 5.6.12
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 615438397
Output for 5.6.11
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 683389649
Output for 5.6.10
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 987155557
Output for 5.6.9
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 2012413187
Output for 5.6.8
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1549851201
Output for 5.6.7
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 144572705
Output for 5.6.6
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1535357923
Output for 5.6.5
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 251137428
Output for 5.6.4
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1787650425
Output for 5.6.3
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 420037714
Output for 5.6.2
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 630572797
Output for 5.6.1
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1848481328
Output for 5.6.0
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1176676952
Output for 5.5.38
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 27836036
Output for 5.5.37
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1735424837
Output for 5.5.36
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1997909154
Output for 5.5.35
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 2048652502
Output for 5.5.34
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 393278659
Output for 5.5.33
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 624095742
Output for 5.5.32
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1204243032
Output for 5.5.31
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 887501151
Output for 5.5.30
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1589606453
Output for 5.5.29
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1378937649
Output for 5.5.28
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1752972182
Output for 5.5.27
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 353152166
Output for 5.5.26
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 928575814
Output for 5.5.25
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1654043647
Output for 5.5.24
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 181055697
Output for 5.5.23
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 661892123
Output for 5.5.22
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 496492068
Output for 5.5.21
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1500112043
Output for 5.5.20
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 681167435
Output for 5.5.19
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 105738186
Output for 5.5.18
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 54167377
Output for 5.5.16
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 884537948
Output for 5.5.15
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1256076098
Output for 5.5.14
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 2076942801
Output for 5.5.13
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1233234345
Output for 5.5.12
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 9516758
Output for 5.5.11
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 869220086
Output for 5.5.10
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 740816987
Output for 5.5.9
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 48841880
Output for 5.5.8
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1798823634
Output for 5.5.7
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1473129608
Output for 5.5.6
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 375504959
Output for 5.5.5
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 93194490
Output for 5.5.4
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1919820557
Output for 5.5.3
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1407755442
Output for 5.5.2
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 927904268
Output for 5.5.1
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1670144113
Output for 5.5.0
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1817792840
Output for 5.4.45
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 126640945
Output for 5.4.44
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 554369319
Output for 5.4.43
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 374612250
Output for 5.4.42
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 370191378
Output for 5.4.41
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 331185228
Output for 5.4.40
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 40994881
Output for 5.4.39
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 269077112
Output for 5.4.38
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1326534575
Output for 5.4.37
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1384831406
Output for 5.4.36
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 550130012
Output for 5.4.35
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1700403063
Output for 5.4.34
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 112454729
Output for 5.4.32
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1458288285
Output for 5.4.31
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 875819400
Output for 5.4.30
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 136480645
Output for 5.4.29
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 100452368
Output for 5.4.28
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 349697932
Output for 5.4.27
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1876814337
Output for 5.4.26
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1153525642
Output for 5.4.25
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 784871462
Output for 5.4.24
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 39179120
Output for 5.4.23
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1297347699
Output for 5.4.22
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1243268412
Output for 5.4.21
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 84553019
Output for 5.4.20
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1699948108
Output for 5.4.19
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1859519051
Output for 5.4.18
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 628946706
Output for 5.4.17
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1333519223
Output for 5.4.16
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1905270203
Output for 5.4.15
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1105524272
Output for 5.4.14
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 850726292
Output for 5.4.13
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 96297618
Output for 5.4.12
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 361708508
Output for 5.4.11
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 627851298
Output for 5.4.10
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1080443413
Output for 5.4.9
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 915023028
Output for 5.4.8
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1332678068
Output for 5.4.7
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 794160965
Output for 5.4.6
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1110576117
Output for 5.4.5
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1507574042
Output for 5.4.4
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1182951972
Output for 5.4.3
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1220447257
Output for 5.4.2
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1183196102
Output for 5.4.1
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1517389809
Output for 5.4.0
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1872649882
Output for 5.3.29
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1590912475
Output for 5.3.28
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1775643454
Output for 5.3.27
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 810059017
Output for 5.3.26
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 120172651
Output for 5.3.25
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 61690555
Output for 5.3.24
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 46725935
Output for 5.3.23
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1891184614
Output for 5.3.22
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 452613003
Output for 5.3.21
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1294566141
Output for 5.3.20
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 624859497
Output for 5.3.19
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 290965221
Output for 5.3.18
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1476571026
Output for 5.3.17
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 720805382
Output for 5.3.16
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 663506049
Output for 5.3.15
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 251630687
Output for 5.3.14
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 792065723
Output for 5.3.13
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 48407791
Output for 5.3.12
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1319136648
Output for 5.3.11
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 18455381
Output for 5.3.10
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 532263962
Output for 5.3.9
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 57164359
Output for 5.3.8
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1025726561
Output for 5.3.7
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 942596554
Output for 5.3.6
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1399758569
Output for 5.3.5
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 297395630
Output for 5.3.4
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 47816287
Output for 5.3.3
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 835848185
Output for 5.3.2
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 228375864
Output for 5.3.1
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1160820129
Output for 5.3.0
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 694844190
Output for 5.2.17
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1204296498
Output for 5.2.16
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 294532034
Output for 5.2.15
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1982874510
Output for 5.2.14
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 545737605
Output for 5.2.13
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 794801489
Output for 5.2.12
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 2120499844
Output for 5.2.11
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1048408613
Output for 5.2.10
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 2144089779
Output for 5.2.9
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1175722569
Output for 5.2.8
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1377168344
Output for 5.2.7
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1407300646
Output for 5.2.6
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1757142056
Output for 5.2.5
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 629131332
Output for 5.2.4
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1356669097
Output for 5.2.3
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 522971241
Output for 5.2.2
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 2065705310
Output for 5.2.1
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 173686938
Output for 5.2.0
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 77184122
Output for 5.1.6
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 2078674653
Output for 5.1.5
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 854729157
Output for 5.1.4
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 657258603
Output for 5.1.3
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 131645985
Output for 5.1.2
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 12901010
Output for 5.1.1
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 1520792207
Output for 5.1.0
Warning: Missing argument 1 for dec::bin(), called in /in/9gmn0 on line 7 and defined in /in/9gmn0 on line 3 222944781
Output for 5.0.5
Warning: Missing argument 1 for dec::bin() in /in/9gmn0 on line 3 2024658330
Output for 5.0.4
Warning: Missing argument 1 for dec::bin() in /in/9gmn0 on line 3 698354576
Output for 5.0.3
Warning: Missing argument 1 for dec::bin() in /in/9gmn0 on line 3 876606250
Output for 5.0.2
Warning: Missing argument 1 for dec::bin() in /in/9gmn0 on line 3 349455866
Output for 5.0.1
Warning: Missing argument 1 for dec::bin() in /in/9gmn0 on line 3 686923346
Output for 5.0.0
Warning: Missing argument 1 for dec::bin() in /in/9gmn0 on line 3 753646776
Output for 4.4.9
Warning: Missing argument 1 for bin() in /in/9gmn0 on line 3 2107980266
Output for 4.4.8
Warning: Missing argument 1 for bin() in /in/9gmn0 on line 3 1041367397
Output for 4.4.7
Warning: Missing argument 1 for bin() in /in/9gmn0 on line 3 818810773
Output for 4.4.6
Warning: Missing argument 1 for bin() in /in/9gmn0 on line 3 2015338442
Output for 4.4.5
Warning: Missing argument 1 for bin() in /in/9gmn0 on line 3 1072628186
Output for 4.4.4
Warning: Missing argument 1 for bin() in /in/9gmn0 on line 3 1610393264
Output for 4.4.3
Warning: Missing argument 1 for bin() in /in/9gmn0 on line 3 281313912
Output for 4.4.2
Warning: Missing argument 1 for bin() in /in/9gmn0 on line 3 476569287
Output for 4.4.1
Warning: Missing argument 1 for bin() in /in/9gmn0 on line 3 38282821
Output for 4.4.0
Warning: Missing argument 1 for bin() in /in/9gmn0 on line 3 722171906
Output for 4.3.11
Warning: Missing argument 1 for bin() in /in/9gmn0 on line 3 571500965
Output for 4.3.10
Warning: Missing argument 1 for bin() in /in/9gmn0 on line 3 9779609
Output for 4.3.9
Warning: Missing argument 1 for bin() in /in/9gmn0 on line 3 1284368469
Output for 4.3.8
Warning: Missing argument 1 for bin() in /in/9gmn0 on line 3 907780320
Output for 4.3.7
Warning: Missing argument 1 for bin() in /in/9gmn0 on line 3 1288356370
Output for 4.3.6
Warning: Missing argument 1 for bin() in /in/9gmn0 on line 3 79481647
Output for 4.3.5
Warning: Missing argument 1 for bin() in /in/9gmn0 on line 3 482329130
Output for 4.3.4
Warning: Missing argument 1 for bin() in /in/9gmn0 on line 3 874308494
Output for 4.3.3
Warning: Missing argument 1 for bin() in /in/9gmn0 on line 3 942744921
Output for 4.3.2
Warning: Missing argument 1 for bin() in /in/9gmn0 on line 3 1361491321
Output for 4.3.1
Warning: Missing argument 1 for bin() in /in/9gmn0 on line 3 1533934289
Output for 4.3.0
Warning: Missing argument 1 for bin() in /in/9gmn0 on line 3 950096976


text:
# Ionisation

### Ionisation

Ionization (or ionisation, see American and British English spelling differences) is the process by which an atom or a molecule acquires a negative or positive charge by gaining or losing electrons.

## Ionization to negatively charged ions

The negatively charged ions are produced when a free electron collides with an atom and is subsequently trapped inside the electric potential barrier, releasing any excess energy. The process is known as Electron capture ionization.

## Ionization to positively charged ions

The positively charged ions are produced by transferring sufficient amount of energy to a bound electron. The threshold amount of the required energy is known as ionization potential.

The ionization energy, generally, is supplied by incident photons (photoionization) or via collision with other particles. The ionization of a material by subjecting it to a high temperature (thermal ionizaion) is a form of collision mediated ionization for which a bound electron acquires energy in multiple steps, jumping into a higher energy level in each step, until reaching to the continuum of free steps.

## Semi-Classical description of ionization

Classical physics and the Bohr model of the atom can qualitatively explain photoionization and collision mediated ionization. In these cases, during the ionization process, the energy of the electron exceeds the energy difference of the potential barrier it is trying to pass. The semi-classical description, however, cannot describe tunnel ionization since the process involves the passage of electron through a classically forbidden potential barrier.

## Quantum mechanical description of ionization

The interaction of atoms and molecules with sufficiently strong laser pulses leads to the ionization to singly or multiply charged ions. The ionization rate, i.e. the ionization probability in unit time, can only be calculated using quantum mechanics. In general, the analytical solutions are not available, and the approximations required for manageable numerical calculations do not provide accurate enough results. However, when the laser intensity is sufficiently high, the detailed structure of the atom or molecule can be ignored and analytical solution for the ionization rate is possible.

### Tunnel ionization

Tunnel ionization is ionization due to quantum tunneling. In classical ionization, an electron must have enough energy to make it over the potential barrier, but quantum tunneling allows the electron simply to go through the potential barrier instead of going all the way over it because of the wave nature of the electron. The probability of an electron's tunneling through the barrier drops off exponentially with the width of the potential barrier. Therefore, an electron with a higher energy can make it further up the potential barrier, leaving a much thinner barrier to tunnel through and, thus, a greater chance to do so. In practice, tunnel ionization is observable when the atom or molecule is interacting with near-infrared strong laser pauses. This process can be understood as a process by which a bounded electron, through the absorption of more than one photon from the laser field, is ionized. This picture is generally known as multiphoton ionization (MPI).

Keldysh[1] modeled the MPI process as a transition of the electron from the ground state of the atom to the Volkov states.[2] In this model the perturbation of the ground state by the laser field is neglected and the details of atomic structure in determining the ionization probability are not taken into account. The major difficulty with Keldysh's model was its neglect of the effects of Coulomb interaction on the final state of the electron. As it is observed from figure, the Coulomb field is not very small in magnitude compared to the potential of the laser at larger distances from the nucleus. This is in contrast to the approximation made by neglecting the potential of the laser at regions near the nucleus. Perelomov et al.[3][4] included the Coulomb interaction at larger internuclear distances. Their model (which we call PPT model) was derived for short range potential and includes the effect of the long range Coulomb interaction through the first order correction in the quasi-classical action. Larochelle et al.[5] have compared the theoretically predicted ion versus intensity curves of rare gas atoms interacting with a Ti:Sapphire laser with experimental measurement. They have shown that the total ionization rate predicted by the PPT model fit very well the experimental ion yields for all rare gases in the intermediate regime of Keldysh parameter.

The rate of MPI on atom with an ionization potential $E_i$ in a linearly polarized laser with frequency $\omega$ is given by

$W_\left\{PPT\right\}=|C_\left\{n^\left\{*\right\}l^\left\{*\right\}\right\}|^\left\{2\right\}\sqrt\left\{\frac\left\{6\right\}\left\{\pi\right\}\right\}f_\left\{lm\right\}E_\left\{i\right\}\left(2\left(2E_i\right)^\left\{\frac\left\{3\right\}\left\{2\right\}\right\}/F\right)^\left\{n^\left\{*2\right\}-|m|-3/2\right\}\left(1+\gamma\right)^\left\{2\right\}\right)^\left\{|m/2|+3/4\right\}A_\left\{m\right\}\left(\omega, \gamma\right)e^\left\{-\left(2\left(2E_i\right)^\left\{\frac\left\{3\right\}\left\{2\right\}\right\}/F\right)g\left(\gamma\right)\right\}$

where $\gamma= \frac\left\{\omega F\right\}\left\{\sqrt \left\{2E_i\right\}\right\}$ is the Keldyshs adiabaticity parameter, $n^\left\{*\right\}=\sqrt\left\{2E_i\right\}/Z^\left\{2\right\}$, $F$ is the peak electric field of laser and $l^\left\{*\right\}=n^\left\{*\right\}-1$. The coefficients $f_\left\{lm\right\}$, $g\left(\gamma\right)$ and $C_\left\{n^\left\{*\right\}l^\left\{*\right\}\right\}$ are given by

$f_\left\{lm\right\}= \frac\left\{\left(2l+1\right)\left(l+|m|\right)^\left\{!\right\}\right\}\left\{2^\left\{m\right\}|m|^\left\{!\right\}\left(l-|m|\right)^\left\{!\right\}\right\}$
$g\left(\gamma\right)=\frac\left\{3\right\}\left\{2\gamma\right\} \left(1+\frac\left\{1\right\}\left\{2\gamma^\left\{2\right\}\right\}\sinh^\left\{-1\right\}\left(\gamma\right)-\frac\left\{\sqrt\left\{1+\gamma^\left\{2\right\}\right\}\right\}\left\{2\gamma\right\}\right)$
$|C_\left\{n^\left\{*\right\}l^\left\{*\right\}\right\}|^\left\{2\right\}= \frac\left\{2^\left\{2n^\left\{*\right\}\right\}\right\}\left\{n^\left\{*\right\}\Gamma\left(n^\left\{*\right\}+l^\left\{*\right\}+1\right)\Gamma\left(n^\left\{*\right\}l^\left\{*\right\}\right)\right\}$

The coefficient $A_\left\{m\right\}\left(\omega, \gamma\right)$ is given by

$A_\left\{m\right\}\left(\omega, \gamma\right)=\frac\left\{4\right\}\left\{3\pi\right\}\frac\left\{1\right\}\left\{|m|^\left\{!\right\}\right\}\frac\left\{\gamma^\left\{2\right\}\right\}\left\{1+\gamma^\left\{2\right\}\right\}\sum_\left\{n>v\right\}^\left\{\infty\right\}w_\left\{m\right\}\left(\sqrt\left\{\frac\left\{2\gamma\right\}\left\{\sqrt\left\{1+\gamma^\left\{2\right\}\right\}\right\}\left(n-v\right)\right\}e^\left\{-\left(n-v\right)\alpha\left(\gamma\right)\right\}\right)$

where

$w_\left\{m\right\}\left(x\right)=e^\left\{-x^\left\{2\right\}\right\}\int_0^x \left(x^2-y^2\right)^m e^\left\{y^2\right\}\,dy$
$\alpha\left(\gamma\right)= 2\left(\sinh^\left\{-1\right\}\left(\gamma\right)-\frac\left\{\gamma\right\}\left\{\sqrt\left\{1+\gamma^\left\{2\right\}\right\}\right\}\right)$
$v= \frac\left\{E_i\right\}\left\{\omega\right\}\left(1+\frac\left\{2\right\}\left\{\gamma^\left\{2\right\}\right\}\right)$

#### Quasi-static tunnel ionization

The quasi-static tunnelling (QST) is the ionization whose rate can be satisfactorily predicted by the ADK model,[6] i.e. the limit of the PPT model when $\gamma$ approaches zero.[7] The rate of QST is given by

$W_\left\{ADK\right\}=|C_\left\{n^\left\{*\right\}l^\left\{*\right\}\right\}|^\left\{2\right\}\sqrt\left\{\frac\left\{6\right\}\left\{\pi\right\}\right\}f_\left\{lm\right\}E_\left\{i\right\}\left(2\left(2E_i\right)^\left\{\frac\left\{3\right\}\left\{2\right\}\right\}/F\right)^\left\{n^\left\{*2\right\}-|m|-3/2\right\}e^\left\{-\left(2\left(2E_i\right)^\left\{\frac\left\{3\right\}\left\{2\right\}\right\}/3F\right)\right\}$

As compared to $W_\left\{PPT\right\}$ the absence of summation over n, which represent different above threshold ionization (ATI) peaks, is remarkable.

## Strong field approximation for the ionization rate

The calculations of PPT are done in the E-gauge, meaning that the laser field is taken as electromagnetic waves. The ionization rate can also be calculated in A-gauge, which emphasis the particle nature of light (absorbing multiple photons during ionization). This approach was adopted by Krainov model[8] based on the earlier works of Faisal[9] and Reiss.[10] The resulting rate is given by

$W_\left\{KRA\right\}= \sum_\left\{n=N\right\}^\left\{n=\infty\right\} 2 \pi \omega^\left\{2\right\} p \left(n-n_\mathrm\left\{osc\right\}\right)^2 \int d \Omega |FT \left(I_\left\{KAR\right\} \Psi \left( \mathbf\left\{r\right\}\right)\right)|^2 J_n^2 \left(n_f, \frac\left\{n_\mathrm\left\{osc\right\}\right\}\left\{2\right\}\right)$

where, $N=\left[n_i + n_\mathrm\left\{osc\right\}\right]$ is the minimum number of photons necessary to ionize the atom, $n_\left\{i\right\} =E_i/\omega$, $n_\mathrm\left\{osc\right\} = U_\left\{p\right\} / \omega$ ( $U_p$ is the ponderomotive energy), $J_\left\{n\right\} \left(u, v\right)$ is the double Bessel function, $p = \sqrt\left\{ 2 \omega \left(n-n_\mathrm\left\{osc\right\}- n_i\right)\right\}$, $n_\left\{f\right\} = 2 \sqrt \left\{ n_\mathrm\left\{osc\right\} / \omega\right\} p cos\left(\theta\right)$ where $\theta$ is the angle between the momentum of the electron, p, and the electric field of the laser, F, and , the symbol FT denotes the three dimensional Fourier transformation. Finally, $I_\left\{KAR\right\} = \left(\frac \left\{2 Z^2\right\}\left\{n^2 F r\right\}\right)^n$ incorporates the Coulomb correction in the SFA model.

### Atomic stabilization/population trapping

In calculating the rate of MPI of atoms only transitions to the continuum states are considered. Such an approximation is acceptable as long as there is no multiphoton resonance between the ground state and some excited states. However, in real situation of interaction with pulsed lasers, during the evolution of laser intensity, due to different Stark shift of the ground and excited states there is a possibility that some excited state go into multiphoton resonance with the ground state. Within the dressed atom picture, the ground state dressed by $m$ photons and the resonant state undergo an avoided crossing at the resonance intensity $I_r$. The minimum distance, $V_m$, at the avoided crossing is proportional to the generalized Rabi frequency, $\Gamma\left(t\right)=\Gamma_m I\left(t\right)^\left\{m/2\right\}$ coupling the two states. According to Story et al.,[11] the probability of remaining in the ground state, $P_g$, is given by

$P_g=\exp\left\left(-\frac\left\{2\pi W_m^2\right\}\left\{dW/dt\right\}\right\right)$

where $W$ is the time-dependent energy difference between the two dressed states. In interaction with a short pulse, if the dynamic resonance is reached in the rising or the falling part of the pulse, the population practically remains in the ground state and the effect of multiphoton resonances may be neglected. However, if the states go onto resonance at the peak of the pulse, where $dW/dt=0$, then the excited state is populated. After being populated, since the ionization potential of the excited state is small, it is expected that the electron will be instantly ionized.

In 1992, de Boer and Muller [12] showed that Xe atoms subjected to short laser pulses could survive in the highly excited states 4f, 5f, and 6f . These states were believed to have been excited by the dynamic Stark shift of the levels into multiphoton resonance with the field during the rising part of the laser pulse. Subsequent evolution of the laser pulse did not ionize completely these states leaving behind some highly excited atoms. We shall refer to this phenomenon as "population trapping".

We mention the theoretical calculation that incomplete ionization occurs whenever there is parallel resonant excitation into a common level with ionization loss.[13] We consider a state such as 6f of Xe whidi consists of 7 quasi-degnerate levels in the range of the laser bandwidth. These levels along with the continuum constitute a lambda system. The mechanism of the lambda type trapping is schematically presented in figure. At the rising part of the pulse (a) the excited state (with two degenerate levels 1 and 2) are not in multiphoton resonance with the ground state. The electron is ionized through multiphoton coupling with the continuum. As the intensity of the pulse is increased the excited state and the continuum are shifted in energy due to the Stark shift. At the peak of the pulse (b) the excited states go into multiphoton resonance with the ground state. As the intensity statrts to decrease (c), the two state are coupled through continuum and the population is trapped in a coherent superposition of the two states. Under subsequent action of the same pulse, due to interference in the transition amplitudes of the lambda system, the field cannot ionize the population completely and a fraction of the population will be trapped in a coherent superposition of the quasi degenerate levels. According to this explanation the states with higher angular momentum- with more sublevels- would have a higher probability of trapping the population. In general the strength of the trapping will be determined by the strength of the two photon coupling between the quasi-degenerate levels via the continuum.In 1996, using the very stable laser and by minimizing the masking effects of the focal region expansion with increasing intensity, Talebpour et al.[14] observed structures on the curves of singly charged ions of Xe, Kr and Ar. These structures were attributed to electron trapping in the strong laser field. A more unambiguous demonstration of population trapping has been reported by T. Morishita and C. D. Lin.[15]

### Non-sequential multiple ionization

The phenomenon of non-sequential ionization (NSI) of atoms  exposed  to intense laser fields has been a subject ofmany theoretical and experimental studies since 1983. The pioneering work began with the observation of a “knee” structure on the Xe2+ ion signal versus intensity curve by L’Huillier et al.[16] From the experimental point of view, the NS double ionization refers to processes which somehow enhance the rate of production of doubly charged ions by a huge factor at intensities below the saturation intensity of the singly charged ion. Many, on the other hand, prefer to define the NSI as a process by which two electrons are ionized nearly simultaneously. This definition implies that apart from the sequential channel $A+L -> A^+ + L -> A^\left\{++\right\}$ there is another channel $A+L-> A^\left\{++\right\}$ which is the main contribution to the production of doubly charged ions at lower intensities. The first observation of triple NSI in Ar interacting with a 1 micron laser was reported by Augst et al.[17] Later, systematically studying the NSI of all rare gas atoms, the quadruple NSI of Xe was observed.[18] The most important conclusion of this study was the observation of the following relation between the rate of NSI to any charge state and the rate of tunnel ionization (predicted by the ADK formula) to the previous charge states;

$W_\left\{NS\right\}\left(A^\left\{n+\right\}\right)=\sum_\left\{i=1\right\}^\left\{n-1\right\}\alpha_n\left(\lambda\right)W_\left\{ADK\right\}\left(A^\left\{i+\right\}\right)$

where $W_\left\{ADK\right\}\left(A^\left\{i+\right\}\right)$ is the rate of quasi-static tunneling to i’th charge state and $\alpha_n\left(\lambda\right)$ are some constants depending on the wavelength of the laser (but not on the pulse duration).

Two models have been proposed to explain the non-sequential ionization; the shake-off model and electron re-scattering model. The shake-off (SO) model, first proposed by Fittinghoff et al.,[19] is adopted from the field of ionization of atoms by X rays and electron projectiles where the SO process is one of the major mechanisms responsible for the multiple ionization of atoms. The SO model describes the NS process as a mechanism where one electron is ionized by the laser field and the departure of this electron is so rapid that the remaining electrons do not have enough time to adjust themselves to the new energy states. Therefore, there is a certain probability that, after the ionization of the first electron, a second electron is excited to states with higher energy (shake-up) or even ionized (shake-off). We should mention that, until now, there has been no quantitative calculation based on the SO model, and the model is still qualitative.

The electron rescattering model was independently developed by Kuchiev,[20] Corkum,[21] Becker and Faisal [22] and Faisal and Becker.[23] The principle features of the model can be understood easily from Corkums version. Corkums model describes the NS ionization as a process whereby an electron is tunnel ionized. The electron then interacts with the laser field where it is accelerated away from the nuclear core. If the electron has been ionized at an appropriate phase of the field, it will pass by the position of the remaining ion half a cycle later, where it can free an additional electron by electron impact. Only half of the time the electron is released with the appropriate phase and the other half it never return to the nuclear core. The maximum kinetic energy that the returning electron can have is 3.17 times the ponderomotive potential ($U_p$) of the laser. Corkums model places a cut-off limit on the minimum intensity ($U_p$ is proportional to intensity) where ionization due to re-scattering can occur.

The re-scattering model in Kuchievs version (Kuchievs model) is quantum mechanical. The basic idea of the model is illustrated by Feynman diagrams in figure a. First both electrons are in the ground state of an atom. The lines marked a and b describe the corresponding atomic states. Then the electron a is ionized. The beginning of the ionization process is shown by the intersection with a sloped dashed line. where the MPI occurs. The propagation of the ionized electron in the laser field, during which it absorbs other photons (ATI), is shown by the full thick line. The collision of this electron with the parent atomic ion is shown by a vertical dotted line representing the Coulomb interaction between the electrons. The state marked with c describes the ion excitation to a discrete or continuum state. Figure b describes the exchange process. Kuchievs model, contrary to Corkums model, does not predict any threshold intensity for the occurrence of NS ionization.

Kuciev did not include the Coulomb effects on the dynamics of the ionized electron. This resulted in the underestimation of the double ionization rate by a huge factor. Obviously, in the approach of Becker and Faisal (which is equivalent to Kuchiev`s model in spirit), this drawback does not exist. In fact, their model is more exact and does not suffer from the large number of approximations made by Kuchiev. Their calculation results perfectly fit with the experimental results of Walker et al.[24] Becker and Faisal[25] have been able to fit the experimental results on the multiple NSI of rare gas atoms using their model. As a result the electron re-scattering can be taken as the main mechanism for the occurrence of the NSI process.

### Multiphoton ionization of inner-valence electrons and fragmentation of polyatomic molecules

The ionization of inner valance electrons are responsible for the fragmentation of polyatomic molecules in strong laser fields. According to a qualitative model[26][27] the dissociation of the molecules occurs through a three-step mechanism:

• MPI of electrons from the inner orbitals of the molecule which results in a molecular ion in ro-vibrational levels of an excited electronic state;
• Rapid radiationless transition to the high-lying ro-vibrational levels of a lower electronic state; and
• Subsequent dissociation of the ion to different fragments through various fragmentation channels.

The short pulse induced molecular fragmentation may be used as an ion source for high performance mass spectroscopy. The selectivity provided by a short pulse based source is superior to that expected when using the conventional electron ionization based sources, in particular when the identification of optical isomers is required.[28][29]

## Dissociation – distinction

A substance may dissociate without necessarily producing ions. As an example, the molecules of table sugar dissociate in water (sugar is dissolved) but exist as intact neutral entities. Another subtle event is the dissociation of sodium chloride (table salt) into sodium and chlorine ions. Although it may seem as a case of ionization, in reality the ions already exist within the crystal lattice. When salt is dissociated, its constituent ions are simply surrounded by water molecules and their effects are visible (e.g. the solution becomes electrolytic). However, no transfer or displacement of electrons occurs. Actually, the chemical synthesis of salt involves ionization. This is a chemical reaction.

format: plain


text:
## Wikipedia - Scambio di materia (it) transfer

https://doi.org/10.1351/goldbook.T06439
Movement of a component within a system or across its boundary. It may be expressed using different kinds of quantities, e.g rates of change $$\frac{\text{d}Q}{\text{d}t}$$ or $$\frac{\Delta Q}{\Delta t}$$. Examples are: mass rate, $$\frac{\text{d}m_{\text{B}}}{\text{d}t}$$ or $$\frac{\Delta m_{\text{B}}}{\Delta t}$$; substance rate, $$\frac{\text{d}n_{\text{B}}}{\text{d}t}$$ or $$\frac{\Delta n_{\text{B}}}{\Delta t}$$.
Source:
PAC, 1992, 64, 1569. (Quantities and units for metabolic processes as a function of time (IUPAC Recommendations 1992)) on page 1573 [Terms] [Paper]


text:
# How to convert LaTeX to mathml code directly? [duplicate]

In microsoft word it can't understand every mathematical equation that I input in latex correctly.the only language that it understands correctly is the mathml code. But to do so I have to input the equation in latex to this script in a notepad [example]

<html>
<title>MathJax TeX Test Page</title>
<script type="text/javascript" async
src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS_CHTML">
</script>
<body>
Your test: $$\mathbf{F}_{2}=k \frac{q_{1} q_{2} \hat{\mathbf{r}}_{21}}{r_{21}^{2}}$$
</body>
</html>

then I have to save this file as a HTML file.Then I have to open the file in chrome and when the equation appears I have to extract the mathml code[by right clicking the equation and selecting to show the code].Then when I paste the code in word as keep text only,then it gives the exact equation. Is there any easier way like a software or an online converter to convert latex into Mathml code[not mathml]? thank you


text:
# How do you find the reciprocal of 1/22?

Feb 20, 2017

Flip the number upside down. The numerator becomes the denominator and the denominator becomes numerator.

#### Explanation:

$\frac{1}{22}$ becomes $\frac{22}{1}$ as the reciprocal .

$\frac{1}{22} \times \frac{22}{1} = 1$

A number multiplied by its reciprocal is always equal to 1 .

(The reciprocal is also called the multiplicative inverse.)

Feb 20, 2017

You just have to flip $\setminus \frac{1}{22}$ which gives you 22. When you multiply $\setminus \frac{1}{22}$ and 22, you must get 1.


text:
## Bulletin of the Belgian Mathematical Society - Simon Stevin

### Kazhdan's Property T for the Symplectic Group over a Ring

Markus Neuhauser

#### Abstract

Kazhdan constants for $\mathop{\rm {Sp}}\left( n,\Omega \right)$ where $\Omega$ is a commutative topological ring with dense finitely generated subring with unity are determined. This implies Kazhdan's property T for these groups. As application explicit Kazhdan constants are determined for the loop groups corresponding to $\mathop{\rm {Sp}}\left( n,\mathbf{C}\right)$. These are further examples of groups with property T which are infinite dimensional Lie groups and not locally compact.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 10, Number 4 (2003), 537-550.

Dates
First available in Project Euclid: 5 December 2003

https://projecteuclid.org/euclid.bbms/1070645800

Digital Object Identifier
doi:10.36045/bbms/1070645800

Mathematical Reviews number (MathSciNet)
MR2040529

Zentralblatt MATH identifier
1063.22021

#### Citation

Neuhauser, Markus. Kazhdan's Property T for the Symplectic Group over a Ring. Bull. Belg. Math. Soc. Simon Stevin 10 (2003), no. 4, 537--550. doi:10.36045/bbms/1070645800. https://projecteuclid.org/euclid.bbms/1070645800


text:
# Why does Lenstra ECM work?

I came across Lenstra ECM algorithm and I wonder why it works. Please refer for simplicity to Wikipedia section Why does the algorithm work

I NOT a math expert but I understood first part well enough (I suppose), what I miss is

When this is the case, $kP$ does not exist on the original curve, and in the computations we found some $v$ with either $\text{gcd}(v,p)=p$ or $\text{gcd}(v,q)=q$, but not both. That is, $\text{gcd}(v,n)$ gave a non-trivial factor of $n$.

As far as I know this has to do with the fact that $E(\mathbb Z/n\mathbb Z)$ is not a group if $n$ is not prime so some element (i.e. $x_1-x_2$) is not invertible but what is the link between non invertible elements and $n$ factors?

Thanks to everyone

-
Have you studied or worked with some of the simpler factoring methods like Pollard's rho? It would help to have some clearer idea of your math background, e.g. understanding of elementary number theory. –  hardmath Dec 5 '12 at 17:26
Yes, I studied Pollard's Rho (with Pollard p-1 and William p+1) –  knm241 Dec 5 '12 at 18:02
I'm fond of David Bressoud's book, Factorization and Primality Testing. If you are interested in arithmetic of elliptic curves, then another favorite is Silverman and Tate's Rational Points on Elliptic Curves. You can look for reviews, etc. in Google Books or at bookseller websites. –  hardmath Dec 7 '12 at 0:23


text:
# sklearn¶

class openTSNE.sklearn.TSNE(n_components=2, perplexity=30, learning_rate='auto', early_exaggeration_iter=250, early_exaggeration=12, n_iter=500, exaggeration=None, dof=1, theta=0.5, n_interpolation_points=3, min_num_intervals=50, ints_in_interval=1, initialization='pca', metric='euclidean', metric_params=None, initial_momentum=0.5, final_momentum=0.8, max_grad_norm=None, max_step_norm=5, n_jobs=1, neighbors='auto', negative_gradient_method='auto', callbacks=None, callbacks_every_iters=50, random_state=None, verbose=False)[source]

t-Distributed Stochastic Neighbor Embedding.

Parameters: n_components (int) – The dimension of the embedding space. This deafults to 2 for easy visualization, but sometimes 1 is used for t-SNE heatmaps. t-SNE is not designed to embed into higher dimension and please note that acceleration schemes break down and are not fully implemented. perplexity (float) – Perplexity can be thought of as the continuous $$k$$ number of nearest neighbors, for which t-SNE will attempt to preserve distances. learning_rate (Union[str, float]) – The learning rate for t-SNE optimization. When learning_rate="auto" the appropriate learning rate is selected according to max(200, N / 12), as determined in Belkina et al. “Automated optimized parameters for T-distributed stochastic neighbor embedding improve visualization and analysis of large datasets”, 2019. early_exaggeration_iter (int) – The number of iterations to run in the early exaggeration phase. early_exaggeration (float) – The exaggeration factor to use during the early exaggeration phase. Typical values range from 12 to 32. n_iter (int) – The number of iterations to run in the normal optimization regime. exaggeration (float) – The exaggeration factor to use during the normal optimization phase. This can be used to form more densely packed clusters and is useful for large data sets. dof (float) – Degrees of freedom as described in Kobak et al. “Heavy-tailed kernels reveal a finer cluster structure in t-SNE visualisations”, 2019. theta (float) – Only used when negative_gradient_method="bh" or its other aliases. This is the trade-off parameter between speed and accuracy of the tree approximation method. Typical values range from 0.2 to 0.8. The value 0 indicates that no approximation is to be made and produces exact results also producing longer runtime. Alternatively, you can use auto to approximately select the faster method. n_interpolation_points (int) – Only used when negative_gradient_method="fft" or its other aliases. The number of interpolation points to use within each grid cell for interpolation based t-SNE. It is highly recommended leaving this value at the default 3. min_num_intervals (int) – Only used when negative_gradient_method="fft" or its other aliases. The minimum number of grid cells to use, regardless of the ints_in_interval parameter. Higher values provide more accurate gradient estimations. ints_in_interval (float) – Only used when negative_gradient_method="fft" or its other aliases. Indicates how large a grid cell should be e.g. a value of 3 indicates a grid side length of 3. Lower values provide more accurate gradient estimations. initialization (Union[np.ndarray, str]) – The initial point positions to be used in the embedding space. Can be a precomputed numpy array, pca, spectral or random. Please note that when passing in a precomputed positions, it is highly recommended that the point positions have small variance (std(Y) < 0.0001), otherwise you may get poor embeddings. metric (Union[str, Callable]) – The metric to be used to compute affinities between points in the original space. metric_params (dict) – Additional keyword arguments for the metric function. initial_momentum (float) – The momentum to use during the early exaggeration phase. final_momentum (float) – The momentum to use during the normal optimization phase. max_grad_norm (float) – Maximum gradient norm. If the norm exceeds this value, it will be clipped. This is most beneficial when adding points into an existing embedding and the new points overlap with the reference points, leading to large gradients. This can make points “shoot off” from the embedding, causing the interpolation method to compute a very large grid, and leads to worse results. max_step_norm (float) – Maximum update norm. If the norm exceeds this value, it will be clipped. This prevents points from “shooting off” from the embedding. n_jobs (int) – The number of threads to use while running t-SNE. This follows the scikit-learn convention, -1 meaning all processors, -2 meaning all but one, etc. neighbors (str) – Specifies the nearest neighbor method to use. Can be exact, annoy, pynndescent, hnsw, approx, or auto (default). approx uses Annoy if the input data matrix is not a sparse object and if Annoy supports the given metric. Otherwise it uses Pynndescent. auto uses exact nearest neighbors for N<1000 and the same heuristic as approx for N>=1000. negative_gradient_method (str) – Specifies the negative gradient approximation method to use. For smaller data sets, the Barnes-Hut approximation is appropriate and can be set using one of the following aliases: bh, BH or barnes-hut. For larger data sets, the FFT accelerated interpolation method is more appropriate and can be set using one of the following aliases: fft, FFT or ìnterpolation. Alternatively, you can use auto to approximately select the faster method. callbacks (Union[Callable, List[Callable]]) – Callbacks, which will be run every callbacks_every_iters iterations. callbacks_every_iters (int) – How many iterations should pass between each time the callbacks are invoked. random_state (Union[int, RandomState]) – If the value is an int, random_state is the seed used by the random number generator. If the value is a RandomState instance, then it will be used as the random number generator. If the value is None, the random number generator is the RandomState instance used by np.random. verbose (bool) –
fit(X, y=None)[source]

Fit X into an embedded space.

Parameters: X (np.ndarray) – The data matrix to be embedded. y (ignored) –
fit_transform(X, y=None)[source]

Fit X into an embedded space and return that transformed output.

Parameters: X (np.ndarray) – The data matrix to be embedded. y (ignored) – Embedding of the training data in low-dimensional space. np.ndarray
transform(X, *args, **kwargs)[source]

Apply dimensionality reduction to X.

See openTSNE.TSNEEmbedding.transform() for additional parameters.

Parameters: X (np.ndarray) – The data matrix to be embedded. Embedding of the training data in low-dimensional space. np.ndarray


text:
# NCERT SOLUTIONS FOR CLASS 11 MATHS CHAPTER 5 / COMPLEX NUMBERS AND QUADRATIC EQUATIONS EX 5.2 /

Find the modulus and the arguments of each of the complex numbers in Exercises 1 to 2.

Question 1.

$z=-1-i\sqrt { 3 }$

Solution.

We have, $z=-1-i\sqrt { 3 }$

Question 2.

$z=-\sqrt { 3 } +i$

Solution.

We have, $z=-\sqrt { 3 } +i$

Convert each of the complex numbers given in Exercises 3 to 8 in the polar form:

Question 3.

1 – i

Solution.

We have, z = 1 – i

Question 4. -1 + i

Solution.

We have, z = -1 + i

Question 5. -1 – i

Solution.

We have, z = -1 – i

Question 6. -3

Solution.

We have, z = -3, i.e., z = -3 + 0i

Question 7.

$\sqrt { 3 } +i$

Solution.

We have, $z=\sqrt { 3 } +i$

Question 8. i

Solution.

We have, z = i, i.e., z = 0 + 1.i


text:
Leonhard Euler

(Redirected from Euler)
Although to penetrate into the intimate mysteries of nature and thence to learn the true causes of phenomena is not allowed to us, nevertheless it can happen that a certain fictive hypothesis may suffice for explaining many phenomena.

Leonhard Euler (15 April 170718 September 1783) Swiss mathematician and physicist, considered to be one of the greatest mathematicians of all time.

Sourced

Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.
To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as they are usually believed to be.
• Madam, I have come from a country where people are hanged if they talk.
• In Berlin, to the Queen Mother of Prussia, on his lack of conversation in his meeting with her, on his return from Russia; as quoted in Science in Russian Culture : A History to 1860 (1963) Alexander Vucinich
• Variant: Madame... I have come from a country where one can be hanged for what one says.
• Now I will have less distraction.
• Upon losing the use of his right eye; as quoted in In Mathematical Circles (1969) by H. Eves
• Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.
• As quoted in Calculus Gems (1992) by G. Simmons
• Since the fabric of the universe is most perfect and the work of a most wise Creator, nothing at all takes place in the universe in which some rule of maximum or minimum does not appear ... there is absolutely no doubt that every affect in the universe can be explained satisfactorily from final causes, by the aid of the method of maxima and minima, as it can be from the effective causes themselves ... Of course, when the effective causes are too obscure, but the final causes are readily ascertained, the problem is commonly solved by the indirect method...
• As quoted in The Anthropic Cosmological Principle (1986) by John D. Barrow and Frank J. Tipler, p. 150
• To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as they are usually believed to be.
• As quoted in Fundamentals of Teaching Mathematics at University Level (2000) by Benjamin Baumslag, p. 214
• La construction d'une machine propre à exprimer tous les sons de nos paroles , avec toutes les articulations , seroit sans-doute une découverte bien importante. ... La chose ne me paroît pas impossible.
• It would be a considerable invention indeed, that of a machine able to mimic speech, with its sounds and articulations. ... I think it is not impossible.
• Letter to Friederike Charlotte of Brandenburg-Schwedt (16 June 1761)
• Lettres à une Princesse d'Allemagne sur différentes questions de physique et de philosophie, Royer, 1788, p. 265
• As quoted in An Introduction to Text-to-Speech Synthesis (2001) by Thierry Dutoit, p. 27; also in Fabian Brackhane and Jürgen Trouvain "Zur heutigen Bedeutung der Sprechmaschine Wolfgang von Kempelens" (in: Bernd J. Kröger (ed.): Elektronische Sprachsignalverarbeitung 2009, Band 2 der Tagungsbände der 20. Konferenz "Elektronische Sprachsignalverarbeitung" (ESSV), Dresden: TUDpress, 2009, pp. 97–107)

Introduction to the Analysis of the Infinite (Introductio in analysin infinitorum) (1748)

Translated as Introduction to Analysis of the Infinite (1988-89) by John Blanton (Book I ISBN 0387968245; Book II ISBN 0387971327).
Online version
• A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities.
• §4.

Conjecture regarding the nature of the air, so as to explain the phenomena observed in the atmosphere (Conjectura circa naturam aeris, pro explicandis phaenomenis in atmosphaera observatis) (1780)

Online version
• Quanquam nobis in intima naturae mysteria penetrare, indeque veras caussas Phaenomenorum agnoscere neutiquam est concessum: tamen evenire potest, ut hypothesis quaedam ficta pluribus phaenomenis explicandis aeque satisfaciat, ac si vera caussa nobis esset perspecta.
• Although to penetrate into the intimate mysteries of nature and thence to learn the true causes of phenomena is not allowed to us, nevertheless it can happen that a certain fictive hypothesis may suffice for explaining many phenomena.
• §1.

Unsourced

He calculated without any apparent effort, just as men breathe, as eagles sustain themselves in the air. ~ François Arago
Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence. ~ Keith Devlin
Alphabetized by author
• He calculated without any apparent effort, just as men breathe, as eagles sustain themselves in the air.
• François Arago; Variant: Euler calculated without apparent effort, as men breathe, or as eagles sustain themselves in the wind.
• The most influential mathematics textbook of ancient times is easily named, for the Elements of Euclid has set the pattern in elementary geometry ever since. The most effective textbook of the medieval age is less easily designated; but a good case can be made out for the Al-jabr of Al-Khwarizmi, from which algebra arose and took its name. Is it possible to indicate a modern textbook of comparable influence and prestige? Some would mention the Géométrie of Descartes or the Principia of Newton or the Disquisitiones of Gauss; but in pedagogical significance these classics fell short of a work by Euler titled Introductio in analysin infinitorum.
• Carl B. Boyer on Euler's Introduction to the Analysis of the Infinite in "The Foremost Textbook of Modern Times" (1950)
• The Introductio does not boast an impressive number of editions, yet its influence was pervasive. In originality and in the richness of its scope it ranks among the greatest of textbooks; but it is outstanding also for clarity of exposition. Published two hundred and two years ago, it nevertheless possesses a remarkable modernity of terminology and notation, as well as of viewpoint. Imitation is indeed the sincerest form of flattery.
• Carl B. Boyer in "The Foremost Textbook of Modern Times" (1950)
• Euler calculated the force of the wheels necessary to raise the water in a reservoir … My mill was carried out geometrically and could not raise a drop of water fifty yards from the reservoir. Vanity of vanities! Vanity of geometry!
• Read Euler: he is our master in everything.
• Pierre Simon de Laplace, as quoted in Calculus Gems (1992) variant: Read Euler, read Euler. He is the master of us all.
• He was later to write that he had made some of his best discoveries while holding a baby in his arms surrounded by playing children.
• Richard Mankiewicz, in The Story of Mathematics (2000), p. 142
• Euler and Ramanujan are mathematicians of the greatest importance in the history of constants (and of course in the history of Mathematics ...)
• E. W. Middlemast
• It is the invaluable merit of the great Basle mathematician Leonard Euler, to have freed the analytical calculus from all geometric bounds, and thus to have established analysis as an independent science, which from his time on has maintained an unchallenged leadership in the field of mathematics.
• Thomas Reid, as quoted in Mathematical Maxims and Minims (1988) by N. Rose
• Perhaps the most surprising thing about mathematics is that it is so surprising. The rules which we make up at the beginning seem ordinary and inevitable, but it is impossible to foresee their consequences. These have only been found out by long study, extending over many centuries. Much of our knowledge is due to a comparatively few great mathematicians such as Newton, Euler, Gauss, or Riemann; few careers can have been more satisfying than theirs. They have contributed something to human thought even more lasting than great literature, since it is independent of language.

On "Euler's identity"

In mathematical analysis, Euler's identity is the equation "$e^{i \pi} + 1 = 0. \,\!$"
Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means. But we have proved it, and therefore we know it must be the truth. ~ Benjamin Peirce
• One of the most frequently mentioned equations was Euler's equation, $e^{i \pi} + 1 = 0. \,\!$ Respondents called it "the most profound mathematical statement ever written"; "uncanny and sublime"; "filled with cosmic beauty"; and "mind-blowing". Another asked: "What could be more mystical than an imaginary number interacting with real numbers to produce nothing?" The equation contains nine basic concepts of mathematics — once and only once — in a single expression. These are: e (the base of natural logarithms); the exponent operation; π; plus (or minus, depending on how you write it); multiplication; imaginary numbers; equals; one; and zero.
• Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence.
• Our jewel ... one of the most remarkable, almost astounding, formulas in all of mathematics.
• There is a famous formula, perhaps the most compact and famous of all formulas — developed by Euler from a discovery of de Moivre: $e^{i \pi} + 1 = 0. \,\!$ It appeals equally to the mystic, the scientist, the philosopher, the mathematician.
• Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means. But we have proved it, and therefore we know it must be the truth.
• Benjamin Peirce, as quoted in notes by W. E. Byerly, published in Benjamin Peirce, 1809-1880 : Biographical Sketch and Bibliography (1925) by R. C. Archibald; also in Mathematics and the Imagination (1940) by Edward Kasner and James Newman


text:
A Domain object is a representation of a single dimension of a ParamSet. Domain objects are used to construct ParamSets, either through the ps() short form, or through the ParamSet$search_space() mechanism (see to_tune()). Domain corresponds to a Param object, except it does not have an $id, and it does have a trafo and dependencies (depends) associated with it. For each of the basic Param classes (ParamInt, ParamDbl, ParamLgl, ParamFct, and ParamUty) there is a function constructing a Domain object (p_int(), p_dbl(), p_lgl(), p_fct(), p_uty()). They each have the same arguments as the corresponding Param $new() function, except without the id argument, and with the the additional parameters trafo, and depends. Domain objects are representations of parameter ranges and are intermediate objects to be used in short form constructions in to_tune() and ps(). Because of their nature, they should not be modified by the user. The Domain object's internals are subject to change and should not be relid upon. p_int( lower = -Inf, upper = Inf, special_vals = list(), default = NO_DEF, tags = character(), depends = NULL, trafo = NULL, logscale = FALSE ) p_dbl( lower = -Inf, upper = Inf, special_vals = list(), default = NO_DEF, tags = character(), tolerance = sqrt(.Machine$double.eps),
depends = NULL,
trafo = NULL,
logscale = FALSE
)

p_uty(
default = NO_DEF,
tags = character(),
custom_check = NULL,
depends = NULL,
trafo = NULL
)

p_lgl(
special_vals = list(),
default = NO_DEF,
tags = character(),
depends = NULL,
trafo = NULL
)

p_fct(
levels,
special_vals = list(),
default = NO_DEF,
tags = character(),
depends = NULL,
trafo = NULL
)

## Arguments

lower (numeric(1)) Lower bound, can be -Inf. (numeric(1)) Upper bound can be +Inf. (list()) Arbitrary special values this parameter is allowed to take, to make it feasible. This allows extending the domain of the parameter. Note that these values are only used in feasibility checks, neither in generating designs nor sampling. (any) Default value. Can be from the domain of the parameter or an element of special_vals. Has value NO_DEF if no default exists. NULL can be a valid default. The value has no effect on ParamSet$values or the behavior of ParamSet$check(), $test() or $assert(). The default is intended to be used for documentation purposes.  (character()) Arbitrary tags to group and subset parameters. Some tags serve a special purpose: "required" implies that the parameters has to be given when setting values in ParamSet. (call | expression) An expression indicating a requirement for the parameter that will be constructed from this. Can be given as an expression (using quote()), or the expression can be entered directly and will be parsed using NSE (see examples). The expression may be of the form  ==  or  %in% , which will result in dependencies according to ParamSet$add_dep(on = "", cond = CondEqual$new()) or ParamSet$add_dep(on = "", cond = CondAnyOf$new()), respectively (see CondEqual, CondAnyOf). The expression may also contain multiple conditions separated by &&. (function) Single argument function performing the transformation of a parameter. When the Domain is used to construct a ParamSet, this transformation will be applied to the corresponding parameter as part of the $trafo function. Note that the trafo is not inherited by TuneTokens! Defining a parameter with e.g. p_dbl(..., trafo = ...) will not automatically give the to_tune() assigned to it a transformation. trafo only makes sense for ParamSets that get used as search spaces for optimization or tuning, it is not useful when defining domains or hyperparameter ranges of learning algorithms, because these do not use trafos. (logical(1)) Put numeric domains on a log scale. Default FALSE. Log-scale Domains represent parameter ranges where lower and upper bounds are logarithmized, and where a trafo is added that exponentiates sampled values to the original scale. This is not the same as setting trafo = exp, because logscale = TRUE will handle parameter bounds internally: a p_dbl(1, 10, logscale = TRUE) results in a ParamDbl that has lower bound 0, upper bound log(10), and uses exp transformation on these. Therefore, the given bounds represent the bounds after the transformation. (see examples). p_int() with logscale = TRUE results in a ParamDbl, not a ParamInt, but with bounds log(max(lower, 0.5)) ... log(upper + 1) and a trafo similar to "as.integer(exp(x))" (with additional bounds correction). The lower bound is lifted to 0.5 if lower 0 to handle the lower == 0 case. The upper bound is increased to log(upper + 1) because the trafo would otherwise almost never generate a value of upper. When logscale is TRUE, then upper bounds may be infinite, but lower bounds should be greater than 0 for p_dbl() or greater or equal 0 for p_int(). Note that "logscale" is not inherited by TuneTokens! Defining a parameter with p_dbl(... logscale = TRUE) will not automatically give the to_tune() assigned to it log-scale. logscale only makes sense for ParamSets that get used as search spaces for optimization or tuning, it is not useful when defining domains or hyperparameter ranges of learning algorithms, because these do not use trafos. logscale happens on a natural (e == 2.718282...) basis. Be aware that using a different base (log10()/10^, log2()/2^) is completely equivalent and does not change the values being sampled after transformation. (numeric(1)) Initializes the $tolerance field that determines the (function()) Custom function to check the feasibility. Function which checks the input. Must return 'TRUE' if the input is valid and a character(1) with the error message otherwise. This function should not throw an error. Defaults to NULL, which means that no check is performed. (character | atomic | list) Allowed categorical values of the parameter. If this is not a character, then a trafo is generated that converts the names (if not given: as.character() of the values) of the levels argument to the values. This trafo is then performed before the function given as the trafo argument.

## Value

A Domain object.

The p_fct function admits a levels argument that goes beyond the levels accepted by ParamFct$new(). Instead of a character vector, any atomic vector or list (optionally named) may be given. (If the value is a list that is not named, the names are inferred using as.character() on the values.) The resulting Domain will correspond to a range of values given by the names of the levels argument with a trafo that maps the character names to the arbitrary values of the levels argument. ## See also Other ParamSet construction helpers: ps(), to_tune() ## Examples params = ps( unbounded_integer = p_int(), bounded_double = p_dbl(0, 10), half_bounded_integer = p_dbl(1), half_bounded_double = p_dbl(upper = 1), double_with_trafo = p_dbl(-1, 1, trafo = exp), extra_double = p_dbl(0, 1, special_vals = list("xxx"), tags = "tagged"), factor_param = p_fct(c("a", "b", "c")), factor_param_with_implicit_trafo = p_fct(list(a = 1, b = 2, c = list())) ) print(params) #> <ParamSet> #> id class lower upper nlevels default #> 1: unbounded_integer ParamInt -Inf Inf Inf <NoDefault[3]> #> 2: bounded_double ParamDbl 0 10 Inf <NoDefault[3]> #> 3: half_bounded_integer ParamDbl 1 Inf Inf <NoDefault[3]> #> 4: half_bounded_double ParamDbl -Inf 1 Inf <NoDefault[3]> #> 5: double_with_trafo ParamDbl -1 1 Inf <NoDefault[3]> #> 6: extra_double ParamDbl 0 1 Inf <NoDefault[3]> #> 7: factor_param ParamFct NA NA 3 <NoDefault[3]> #> 8: factor_param_with_implicit_trafo ParamFct NA NA 3 <NoDefault[3]> #> value #> 1: #> 2: #> 3: #> 4: #> 5: #> 6: #> 7: #> 8: #> Trafo is set. params$trafo(list(
bounded_double = 1,
double_with_trafo = 1,
factor_param = "c",
factor_param_with_implicit_trafo = "c"
))
#> $bounded_double #> [1] 1 #> #>$double_with_trafo
#> [1] 2.718282
#>
#> $factor_param #> [1] "c" #> #>$factor_param_with_implicit_trafo
#> list()
#>
# logscale:
params = ps(x = p_dbl(1, 100, logscale = TRUE))

# The ParamSet has bounds log(1) .. log(100):
print(params)
#> <ParamSet>
#>    id    class lower   upper nlevels        default value
#> 1:  x ParamDbl     0 4.60517     Inf <NoDefault[3]>
#> Trafo is set.
# When generating a equidistant grid, it is equidistant within log values
grid = generate_design_grid(params, 3)
print(grid)
#> <Design> with 3 rows:
#>           x
#> 1: 0.000000
#> 2: 2.302585
#> 3: 4.605170
# But the values are on a log scale with desired bounds after trafo
print(grid$transpose()) #> [[1]] #> [[1]]$x
#> [1] 1
#>
#>
#> [[2]]
#> [[2]]$x #> [1] 10 #> #> #> [[3]] #> [[3]]$x
#> [1] 100
#>
#>
# Integer parameters with logscale are ParamDbls pre-trafo
params = ps(x = p_int(0, 10, logscale = TRUE))
print(params)
#> <ParamSet>
#>    id    class      lower    upper nlevels        default value
#> 1:  x ParamDbl -0.6931472 2.397895     Inf <NoDefault[3]>
#> Trafo is set.
grid = generate_design_grid(params, 4)
print(grid)
#> <Design> with 4 rows:
#>             x
#> 1: -0.6931472
#> 2:  0.3372003
#> 3:  1.3675478
#> 4:  2.3978953
# ... but get transformed to integers.
print(grid$transpose()) #> [[1]] #> [[1]]$x
#> [1] 0
#>
#>
#> [[2]]
#> [[2]]$x #> [1] 1 #> #> #> [[3]] #> [[3]]$x
#> [1] 3
#>
#>
#> [[4]]
#> [[4]]\$x
#> [1] 10
#>
#>
`


text:
## anonymous one year ago if x+1/x=7, what is x^3+1/x^2?

• This Question is Open
1. zepdrix

Is it $$\large\rm \frac{x+1}{x}=7$$ or $$\large\rm x+\frac{1}{x}=7$$ ?

2. anonymous

(x) + (1/x) = 7, the second one

3. zepdrix

Fractions are bad. No bueno. So let's multiply through to get rid of the x in the denominator. Multiplying by x gives us:$\large\rm x^2+1=7x$Ok with that step? What should we do next, what do you think? :)

4. zepdrix

There is probably a cool way to relate the first equation to the second one... but I'm not seeing it. So instead what we can do is.. simply solve for the value of x in the first equation, and plug that value into the second equation.

5. anonymous

I do not think that is the way my teacher would like for me to go about it. I have been staring at this problem for a good 35 minutes and I'm stumped.

6. zepdrix

No? Hmm. Ok lemme make sure I'm understanding the second equation then, is it $$\large\rm x^3+\frac{1}{x^2}$$ ?

7. anonymous

Yes it is.

8. Mertsj

$x+\frac{1}{x}=7$ $x=7-\frac{1}{x}$ $\frac{1}{x}=7-x$

9. zepdrix

I was thinking maybe we could do something like this:$\large\rm x+\frac{1}{x}=7$Then,$\large\rm \left(x+\frac{1}{x}\right)^2=49$and$\large\rm x^2+\frac{1}{x}=47$But that doesn't quite get us the third power :( Darn...

10. zepdrix

Woops typo,$\large\rm x^2+\frac{1}{x^2}=47$

11. Mertsj

$x^3+\frac{1}{x^2}=(7-\frac{1}{x})^3+\frac{1}{(7-x)^2}$

12. freckles

hero had a good way on a previous post like this one something about squaring both sides and also cubing both sides of that one equation

13. freckles

$(x+\frac{1}{x})^2=7^2 \\ (x+\frac{1}{x})^3=7^3$ some expanding will be involved

14. freckles

you will get values for both $x^2+\frac{1}{x^2} \text{ and } x^3+\frac{1}{x^3} \\ \text{ then add those you get } \\ x^3+\frac{1}{x^2}+x^2+\frac{1}{x^3} \\ =x^3+\frac{1}{x^2}+\frac{1}{x}(x^3+\frac{1}{x^2}) \\ =(x^3+\frac{1}{x^2})(1+\frac{1}{x}) \\ \text{ so you have } \\ x^3+\frac{1}{x^2}+x^2+\frac{1}{x^3}=(x^3+\frac{1}{x^2})(1+\frac{1}{x}) \\ \text{ solving for } x^3+\frac{1}{x^2} \\ \text{ you have } \\ x^3+\frac{1}{x^2}=\frac{x^3+\frac{1}{x^2}+x^2+\frac{1}{x^3}}{1+\frac{1}{x}}$ and remember you found the numerator above from doing what hero suggested and then you are given the denominator

15. anonymous

seems like there is a more prosaic way of doing this

16. freckles

you know what I think I want to re-express what I said just a bit: $x^3+\frac{1}{x^2}=\frac{(x^3+\frac{1}{x^3})+(x^2+\frac{1}{x^2})}{1+\frac{1}{x}}$ just wanted to group together what we actually will find from the above equations that I mentioned (or I mean hero had mentioned in a previous post)

17. anonymous

first equation tells you $x=\frac{7\pm3\sqrt{5}}{2}$ it is much less fun, but i am sure will get an answer


text:
# Subliming

After months of abortive attempts, I’ve taken up Sublime Text 2.

I did that partly to stop people from recommending it to me (seriously people, you were getting on my nerves), and partly because it’s a “better vim than vim” when doing code refactoring – and yes, I’m using the “Vintage” emulation mode. It’s good enough (even does split windows), and I don’t have to switch mental gears when going between local and remote editing.

It wasn’t all roses – I loathe Sublime Text’s default look and feel as well as its overall UI choices (the TextMate alpha works fine and is so much nicer that you nearly weep when you’re using it), but vim emulation and its Quicksilver-like command completion eventually won me over1.

So once I found a decent enough visual theme (a variant of the Pastie theme I use in TextMate and on this site’s code samples) I could mostly ignore minor irritations and just get on with it2.

The only two exceptions are that I utterly loathe the way it handles a) search results and b) git (alas, I’m told I’m using the best git plugin currently available).

Seriously, the user experience in either regard is so bad that it’s almost charmingly gauche, but it sort of makes up for that with multiple cursor support (which I use seldom, but effectively) and its extensibility.

It bears saying that being readily extensible in Python was never a factor – just because it’s my favourite language doesn’t mean I feel like tweaking my editor – but it beats vimscript hands down, dovetails nicely with my penchant for using JSON configuration files for my own stuff, and I concede the point that having an interpreter prompt ready at the tap of a key is quite handy.

But, overall, the main reason I’m currently using it is that text editors (especially programming editors) should just work, and as efficiently as possible at that.

vim is tremendously powerful and I amassed a number of plugins for it over the years (including a native sidebar for the Mac), but Sublime has a slight edge right now – the Command Palette makes it easier to invoke seldom-used commands, and as a result a number of the irritating little “how in blazes do I do that again?” moments have vanished and I can stay in flow longer.

There also seems to be an awful lot of modern plugins for it – whereby “modern” I mean all the latest fads. And being able to rsync my whole environment to Ubuntu and just keep going (with the usual minor niggles regarding keyboard layouts) is a nice plus.

If only it could be as slick as Coda 2 for front-end development, I’d be set.

1. I was previously using Quicksilver itself to trigger TextMate menus (via the Current Application proxy object), which works fine as long as what you want is actually present on the menu tree. I’ve been using this hidden gem for seven years now, and it’s one of the reasons I’ve stuck with Quicksilver instead of using newer, stupider launchers. ↩

2. Like its minimap, which I find ridiculous and switched off right away. Seriously, if you need that kind of frippery to find your way around your own code then you probably shouldn’t be coding anyway. ↩


text:
# Visualizing Newton’s Method

Well for most of us winter break is coming to an end, or for the unlucky, might have already ended, which means its time to start thinking about teaching again. One of the topics commonly covered in a first or second semester of calculus is the use of Newton’s method to approximate roots of functions. Recall the basic approach of Newton’s method is that given a real valued differentiable function $f$ and an initial condition $x_0$ we let

$$x_n=x_{n-1}-\frac{f(x_{n-1})}{f’(x_{n-1})},$$

and then, under certain assumptions, $x_n$ should converge to a root of $f$.

When I’ve taught Newton’s method, I tried to stress that this method is not guaranteed to always work and can be fairly sensitive to the initial condition $x_0$. Going into the precise details of when the method will in fact work is well beyond my first semester calculus students. That said, I am not sure they’ve ever really appreciated these points. In fact, to be completely honest, I am not sure I completely appreciated some of the aspects of these things before I started teaching.

That said, one way to visualize some of these complexities is via a cool program called FractalStream. For example, using FractalStream, we can run the following script:

iterate z – (z^3 – 1)/(3*z^2) until z stops.

and then turn on autocoloring – under the color settings – and we get the following interesting picture of the complex plane colored red, blue, and green:

Newton’s method plot for z^3-1.

Given the above script, FractalStream takes each point in the complex plane and iterates it under the given map until the sequence seems to stops. It then colors that initial point depending on where the sequence of iterates ended. That is to say, since we chose to iterate the function:

$$N(z)=z-\frac{z^3-1}{3z^2}$$

the above script actually performs Newton’s method using the polynomial $z^3-1$ for each initial value $x_0$ and labels each value by which root we end at (assuming we end at a root).

In the above example, we see there are three colors since $z^3-1$ has three roots. Specifically green corresponds to the root 1, blue to $(-1)^{2/3}$, and finally red to the root $-\sqrt[3]{-1}$. Notice that while there is a large area round each root in which Newton’s method converges quickly to that root the areas sort of between each root show more complex behavior. In particular, in this region we see just how sensitive to the initial condition Newton’s method becomes. Additionally, using the orbit tool in FractalStream we get a sense that points in this region take longer to converge:

Forward iterates of a point converge to a root of z^3-1 under Newton’s method.

In this picture the white dot marks each forward iterate under the map $N(z)$ of the point chosen.

We can create similar plots for other polynomials as well. For example, the plot for $z^3-3z$ looks quite a bit different than the one for $z^3-1$ — in part because all the roots are real.

Newton’s method plot for z^3-3z.

It’s pretty fun just playing around with what plots various polynomials produce!

Why these pictures look the way they do has to do with complex dynamics, which is a really awesome subject. If you want to learn more about these pictures and what is going on you might want to check out either Dynamics in One Complex Variable by Milnor or Complex Dynamics by Carleson and Gamelin. However, even for those not interested in complex dynamics these pictures might be something fun to show your students next time you teach Newton’s method!


text:
Article

Finite Size Corrections to Entanglement in Quantum Critical Systems

• F. C. Alcaraz
Physical Review A (Impact Factor: 3.04). 09/2008; 78:032319. DOI: 10.1103/PhysRevA.78.032319
Source: arXiv

ABSTRACT We analyze the finite size corrections to entanglement in quantum critical systems. By using conformal symmetry and density functional theory, we discuss the structure of the finite size contributions to a general measure of ground state entanglement, which are ruled by the central charge of the underlying conformal field theory. More generally, we show that all conformal towers formed by an infinite number of excited states (as the size of the system $L \to \infty$) exhibit a unique pattern of entanglement, which differ only at leading order $(1/L)^2$. In this case, entanglement is also shown to obey a universal structure, given by the anomalous dimensions of the primary operators of the theory. As an illustration, we discuss the behavior of pairwise entanglement for the eigenspectrum of the spin-1/2 XXZ chain with an arbitrary length $L$ for both periodic and twisted boundary conditions.

0 Bookmarks
·
77 Views
• Source
Article: Two particle excited states entanglement entropy in a one-dimensional ring
[Hide abstract]
ABSTRACT: The properties of the entanglement entropy (EE) of two particle excited states in a one-dimensional ring are studied. For a clean system we show analytically that as long as the momenta of the two particles are not close, the EE is twice the value of the EE of any single particle state. For almost identical momenta the EE is lower than this value. The introduction of disorder is numerically shown to lead to a decrease in the median EE of a two particle excited state, while interactions (which have no effect for the clean case) mitigate the decrease. For a ring which is of the same size as the localization length, interaction increase the EE of a typical two particle excited state above the clean system EE value.
Physical review. B, Condensed matter 02/2013; 87(7). · 3.77 Impact Factor
• Source
Article: Entanglement entropy of low-lying excitation in localized interacting system: Signature of Fock space delocalization
[Hide abstract]
ABSTRACT: The properties of the entanglement entropy (EE) of low-lying excitations in one-dimensional disordered interacting systems are studied. The ground state EE shows a clear signature of localization, while low-lying excitation shows a crossover from metallic behavior at short sample sizes to localized at longer length. The dependence of the crossover as function of interaction strength and sample length is studied using the density matrix renormalization group (DMRG). This behavior corresponds to the presence of the predicted many particle critical energy in the vicinity of the Fermi energy. Implications of these results to experiments are discussed.
10/2013;
• Source
Article: Excited states entanglement in homogeneous fermionic chains
[Hide abstract]
ABSTRACT: We study the Renyi entanglement entropy of an interval in a periodic spin chain, for a general eigenstate of a free, translational invariant Hamiltonian. In order to compute analytically the entropy we use two technical tools. The first one is used to reduce logarithmically the complexity of the problem and the second one to compute the R\'enyi entropy of the chosen subsystem. We introduce new strategies to perform the computations and derive new expressions for the entropy of these general states. Finally we show the perfect agreement of the analytical computations and the numerical results.
Journal of Physics A Mathematical and Theoretical 01/2014; 47(24). · 1.77 Impact Factor


text:
# convert a quantum-number MPS to a no-quantum-number MPS

+1 vote

What I want to do is to compute <psi1|OP|psi2>,

inner (psi1, OP, psi2);

where psi1 and psi2 are MPS and OP is an MPO. Here psi1 and psi2 have conserved but different quantum numbers (particle number), but OP does not. For example psi1 and psi2 have 1 and 2 electrons respectively, and OP is the annihilation operator.

In my task, psi1 and psi2 are loaded from files, and OP is generated by the program. Since OP does not conserve quantum number, I cannot use the SiteSet from psi1 to create OP, while the function "inner" requires psi1, psi2 and OP have the same Site indices. What is the proper way to do such tasks? Thanks!

commented by (46.8k points)
Hi Chiamin,
One question to clarify: is OP just a single-site or few-site operator, or is it a sum of many terms?

Miles
commented by (250 points)
Hi Miles,

OP is an MPO for sum of many terms. In my case, it is an MPO for superconducting order parameter.
commented by (46.8k points)
Thanks - ok just wanted to make sure an MPO was necessary. If it was a single or few-site operator there might be a simpler approach.

My other question is this: I see that your MPO does not conserve the quantum number, but does it change it by a definite amount? That’s actually all that is required to still use the QN system in ITensor. So for example if one is conserving particle number and the MPO has the property of always lowering the particle number by exactly 1, then even though the MPO does not conserve particle number you can still make it with AutoMPO and using QN ITensors.

But maybe for your case your MPO creates a superposition of states with many different particle numbers? I’m not sure.

Thanks,
Miles
commented by (250 points)
Yes. The MPO is the sum of creation and annihilation operators of a singlet on every bonds, so it change the particle number for +2 or -2. And psi1 and psi2 have particle number difference 2.

+1 vote

Hi Chiamin,
Thanks for the question. I realized we hadn't documented this feature well enough so I just added some information about it to the website.

The feature I'd recommend you use for this is the function removeQNs(psi) where psi is an MPS. You can call a similar function on an MPO. The removeQNs function returns an MPS (or MPO) with all of the zeros filled in and QN information removed from the indices.

So if you are able to make your operator as an MPO and your MPS as well, then you can call removeQNs on all of them to get dense versions and compute the quantity you want with inner. Note that depending on how you made each MPS or the MPO you may need to modify their site indices to be the same. But if e.g. you made the MPO with AutoMPO using the same site set as for the MPS then you should be fine.

Best,
Miles

commented by (250 points)
Thanks Miles! Is it possible to remove only one of the quantum number? For example, remove the quantum number of particle number but remain the total Sz conserved? It is not necessary but just wonder.
commented by (46.8k points)
Hi Chiamin,
Good question. It is not possible right now, but it would be a really good feature to have in the future, and ideal for a case like this, since your operator conserves fermion parity. We may add this feature first in the Julia version as the block-sparse storage system there has a more flexible and general-purpose design (see NDTensors.jl) and because we are planning to implement ITensor slicing there which may involve some of the same pieces of code.

Best,
Miles