Content-Type: multipart/mixed; boundary="-------------0003100813818" This is a multi-part message in MIME format. ---------------0003100813818 Content-Type: text/plain; name="00-107.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-107.keywords" Stochastic equations, Random dynamical systems, Symmetry ---------------0003100813818 Content-Type: application/postscript; name="SDE2.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="SDE2.ps" %!PS-Adobe-3.0 %%Title: (SDE2.tex) %%Creator: (Textures\252: LaserWriter 8 8.3.3) %%CreationDate: (4:55 PM Wednesday, March 8, 2000) %%For: (degasperis) %%Pages: 26 %%DocumentFonts: %%DocumentNeededFonts: %%DocumentSuppliedFonts: %%DocumentData: Clean7Bit %%PageOrder: Ascend %%Orientation: Portrait %%DocumentMedia: Default 612 792 0 () () %ADO_ImageableArea: 30 31 582 761 %%EndComments userdict begin/dscInfo 5 dict dup begin /Title(SDE2.tex)def /Creator(Textures\252: LaserWriter 8 8.3.3)def /CreationDate(4:55 PM Wednesday, March 8, 2000)def /For(degasperis)def /Pages 26 def end def end /md 200 dict def md begin/currentpacking where {pop /sc_oldpacking currentpacking def true setpacking}if %%BeginFile: adobe_psp_basic %%Copyright: Copyright 1990-1993 Adobe Systems Incorporated. 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(k)s 1 f 152.417 y 265.189 (\050)s 0 f 269.856 (x)s 5 f 277.139 (;)s 282.473 (t)s 1 f 286.806 (\051)s 293.473 (d)s 5 f 300.139 (w)s 6 f 147.463 y 309.053 (k)s 1 f 152.417 y 314.957 (\050)s 5 f 319.624 (t)s 1 f 323.957 (\051)s 5 f 332.624 (:)s 1 f 445.087 (\0501)s 5 f 455.754 (:)s 1 f 459.087 (1\051)s 186.239 y 0 (In)s 15 (other)s 46.367 (w)s 54.7 (ords,)s 84.133 (w)s 92.467 (e)s 101.8 (will)s 124.467 (consider)s 171.234 (in\257nitesimal)s 239.3 (transformations)s 325.834 (of)s 339.501 (the)s 360.167 (form)s 9 f 197.193 y 150.495 (8)s 207.993 y 150.495 (\074)s 229.593 y 150.495 (:)s 5 f 207.85 y 163.162 (x)s 6 f 203.495 y 170.02 (i)s 7 f 207.85 y 177.349 (!)s 5 f 192.682 (y)s 6 f 203.495 y 198.996 (i)s 1 f 207.85 y 210.324 (=)s 5 f 222.991 (x)s 6 f 203.495 y 229.849 (i)s 1 f 207.85 y 236.511 (+)s 5 f 248.511 (\042\273)s 6 f 203.495 y 259.909 (i)s 1 f 207.85 y 263.904 (\050)s 5 f 268.571 (x;)s 280.762 (t)s 1 f 285.096 (\051)s 5 f 222.25 y 163.162 (t)s 7 f 170.829 (!)s 5 f 186.162 (s)s 1 f 199.12 (=)s 5 f 211.787 (t)s 1 f 218.787 (+)s 5 f 230.787 (\042\277)s 1 f 242.987 (\050)s 5 f 247.653 (t)s 1 f 251.987 (\051)s 5 f 236.65 y 163.162 (w)s 6 f 232.295 y 172.076 (i)s 7 f 236.65 y 179.404 (!)s 5 f 194.738 (z)s 6 f 232.295 y 200.846 (i)s 1 f 236.65 y 212.174 (=)s 5 f 224.841 (w)s 6 f 232.295 y 233.755 (i)s 1 f 236.65 y 240.417 (+)s 5 f 252.417 (\042\271)s 6 f 232.295 y 265.243 (i)s 1 f 236.65 y 269.238 (\050)s 5 f 273.905 (w)s 282.819 (;)s 288.152 (t)s 1 f 292.485 (\051)s 309.152 (.)s 221.793 y 445.087 (\0501)s 5 f 455.754 (:)s 1 f 459.087 (2\051)s 257.348 y 0 (W)s 11.333 (e)s 20.667 (also)s 44.733 (call)s 66.733 (symmetry)s 122.834 (generators)s 180.967 (of)s 194.634 (this)s 218.034 (form,)s 249.734 (\134)s 2 f 255.734 (W-symmetries)s 1 f 331.047 (\042.)s 283.748 y 0 (Note)s 29.184 (that)s 55.367 (this)s 78.951 (is)s 91.201 (not)s 112.718 (the)s 133.568 (most)s 163.152 (general)s 204.702 (p)s 211.702 (ossible)s 250.019 (form)s 278.569 (of)s 292.42 (a)s 302.603 (transformation)s 384.587 (for)s 403.137 (the)s 423.987 (v)s 429.654 (ariables)s 298.148 y 0 (in)s 9.667 (v)s 15.667 (olv)s 31 (ed)s 47 (in)s 61 (\0501.1\051;)s 93 (a)s 103 (few)s 124.667 (w)s 133 (ords)s 159.1 (on)s 175.767 (suc)s 192.167 (h)s 202.834 (a)s 212.834 (restriction)s 270.3 (are)s 290.334 (in)s 304.334 (order.)s 324.548 y 0 (The)s 26.258 (restriction)s 85.316 (on)s 5 f 103.574 (\277)s 1 f 110.178 (,)s 119.5 (i.e.)s 144.941 (the)s 167.199 (fact)s 192.457 (that)s 220.048 (the)s 242.306 (transformation)s 325.698 (of)s 340.956 (the)s 363.214 (time)s 392.138 (v)s 397.805 (ariable)s 438.763 (is)s 452.421 (not)s 338.948 y 0 (allo)s 18.333 (w)s 26.667 (ed)s 42.91 (to)s 57.819 (dep)s 76.819 (end)s 99.729 (on)s 116.639 (the)s 137.549 (p)s 144.549 (osition)s 0 f 183.525 (x)s 1 f 190.808 (\050)s 5 f 195.475 (t)s 1 f 199.808 (\051)s 208.718 (nor)s 230.328 (on)s 247.237 (the)s 268.147 (realization)s 327.09 (of)s 341 (the)s 361.91 (Wiener)s 403.853 (pro)s 421.553 (cess)s 0 f 445.929 (w)s 1 f 456.087 (\050)s 5 f 460.754 (t)s 1 f 465.087 (\051)s 353.348 y 0 (w)s 8.333 (as)s 23.618 (already)s 66.536 (in)s 76.203 (tro)s 91.903 (duced)s 127.121 (in)s 141.672 ([4])s 158.89 (and)s 182.775 (is)s 195.392 (en)s 207.059 (tirely)s 239.31 (natural)s 281.895 (if)s 293.446 (w)s 301.78 (e)s 311.664 (w)s 319.998 (an)s 332.331 (t)s 341.549 (to)s 356.767 (tak)s 373.434 (e)s 383.318 (in)s 392.985 (to)s 408.203 (accoun)s 443.87 (t)s 453.088 (the)s 367.748 y 0 (ph)s 13 (ysical)s 45.532 (meaning)s 92.998 (of)s 106.13 (the)s 126.263 (time)s 153.062 (v)s 158.728 (ariable)s 5 f 197.561 (t)s 1 f 201.894 (:)s 210.293 (only)s 236.092 (time)s 262.891 (reparametrizations)s 364.524 (are)s 384.023 (allo)s 402.356 (w)s 410.689 (ed)s 426.155 (\050see)s 449.687 (also)s 382.148 y 0 (app)s 19.667 (endix)s 52 (A)s 65 (of)s 78.667 ([4]\051.)s 408.548 y 0 (In)s 15.292 (\0501.2\051,)s 47.657 (w)s 55.991 (e)s 65.616 (ha)s 77.949 (v)s 83.949 (e)s 93.575 (also)s 117.933 (allo)s 136.267 (w)s 144.6 (ed)s 160.892 (the)s 181.851 (transformation)s 263.943 (of)s 277.902 (the)s 298.861 (v)s 304.861 (ector)s 335.186 (Wiener)s 377.178 (pro)s 394.878 (cess)s 0 f 419.304 (w)s 1 f 429.462 (\050)s 5 f 434.129 (t)s 1 f 438.462 (\051)s 447.421 (only)s 422.948 y 0 (as)s 15.881 (an)s 33.695 (\134in)s 49.362 (ternal\042)s 91.21 (one:)s 120.172 (i.e.,)s 144.273 (it)s 157.42 (cannot)s 197.901 (dep)s 216.902 (end)s 240.716 (on)s 258.53 (the)s 280.344 (p)s 287.344 (osition)s 0 f 327.225 (x)s 1 f 334.509 (\050)s 5 f 339.175 (t)s 1 f 343.509 (\051;)s 357.23 (the)s 379.045 (reason)s 417.626 (for)s 437.14 (this)s 461.688 (is)s 437.348 y 0 (analogous)s 55.994 (to)s 71.254 (the)s 92.514 (one)s 115.107 (men)s 136.774 (tioned)s 174.034 (ab)s 187.034 (o)s 192.701 (v)s 198.701 (e:)s 213.888 (as)s 229.215 (w)s 237.548 (e)s 247.475 (think)s 279.735 (of)s 293.995 (the)s 315.255 (sto)s 330.989 (c)s 335.989 (hastic)s 371.315 (pro)s 389.015 (cess)s 0 f 413.742 (w)s 1 f 423.901 (\050)s 5 f 428.567 (t)s 1 f 432.901 (\051)s 442.161 (to)s 457.421 (b)s 464.421 (e)s 451.748 y 0 (indep)s 29 (enden)s 59.333 (t)s 69.654 (of)s 84.974 (the)s 107.294 (ev)s 118.627 (olution)s 160.948 (of)s 176.268 (the)s 0 f 198.588 (x)s 1 f 205.871 (\050)s 5 f 210.538 (t)s 1 f 214.871 (\051,)s 228.938 (w)s 237.272 (e)s 248.259 (w)s 256.592 (an)s 268.925 (t)s 2 f 279.245 (a)s 291.193 (fortiori)s 1 f 334.26 (its)s 352.647 (transformation)s 436.1 (not)s 459.087 (to)s 466.148 y 0 (dep)s 19 (end)s 41.667 (on)s 58.333 (this.)s 492.548 y 0 (Finally)s 35.833 (,)s 42.648 (w)s 50.981 (e)s 59.666 (ha)s 72 (v)s 78 (e)s 86.685 (allo)s 105.018 (w)s 113.352 (ed)s 128.703 (the)s 148.722 (transformation)s 229.874 (on)s 245.892 (the)s 265.91 (spatial)s 303.996 (co)s 315.662 (ordinates)s 0 f 367.114 (x)s 1 f 374.397 (\050)s 5 f 379.064 (t)s 1 f 383.397 (\051)s 391.416 (to)s 405.434 (dep)s 424.434 (end)s 446.453 (on)s 0 f 462.471 (x)s 1 f 506.948 y 0 (and)s 5 f 22.598 (t)s 1 f 26.932 (,)s 33.677 (but)s 54.942 (not)s 75.54 (on)s 91.472 (the)s 0 f 111.404 (w)s 1 f 121.562 (;)s 128.405 (this)s 151.07 (means)s 187.069 (that)s 212.334 (w)s 220.667 (e)s 229.265 (do)s 245.197 (not)s 265.795 (w)s 274.129 (an)s 286.462 (t)s 294.394 (to)s 308.326 (consider)s 354.357 (transformations)s 440.156 (of)s 453.087 (the)s 521.348 y 0 (spatial)s 39.283 (co)s 50.95 (ordinates)s 103.6 (whic)s 127.266 (h)s 138.483 (dep)s 157.483 (end)s 180.699 (on)s 197.915 (the)s 219.132 (realization)s 278.381 (of)s 292.598 (the)s 313.814 (sto)s 329.547 (c)s 334.547 (hastic)s 369.83 (pro)s 387.53 (cess)s 0 f 412.213 (w)s 1 f 422.372 (\050)s 5 f 427.038 (t)s 1 f 431.372 (\051.)s 446.354 (This)s 535.748 y 0 (again)s 32.284 (is)s 44.635 (a)s 54.92 (natural)s 97.237 (requiremen)s 155.304 (t)s 164.255 (in)s 178.539 (ph)s 191.54 (ysical)s 224.891 (terms,)s 262.013 (but)s 284.297 (it)s 296.581 (should)s 334.932 (b)s 341.932 (e)s 351.55 (noted)s 385.168 (that)s 411.452 (in)s 425.736 (this)s 449.421 (w)s 457.754 (a)s 463.421 (y)s 550.148 y 0 (w)s 8.333 (e)s 18.702 (are)s 39.77 (also)s 64.872 (discarding)s 123.34 (the)s 145.042 (transformations)s 232.61 (needed)s 273.645 (to)s 289.347 (obtain)s 327.715 (normal)s 369.451 (forms)s 403.586 (of)s 418.287 (sto)s 434.021 (c)s 439.021 (hastic)s 564.548 y 0 (di\256eren)s 38.7 (tial)s 60.033 (equations,)s 117.1 (see)s 136.5 ([1,2])s 162.5 (and)s 185.834 (references)s 240.967 (therein.)s 590.948 y 0 (Let)s 22.695 (us)s 39.291 (no)s 51.624 (w)s 65.486 (compute)s 115.348 (ho)s 127.681 (w)s 141.543 (\0501.1\051)s 171.405 (c)s 176.405 (hanges)s 217.001 (under)s 252.229 (\0501.2\051.)s 289.149 (W)s 300.482 (e)s 311.011 (can)s 334.206 (\257rst)s 360.168 (of)s 375.03 (all)s 392.892 (implemen)s 443.225 (t)s 453.087 (the)s 605.348 y 0 (c)s 5 (hange)s 40.384 (on)s 0 f 57.768 (x)s 1 f 69.769 (and)s 5 f 93.82 (t)s 1 f 98.153 (;)s 106.562 (rep)s 123.596 (eating)s 160.313 (the)s 181.697 (computations)s 257.148 (giv)s 272.481 (en)s 289.199 (in)s 303.916 ([4])s 321.3 (\173)s 332.018 (see)s 352.135 (in)s 366.852 (particular)s 422.97 (section)s 463.754 (3)s 619.748 y 0 (there)s 30.7 (\173)s 40.7 (w)s 49.033 (e)s 58.367 (ha)s 70.7 (v)s 76.7 (e)s 86.033 (that)s 112.033 (\0501.1\051)s 140.7 (is)s 152.767 (c)s 157.767 (hanged)s 199.1 (in)s 208.767 (to)s 223.434 (a)s 233.434 (new)s 258.1 (Ito)s 277.1 (equation)s 326.1 (for)s 0 f 344.467 (y)s 1 f 351.942 (\050)s 5 f 356.609 (t)s 1 f 360.942 (\051,)s 372.942 (whic)s 396.609 (h)s 407.276 (w)s 415.609 (e)s 424.942 (write)s 455.642 (as)s 643.202 y 143.601 (d)s 5 f 150.267 (y)s 6 f 638.247 y 156.581 (i)s 1 f 643.202 y 167.91 (=)s 9 f 640.035 y 186.826 (e)s 5 f 643.202 y 184.576 (f)s 6 f 638.247 y 191.743 (i)s 1 f 643.202 y 195.738 (\050)s 5 f 200.405 (y)s 206.719 (;)s 212.052 (s)s 1 f 217.677 (\051)s 224.344 (d)s 5 f 231.01 (s)s 1 f 243.302 (+)s 9 f 259.612 (e)s 5 f 259.302 (\276)s 6 f 638.247 y 266.589 (i)s 646.169 y 270.159 (k)s 1 f 643.202 y 276.063 (\050)s 5 f 280.73 (y)s 287.044 (;)s 292.377 (s)s 1 f 298.002 (\051)s 304.669 (d)s 5 f 311.335 (w)s 6 f 638.247 y 320.249 (k)s 1 f 643.202 y 445.087 (\0501)s 5 f 455.754 (:)s 1 f 459.087 (3\051)s 672.002 y 231.877 (2)s end endp %%Page: 5 5 %%BeginPageSetup initializepage (degasperis; page: 5 of 26)setjob %%EndPageSetup gS 0 0 552 730 rC 0 0 :M f0_12 sf ( )S 4 0 :M f1_12 sf ( )S f2_12 sf ( )S f8_12 sf ( )S 14 0 :M f4_12 sf .136 .014( )J 22 0 :M f5_12 sf .138 .014( )J 31 0 :M f6_12 sf ( )S f7_12 sf ( )S 0 0 :M initclip TeXdict begin 0 10 1200 1000 div mul /CMBX10 findfont fdef 1 10 1200 1000 div mul /CMR10 findfont fdef 2 10 1200 1000 div mul /CMTI10 findfont fdef 4 7 1200 1000 div mul /CMR7 findfont fdef 5 10 1200 1000 div mul /CMMI10 findfont fdef 6 7 1200 1000 div mul /CMMI7 findfont fdef 7 10 1200 1000 div mul /CMSY10 findfont fdef 9 10 1200 1000 div mul /CMEX10 findfont fdef 42 41 translate 72 72.27 div dup scale 1 f 12 y 0 (where)s 34.7 (the)s 55.367 (tilded)s 89.367 (functions)s 141.1 (are)s 161.134 (giv)s 176.467 (en)s 192.467 (b)s 198.8 (y)s 9 f 37.328 y 80.1 (e)s 5 f 40.494 y 77.85 (f)s 6 f 35.54 y 85.016 (i)s 1 f 40.494 y 89.012 (\050)s 5 f 93.678 (y)s 99.992 (;)s 105.325 (s)s 1 f 110.95 (\051)s 118.95 (=)s 5 f 131.617 (f)s 6 f 35.54 y 138.784 (i)s 1 f 40.494 y 142.779 (\050)s 5 f 147.446 (y)s 153.759 (;)s 159.093 (s)s 1 f 164.718 (\051)s 172.051 (+)s 5 f 184.051 (\042)s 1 f 189.647 (\050)s 5 f 194.314 (\261)s 200.101 (f)s 1 f 207.268 (\051)s 6 f 35.54 y 211.934 (i)s 5 f 40.494 y 219.93 (;)s 9 f 229.573 (e)s 5 f 229.263 (\276)s 6 f 35.54 y 236.55 (i)s 43.461 y 240.12 (k)s 1 f 40.494 y 246.024 (\050)s 5 f 250.691 (y)s 257.005 (;)s 262.338 (s)s 1 f 267.963 (\051)s 275.963 (=)s 5 f 288.629 (\276)s 6 f 35.54 y 295.917 (i)s 43.461 y 299.486 (k)s 1 f 40.494 y 305.391 (\050)s 5 f 310.057 (y)s 316.371 (;)s 321.704 (s)s 1 f 327.329 (\051)s 334.663 (+)s 5 f 346.663 (\042)s 1 f 352.259 (\050)s 5 f 356.925 (\261)s 362.713 (\276)s 1 f 370 (\051)s 6 f 35.54 y 374.667 (i)s 43.461 y 378.667 (k)s 5 f 40.494 y 388.571 (:)s 1 f 445.087 (\0501)s 5 f 455.754 (:)s 1 f 459.087 (4\051)s 68.988 y 0 (With)s 31.006 (the)s 51.678 (notation)s 5 f 99.683 (A)s 1 f 112.026 (=)s 124.702 (\0501)s 5 f 135.368 (=)s 1 f 141.368 (2\051\050)s 5 f 156.702 (\276)s 163.989 (\276)s 6 f 64.634 y 171.277 (T)s 1 f 68.988 y 178.836 (\051,)s 190.843 (the)s 211.515 (v)s 217.182 (ariations)s 266.621 (are)s 286.66 (explicitely)s 343.999 (giv)s 359.332 (en)s 375.338 (b)s 381.671 (y)s 392.01 (\050see)s 416.082 (eq.\0503.4\051)s 459.754 (in)s 83.388 y 0 ([4]\051)s 98.376 y 74.315 (\050)s 5 f 78.982 (\261)s 84.769 (f)s 1 f 91.936 (\051)s 6 f 93.422 y 96.602 (i)s 1 f 98.376 y 111.931 (=)s 5 f 125.264 (@)s 6 f 100.176 y 131.635 (t)s 5 f 98.376 y 135.86 (\273)s 6 f 93.422 y 141.662 (i)s 1 f 98.376 y 148.324 (+)s 160.324 ([\050)s 5 f 168.324 (f)s 6 f 93.422 y 175.491 (j)s 7 f 98.376 y 183.213 (\242)s 5 f 189.213 (@)s 6 f 100.176 y 195.584 (j)s 1 f 98.376 y 200.641 (\051)s 5 f 205.307 (\273)s 6 f 93.422 y 211.109 (i)s 7 f 98.376 y 217.771 (\241)s 1 f 229.771 (\050)s 5 f 234.438 (\273)s 6 f 93.422 y 240.24 (j)s 7 f 98.376 y 247.963 (\242)s 5 f 253.963 (@)s 6 f 100.176 y 260.333 (j)s 1 f 98.376 y 265.39 (\051)s 5 f 270.056 (f)s 6 f 93.422 y 277.223 (i)s 1 f 98.376 y 281.218 (])s 7 f 287.218 (\241)s 5 f 299.218 (@)s 6 f 100.176 y 305.589 (t)s 1 f 98.376 y 309.814 (\050)s 5 f 314.481 (f)s 6 f 93.422 y 321.647 (i)s 5 f 98.376 y 325.643 (\277)s 1 f 332.247 (\051)s 339.58 (+)s 5 f 351.58 (A)s 6 f 93.422 y 360.58 (j)s 365.036 (k)s 5 f 98.376 y 370.94 (@)s 4 f 93.422 y 377.978 (2)s 6 f 101.343 y 377.311 (j)s 381.768 (k)s 5 f 98.376 y 387.672 (\273)s 6 f 93.422 y 393.474 (i)s 1 f 119.101 y 72.285 (\050)s 5 f 76.952 (\261)s 82.739 (\276)s 1 f 90.027 (\051)s 6 f 114.147 y 94.693 (i)s 122.068 y 98.693 (k)s 1 f 119.101 y 111.931 (=)s 125.264 ([\050)s 5 f 133.264 (\276)s 6 f 113.335 y 140.552 (j)s 122.721 y 144.121 (k)s 7 f 119.101 y 152.692 (\242)s 5 f 158.692 (@)s 6 f 120.901 y 165.063 (j)s 1 f 119.101 y 170.119 (\051)s 5 f 174.786 (\273)s 6 f 114.147 y 180.588 (i)s 7 f 119.101 y 187.25 (\241)s 1 f 199.25 (\050)s 5 f 203.916 (\273)s 6 f 114.147 y 209.718 (j)s 7 f 119.101 y 217.441 (\242)s 5 f 223.441 (@)s 6 f 120.901 y 229.812 (j)s 1 f 119.101 y 234.868 (\051)s 5 f 239.535 (\276)s 6 f 114.147 y 246.822 (i)s 122.068 y 250.392 (k)s 1 f 119.101 y 256.296 (])s 7 f 262.296 (\241)s 5 f 274.296 (\277)s 280.9 (@)s 6 f 120.901 y 287.271 (t)s 5 f 119.101 y 291.496 (\276)s 6 f 114.147 y 298.784 (i)s 122.068 y 302.353 (k)s 7 f 119.101 y 310.924 (\241)s 1 f 322.924 (\0501)s 5 f 333.59 (=)s 1 f 339.59 (2\051)s 5 f 350.257 (\276)s 6 f 114.147 y 357.545 (i)s 122.068 y 361.114 (k)s 5 f 119.101 y 367.018 (@)s 6 f 120.901 y 373.389 (t)s 5 f 119.101 y 377.614 (\277)s 388.218 (:)s 1 f 108.445 y 445.087 (\0501)s 5 f 455.754 (:)s 1 f 459.087 (5\051)s 154.678 y 0 (Note)s 28.886 (that)s 54.773 (\0501.3\051)s 83.326 (is)s 95.279 (still)s 118.565 (expressed)s 172.285 (in)s 186.171 (terms)s 219.491 (of)s 233.044 (the)s 253.597 (\134old\042)s 285.484 (Wiener)s 327.07 (pro)s 344.77 (cesses)s 5 f 378.856 (w)s 6 f 150.324 y 387.77 (i)s 1 f 154.678 y 391.765 (\050)s 5 f 396.432 (s)s 1 f 402.057 (\051;)s 413.981 (w)s 422.315 (e)s 431.534 (ha)s 443.868 (v)s 449.868 (e)s 459.087 (to)s 169.078 y 0 (express)s 40.845 (it)s 51.857 (in)s 64.869 (terms)s 97.314 (of)s 109.992 (the)s 129.671 (\134new\042)s 165.349 (Wiener)s 206.061 (pro)s 223.761 (cesses)s 5 f 256.973 (z)s 6 f 164.724 y 263.081 (i)s 1 f 169.078 y 267.076 (\050)s 5 f 271.743 (s)s 1 f 277.368 (\051.)s 290.372 (W)s 301.705 (e)s 310.05 (also)s 333.129 (stress)s 365.04 (that)s 390.052 (the)s 409.731 (e\256ect)s 440.409 (of)s 453.088 (the)s 183.478 y 0 (c)s 5 (hange)s 38.473 (of)s 50.946 (the)s 70.419 (time)s 96.558 (v)s 102.225 (ariable)s 140.398 (\050)s 5 f 145.064 (t)s 7 f 152.731 (!)s 5 f 168.064 (s)s 1 f 173.689 (\051)s 181.162 (in)s 193.968 (the)s 0 f 213.441 (w)s 1 f 226.405 (has)s 246.612 (already)s 287.784 (b)s 294.784 (e)s 302.924 (tak)s 319.591 (en)s 334.397 (in)s 344.063 (to)s 357.536 (accoun)s 393.203 (t,)s 404.248 (conformly)s 459.088 (to)s 197.878 y 0 (the)s 19.704 (discussion)s 74.941 (in)s 87.978 (app)s 107.645 (endix)s 139.016 (A)s 151.052 (of)s 163.756 ([4],)s 182.986 (where)s 216.723 (it)s 227.76 (is)s 238.864 (sho)s 255.93 (wn)s 274.301 (that)s 299.338 (this)s 321.775 (amoun)s 356.776 (ts)s 369.213 (to)s 382.917 (a)s 391.954 (transformation)s 212.278 y 0 (of)s 14.291 (the)s 5 f 35.583 (w)s 6 f 207.924 y 44.497 (i)s 1 f 212.278 y 53.117 (in)s 62.783 (to)s 78.075 (new)s 103.366 (pro)s 121.066 (cesses)s 9 f 158.014 (e)s 5 f 155.891 (w)s 6 f 207.924 y 164.805 (i)s 1 f 212.278 y 173.424 (di\256ering)s 221.083 (only)s 248.041 (b)s 254.374 (y)s 265.332 (a)s 275.957 (scalar)s 310.682 (term,)s 5 f 343.496 (dw)s 6 f 207.924 y 358.656 (i)s 1 f 212.278 y 367.025 (=)s 380.733 (\0501)s 7 f 394.483 (\241)s 5 f 406.899 (\042@)s 6 f 214.078 y 418.866 (s)s 5 f 212.278 y 423.995 (\277)s 430.599 (=)s 1 f 436.599 (2\051)s 5 f 447.266 (d)s 9 f 455.635 (e)s 5 f 453.512 (w)s 6 f 207.924 y 462.426 (i)s 1 f 212.278 y 466.421 (.)s 226.678 y 0 (The)s 24.411 (scalar)s 58.255 (term)s 86.699 (can)s 108.443 (th)s 119.443 (us)s 134.587 (b)s 141.587 (e)s 150.664 (absorb)s 185.764 (ed)s 201.508 (in)s 211.175 (to)s 225.586 (the)s 5 f 245.996 (\276)s 1 f 257.028 (matrix,)s 299.19 (as)s 313.667 (w)s 322.001 (e)s 331.078 (ha)s 343.411 (v)s 349.411 (e)s 358.489 (done)s 386.899 (in)s 400.643 (writing)s 441.754 (\0501.5\051.)s 253.078 y 0 (Let)s 22.86 (us)s 39.619 (no)s 51.953 (w)s 65.979 (fo)s 75.979 (cus)s 98.072 (on)s 116.098 (the)s 138.125 (transformation)s 5 f 221.285 (w)s 6 f 248.724 y 230.199 (k)s 7 f 253.078 y 241.702 (!)s 5 f 259.301 (z)s 6 f 248.724 y 265.41 (k)s 1 f 253.078 y 271.314 (;)s 280.687 (in)s 296.046 (\0501.2\051)s 326.073 (ab)s 339.073 (o)s 344.739 (v)s 350.739 (e)s 361.432 (w)s 369.766 (e)s 380.459 (ha)s 392.792 (v)s 398.792 (e)s 5 f 409.485 (z)s 6 f 248.724 y 415.593 (k)s 1 f 253.078 y 427.097 (=)s 5 f 442.03 (w)s 6 f 248.724 y 450.944 (k)s 1 f 253.078 y 460.421 (+)s 5 f 267.478 y 0 (\042\271)s 6 f 263.124 y 12.826 (k)s 1 f 267.478 y 18.731 (\050)s 0 f 23.397 (w)s 5 f 33.556 (;)s 38.889 (t)s 1 f 43.222 (\051,)s 57.22 (but)s 80.818 (actually)s 5 f 128.084 (\271)s 1 f 135.314 (\050)s 0 f 139.981 (w)s 5 f 150.139 (;)s 155.472 (t)s 1 f 159.806 (\051)s 170.071 (cannot)s 211.003 (b)s 218.003 (e)s 228.934 (an)s 247.199 (arbitrary)s 299.898 (function.)s 356.359 (Indeed,)s 0 f 400.691 (z)s 1 f 406.824 (\050)s 5 f 411.491 (t)s 1 f 415.824 (\051)s 426.089 (m)s 435.756 (ust)s 457.421 (b)s 464.421 (e)s 281.878 y 0 (again)s 32.358 (a)s 42.715 (v)s 48.715 (ector)s 79.106 (Wiener)s 121.164 (pro)s 138.864 (cess)s 163.355 (\050in)s 182.379 (particular,)s 241.56 (all)s 258.584 (the)s 279.608 (momen)s 317.275 (ta)s 332.299 (of)s 0 f 346.324 (z)s 1 f 356.815 (and)s 380.506 (d)s 0 f 387.172 (z)s 1 f 397.663 (m)s 407.33 (ust)s 427.754 (coincide)s 296.278 y 0 (with)s 28.573 (those)s 61.212 (of)s 0 f 76.118 (w)s 1 f 91.516 (and)s 116.088 (d)s 0 f 122.755 (w)s 1 f 132.913 (\051;)s 146.772 (th)s 157.772 (us)s 174.412 (the)s 196.318 (only)s 223.89 (p)s 230.89 (ossibilit)s 270.69 (y)s 282.263 (is)s 295.569 (pro)s 312.603 (vided)s 346.175 (b)s 352.509 (y)s 364.081 (constan)s 403.815 (t)s 413.721 (orthogonal)s 310.678 y 0 (transformations,)s 5 f 325.078 y 143.495 (z)s 6 f 320.124 y 149.604 (k)s 1 f 325.078 y 155.508 (\050)s 5 f 160.174 (t)s 1 f 164.508 (\051)s 176.508 (=)s 5 f 193.174 (M)s 6 f 320.124 y 206.124 (k)s 328.045 y 208.816 (j)s 5 f 325.078 y 213.872 (w)s 6 f 320.124 y 222.786 (j)s 1 f 325.078 y 227.842 (\050)s 5 f 232.509 (t)s 1 f 236.842 (\051)s 5 f 249.509 (;)s 262.842 (M)s 275.792 (M)s 4 f 320.124 y 288.742 (+)s 1 f 325.078 y 300.042 (=)s 5 f 312.709 (I)s 322.926 (:)s 1 f 445.087 (\0501)s 5 f 455.754 (:)s 1 f 459.087 (6\051)s 347.743 y 0 (\050The)s 29.333 (constancy)s 85.067 (in)s 99.067 (time)s 126.4 (of)s 5 f 140.067 (M)s 1 f 157.017 (is)s 169.084 (easily)s 202.15 (deriv)s 228.184 (ed)s 244.184 (considering)s 7 f 306.95 (h)s 5 f 311.617 (z)s 1 f 317.726 (\050)s 5 f 322.392 (t)s 1 f 326.726 (\051)s 5 f 331.392 (;)s 336.725 (z)s 1 f 342.834 (\050)s 5 f 347.501 (t)s 1 f 354.5 (+)s 5 f 366.5 (\261)s 372.288 (t)s 1 f 376.621 (\051)s 7 f 381.288 (i)s 1 f 385.955 (.\051)s 374.143 y 0 (As)s 17.841 (w)s 26.174 (e)s 35.614 (w)s 43.948 (an)s 56.281 (t)s 65.055 (to)s 79.829 (consider)s 126.703 (in\257nitesimal)s 194.877 (generators)s 253.117 (of)s 266.891 (suc)s 283.291 (h)s 294.065 (transformations,)s 384.066 (and)s 407.506 (to)s 422.28 (write)s 453.087 (the)s 388.543 y 0 (transformation)s 81.8 (from)s 0 f 110.167 (w)s 1 f 124.325 (to)s 0 f 138.992 (z)s 1 f 149.125 (in)s 163.125 (the)s 183.792 (form)s 212.159 (\0501.2\051,)s 244.159 (this)s 267.559 (will)s 290.225 (b)s 297.226 (e)s 5 f 417.037 y 150.296 (z)s 6 f 412.082 y 156.404 (k)s 1 f 417.037 y 162.309 (\050)s 5 f 166.975 (t)s 1 f 171.309 (\051)s 183.309 (=)s 5 f 199.975 (w)s 6 f 412.082 y 208.889 (k)s 1 f 417.037 y 214.793 (\050)s 5 f 219.46 (t)s 1 f 223.793 (\051)s 233.127 (+)s 5 f 247.126 (\042)s 254.722 (B)s 6 f 412.082 y 264.426 (k)s 420.003 y 263.824 (p)s 1 f 417.037 y 270.331 (\050)s 5 f 274.997 (t)s 1 f 279.331 (\051)s 5 f 283.997 (w)s 6 f 412.082 y 292.911 (p)s 1 f 417.037 y 298.458 (\050)s 5 f 303.125 (t)s 1 f 307.458 (\051)s 5 f 316.125 (;)s 1 f 445.087 (\0501)s 5 f 455.754 (:)s 1 f 459.087 (7\051)s 445.531 y 0 (with)s 5 f 27.333 (B)s 1 f 37.038 (\050)s 5 f 41.704 (t)s 1 f 46.038 (\051)s 54.704 (a)s 64.704 (real)s 88.071 (an)s 100.404 (tisymmetric)s 166.838 (matrix.)s 210.538 (Therefore,)s 474.025 y 172.032 (d)s 5 f 178.699 (w)s 6 f 469.07 y 187.613 (k)s 1 f 474.025 y 200.85 (=)s 217.517 (d)s 5 f 224.183 (z)s 6 f 469.07 y 230.292 (k)s 7 f 474.025 y 238.863 (\241)s 5 f 250.863 (\042B)s 6 f 469.07 y 266.163 (k)s 476.992 y 265.561 (p)s 1 f 474.025 y 272.067 (d)s 5 f 278.733 (z)s 6 f 469.07 y 284.842 (p)s 5 f 474.025 y 294.389 (:)s 1 f 445.087 (\0501)s 5 f 455.754 (:)s 1 f 459.087 (8\051)s 502.519 y 0 (W)s 11.333 (e)s 21.604 (can)s 44.542 (insert)s 78.913 (this)s 103.251 (in)s 112.918 (to)s 128.522 (\0501.3\051)s 158.127 (and,)s 185.966 (using)s 218.304 (also)s 243.308 (\0501.4\051)s 272.913 (and)s 297.184 (\0501.5\051)s 326.788 (\050i.e.)s 354.935 (the)s 376.54 (results)s 415.644 (of)s 430.248 ([4]\051,)s 456.087 (w)s 464.421 (e)s 516.919 y 0 (obtain)s 38.757 (the)s 60.848 (complete)s 112.938 (expression)s 172.196 (for)s 5 f 191.986 (\261)s 197.774 (f)s 1 f 210.364 (and)s 5 f 235.121 (\261)s 240.909 (\276)s 1 f 248.196 (.)s 261.135 (This)s 289.958 (is)s 303.449 (summarized)s 371.639 (in)s 387.063 (the)s 409.154 (Prop)s 435.021 (osition)s 531.319 y 0 (b)s 7 (elo)s 21.333 (w,)s 37.333 (where)s 72.033 (w)s 80.367 (e)s 89.7 (also)s 113.767 (use)s 134.5 (the)s 155.167 (notation,)s 206.5 (already)s 248.867 (in)s 258.534 (tro)s 274.234 (duced)s 308.901 (in)s 322.901 ([4],)s 7 f 559.814 y 141.42 (f)s 5 f 147.42 (f)s 153.92 (;)s 159.253 (g)s 7 f 165.407 (g)s 6 f 554.859 y 171.407 (i)s 1 f 559.814 y 182.736 (:=)s 202.736 (\050)s 5 f 207.402 (f)s 6 f 554.859 y 214.569 (j)s 7 f 559.814 y 222.292 (\242)s 5 f 228.292 (@)s 6 f 561.614 y 234.663 (j)s 1 f 559.814 y 239.719 (\051)s 5 f 244.386 (g)s 6 f 554.859 y 250.54 (i)s 7 f 559.814 y 257.202 (\241)s 1 f 269.202 (\050)s 5 f 273.868 (g)s 6 f 554.859 y 280.022 (j)s 7 f 559.814 y 287.745 (\242)s 5 f 293.745 (@)s 6 f 561.614 y 300.116 (j)s 1 f 559.814 y 305.172 (\051)s 5 f 309.839 (f)s 6 f 554.859 y 317.006 (i)s 5 f 559.814 y 325.001 (:)s 1 f 445.087 (\0501)s 5 f 455.754 (:)s 1 f 459.087 (8\051)s 0 f 600.308 y 0 (Prop)s 30.067 (osition)s 74.529 (1.)s 2 f 90.572 (Under)s 127.18 (the)s 147.048 (in\257nitesimal)s 215.983 (tr)s 224.416 (ansformation)s 5 f 297.798 (x)s 6 f 595.953 y 304.656 (i)s 7 f 600.308 y 311.984 (!)s 5 f 327.318 (y)s 6 f 595.953 y 333.632 (i)s 1 f 600.308 y 340.96 (=)s 5 f 353.627 (x)s 6 f 595.953 y 360.485 (i)s 1 f 600.308 y 367.006 (+)s 5 f 378.865 (\042\273)s 6 f 595.953 y 390.263 (i)s 1 f 600.308 y 394.258 (\050)s 0 f 398.925 (x)s 5 f 406.208 (;)s 411.541 (t)s 1 f 415.875 (\051)s 2 f 420.541 (,)s 5 f 428.463 (t)s 7 f 436.129 (!)s 5 f 451.462 (s)s 1 f 460.421 (=)s 5 f 614.708 y 0 (t)s 1 f 6.707 (+)s 5 f 18.415 (\042\277)s 1 f 30.615 (\050)s 5 f 35.281 (t)s 1 f 39.615 (\051)s 2 f 44.281 (,)s 5 f 52.147 (w)s 6 f 610.353 y 61.061 (i)s 7 f 614.708 y 68.389 (!)s 5 f 83.722 (z)s 6 f 610.353 y 89.831 (i)s 1 f 614.708 y 97.159 (=)s 5 f 109.826 (w)s 6 f 610.353 y 118.74 (i)s 1 f 614.708 y 125.109 (+)s 5 f 136.816 (\042B)s 6 f 610.353 y 152.116 (i)s 618.106 y 151.514 (k)s 1 f 614.708 y 157.418 (\050)s 5 f 162.085 (t)s 1 f 166.418 (\051)s 5 f 171.085 (w)s 6 f 610.353 y 179.999 (k)s 2 f 614.708 y 185.903 (,)s 193.769 (with)s 5 f 219.7 (B)s 2 f 233.563 (a)s 243.855 (r)s 248.302 (e)s 253.208 (al)s 266.567 (antisymmetric)s 345.399 (matrix,)s 387.504 (the)s 407.302 (Ito)s 426.208 (e)s 431.114 (quation)s 1 f 643.202 y 156.297 (d)s 5 f 162.963 (x)s 6 f 638.247 y 169.822 (i)s 1 f 643.202 y 177.15 (=)s 5 f 189.817 (f)s 6 f 638.247 y 196.983 (i)s 1 f 643.202 y 200.978 (\050)s 0 f 205.645 (x)s 5 f 212.928 (;)s 218.262 (t)s 1 f 222.595 (\051d)s 5 f 233.928 (t)s 1 f 240.928 (+)s 5 f 252.928 (\276)s 6 f 638.247 y 260.216 (i)s 646.169 y 259.785 (k)s 1 f 643.202 y 265.689 (\050)s 0 f 270.356 (x)s 5 f 277.639 (;)s 282.973 (t)s 1 f 287.306 (\051d)s 5 f 298.639 (w)s 6 f 638.247 y 307.553 (k)s 1 f 672.002 y 231.877 (3)s end endp %%Page: 6 6 %%BeginPageSetup initializepage (degasperis; page: 6 of 26)setjob %%EndPageSetup gS 0 0 552 730 rC 0 0 :M f0_12 sf ( )S 4 0 :M f1_12 sf ( )S 8 0 :M f2_12 sf ( )S f8_12 sf ( )S f4_12 sf .095 .01( )J 22 0 :M f5_12 sf .138 .014( )J 31 0 :M f6_12 sf ( )S 0 0 :M initclip TeXdict begin 0 10 1200 1000 div mul /CMBX10 findfont fdef 1 10 1200 1000 div mul /CMR10 findfont fdef 2 10 1200 1000 div mul /CMTI10 findfont fdef 4 7 1200 1000 div mul /CMR7 findfont fdef 5 10 1200 1000 div mul /CMMI10 findfont fdef 6 7 1200 1000 div mul /CMMI7 findfont fdef 7 10 1200 1000 div mul /CMSY10 findfont fdef 42 41 translate 72 72.27 div dup scale 2 f 12 y 0 (is)s 12.88 (change)s 47.84 (d)s 58.266 (into)s 83.106 (a)s 93.533 (new)s 118.066 (Ito)s 137.106 (e)s 142.012 (quation)s 1 f 40.629 y 154.483 (d)s 5 f 161.15 (y)s 6 f 35.674 y 167.464 (i)s 1 f 40.629 y 174.792 (=)s 5 f 187.459 (F)s 6 f 35.674 y 196.842 (i)s 1 f 40.629 y 200.837 (\050)s 0 f 205.504 (y)s 5 f 212.979 (;)s 218.312 (s)s 1 f 223.937 (\051d)s 5 f 235.271 (s)s 1 f 243.562 (+)s 5 f 255.562 (S)s 6 f 35.674 y 263.612 (i)s 43.596 y 262.921 (k)s 1 f 40.629 y 268.825 (\050)s 0 f 273.492 (y)s 5 f 280.967 (;)s 286.3 (s)s 1 f 291.925 (\051d)s 5 f 303.258 (z)s 6 f 35.674 y 309.367 (k)s 2 f 69.258 y 0 (with)s 5 f 26.067 (F)s 6 f 64.903 y 35.45 (i)s 1 f 69.258 y 42.778 (=)s 5 f 55.445 (f)s 6 f 64.903 y 62.612 (i)s 1 f 69.258 y 69.273 (+)s 5 f 81.273 (\042)s 1 f 86.869 (\050)s 5 f 91.536 (\261)s 97.323 (f)s 1 f 104.49 (\051)s 6 f 64.903 y 109.157 (i)s 2 f 69.258 y 113.152 (,)s 5 f 121.125 (S)s 6 f 64.903 y 129.175 (i)s 72.656 y 128.483 (k)s 1 f 69.258 y 137.721 (=)s 5 f 150.387 (\276)s 6 f 64.903 y 157.675 (i)s 72.656 y 157.244 (k)s 1 f 69.258 y 165.815 (+)s 5 f 177.815 (\042)s 1 f 183.411 (\050)s 5 f 188.078 (\261)s 193.865 (\276)s 1 f 201.153 (\051)s 6 f 64.903 y 205.819 (i)s 72.656 y 205.819 (k)s 2 f 69.258 y 211.724 (,)s 219.697 (the)s 239.63 (variations)s 295.903 (b)s 300.81 (eing)s 326.569 (given)s 357.849 (by)s 1 f 98.873 y 103.339 (\050)s 5 f 108.006 (\261)s 113.793 (f)s 1 f 120.96 (\051)s 6 f 93.919 y 125.626 (i)s 1 f 98.873 y 137.248 (=)s 5 f 150.875 (@)s 6 f 100.673 y 157.246 (s)s 5 f 98.873 y 162.375 (\273)s 6 f 93.919 y 168.177 (i)s 1 f 98.873 y 176.839 (+)s 7 f 190.839 (f)s 5 f 196.839 (f)s 203.339 (;)s 208.672 (\273)s 7 f 214.474 (g)s 6 f 93.919 y 220.474 (i)s 7 f 98.873 y 229.136 (\241)s 5 f 243.136 (@)s 6 f 100.673 y 249.506 (s)s 1 f 98.873 y 254.636 (\050)s 5 f 259.302 (\277)s 265.906 (f)s 6 f 93.919 y 273.073 (i)s 1 f 98.873 y 277.068 (\051)s 286.402 (+)s 5 f 300.401 (A)s 6 f 93.919 y 309.402 (j)s 313.858 (k)s 5 f 98.873 y 319.762 (@)s 4 f 93.919 y 326.799 (2)s 6 f 101.84 y 326.133 (j)s 330.589 (k)s 5 f 98.873 y 336.493 (\273)s 6 f 93.919 y 342.295 (i)s 1 f 118.786 y 101.309 (\050)s 5 f 105.976 (\261)s 111.763 (\276)s 1 f 119.051 (\051)s 6 f 113.832 y 123.717 (i)s 121.753 y 123.717 (k)s 1 f 118.786 y 137.248 (=)s 7 f 150.875 (f)s 5 f 156.875 (\276)s 6 f 120.586 y 163.732 (k)s 5 f 118.786 y 169.636 (;)s 174.969 (\273)s 7 f 180.771 (g)s 6 f 113.832 y 186.771 (i)s 7 f 118.786 y 193.433 (\241)s 5 f 205.433 (\277)s 212.037 (@)s 6 f 120.586 y 218.408 (s)s 5 f 118.786 y 223.537 (\276)s 6 f 113.832 y 230.825 (i)s 121.753 y 230.394 (k)s 7 f 118.786 y 238.965 (\241)s 1 f 250.965 (\0501)s 5 f 261.632 (=)s 1 f 267.632 (2\051\050)s 5 f 282.965 (@)s 6 f 120.586 y 289.336 (s)s 5 f 118.786 y 294.465 (\277)s 1 f 301.069 (\051)s 5 f 305.736 (\276)s 6 f 113.832 y 313.023 (i)s 121.753 y 312.593 (k)s 7 f 118.786 y 321.164 (\241)s 5 f 333.164 (\276)s 6 f 113.832 y 340.451 (i)s 121.753 y 340.021 (p)s 5 f 118.786 y 345.567 (B)s 6 f 113.02 y 355.272 (p)s 122.406 y 354.67 (k)s 5 f 118.786 y 365.112 (:)s 1 f 108.736 y 445.087 (\0501)s 5 f 455.754 (:)s 1 f 459.087 (9\051)s 165.215 y 0 (Needless)s 49.439 (to)s 65.077 (sa)s 75.477 (y)s 80.811 (,)s 89.359 (\0501.2\051)s 118.998 (will)s 142.636 (b)s 149.636 (e)s 159.942 (a)s 170.914 (symmetry)s 227.986 (of)s 242.624 (\0501.1\051)s 272.263 (only)s 299.568 (if)s 311.54 (it)s 324.512 (lea)s 338.846 (v)s 344.846 (es)s 359.884 (this)s 384.256 (in)s 393.923 (v)s 399.59 (arian)s 425.956 (t,)s 439.171 (i.e.)s 462.754 (if)s 179.615 y 0 (\050)s 5 f 4.667 (\261)s 10.454 (f)s 1 f 17.621 (\051)s 6 f 175.261 y 22.288 (i)s 1 f 179.615 y 30.283 (and)s 53.616 (\050)s 5 f 58.283 (\261)s 64.07 (\276)s 1 f 71.358 (\051)s 6 f 175.261 y 76.024 (i)s 183.014 y 76.024 (k)s 1 f 179.615 y 85.929 (v)s 91.595 (anish)s 122.995 (iden)s 144.662 (tically)s 180.995 (for)s 199.362 (all)s 5 f 216.029 (i)s 1 f 224.163 (and)s 5 f 247.496 (k)s 1 f 258.121 ([4].)s 0 f 206.015 y 0 (Corollary)s 55.641 (.)s 2 f 69.18 (The)s 95.054 (ve)s 105.481 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(will)s 23.904 (b)s 30.904 (e)s 41.474 (the)s 63.377 (same.)s 101.821 (Th)s 116.822 (us)s 133.458 (all)s 151.362 (the)s 173.266 (discussion)s 230.703 (conducted)s 289.273 (in)s 304.51 ([4])s 322.413 (ab)s 335.413 (out)s 357.984 (relations)s 407.987 (in)s 417.654 (tercurring)s 600.002 y 0 (b)s 7 (et)s 16.667 (w)s 25 (en)s 41.111 (symmetries)s 104.388 (of)s 118.165 (an)s 134.942 (Ito)s 154.053 (equation)s 203.163 (and)s 226.607 (of)s 240.384 (the)s 261.161 (asso)s 282.961 (ciated)s 318.405 (F)s 325.239 (okk)s 343.572 (er-Planc)s 386.772 (k)s 397.216 (equation,)s 449.687 (also)s 614.402 y 0 (applies)s 40.864 (to)s 56.328 (the)s 77.792 (presen)s 110.892 (t)s 120.356 (case)s 146.553 (\173)s 157.35 (i.e.)s 180.409 (to)s 195.873 (transformations)s 283.204 (of)s 297.668 (the)s 319.132 (form)s 348.296 (\0501.2\051)s 377.76 (\173)s 388.557 (with)s 416.687 (absolutely)s 628.802 y 0 (no)s 17.28 (c)s 22.28 (hanges.)s 68.187 (Needless)s 117.267 (to)s 132.547 (sa)s 142.947 (y)s 148.28 (,)s 156.38 (for)s 175.36 (more)s 206.007 (general)s 247.987 (transformations)s 335.134 (\173)s 345.747 (for)s 364.727 (whic)s 388.394 (h)s 399.674 (the)s 420.954 (t)s 425.287 (w)s 433.621 (o-steps)s 643.202 y 0 (pro)s 17.7 (cedure)s 55.733 (cannot)s 95.067 (b)s 102.067 (e)s 111.4 (applied)s 153.4 (\173)s 163.4 (this)s 186.8 (w)s 195.134 (ould)s 221.8 (not)s 243.134 (b)s 250.134 (e)s 259.467 (true.)s 7 f 289.501 (\246)s 1 f 672.002 y 231.877 (4)s end endp %%Page: 7 7 %%BeginPageSetup initializepage (degasperis; page: 7 of 26)setjob %%EndPageSetup gS 0 0 552 730 rC 0 0 :M f0_12 sf ( )S f1_12 sf ( )S 7 0 :M f2_12 sf ( )S f8_12 sf ( )S 14 0 :M f4_12 sf .116 .012( )J f5_12 sf .148 .015( )J 31 0 :M f6_12 sf ( )S 0 0 :M initclip TeXdict begin 0 10 1200 1000 div mul /CMBX10 findfont fdef 1 10 1200 1000 div mul /CMR10 findfont fdef 2 10 1200 1000 div mul /CMTI10 findfont fdef 4 7 1200 1000 div mul /CMR7 findfont fdef 5 10 1200 1000 div mul /CMMI10 findfont fdef 6 7 1200 1000 div mul /CMMI7 findfont fdef 7 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(realization)s 323.833 (of)s 338.261 (the)s 5 f 359.689 (n)s 1 f 366.892 (-dimensional)s 437.721 (v)s 443.721 (ector)s 138.936 y 0 (Wiener)s 40.957 (pro)s 58.657 (cess)s 0 f 82.048 (w)s 1 f 92.206 (\050)s 5 f 96.873 (t)s 1 f 101.206 (\051.)s 114.292 (In)s 128.549 (more)s 157.84 (precise)s 196.531 (terms,)s 232.703 (an)s 248.627 (Ito)s 266.884 (equation)s 315.142 (also)s 338.466 (de\257nes)s 377.123 (a)s 5 f 386.38 (N)s 1 f 397.33 (-p)s 408.33 (oin)s 423.997 (t)s 431.921 (pro)s 449.621 (cess)s 153.336 y 0 (in)s 15.036 (a)s 26.073 (realization)s 85.809 (of)s 100.512 (whic)s 124.179 (h)s 5 f 135.882 (N)s 1 f 151.869 (particles)s 201.005 (with)s 229.375 (di\256eren)s 268.075 (t)s 277.778 (initial)s 313.481 (p)s 320.481 (osition)s 360.251 (mo)s 375.918 (v)s 381.918 (e)s 392.287 (sim)s 410.021 (ultaneously)s 167.736 y 0 (under)s 34.033 (the)s 54.7 (Ito)s 73.7 (equation)s 122.7 (with)s 150.034 (the)s 2 f 170.7 (same)s 1 f 201.073 (realization)s 259.774 (of)s 0 f 273.44 (w)s 1 f 283.599 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264.799 y 312.625 (A)s 1 f 321.625 (\050)s 5 f 326.292 (x;)s 338.484 (x)s 1 f 345.342 (\051)s 5 f 362.009 (A)s 1 f 371.009 (\050)s 5 f 375.675 (x;)s 387.867 (y)s 1 f 394.181 (\051)s 5 f 279.199 y 312.898 (A)s 1 f 321.898 (\050)s 5 f 326.564 (y)s 332.878 (;)s 338.211 (x)s 1 f 345.07 (\051)s 5 f 362.281 (A)s 1 f 371.281 (\050)s 5 f 375.948 (y)s 382.262 (;)s 387.595 (y)s 1 f 393.909 (\051)s 9 f 255.079 y 400.848 (\266)s 5 f 271.999 y 415.681 (:)s 1 f 445.087 (\0502)s 5 f 455.754 (:)s 1 f 459.087 (5\051)s 319.945 y 0 (W)s 11.333 (e)s 22.159 (can)s 45.651 (no)s 57.984 (w)s 72.142 (lo)s 81.809 (ok)s 99.634 (for)s 119.493 (symmetries)s 184.152 (of)s 199.31 (\0502.3\051)s 229.469 (using)s 262.361 (the)s 284.52 (equations)s 339.745 (for)s 359.604 (symmetries)s 424.262 (of)s 439.421 (\050OPP)s 334.345 y 0 (de\257ned)s 42.483 (b)s 48.816 (y\051)s 64.966 (an)s 82.783 (Ito)s 102.932 (equation)s 153.082 (giv)s 168.415 (en)s 185.565 (in)s 200.715 ([4])s 218.531 (or,)s 238.002 (more)s 269.185 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(\0502)s 5 f 455.754 (:)s 1 f 459.087 (6\051)s 449.455 y 0 (In)s 16.706 (the)s 39.078 (follo)s 61.078 (wing)s 91.45 (w)s 99.783 (e)s 110.822 (will)s 135.194 (k)s 141.194 (eep)s 164.233 (trac)s 184.6 (k)s 196.639 (of)s 5 f 212.011 (x)s 1 f 224.575 (and)s 5 f 249.614 (y)s 1 f 261.633 (dep)s 280.633 (endencies)s 335.739 (but,)s 363.204 (for)s 383.276 (ease)s 410.382 (of)s 425.754 (notation)s 463.855 y 0 (and)s 24.366 (since)s 54.799 (all)s 72.499 (quan)s 97.832 (tities)s 128.932 (dep)s 147.932 (end)s 171.632 (on)s 5 f 189.331 (t)s 1 f 193.664 (,)s 202.289 (will)s 225.989 (drop)s 255.055 (the)s 5 f 276.754 (t)s 1 f 286.121 (dep)s 305.121 (endencies.)s 366.286 (W)s 377.62 (e)s 387.986 (will)s 411.685 (also)s 436.785 (use)s 5 f 458.551 (n)s 1 f 465.754 (-)s 478.255 y 0 (v)s 6 (ector)s 35.879 (notation)s 83.724 (throughout;)s 7 f 149.655 (r)s 6 f 480.055 y 159.655 (x)s 1 f 478.255 y 169.542 (will)s 192.054 (b)s 199.054 (e)s 208.233 (the)s 228.745 (gradien)s 267.111 (t)s 275.623 (v)s 281.623 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f 281.172 (y)s 287.486 (;)s 292.819 (x)s 1 f 299.677 (\051)s 7 f 304.344 (r)s 6 f 612.002 y 314.344 (y)s 5 f 610.202 y 320.112 (\301)s 1 f 327.262 (\050)s 5 f 331.929 (y)s 1 f 338.243 (\051)s 9 f 586.082 y 345.318 (\266)s 1 f 603.002 y 357.485 (=)s 636.602 y 70.47 (=)s 9 f 619.682 y 83.136 (\265)s 7 f 629.402 y 93.97 (f)s 5 f 99.97 (\301)s 1 f 107.12 (\050)s 5 f 111.786 (x)s 1 f 118.645 (\051)s 5 f 123.311 (;)s 128.645 (\264)s 1 f 135.034 (\050)s 5 f 139.7 (x;)s 151.892 (y)s 1 f 158.206 (\051)s 7 f 162.872 (g)s 6 f 631.202 y 168.872 (x)s 1 f 629.402 y 177.581 (+)s 5 f 189.581 (\301)s 1 f 196.731 (\050)s 5 f 201.397 (y)s 1 f 207.711 (\051)s 7 f 212.378 (r)s 6 f 631.202 y 222.378 (y)s 5 f 629.402 y 228.146 (\264)s 1 f 234.535 (\050)s 5 f 239.202 (x;)s 251.393 (y)s 1 f 257.707 (\051)s 7 f 643.802 y 93.97 (f)s 5 f 99.97 (\301)s 1 f 107.12 (\050)s 5 f 111.786 (y)s 1 f 118.1 (\051)s 5 f 122.767 (;)s 128.1 (\264)s 1 f 134.489 (\050)s 5 f 139.156 (y)s 145.47 (;)s 150.803 (x)s 1 f 157.661 (\051)s 7 f 162.328 (g)s 6 f 645.602 y 168.328 (y)s 1 f 643.802 y 176.763 (+)s 5 f 188.763 (\301)s 1 f 195.913 (\050)s 5 f 200.579 (x)s 1 f 207.438 (\051)s 7 f 212.104 (r)s 6 f 645.602 y 222.104 (x)s 5 f 643.802 y 228.146 (\264)s 1 f 234.535 (\050)s 5 f 239.202 (y)s 245.516 (;)s 250.849 (x)s 1 f 257.707 (\051)s 9 f 619.682 y 264.374 (\266)s 5 f 636.602 y 283.207 (:)s 1 f 603.002 y 445.087 (\0502)s 5 f 455.754 (:)s 1 f 459.087 (7\051)s 672.002 y 231.877 (6)s end endp %%Page: 9 9 %%BeginPageSetup initializepage (degasperis; page: 9 of 26)setjob %%EndPageSetup gS 0 0 552 730 rC 0 0 :M f1_12 sf ( )S f8_12 sf ( )S 7 0 :M f4_12 sf .136 .014( )J 15 0 :M f5_12 sf .138 .014( )J 24 0 :M f6_12 sf ( )S f7_12 sf ( )S 0 0 :M initclip TeXdict begin 1 10 1200 1000 div mul /CMR10 findfont fdef 4 7 1200 1000 div mul /CMR7 findfont fdef 5 10 1200 1000 div mul /CMMI10 findfont fdef 6 7 1200 1000 div mul /CMMI7 findfont fdef 7 10 1200 1000 div mul /CMSY10 findfont fdef 9 10 1200 1000 div mul /CMEX10 findfont fdef 42 41 translate 72 72.27 div dup scale 1 f 12 y 0 (W)s 11.333 (e)s 20.667 (also)s 44.733 (ha)s 57.067 (v)s 63.067 (e)s 72.4 (immediately)s 5 f 48.174 y 111.812 (@)s 6 f 49.974 y 118.183 (t)s 5 f 48.174 y 122.408 (\273)s 1 f 131.543 (=)s 9 f 31.254 y 144.21 (\265)s 5 f 40.974 y 155.043 (@)s 6 f 42.774 y 161.414 (t)s 5 f 40.974 y 165.639 (\264)s 1 f 172.028 (\050)s 5 f 176.695 (x;)s 188.886 (y)s 1 f 195.2 (\051)s 5 f 55.374 y 155.043 (@)s 6 f 57.174 y 161.414 (t)s 5 f 55.374 y 165.639 (\264)s 1 f 172.028 (\050)s 5 f 176.695 (y)s 183.009 (;)s 188.342 (x)s 1 f 195.2 (\051)s 9 f 31.254 y 201.867 (\266)s 1 f 48.174 y 216.7 (;)s 5 f 226.034 (@)s 6 f 49.974 y 232.404 (t)s 1 f 48.174 y 236.629 (\050)s 5 f 241.296 (\277)s 247.9 (f)s 1 f 255.067 (\051)s 263.067 (=)s 9 f 31.254 y 275.734 (\265)s 5 f 40.974 y 286.567 (@)s 6 f 42.774 y 292.938 (t)s 1 f 40.974 y 297.163 ([)s 5 f 300.496 (\277)s 307.1 (\301)s 1 f 314.25 (\050)s 5 f 318.917 (x)s 1 f 325.775 (\051])s 5 f 55.374 y 286.839 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(e)s 323.174 (ha)s 335.507 (v)s 341.507 (e)s 351.527 (to)s 366.88 (p)s 373.88 (erform)s 412.967 (some)s 443.721 (more)s 588.002 y 0 (computation)s 70.974 (in)s 85.948 (the)s 107.589 (case)s 5 f 133.963 (k)s 7 f 145.545 (\267)s 5 f 159.835 (n)s 1 f 167.038 (.)s 178.627 (This)s 207.001 (condition)s 260.642 (will)s 284.283 (b)s 291.283 (e)s 301.591 (understo)s 347.357 (o)s 353.691 (d)s 365.331 (from)s 394.672 (no)s 407.006 (w)s 420.646 (on)s 438.287 (unless)s 602.402 y 0 (di\256eren)s 38.7 (tly)s 57.033 (sp)s 68.767 (eci\257ed.)s 628.802 y 0 (W)s 11.333 (e)s 21.123 (use)s 42.313 (the)s 63.437 (explicit)s 106.227 (expressions)s 169.25 (for)s 5 f 188.073 (\276)s 1 f 199.817 (and)s 5 f 223.607 (\273)s 1 f 233.866 (in)s 248.323 (terms)s 282.213 (of)s 5 f 296.336 (\275)s 1 f 306.997 (and)s 5 f 330.787 (\264)s 1 f 337.176 (,)s 345.08 (see)s 364.937 (\0502.4\051)s 394.06 (and)s 417.85 (\0502.5\051,)s 450.421 (and)s 643.202 y 0 (write)s 5 f 30.7 (\275)s 6 f 645.002 y 36.904 (k)s 1 f 643.202 y 46.808 (for)s 65.175 (the)s 5 f 85.842 (n)s 1 f 97.045 (v)s 103.045 (ector)s 133.078 (of)s 146.745 (comp)s 175.078 (onen)s 199.411 (ts)s 212.811 (\050)s 5 f 217.478 (\275)s 4 f 638.847 y 223.682 (1)s 6 f 646.601 y 223.682 (k)s 5 f 643.202 y 229.587 (;)s 234.92 (:::;)s 250.253 (\275)s 6 f 638.847 y 256.457 (n)s 646.601 y 256.457 (k)s 1 f 643.202 y 262.989 (\051,)s 274.989 (so)s 289.723 (that)s 315.723 (e.g.)s 5 f 339.056 (\275)s 6 f 645.002 y 345.26 (k)s 1 f 643.202 y 351.165 (\050)s 5 f 355.831 (x)s 1 f 362.69 (\051)s 7 f 367.356 (r)s 6 f 645.002 y 377.356 (x)s 1 f 643.202 y 386.731 (:=)s 5 f 402.731 (\275)s 6 f 637.435 y 408.935 (j)s 646.822 y 408.935 (k)s 1 f 643.202 y 414.84 (\050)s 5 f 419.506 (@)s 426.544 (=@)s 439.581 (x)s 6 f 638.847 y 446.44 (j)s 1 f 643.202 y 451.496 (\051.)s 672.002 y 231.877 (7)s end endp %%Page: 10 10 %%BeginPageSetup initializepage (degasperis; page: 10 of 26)setjob %%EndPageSetup gS 0 0 552 730 rC 0 0 :M f0_12 sf ( )S 4 0 :M f1_12 sf ( )S f8_12 sf ( )S 11 0 :M f4_12 sf .136 .014( )J 19 0 :M f5_12 sf .138 .014( )J 28 0 :M f6_12 sf ( )S f9_12 sf 11.923 1.192( )J 53 0 :M f7_12 sf ( )S 0 0 :M initclip TeXdict begin 0 10 1200 1000 div mul /CMBX10 findfont fdef 1 10 1200 1000 div mul /CMR10 findfont fdef 4 7 1200 1000 div mul /CMR7 findfont fdef 5 10 1200 1000 div mul /CMMI10 findfont fdef 6 7 1200 1000 div mul /CMMI7 findfont fdef 7 10 1200 1000 div mul /CMSY10 findfont fdef 8 7 1200 1000 div mul /CMSY7 findfont fdef 9 10 1200 1000 div mul /CMEX10 findfont fdef 42 41 translate 72 72.27 div dup scale 1 f 12 y 0 (W)s 11.333 (e)s 20.297 (ha)s 32.631 (v)s 38.631 (e)s 47.594 (that)s 73.225 (\050)s 5 f 77.892 (\276)s 6 f 6.233 y 85.179 (j)s 15.62 y 84.749 (k)s 7 f 12 y 92.58 (\242)s 5 f 97.841 (@)s 6 f 13.8 y 104.212 (j)s 1 f 12 y 109.269 (\051)s 117.269 (=)s 5 f 129.935 (\275)s 6 f 13.8 y 136.139 (k)s 1 f 12 y 142.044 (\050)s 5 f 146.71 (x)s 1 f 153.569 (\051)s 7 f 158.235 (r)s 6 f 13.8 y 168.235 (x)s 1 f 12 y 176.205 (+)s 5 f 187.466 (\275)s 6 f 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(@)s 6 f 224.906 y 170.124 (t)s 5 f 223.106 y 174.349 (\275)s 6 f 224.906 y 180.553 (k)s 1 f 223.106 y 186.458 (\050)s 5 f 191.124 (y)s 1 f 197.438 (\051)s 9 f 198.986 y 204.377 (\266)s 5 f 215.906 y 219.21 (;)s 1 f 228.544 (\050)s 5 f 233.21 (@)s 6 f 217.706 y 239.581 (t)s 5 f 215.906 y 243.806 (\277)s 1 f 250.41 (\051)s 5 f 255.077 (\276)s 6 f 210.951 y 262.365 (i)s 218.872 y 261.934 (k)s 1 f 215.906 y 271.171 (=)s 9 f 198.986 y 283.838 (\265)s 1 f 208.706 y 294.671 (\050)s 5 f 299.338 (@)s 6 f 210.506 y 305.709 (t)s 5 f 208.706 y 309.934 (\277)s 1 f 316.538 (\051)s 5 f 321.205 (\275)s 6 f 210.506 y 327.409 (k)s 1 f 208.706 y 333.313 (\050)s 5 f 337.98 (x)s 1 f 344.838 (\051)s 223.106 y 294.944 (\050)s 5 f 299.61 (@)s 6 f 224.906 y 305.981 (t)s 5 f 223.106 y 310.206 (\277)s 1 f 316.81 (\051)s 5 f 321.477 (\275)s 6 f 224.906 y 327.681 (k)s 1 f 223.106 y 333.585 (\050)s 5 f 338.252 (y)s 1 f 344.566 (\051)s 9 f 198.986 y 351.505 (\266)s 5 f 215.906 y 366.338 (:)s 1 f 439.087 (\0502)s 5 f 449.754 (:)s 1 f 453.087 (14\051)s 261.237 y 0 (As)s 17.204 (for)s 35.041 (the)s 55.178 (last)s 77.381 (term,)s 108.991 (i.e.)s 5 f 129.481 (\276)s 6 f 256.882 y 136.768 (i)s 264.203 y 136.338 (p)s 5 f 261.237 y 141.885 (B)s 6 f 255.47 y 151.589 (p)s 264.857 y 150.987 (k)s 1 f 261.237 y 157.136 (,)s 164.045 (w)s 172.378 (e)s 181.182 (notice)s 215.986 (that)s 241.456 (in)s 254.926 (view)s 282.063 (of)s 295.2 (the)s 315.337 (presen)s 348.437 (t)s 356.574 (framew)s 394.608 (ork,)s 418.55 (w)s 426.884 (e)s 435.687 (should)s 275.637 y 0 (exp)s 18.667 (ect)s 37.758 (that)s 5 f 63.516 (B)s 1 f 76.978 (do)s 89.978 (es)s 103.802 (not)s 124.894 (mix)s 148.318 (the)s 168.743 (\257rst)s 5 f 193.267 (n)s 1 f 204.228 (comp)s 232.561 (onen)s 256.895 (ts)s 270.053 (of)s 283.477 (the)s 303.902 (\0502)s 5 f 314.569 (n)s 1 f 321.771 (\051-dimensional)s 396.263 (v)s 402.263 (ector)s 432.054 (Wiener)s 290.037 y 0 (pro)s 17.7 (cess)s 0 f 41.345 (v)s 1 f 52.332 (with)s 79.178 (the)s 99.357 (other)s 130.235 (ones,)s 159.912 (and)s 182.757 (the)s 202.936 (transformation)s 284.248 (of)s 297.427 (the)s 317.606 (latter)s 349.818 (ones)s 376.063 (can)s 397.575 (b)s 404.575 (e)s 413.421 (completely)s 304.437 y 0 (disregarded)s 64.133 (as)s 78.867 (they)s 105.867 (should)s 143.934 (pla)s 159.6 (y)s 169.934 (no)s 186.6 (role.)s 214.634 (Th)s 229.634 (us)s 245.034 (w)s 253.367 (e)s 262.7 (should)s 300.767 (assume)s 5 f 336.188 y 191.05 (B)s 1 f 204.087 (=)s 9 f 319.268 y 216.754 (\265)s 5 f 328.988 y 227.587 (M)s 1 f 252.537 (0)s 343.388 y 231.062 (0)s 252.537 (0)s 9 f 319.268 y 260.537 (\266)s 5 f 336.188 y 275.371 (:)s 1 f 439.087 (\0502)s 5 f 449.754 (:)s 1 f 453.087 (15\051)s 379.94 y 0 (Indeed,)s 42.333 (if)s 53.333 (w)s 61.667 (e)s 71 (write)s 101.7 (in)s 115.7 (full)s 136.7 (generalit)s 181.734 (y)s 5 f 411.691 y 183.5 (B)s 1 f 196.537 (=)s 9 f 394.771 y 209.204 (\265)s 5 f 404.491 y 222.075 (M)s 249.062 (M)s 4 f 406.291 y 260.704 (2)s 5 f 418.891 y 220.037 (M)s 4 f 420.691 y 231.679 (3)s 5 f 418.891 y 249.062 (M)s 4 f 420.691 y 260.704 (4)s 9 f 394.771 y 268.087 (\266)s 5 f 411.691 y 282.921 (;)s 1 f 439.087 (\0502)s 5 f 449.754 (:)s 1 f 453.087 (16\051)s 443.442 y 0 (w)s 8.333 (e)s 17.667 (obtain)s 5 f 462.042 y 159.498 (\276)s 166.785 (B)s 1 f 179.823 (=)s 9 f 445.122 y 192.489 (\265)s 5 f 454.842 y 203.323 (\275)s 1 f 209.527 (\050)s 5 f 214.194 (x)s 1 f 221.052 (\051)s 5 f 225.719 (M)s 250.669 (\275)s 1 f 256.873 (\050)s 5 f 261.54 (x)s 1 f 268.398 (\051)s 5 f 273.065 (M)s 4 f 456.642 y 284.706 (2)s 5 f 469.242 y 203.595 (\275)s 1 f 209.799 (\050)s 5 f 214.466 (y)s 1 f 220.78 (\051)s 5 f 225.446 (M)s 250.941 (\275)s 1 f 257.145 (\050)s 5 f 261.812 (y)s 1 f 268.126 (\051)s 5 f 272.792 (M)s 4 f 471.042 y 284.434 (2)s 9 f 445.122 y 292.09 (\266)s 5 f 462.042 y 306.923 (:)s 1 f 439.087 (\0502)s 5 f 449.754 (:)s 1 f 453.087 (17\051)s 489.784 y 0 (Notice)s 37.868 (that)s 64.07 (the)s 84.938 (terms)s 118.573 (in)s 132.774 (the)s 153.643 (last)s 176.577 (blo)s 192.911 (c)s 197.911 (k)s 208.446 (column)s 250.647 (corresp)s 288.447 (ond)s 311.982 (to)s 5 f 326.85 (k)s 337.144 (\076)s 350.147 (n)s 1 f 361.551 (and)s 385.086 (should)s 423.354 (therefore)s 504.184 y 0 (b)s 7 (e)s 15.886 (set)s 34.171 (equal)s 65.39 (to)s 79.609 (zero)s 104.528 (\050as)s 123.48 (discussed)s 175.232 (ab)s 188.232 (o)s 193.899 (v)s 199.899 (e\051)s 213.451 (to)s 227.67 (satisfy)s 264.689 (the)s 284.908 (determining)s 351.16 (equation,)s 403.135 (whic)s 426.802 (h)s 437.021 (means)s 5 f 518.584 y 0 (M)s 4 f 520.384 y 11.642 (2)s 1 f 518.584 y 21.845 (=)s 35.999 (0;)s 50.671 (the)s 72.23 (an)s 84.563 (tisymmetry)s 149.556 (of)s 5 f 164.115 (B)s 1 f 178.711 (implies)s 220.337 (then)s 5 f 248.563 (M)s 4 f 520.384 y 260.204 (3)s 1 f 518.584 y 270.408 (=)s 284.562 (0)s 295.454 (as)s 311.08 (w)s 319.413 (ell.)s 342.757 (The)s 368.316 (\050)s 5 f 372.982 (n)s 7 f 383.447 (\243)s 5 f 396.042 (n)s 1 f 403.244 (\051)s 412.803 (matrix)s 5 f 452.729 (M)s 4 f 520.384 y 464.371 (4)s 1 f 532.984 y 0 (remains)s 45.998 (undetermined,)s 126.904 (but)s 150.136 (it)s 163.367 (corresp)s 201.167 (onds)s 230.465 (to)s 246.363 (an)s 264.261 (in)s 273.928 (ternal)s 309.86 (transformation)s 392.891 (of)s 407.789 (the)s 429.687 (\134ghost\042)s 547.384 y 0 (comp)s 28.333 (onen)s 52.667 (ts)s 5 f 66.185 (v)s 6 f 543.029 y 72.432 (n)s 4 f 78.364 (+1)s 5 f 547.384 y 91.114 (;)s 96.447 (:::;)s 111.781 (v)s 4 f 543.029 y 118.028 (2)s 6 f 122.811 (n)s 1 f 547.384 y 133.461 (of)s 147.246 (the)s 168.03 (v)s 174.03 (ector)s 204.181 (Wiener)s 245.999 (pro)s 263.699 (cess,)s 291.313 (whic)s 314.98 (h)s 325.765 (pla)s 341.431 (y)s 351.882 (no)s 368.667 (role)s 392.152 (in)s 406.269 (the)s 427.054 (problem)s 561.784 y 0 (and)s 22.124 (should)s 58.981 (b)s 65.981 (e)s 74.105 (disregarded)s 137.029 (to)s 150.486 (a)s 156.153 (v)s 162.153 (oid)s 180.944 (ha)s 193.277 (ving)s 218.401 (spurious)s 264.692 (symmetries)s 326.649 (\050whic)s 354.982 (h)s 364.44 (are)s 383.263 (ho)s 395.597 (w)s 403.93 (ev)s 415.264 (er)s 428.087 (mapp)s 457.754 (ed)s 576.184 y 0 (to)s 14.667 (the)s 35.333 (iden)s 57 (tit)s 69.333 (y)s 79.667 (transformation)s 161.467 (in)s 175.467 (terms)s 208.9 (of)s 222.567 (the)s 243.234 (original)s 286.601 (t)s 290.934 (w)s 299.267 (o-particle)s 352.634 (problem\051.)s 602.584 y 0 (Th)s 15 (us,)s 33.733 (the)s 54.4 (term)s 5 f 83.1 (\276)s 6 f 598.229 y 90.388 (i)s 605.551 y 89.957 (p)s 5 f 602.584 y 95.504 (B)s 6 f 596.817 y 105.208 (p)s 606.204 y 104.606 (k)s 1 f 602.584 y 114.755 (reads,)s 149.522 (for)s 5 f 167.888 (k)s 7 f 177.847 (\267)s 5 f 190.513 (n)s 1 f 201.716 (and)s 225.049 (in)s 239.049 (the)s 259.716 (ab)s 272.716 (o)s 278.383 (v)s 284.383 (e)s 293.716 (notation,)s 5 f 636.602 y 174.821 (\276)s 6 f 631.647 y 182.108 (i)s 639.569 y 181.678 (p)s 5 f 636.602 y 187.224 (B)s 6 f 630.835 y 196.929 (p)s 640.222 y 196.326 (k)s 1 f 636.602 y 205.809 (=)s 9 f 619.682 y 218.475 (\265)s 5 f 628.529 y 229.309 (\275)s 6 f 630.329 y 235.513 (j)s 1 f 628.529 y 240.569 (\050)s 5 f 245.236 (x)s 1 f 252.094 (\051)s 5 f 256.761 (M)s 6 f 622.763 y 269.711 (j)s 632.149 y 268.402 (k)s 5 f 644.674 y 229.581 (\275)s 6 f 646.474 y 235.785 (j)s 1 f 644.674 y 240.841 (\050)s 5 f 245.508 (y)s 1 f 251.822 (\051)s 5 f 256.488 (M)s 6 f 638.908 y 269.438 (j)s 648.294 y 268.13 (k)s 9 f 619.682 y 276.767 (\266)s 5 f 636.602 y 291.6 (:)s 1 f 439.087 (\0502)s 5 f 449.754 (:)s 1 f 453.087 (18\051)s 672.002 y 231.877 (8)s end endp %%Page: 11 11 %%BeginPageSetup initializepage (degasperis; page: 11 of 26)setjob %%EndPageSetup gS 0 0 552 730 rC 0 0 :M f0_12 sf ( )S 4 0 :M f1_12 sf ( )S f2_12 sf ( )S f8_12 sf ( )S 14 0 :M f4_12 sf .136 .014( )J 22 0 :M f5_12 sf .138 .014( )J 31 0 :M f6_12 sf ( )S f7_12 sf ( )S 0 0 :M initclip TeXdict begin 0 10 1200 1000 div mul /CMBX10 findfont fdef 1 10 1200 1000 div mul /CMR10 findfont fdef 2 10 1200 1000 div mul /CMTI10 findfont fdef 4 7 1200 1000 div mul /CMR7 findfont fdef 5 10 1200 1000 div mul /CMMI10 findfont fdef 6 7 1200 1000 div mul /CMMI7 findfont fdef 7 10 1200 1000 div mul /CMSY10 findfont fdef 9 10 1200 1000 div mul /CMEX10 findfont fdef 42 41 translate 72 72.27 div dup scale 1 f 12 y 0 (Putting)s 44.178 (together)s 91.557 (\0502.13\051,)s 129.571 (\0502.14\051)s 164.249 (and)s 187.594 (\0502.18\051)s 222.273 (w)s 230.606 (e)s 239.951 (\257nally)s 276.296 (obtain)s 313.641 (the)s 334.319 (second)s 373.064 (equation)s 422.076 (of)s 435.754 (\0501.10\051.)s 26.4 y 0 (Notice)s 37.69 (that)s 63.714 (with)s 91.071 (\0502.15\051)s 125.762 (for)s 5 f 144.152 (B)s 1 f 153.856 (,)s 161.219 (the)s 181.91 (equations)s 235.667 (for)s 5 f 254.057 (k)s 264.055 (\076)s 276.761 (n)s 1 f 287.988 (are)s 308.045 (iden)s 329.712 (tically)s 366.069 (satis\257ed.)s 416.94 (As)s 434.697 (for)s 453.087 (the)s 40.8 y 0 (other)s 30.421 (ones,)s 59.731 (notice)s 94.119 (that)s 119.173 (\050for)s 5 f 141.261 (k)s 1 f 150.941 (\257xed\051)s 183.662 (those)s 214.116 (for)s 5 f 231.537 (i)s 1 f 239.005 (=)s 5 f 251.671 (n)s 1 f 259.649 (+)s 269.758 (1)s 5 f 275.758 (;)s 281.091 (:::;)s 1 f 296.425 (2)s 5 f 302.425 (n)s 1 f 312.682 (can)s 333.736 (b)s 340.736 (e)s 349.124 (obtained)s 397.512 (from)s 424.933 (those)s 455.387 (for)s 5 f 55.2 y 0 (i)s 1 f 7.467 (=)s 20.134 (1)s 5 f 26.134 (;)s 31.467 (:::;)s 46.801 (n)s 1 f 57.415 (simply)s 95.227 (b)s 101.56 (y)s 111.305 (exc)s 127.972 (hanging)s 5 f 172.717 (x)s 1 f 182.987 (and)s 5 f 205.732 (y)s 1 f 212.046 (,)s 218.909 (as)s 233.054 (can)s 254.466 (b)s 261.466 (e)s 270.211 (c)s 275.211 (hec)s 292.211 (k)s 298.211 (ed)s 313.622 (from)s 341.401 (\0502.13\051,)s 378.93 (\0502.14\051)s 413.009 (and)s 435.754 (\0502.18\051.)s 69.6 y 0 (Th)s 15 (us)s 30.357 (w)s 38.69 (e)s 47.98 (can)s 69.937 (write)s 100.594 (do)s 112.927 (wn)s 132.217 (only)s 158.507 (the)s 179.131 (\257rst)s 5 f 203.854 (n)s 1 f 211.057 (,)s 218.356 (whic)s 242.022 (h)s 252.646 (w)s 260.979 (e)s 270.269 (do)s 286.893 (\050recalling)s 339.549 (that)s 5 f 365.506 (\275)s 6 f 71.4 y 371.71 (k)s 7 f 69.6 y 377.615 (r)s 6 f 71.4 y 387.615 (x)s 1 f 69.6 y 396.99 (:=)s 5 f 412.99 (\275)s 6 f 63.833 y 419.194 (j)s 73.22 y 419.194 (k)s 1 f 69.6 y 425.098 (\050)s 5 f 429.765 (@)s 436.802 (=@)s 449.84 (x)s 6 f 65.245 y 456.698 (j)s 1 f 69.6 y 461.754 (\051,)s 84 y 0 (to)s 14.667 (a)s 20.333 (v)s 26.333 (oid)s 46.333 (an)s 58.667 (y)s 69 (confusion\051)s 126.734 (in)s 140.734 (the)s 161.4 (notation)s 209.4 (emplo)s 240.401 (y)s 246.401 (ed)s 262.401 (ab)s 275.401 (o)s 281.067 (v)s 287.067 (e:)s 9 f 103.08 y 6 (h)s 7 f 116.4 y 11.667 (f)s 5 f 17.667 (\275)s 6 f 118.2 y 23.871 (k)s 1 f 116.4 y 29.775 (\050)s 5 f 34.442 (x)s 1 f 41.3 (\051)s 5 f 45.967 (;)s 51.3 (\264)s 1 f 57.689 (\050)s 5 f 62.356 (x;)s 74.547 (y)s 1 f 80.861 (\051)s 7 f 85.528 (g)s 6 f 118.2 y 91.528 (x)s 7 f 116.4 y 100.236 (\241)s 5 f 112.236 (\277)s 118.84 (@)s 6 f 118.2 y 125.211 (t)s 5 f 116.4 y 129.436 (\275)s 6 f 118.2 y 135.64 (k)s 1 f 116.4 y 141.545 (\050)s 5 f 146.211 (x)s 1 f 153.07 (\051)s 7 f 160.403 (\241)s 1 f 172.403 (\0501)s 5 f 183.069 (=)s 1 f 189.07 (2\051)s 5 f 199.736 (\275)s 6 f 118.2 y 205.94 (k)s 1 f 116.4 y 211.845 (\050)s 5 f 216.511 (x)s 1 f 223.37 (\051)s 5 f 228.036 (@)s 6 f 118.2 y 234.407 (t)s 5 f 116.4 y 238.632 (\277)s 7 f 247.903 (\241)s 5 f 259.903 (\275)s 6 f 118.2 y 266.107 (j)s 1 f 116.4 y 271.163 (\050)s 5 f 275.83 (x)s 1 f 282.688 (\051)s 5 f 287.355 (M)s 6 f 110.633 y 300.305 (j)s 120.02 y 298.997 (k)s 9 f 103.08 y 305.361 (i)s 1 f 116.4 y 311.497 (+)s 5 f 321.299 (\275)s 6 f 118.2 y 327.503 (k)s 1 f 116.4 y 333.408 (\050)s 5 f 338.074 (y)s 1 f 344.388 (\051)s 7 f 349.055 (r)s 6 f 118.2 y 359.055 (y)s 5 f 116.4 y 364.823 (\264)s 1 f 371.212 (\050)s 5 f 375.879 (x;)s 388.07 (y)s 1 f 394.384 (\051)s 405.285 (=)s 420.853 (0)s 5 f 429.754 (:)s 1 f 439.087 (\0502)s 5 f 449.754 (:)s 1 f 453.087 (19\051)s 160.134 y 0 (Notice)s 39.363 (that)s 67.059 (the)s 89.422 (square)s 128.885 (brac)s 151.251 (k)s 157.251 (et)s 172.947 (corresp)s 210.748 (onds)s 240.51 (to)s 256.873 (the)s 279.236 (determining)s 347.632 (equation)s 398.328 (for)s 418.391 (the)s 440.754 (OPP;)s 174.534 y 0 (indeed,)s 41.287 (if)s 52.228 (w)s 60.561 (e)s 69.836 (assume)s 5 f 111.244 (\264)s 1 f 117.633 (\050)s 5 f 122.3 (x;)s 134.492 (y)s 140.805 (;)s 146.139 (t)s 1 f 150.472 (\051)s 159.08 (do)s 172.08 (es)s 186.088 (actually)s 231.697 (not)s 252.971 (dep)s 271.971 (end)s 294.58 (on)s 5 f 311.188 (y)s 1 f 317.502 (,)s 324.788 (the)s 345.396 (term)s 374.038 (outside)s 415.379 (the)s 435.987 (square)s 188.934 y 0 (brac)s 22.367 (k)s 28.367 (et)s 42.367 (cancels)s 83.1 (out)s 104.433 (\050as)s 123.833 (it)s 135.834 (should)s 173.9 (b)s 180.9 (e\051)s 194.9 (and)s 218.234 (w)s 226.567 (e)s 235.9 (are)s 255.934 (reduced)s 300.634 (to)s 315.3 (the)s 335.967 (one-particle)s 401.334 (situation.)s 215.334 y 0 (W)s 11.333 (e)s 21.292 (summarize)s 82.018 (the)s 103.31 (result)s 137.369 (of)s 151.661 (our)s 173.653 (computations)s 249.012 (in)s 263.637 (the)s 284.929 (follo)s 306.929 (wing)s 336.221 (statemen)s 383.288 (t,)s 396.07 (where)s 431.395 (w)s 439.729 (e)s 449.687 (also)s 229.734 y 0 (rein)s 19.7 (tro)s 35.4 (duce)s 63.4 (the)s 5 f 84.067 (t)s 1 f 92.4 (dep)s 111.4 (endence)s 156.734 (for)s 175.1 (completeness.)s 0 f 256.134 y 0 (Prop)s 30.067 (osition)s 74.937 (2.)s 2 f 91.858 (The)s 116.654 (determining)s 183.615 (e)s 188.522 (quations)s 236.624 (for)s 256.052 (symmetry)s 311.361 (gener)s 339.114 (ators)s 369.889 (of)s 384.258 (the)s 404.453 (RDS)s 433.568 (de\257ne)s 463.621 (d)s 270.534 y 0 (by)s 15.64 (the)s 35.573 (Ito)s 54.613 (e)s 59.52 (quation)s 1 f 102.453 (d)s 5 f 109.119 (x)s 6 f 266.179 y 115.978 (i)s 1 f 270.534 y 123.306 (=)s 5 f 135.973 (\301)s 6 f 266.179 y 143.123 (i)s 1 f 270.534 y 147.118 (\050)s 5 f 151.784 (x;)s 163.976 (t)s 1 f 168.309 (\051d)s 5 f 179.643 (t)s 1 f 186.643 (+)s 5 f 198.643 (\275)s 6 f 266.179 y 204.847 (i)s 273.932 y 204.847 (k)s 1 f 270.534 y 210.751 (\050)s 5 f 215.418 (x;)s 227.609 (t)s 1 f 231.943 (\051d)s 5 f 243.276 (w)s 6 f 266.179 y 252.19 (k)s 2 f 270.534 y 262.387 (ar)s 272.967 (e)s 9 f 290.18 y 21.508 (\243)s 7 f 299.9 y 26.508 (f)s 5 f 32.508 (\301)s 1 f 39.658 (\050)s 5 f 44.325 (x;)s 56.517 (t)s 1 f 60.85 (\051)s 5 f 65.517 (;)s 70.85 (\264)s 1 f 77.239 (\050)s 5 f 81.906 (x;)s 94.097 (y)s 100.411 (;)s 105.744 (t)s 1 f 110.078 (\051)s 7 f 114.744 (g)s 6 f 301.7 y 120.744 (x)s 1 f 299.9 y 129.453 (+)s 5 f 141.453 (@)s 6 f 301.7 y 147.824 (t)s 5 f 299.9 y 152.049 (\264)s 1 f 158.437 (\050)s 5 f 163.104 (x;)s 175.296 (y)s 181.61 (;)s 186.943 (t)s 1 f 191.276 (\051)s 7 f 198.61 (\241)s 5 f 210.61 (@)s 6 f 301.7 y 216.98 (t)s 1 f 299.9 y 223.205 (\050)s 5 f 227.872 (\277)s 1 f 234.476 (\050)s 5 f 239.143 (t)s 1 f 243.476 (\051)s 5 f 248.143 (\301)s 1 f 255.293 (\050)s 5 f 259.96 (x;)s 272.151 (t)s 1 f 276.485 (\051\051)s 288.485 (+)s 5 f 300.485 (A)s 1 f 309.485 (\050)s 5 f 314.151 (x;)s 326.343 (x)s 1 f 333.201 (\051)s 7 f 337.868 (r)s 4 f 294.945 y 347.868 (2)s 6 f 302.867 y 347.868 (xx)s 5 f 299.9 y 359.351 (\264)s 1 f 365.74 (\050)s 5 f 370.407 (x;)s 382.598 (y)s 388.912 (;)s 394.246 (t)s 1 f 398.579 (\051)s 9 f 290.18 y 403.246 (\244)s 1 f 299.9 y 410.246 (+)s 319.268 y 30.762 (+)s 42.762 ([)s 5 f 46.095 (\301)s 1 f 53.245 (\050)s 5 f 57.912 (y)s 64.226 (;)s 69.559 (t)s 1 f 73.892 (\051)s 7 f 78.559 (r)s 6 f 321.068 y 88.559 (y)s 5 f 319.268 y 94.327 (\264)s 1 f 100.716 (\050)s 5 f 105.383 (x;)s 117.574 (y)s 123.888 (;)s 129.221 (t)s 1 f 133.555 (\051)s 140.888 (+)s 5 f 152.888 (A)s 1 f 161.888 (\050)s 5 f 166.555 (y)s 172.869 (;)s 178.202 (y)s 1 f 184.516 (\051)s 7 f 189.182 (r)s 4 f 314.313 y 199.182 (2)s 6 f 322.235 y 199.182 (y)s 204.351 (y)s 5 f 319.268 y 210.119 (\264)s 1 f 216.508 (\050)s 5 f 221.174 (x;)s 233.366 (y)s 239.68 (;)s 245.013 (t)s 1 f 249.346 (\051+)s 339.036 y 35.055 (+)s 5 f 47.055 (A)s 1 f 56.055 (\050)s 5 f 60.722 (x;)s 72.913 (y)s 1 f 79.227 (\051)s 7 f 83.894 (r)s 4 f 334.081 y 93.894 (2)s 6 f 342.003 y 93.894 (xy)s 5 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107.008 (x;)s 119.199 (y)s 125.513 (;)s 130.846 (t)s 1 f 135.18 (\051)s 147.473 (=)s 164.433 (0)s 5 f 174.726 (:)s 1 f 342.384 y 439.087 (\0502)s 5 f 449.754 (:)s 1 f 453.087 (20\051)s 0 f 431.236 y 0 (Remark)s 52.953 (6.)s 1 f 70.333 (The)s 95.438 (terms)s 129.309 (in)s 143.747 (square)s 181.951 (brac)s 204.318 (k)s 210.318 (ets)s 229.489 (coincide)s 275.927 (with)s 303.698 (the)s 324.803 (left)s 346.24 (hand)s 376.678 (sides)s 405.916 (of)s 420.021 (equations)s 445.636 y 0 (\0501.10\051.)s 7 f 39.333 (\246)s 1 f 472.036 y 0 (It)s 12.146 (is)s 23.358 (ob)s 35.692 (vious)s 65.904 (that,)s 94.554 (as)s 108.433 (can)s 129.579 (also)s 152.792 (b)s 159.792 (e)s 168.271 (c)s 173.271 (hec)s 190.271 (k)s 196.271 (ed)s 211.417 (b)s 217.75 (y)s 227.229 (setting)s 7 f 265.775 (r)s 6 f 473.836 y 275.775 (y)s 5 f 472.036 y 281.543 (\264)s 1 f 287.932 (\050)s 5 f 292.599 (x;)s 304.79 (y)s 311.104 (;)s 316.438 (t)s 1 f 320.771 (\051)s 328.771 (=)s 341.437 (0)s 350.583 (in)s 363.729 (the)s 383.542 (ab)s 396.542 (o)s 402.208 (v)s 408.208 (e)s 416.687 (equations,)s 486.436 y 0 (w)s 8.333 (e)s 17.667 (ha)s 30 (v)s 36 (e)s 45.333 (the)s 66 (follo)s 88 (wing:)s 0 f 512.836 y 0 (Corollary)s 55.641 (.)s 2 f 66.27 (A)s 74.883 (ny)s 92.197 (symmetry)s 147.692 (of)s 162.247 (the)s 182.628 (OPP)s 212.849 (de\257ne)s 242.902 (d)s 253.777 (by)s 269.865 (an)s 287.486 (Ito)s 306.974 (e)s 311.881 (quation)s 355.262 (is)s 368.59 (also)s 393.572 (a)s 404.446 (symmetry)s 459.941 (of)s 527.236 y 0 (the)s 19.933 (RDS)s 48.786 (de\257ne)s 78.839 (d)s 89.266 (by)s 104.906 (the)s 124.839 (same)s 155.506 (Ito)s 174.545 (e)s 179.452 (quation.)s 1 f 553.636 y 0 (Ho)s 14.667 (w)s 23 (ev)s 34.333 (er,)s 53.797 (it)s 67.475 (app)s 87.142 (ears)s 113.587 (from)s 143.631 (\0502.20\051)s 179.976 (that)s 207.654 (w)s 215.987 (e)s 226.999 (can)s 250.677 (ha)s 263.01 (v)s 269.01 (e)s 280.021 (symmetry)s 337.799 (of)s 353.144 (a)s 364.822 (RDS)s 395.165 (whic)s 418.832 (h)s 431.176 (are)s 2 f 452.888 (not)s 1 f 568.036 y 0 (symmetries)s 63.221 (of)s 76.943 (the)s 97.664 (OPP)s 127.386 (de\257ned)s 168.774 (b)s 175.107 (y)s 185.495 (the)s 206.216 (same)s 236.338 (Ito)s 255.392 (equation.)s 309.223 (In)s 324.278 (particular,)s 383.08 (if)s 394.134 (a)s 404.189 (solution)s 5 f 449.644 (\264)s 1 f 460.087 (of)s 582.436 y 0 (\0502.20\051)s 34.667 (dep)s 53.667 (ends)s 81.067 (on)s 97.734 (the)s 5 f 118.4 (y)s 1 f 128.714 (v)s 134.381 (ariables,)s 181.814 (this)s 205.214 (cannot)s 244.548 (b)s 251.548 (e)s 260.881 (in)s 270.548 (terpreted)s 322.614 (in)s 336.615 (terms)s 370.048 (of)s 383.715 (the)s 404.381 (OPP)s 429.048 (.)s 608.836 y 0 (The)s 24.27 (simple)s 61.274 (examples)s 112.611 (giv)s 127.944 (en)s 143.548 (b)s 150.548 (elo)s 164.881 (w)s 177.152 (\050in)s 195.422 (section)s 235.092 (4\051)s 249.362 (sho)s 266.429 (w)s 278.699 (that)s 304.303 (indeed)s 341.907 (a)s 351.51 (RDS)s 379.78 (admits)s 418.784 (in)s 432.387 (general)s 623.236 y 0 (more)s 30.033 (symmetries)s 93.2 (than)s 121.2 (the)s 141.867 (corresp)s 179.667 (onding)s 219 (OPP)s 243.667 (.)s 672.002 y 231.877 (9)s end endp %%Page: 12 12 %%BeginPageSetup initializepage (degasperis; page: 12 of 26)setjob %%EndPageSetup gS 0 0 552 730 rC 0 0 :M f0_12 sf ( )S 4 0 :M f1_12 sf ( )S f2_12 sf ( )S f8_12 sf ( )S 14 0 :M f4_12 sf .136 .014( )J 22 0 :M f5_12 sf .138 .014( )J 31 0 :M f6_12 sf ( )S f7_12 sf ( )S 0 0 :M initclip TeXdict begin 0 10 1200 1000 div mul /CMBX10 findfont fdef 1 10 1200 1000 div mul /CMR10 findfont fdef 2 10 1200 1000 div mul /CMTI10 findfont fdef 4 7 1200 1000 div mul /CMR7 findfont fdef 5 10 1200 1000 div mul /CMMI10 findfont fdef 6 7 1200 1000 div mul /CMMI7 findfont fdef 7 10 1200 1000 div mul /CMSY10 findfont fdef 9 10 1200 1000 div mul /CMEX10 findfont fdef 42 41 translate 72 72.27 div dup scale 2 f 12 y 20 (2.2.)s 45.146 (Symmetry)s 102.033 (of)s 5 f 116.139 (N)s 2 f 127.089 (-p)s 136.902 (article)s 174.162 (pr)s 184.742 (o)s 190.262 (c)s 195.169 (esses.)s 1 f 46.116 y 0 (The)s 25.243 (whole)s 59.82 (previous)s 108.163 (discussion)s 164.94 (concerned)s 221.55 (t)s 225.883 (w)s 234.216 (o-particle)s 288.16 (pro)s 305.86 (cesses;)s 344.258 (w)s 352.591 (e)s 362.501 (should)s 401.144 (ho)s 413.478 (w)s 421.811 (ev)s 433.144 (er)s 447.754 (con-)s 60.516 y 0 (sider)s 5 f 28.122 (N)s 1 f 39.072 (-particle)s 85.794 (pro)s 103.494 (cesses,)s 140.511 (with)s 5 f 167.2 (N)s 181.483 (\076)s 1 f 194.149 (1)s 203.505 (an)s 219.526 (arbitrary)s 269.982 (in)s 279.648 (teger.)s 314.133 (The)s 338.155 (computations)s 412.244 (in)s 425.599 (this)s 448.354 (case)s 74.916 y 0 (are)s 20.32 (actually)s 66.274 (completely)s 126.894 (analogous)s 182.581 (to)s 197.535 (those)s 229.222 (for)s 247.876 (the)s 268.829 (t)s 273.163 (w)s 281.496 (o-particle)s 335.15 (pro)s 352.85 (cesses;)s 390.813 (the)s 411.767 (main)s 442.054 (e\256ort)s 89.316 y 0 (will)s 22.434 (b)s 29.434 (e)s 38.534 (dev)s 56.534 (oted)s 82.968 (to)s 97.402 (setting)s 136.569 (a)s 146.336 (suitable)s 190.836 (notation,)s 241.984 (and)s 265.084 (once)s 292.185 (this)s 315.352 (is)s 327.185 (done)s 355.619 (w)s 363.952 (e)s 373.053 (will)s 395.487 (brie\260)s 422.187 (y)s 432.287 (go)s 448.054 (o)s 453.721 (v)s 459.721 (er)s 103.716 y 0 (the)s 20.667 (computations)s 95.4 (without)s 140.067 (repro)s 167.8 (ducing)s 206.467 (all)s 223.134 (the)s 243.801 (details.)s 130.116 y 0 (W)s 11.333 (e)s 21.508 (start)s 51.116 (again)s 83.958 (b)s 90.291 (y)s 101.466 (\0502.1\051;)s 134.728 (w)s 143.061 (e)s 153.236 (will)s 176.744 (consider)s 5 f 224.352 (N)s 1 f 240.143 (copies)s 276.385 (of)s 290.893 (it,)s 307.278 (and)s 331.453 (v)s 337.119 (ariables)s 0 f 382.061 (x)s 4 f 132.283 y 389.344 (\050)s 6 f 393.094 (\256)s 4 f 399.333 (\051)s 7 f 130.116 y 408.418 (2)s 0 f 421.154 (R)s 6 f 125.761 y 431.504 (n)s 1 f 130.116 y 438.036 (,)s 446.421 (with)s 5 f 144.516 y 0 (\256)s 1 f 11.054 (=)s 23.721 (1)s 5 f 29.721 (;)s 35.054 (:::;)s 50.387 (N)s 1 f 65.192 (and)s 0 f 88.381 (x)s 4 f 146.683 y 95.664 (\050)s 6 f 99.414 (\256)s 4 f 105.652 (\051)s 1 f 144.516 y 113.336 (=)s 126.002 (\050)s 5 f 130.669 (x)s 4 f 140.161 y 137.527 (1)s 148.381 y 137.527 (\050)s 6 f 141.277 (\256)s 4 f 147.516 (\051)s 5 f 144.516 y 151.866 (;)s 157.199 (:::;)s 172.532 (x)s 6 f 140.161 y 179.391 (n)s 4 f 148.381 y 179.391 (\050)s 6 f 183.141 (\256)s 4 f 189.379 (\051)s 1 f 144.516 y 193.729 (\051.)s 207.014 (All)s 226.536 (the)s 247.058 (equations)s 300.646 (\050and)s 328.501 (results\051)s 371.189 (to)s 385.711 (b)s 392.711 (e)s 401.899 (obtained)s 451.087 (will)s 160.681 y 0 (b)s 7 (e)s 16.724 (co)s 27.724 (v)s 33.391 (arian)s 59.758 (t)s 68.815 (under)s 103.24 (an)s 120.297 (arbitrary)s 171.788 (p)s 178.788 (erm)s 198.488 (utation)s 240.879 (of)s 254.937 (the)s 0 f 275.994 (x)s 4 f 162.848 y 283.278 (\050)s 6 f 287.028 (\256)s 4 f 293.266 (\051)s 1 f 160.681 y 302.007 (v)s 307.674 (ariables)s 352.165 (\050recall)s 389.256 (the)s 5 f 410.313 (N)s 1 f 425.654 (particles)s 175.081 y 0 (do)s 16.109 (not)s 36.884 (in)s 46.551 (teract)s 80.693 (in)s 94.135 (an)s 106.468 (y)s 116.244 (w)s 124.577 (a)s 130.244 (y\051,)s 148.131 (and)s 170.906 (w)s 179.239 (e)s 188.015 (could)s 219.457 (therefore)s 269.299 (just)s 292.474 (fo)s 302.474 (cus)s 322.65 (on)s 338.758 (one)s 360.2 (blo)s 376.534 (c)s 381.534 (k,)s 394.754 (con)s 412.421 (v)s 418.421 (en)s 430.087 (tionally)s 189.481 y 0 (called)s 34 (the)s 54.667 (\257rst,)s 82.767 (of)s 96.433 (these.)s 215.881 y 0 (W)s 11.333 (e)s 23.17 (will)s 48.34 (consider)s 97.61 (a)s 110.113 (\050)s 5 f 114.78 (N)s 125.73 (n)s 1 f 132.933 (\051-dimensional)s 210.17 (v)s 216.17 (ector)s 0 f 248.706 (z)s 1 f 262.345 (=)s 279.184 (\050)s 0 f 283.85 (x)s 4 f 218.048 y 291.134 (\0501\051)s 5 f 215.881 y 304.017 (;)s 309.35 (:::;)s 0 f 324.684 (x)s 4 f 218.048 y 331.967 (\050)s 6 f 335.717 (N)s 4 f 344.219 (\051)s 1 f 215.881 y 348.569 (\051,)s 363.698 (a)s 376.201 (\050)s 5 f 380.868 (N)s 391.818 (n)s 1 f 399.021 (\051-dimensional)s 230.281 y 0 (Wiener)s 41.881 (pro)s 59.581 (cess)s 0 f 83.895 (v)s 1 f 95.55 (ha)s 107.884 (ving)s 134.398 (as)s 149.312 (\257rst)s 5 f 174.259 (n)s 1 f 185.642 (comp)s 213.976 (onen)s 238.309 (ts)s 251.89 (the)s 5 f 272.737 (n)s 1 f 279.94 (-dimensional)s 350.187 (Wiener)s 392.068 (pro)s 409.768 (cess)s 0 f 434.082 (w)s 1 f 448.421 (\050the)s 244.681 y 0 (other)s 32.951 (comp)s 61.284 (onen)s 85.618 (ts)s 100.602 (will)s 124.853 (b)s 131.853 (e)s 142.77 (unessen)s 182.57 (tial\051,)s 213.884 (and,)s 242.531 (similarly)s 293.215 (to)s 309.466 (what)s 341.05 (w)s 349.384 (e)s 360.301 (ha)s 372.635 (v)s 378.635 (e)s 389.552 (done)s 419.803 (ab)s 432.803 (o)s 438.47 (v)s 444.47 (e)s 455.387 (for)s 5 f 259.081 y 0 (N)s 1 f 14.283 (=)s 26.95 (2,)s 40.283 (rewrite)s 81.017 (the)s 101.683 (system)s 288.655 y 82.786 (d)s 5 f 89.453 (x)s 6 f 283.701 y 96.311 (i)s 4 f 291.92 y 96.311 (\050)s 6 f 100.061 (\256)s 4 f 106.299 (\051)s 1 f 288.655 y 117.983 (=)s 5 f 134.649 (\301)s 6 f 283.701 y 141.799 (i)s 9 f 278.935 y 147.794 (\241)s 0 f 288.655 y 153.294 (x)s 4 f 290.822 y 160.578 (\050)s 6 f 164.328 (\256)s 4 f 170.566 (\051)s 5 f 288.655 y 174.916 (;)s 180.25 (t)s 9 f 278.935 y 184.583 (\242)s 1 f 288.655 y 194.083 (d)s 5 f 200.75 (t)s 1 f 211.749 (+)s 5 f 227.749 (\275)s 6 f 283.701 y 233.954 (i)s 291.622 y 233.954 (k)s 9 f 278.935 y 241.858 (\241)s 0 f 288.655 y 247.358 (x)s 4 f 290.822 y 254.641 (\050)s 6 f 258.391 (\256)s 4 f 264.63 (\051)s 5 f 288.655 y 268.98 (;)s 274.313 (t)s 9 f 278.935 y 278.646 (\242)s 1 f 288.655 y 288.146 (d)s 5 f 294.813 (w)s 6 f 283.701 y 303.727 (k)s 5 f 288.655 y 325.631 (\256)s 1 f 336.685 (=)s 349.352 (1)s 5 f 355.352 (;)s 360.685 (:::;)s 376.018 (N)s 1 f 439.087 (\0502)s 5 f 449.754 (:)s 1 f 453.087 (21\051)s 320.582 y 0 (\050notice)s 38.939 (the)s 5 f 58.544 (w)s 6 f 316.228 y 67.458 (k)s 1 f 320.582 y 76.301 (apparing)s 125.273 (in)s 138.212 (the)s 157.817 (di\256eren)s 196.517 (t)s 204.122 (blo)s 220.456 (c)s 225.456 (k)s 234.728 (of)s 247.333 (equations)s 300.005 (are)s 318.977 (the)s 338.583 (same\051)s 372.255 (of)s 5 f 384.86 (N)s 1 f 398.749 (Ito)s 416.687 (equations,)s 334.982 y 0 (eac)s 16.333 (h)s 27 (of)s 40.667 (them)s 5 f 71.333 (n)s 1 f 78.536 (-dimensional,)s 151.936 (as)s 166.67 (a)s 176.67 (single)s 210.07 (\050)s 5 f 214.736 (N)s 225.686 (n)s 1 f 232.889 (\051-dimensional)s 307.623 (Ito)s 326.623 (equation,)s 364.556 y 145.488 (d)s 5 f 152.155 (z)s 6 f 359.602 y 158.263 (i)s 1 f 364.556 y 169.592 (=)s 5 f 186.258 (f)s 6 f 359.602 y 193.425 (i)s 1 f 364.556 y 197.42 (\050)s 0 f 202.087 (z)s 5 f 208.22 (;)s 213.553 (t)s 1 f 217.887 (\051)s 224.553 (d)s 5 f 231.22 (t)s 1 f 242.22 (+)s 5 f 258.22 (\276)s 6 f 359.602 y 265.507 (i)s 367.523 y 265.077 (k)s 1 f 364.556 y 270.981 (\050)s 0 f 275.648 (z)s 5 f 281.781 (;)s 287.114 (t)s 1 f 291.448 (\051)s 298.114 (d)s 5 f 304.781 (v)s 6 f 359.602 y 311.028 (k)s 5 f 364.556 y 320.932 (:)s 1 f 439.087 (\0502)s 5 f 449.754 (:)s 1 f 453.087 (22\051)s 406.131 y 0 (The)s 24.667 (\050)s 5 f 29.333 (N)s 40.283 (n)s 1 f 47.486 (\051-dimensional)s 122.22 (v)s 128.22 (ector)s 5 f 158.253 (f)s 1 f 169.42 (and)s 192.753 (matrix)s 5 f 231.787 (\276)s 1 f 243.074 (app)s 262.741 (earing)s 298.774 (here)s 324.808 (are,)s 348.174 (in)s 5 f 362.174 (n)s 1 f 369.377 (-blo)s 389.71 (c)s 394.71 (k)s 405.044 (notation,)s 5 f 453.705 y 79.755 (f)s 1 f 94.255 (=)s 9 f 425.985 y 110.921 (0)s 447.105 y 110.921 (B)s 454.785 y 110.921 (@)s 5 f 434.738 y 125.281 (\301)s 1 f 132.431 (\050)s 0 f 137.097 (x)s 4 f 436.905 y 144.38 (\0501\051)s 5 f 434.738 y 157.264 (;)s 162.597 (t)s 1 f 166.931 (\051)s 448.671 y 146.772 (.)s 453.471 y 146.772 (.)s 458.271 y 146.772 (.)s 5 f 472.671 y 123.421 (\301)s 1 f 130.571 (\050)s 0 f 135.238 (x)s 4 f 474.838 y 142.521 (\050)s 6 f 146.271 (N)s 4 f 154.773 (\051)s 5 f 472.671 y 159.123 (;)s 164.456 (t)s 1 f 168.79 (\051)s 9 f 425.985 y 175.456 (1)s 447.105 y 175.456 (C)s 454.785 y 175.456 (A)s 5 f 453.705 y 195.956 (;)s 209.29 (\276)s 1 f 223.91 (=)s 9 f 425.985 y 240.577 (0)s 447.105 y 240.577 (B)s 454.785 y 240.577 (@)s 5 f 434.738 y 254.936 (\275)s 1 f 261.14 (\050)s 0 f 265.807 (x)s 4 f 436.905 y 273.09 (\0501\051)s 5 f 434.738 y 285.974 (;)s 291.307 (t)s 1 f 295.64 (\051)s 314.166 (0)s 5 f 332.166 (:)s 337.499 (:)s 342.833 (:)s 1 f 358.166 (0)s 448.671 y 275.955 (.)s 453.471 y 275.955 (.)s 458.271 y 275.955 (.)s 448.671 y 315.499 (.)s 453.471 y 315.499 (.)s 458.271 y 315.499 (.)s 449.871 y 332.833 (.)s 453.471 y 337.499 (.)s 457.071 y 342.166 (.)s 448.671 y 359.499 (.)s 453.471 y 359.499 (.)s 458.271 y 359.499 (.)s 5 f 472.671 y 253.077 (\275)s 1 f 259.281 (\050)s 0 f 263.948 (x)s 4 f 474.838 y 271.231 (\050)s 6 f 274.981 (N)s 4 f 283.483 (\051)s 5 f 472.671 y 287.833 (;)s 293.166 (t)s 1 f 297.499 (\051)s 314.166 (0)s 5 f 332.166 (:)s 337.499 (:)s 342.833 (:)s 1 f 358.166 (0)s 9 f 425.985 y 366.166 (1)s 447.105 y 366.166 (C)s 454.785 y 366.166 (A)s 5 f 453.705 y 386.666 (:)s 1 f 439.087 (\0502)s 5 f 449.754 (:)s 1 f 453.087 (23\051)s 513.279 y 0 (In)s 17.105 (order)s 50.61 (to)s 67.381 (simplify)s 114.886 (notation,)s 168.85 (w)s 177.184 (e)s 188.622 (will)s 213.393 (write)s 5 f 246.198 (\301)s 1 f 253.348 ([)s 5 f 256.681 (\256)s 1 f 264.402 (])s 273.84 (for)s 5 f 294.312 (\301)s 1 f 301.462 (\050)s 0 f 306.128 (x)s 4 f 515.446 y 313.412 (\050)s 6 f 317.162 (\256)s 4 f 323.4 (\051)s 5 f 513.279 y 327.75 (;)s 333.084 (t)s 1 f 337.417 (\051,)s 352.048 (and)s 377.486 (similarly)s 5 f 428.691 (\275)s 1 f 434.895 ([)s 5 f 438.228 (\256)s 1 f 445.949 (])s 455.387 (for)s 5 f 527.679 y 0 (\275)s 1 f 6.204 (\050)s 0 f 10.871 (x)s 4 f 529.846 y 18.154 (\050)s 6 f 21.904 (\256)s 4 f 28.143 (\051)s 5 f 527.679 y 32.493 (;)s 37.826 (t)s 1 f 42.159 (\051.)s 55.493 (In)s 70.493 (this)s 93.893 (notations)s 146.626 (w)s 154.96 (e)s 164.293 (ha)s 176.626 (v)s 182.626 (e)s 5 f 576.52 y 111.69 (f)s 1 f 126.19 (=)s 9 f 548.8 y 142.856 (0)s 569.92 y 142.856 (B)s 577.6 y 142.856 (@)s 5 f 558.187 y 157.831 (\301)s 1 f 164.981 ([1])s 570.853 y 166.073 (.)s 575.653 y 166.073 (.)s 580.453 y 166.073 (.)s 5 f 594.853 y 155.356 (\301)s 1 f 162.506 ([)s 5 f 165.84 (N)s 1 f 176.79 (])s 9 f 548.8 y 182.123 (1)s 569.92 y 182.123 (C)s 577.6 y 182.123 (A)s 5 f 576.52 y 202.623 (;)s 215.956 (\276)s 1 f 226.577 (=)s 9 f 548.8 y 239.244 (0)s 569.92 y 239.244 (B)s 577.6 y 239.244 (@)s 5 f 558.187 y 254.218 (\275)s 1 f 260.423 ([1])s 287.564 (0)s 5 f 305.564 (:)s 310.898 (:)s 316.231 (:)s 1 f 331.564 (0)s 570.853 y 261.987 (.)s 575.653 y 261.987 (.)s 580.453 y 261.987 (.)s 570.853 y 288.898 (.)s 575.653 y 288.898 (.)s 580.453 y 288.898 (.)s 572.053 y 306.231 (.)s 575.653 y 310.898 (.)s 579.253 y 315.564 (.)s 570.853 y 332.898 (.)s 575.653 y 332.898 (.)s 580.453 y 332.898 (.)s 5 f 594.853 y 251.744 (\275)s 1 f 257.948 ([)s 5 f 261.281 (N)s 1 f 272.231 (])s 287.564 (0)s 5 f 305.564 (:)s 310.898 (:)s 316.231 (:)s 1 f 331.564 (0)s 9 f 548.8 y 339.564 (1)s 569.92 y 339.564 (C)s 577.6 y 339.564 (A)s 1 f 576.52 y 439.087 (\0502)s 5 f 449.754 (:)s 1 f 453.087 (24\051)s 625.749 y 0 (W)s 11.333 (e)s 20.843 (will)s 43.685 (also)s 67.928 (denote)s 106.771 (b)s 113.104 (y)s 5 f 123.614 (f)s 4 f 621.394 y 130.78 ([)s 6 f 133.497 (\256)s 4 f 139.736 (])s 1 f 625.749 y 147.228 (the)s 5 f 168.071 (\256)s 1 f 175.792 (-th)s 195.301 (blo)s 211.635 (c)s 216.635 (k)s 227.144 (comp)s 255.477 (onen)s 279.811 (t)s 288.654 (of)s 5 f 302.496 (f)s 1 f 309.663 (,)s 317.216 (and)s 340.726 (similarly)s 390.002 (for)s 408.545 (other)s 440.087 (quan-)s 641.816 y 0 (tities;)s 33.4 (th)s 44.4 (us)s 59.8 (\0502.23\051,)s 97.8 (\0502.24\051)s 132.467 (ab)s 145.467 (o)s 151.134 (v)s 157.134 (e)s 166.467 (read)s 193.167 (simply)s 226.567 (,)s 233.9 (in)s 247.9 (this)s 271.301 (notation,)s 5 f 322.634 (f)s 4 f 637.461 y 329.801 ([)s 6 f 332.517 (\256)s 4 f 338.756 (])s 1 f 641.816 y 345.406 (=)s 5 f 358.072 (\301)s 1 f 365.222 ([)s 5 f 368.556 (\256)s 1 f 376.277 (],)s 5 f 386.943 (\276)s 4 f 635.582 y 394.231 ([)s 6 f 396.947 (\256)s 4 f 403.186 (])s 645.902 y 393.8 ([)s 6 f 396.517 (\257)s 4 f 402.39 (])s 1 f 641.816 y 409.836 (=)s 5 f 422.503 (\275)s 1 f 428.707 ([)s 5 f 432.04 (\256)s 1 f 439.761 (])s 5 f 443.094 (\261)s 6 f 643.616 y 448.428 (\257)s 454.301 (;)s 4 f 457.151 (1)s 1 f 641.816 y 462.534 (.)s 672.002 y 228.877 (10)s end endp %%Page: 13 13 %%BeginPageSetup initializepage (degasperis; page: 13 of 26)setjob %%EndPageSetup gS 0 0 552 730 rC 0 0 :M f0_12 sf ( )S f1_12 sf ( )S 7 0 :M f8_12 sf .1 .01( )J 11 0 :M f4_12 sf .136 .014( )J 19 0 :M f5_12 sf .138 .014( )J 28 0 :M f6_12 sf ( )S f9_12 sf 11.923 1.192( )J 53 0 :M f7_12 sf ( )S 0 0 :M initclip TeXdict begin 0 10 1200 1000 div mul /CMBX10 findfont fdef 1 10 1200 1000 div mul /CMR10 findfont fdef 4 7 1200 1000 div mul /CMR7 findfont fdef 5 10 1200 1000 div mul /CMMI10 findfont fdef 6 7 1200 1000 div mul /CMMI7 findfont fdef 7 10 1200 1000 div mul /CMSY10 findfont fdef 8 7 1200 1000 div mul /CMSY7 findfont fdef 9 10 1200 1000 div mul /CMEX10 findfont fdef 42 41 translate 72 72.27 div dup scale 1 f 12 y 0 (With)s 31 (this)s 54.4 (same)s 84.467 (notation,)s 135.8 (w)s 144.134 (e)s 153.467 (ha)s 165.8 (v)s 171.8 (e)s 181.134 (that)s 50.421 y 128.885 (1)s 6 .48 128.885 55.779 r 66.77 y 128.885 (2)s 5 f 58.539 y 140.325 (\276)s 147.612 (\276)s 6 f 53.584 y 154.9 (T)s 1 f 58.539 y 169.793 (=)s 9 f 30.819 y 186.459 (0)s 51.939 y 186.459 (B)s 59.619 y 186.459 (@)s 5 f 40.206 y 201.101 (A)s 1 f 210.101 ([1)s 5 f 219.434 (;)s 1 f 224.768 (1])s 5 f 248.243 (:)s 253.576 (:)s 258.909 (:)s 276.384 (A)s 1 f 285.384 ([1)s 5 f 294.718 (;)s 300.051 (N)s 1 f 311.001 (])s 52.872 y 215.934 (.)s 57.672 y 215.934 (.)s 62.472 y 215.934 (.)s 54.072 y 248.909 (.)s 57.672 y 253.576 (.)s 61.272 y 258.243 (.)s 52.872 y 293.693 (.)s 57.672 y 293.693 (.)s 62.472 y 293.693 (.)s 5 f 76.872 y 198.959 (A)s 1 f 207.959 ([)s 5 f 211.293 (N)s 221.576 (;)s 1 f 226.909 (1])s 5 f 248.243 (:)s 253.576 (:)s 258.909 (:)s 274.243 (A)s 1 f 283.243 ([)s 5 f 286.576 (N)s 296.859 (;)s 302.193 (N)s 1 f 313.143 (])s 9 f 30.819 y 318.476 (1)s 51.939 y 318.476 (C)s 59.619 y 318.476 (A)s 5 f 58.539 y 338.976 (;)s 1 f 439.087 (\0502)s 5 f 449.754 (:)s 1 f 453.087 (25\051)s 105.078 y 0 (where)s 5 f 122.535 y 170.811 (A)s 1 f 179.811 ([)s 5 f 183.145 (\256)s 190.866 (;)s 196.199 (\257)s 1 f 203.62 (])s 214.286 (:=)s 114.417 y 231.726 (1)s 6 .48 231.726 119.775 r 130.766 y 231.726 (2)s 5 f 122.535 y 243.166 (\275)s 1 f 249.371 ([)s 5 f 252.704 (\256)s 1 f 260.425 (])s 5 f 263.758 (\275)s 6 f 117.58 y 269.962 (T)s 1 f 122.535 y 277.522 ([)s 5 f 280.855 (\257)s 1 f 288.276 (])s 5 f 295.609 (:)s 1 f 439.087 (\0502)s 5 f 449.754 (:)s 1 f 453.087 (26\051)s 161.105 y 0 (W)s 11.333 (e)s 20.663 (will)s 43.326 (also)s 67.389 (write)s 98.085 (the)s 118.748 (di\256eren)s 157.448 (tiation)s 196.111 (op)s 209.111 (erators)s 249.241 (in)s 263.237 (blo)s 279.57 (c)s 284.57 (k)s 294.9 (notation;)s 7 f 346.231 (r)s 6 f 162.905 y 356.231 (\256)s 1 f 161.105 y 367.065 (will)s 389.728 (b)s 396.728 (e)s 406.058 (the)s 426.721 (gradien)s 465.088 (t)s 175.505 y 0 (with)s 27.333 (resp)s 49.1 (ect)s 68.433 (to)s 83.1 (the)s 0 f 103.767 (x)s 4 f 177.672 y 111.05 (\050)s 6 f 114.8 (\256)s 4 f 121.039 (\051)s 1 f 175.505 y 129.389 (v)s 135.055 (ariables,)s 182.489 (and)s 5 f 215.594 y 86.678 (A)s 1 f 95.678 ([)s 5 f 99.012 (\256)s 106.733 (;)s 112.066 (\257)s 1 f 119.487 (])s 7 f 122.82 (r)s 4 f 210.64 y 132.82 (2)s 6 f 218.561 y 132.82 (\256\257)s 1 f 215.594 y 152.865 (:=)s 6 f 200.594 y 178.565 (n)s 9 f 204.194 y 172.865 (X)s 6 f 229.753 y 173.759 (i)s 4 f 177.154 (=1)s 6 f 200.594 y 197.899 (n)s 9 f 204.194 y 192.198 (X)s 6 f 229.753 y 192.562 (j)s 4 f 197.018 (=1)s 1 f 215.594 y 213.531 (\0501)s 5 f 224.198 (=)s 1 f 230.198 (2\051)s 9 f 205.874 y 242.865 (\241)s 5 f 215.594 y 248.365 (\275)s 1 f 254.569 ([)s 5 f 257.902 (\256)s 1 f 265.623 (])s 5 f 268.957 (\275)s 6 f 210.64 y 275.161 (T)s 1 f 215.594 y 282.72 ([)s 5 f 286.054 (\257)s 1 f 293.474 (])s 9 f 205.874 y 296.808 (\242)s 6 f 207.866 y 302.308 (ij)s 5 f 207.476 y 334.04 (@)s 4 f 203.121 y 341.078 (2)s 56.103 .48 318.199 212.834 r 5 f 225.599 y 318.199 (@)s 325.237 (x)s 6 f 221.466 y 332.095 (i)s 4 f 229.686 y 332.095 (\050)s 6 f 335.845 (\256)s 4 f 342.084 (\051)s 5 f 225.599 y 346.434 (@)s 353.471 (x)s 6 f 219.832 y 360.329 (j)s 4 f 229.686 y 360.329 (\050)s 6 f 364.079 (\257)s 4 f 369.952 (\051)s 5 f 215.594 y 379.742 (:)s 1 f 439.087 (\0502)s 5 f 449.754 (:)s 1 f 453.087 (27\051)s 256.925 y 0 (W)s 11.333 (e)s 20.667 (can)s 42.667 (no)s 55 (w)s 67.667 (pro)s 85.367 (ceed)s 112.033 (along)s 144.034 (the)s 164.7 (same)s 194.767 (steps)s 224.9 (as)s 239.634 (in)s 253.634 (the)s 5 f 274.3 (N)s 1 f 288.584 (=)s 301.25 (2)s 311.25 (case.)s 283.325 y 0 (The)s 24.428 (term)s 7 f 52.889 (f)s 5 f 58.889 (f)s 65.389 (;)s 70.723 (\273)s 7 f 76.525 (g)s 1 f 86.286 (is)s 98.114 (readily)s 137.575 (ev)s 148.575 (aluated)s 191.003 (using)s 222.165 (the)s 242.592 (fact)s 266.02 (that)s 7 f 291.782 (r)s 6 f 285.125 y 301.782 (\257)s 5 f 283.325 y 308.255 (\301)s 1 f 315.405 ([)s 5 f 318.738 (\256)s 1 f 326.459 (])s 333.554 (is)s 345.381 (zero)s 370.509 (for)s 5 f 388.637 (\257)s 7 f 399.391 (6)s 1 f 399.391 (=)s 5 f 412.058 (\256)s 1 f 419.779 (:)s 428.326 (in)s 442.087 (blo)s 458.421 (c)s 463.421 (k)s 297.725 y 0 (notation,)s 7 f 327.318 y 27.176 (f)s 5 f 33.176 (f)s 39.676 (;)s 45.009 (\273)s 7 f 50.811 (g)s 4 f 322.364 y 56.811 ([)s 6 f 59.528 (\256)s 4 f 65.766 (])s 1 f 327.318 y 76.416 (=)s 89.75 (\050)s 5 f 94.416 (\301)s 1 f 101.566 ([)s 5 f 104.9 (\257)s 1 f 112.32 (])s 7 f 118.32 (\242)s 124.32 (r)s 6 f 329.118 y 134.32 (\257)s 1 f 327.318 y 140.793 (\051)s 5 f 145.46 (\273)s 4 f 322.364 y 151.262 ([)s 6 f 153.979 (\256)s 4 f 160.217 (])s 7 f 327.318 y 166.201 (\241)s 1 f 178.201 (\050)s 5 f 182.867 (\273)s 4 f 322.364 y 188.669 ([)s 6 f 191.386 (\257)s 4 f 197.259 (])s 7 f 327.318 y 203.242 (\242)s 209.242 (r)s 6 f 329.118 y 219.242 (\257)s 1 f 327.318 y 225.715 (\051)s 5 f 230.382 (\301)s 1 f 237.532 ([)s 5 f 240.865 (\256)s 1 f 248.586 (])s 259.253 (=)s 350.118 y 97.083 (=)s 9 f 336.798 y 113.749 (h)s 1 f 350.118 y 119.416 (\050)s 5 f 124.083 (\301)s 1 f 131.233 ([)s 5 f 134.566 (\256)s 1 f 142.287 (])s 7 f 148.287 (\242)s 154.287 (r)s 6 f 351.918 y 164.287 (\256)s 1 f 350.118 y 171.125 (\051)s 5 f 175.792 (\273)s 4 f 345.164 y 181.594 ([)s 6 f 184.311 (\256)s 4 f 190.549 (])s 7 f 350.118 y 196.533 (\241)s 1 f 208.533 (\050)s 5 f 213.199 (\273)s 4 f 345.164 y 219.001 ([)s 6 f 221.718 (\256)s 4 f 227.957 (])s 7 f 350.118 y 233.94 (\242)s 239.94 (r)s 6 f 351.918 y 249.94 (\256)s 1 f 350.118 y 256.778 (\051)s 5 f 261.445 (\301)s 1 f 268.595 ([)s 5 f 271.928 (\256)s 1 f 279.649 (])s 9 f 336.798 y 282.983 (i)s 1 f 350.118 y 295.316 (+)s 9 f 338.718 y 312.388 (X)s 6 f 364.552 y 311.316 (\257)s 8 f 317.189 (6)s 4 f 317.189 (=)s 6 f 324.555 (\256)s 1 f 350.118 y 330.794 (\050)s 5 f 335.461 (\301)s 1 f 342.611 ([)s 5 f 345.944 (\257)s 1 f 353.365 (])s 7 f 359.365 (\242)s 365.365 (r)s 6 f 351.918 y 375.365 (\257)s 1 f 350.118 y 381.838 (\051)s 5 f 386.504 (\273)s 4 f 345.164 y 392.307 ([)s 6 f 395.023 (\256)s 4 f 401.262 (])s 5 f 350.118 y 408.578 (:)s 1 f 344.724 y 439.087 (\0502)s 5 f 449.754 (:)s 1 f 453.087 (28\051)s 404.391 y 0 (The)s 24.895 (ev)s 35.895 (aluation)s 82.79 (of)s 96.686 (the)s 117.581 (terms)s 5 f 151.242 (@)s 6 f 406.191 y 157.613 (t)s 5 f 404.391 y 161.838 (\273)s 1 f 171.869 (and)s 5 f 195.431 (@)s 6 f 406.191 y 201.802 (t)s 1 f 404.391 y 206.027 (\050)s 5 f 210.693 (\277)s 217.297 (f)s 1 f 224.464 (\051)s 233.359 (do)s 246.359 (es)s 260.654 (not)s 282.216 (presen)s 315.316 (t)s 324.211 (an)s 336.545 (y)s 347.107 (problem,)s 397.426 (giving)s 433.321 (resp)s 455.087 (ec-)s 418.791 y 0 (tiv)s 14 (ely)s 5 f 31.931 (@)s 6 f 420.591 y 38.302 (t)s 5 f 418.791 y 42.527 (\273)s 4 f 414.436 y 48.329 ([)s 6 f 51.046 (\256)s 4 f 57.284 (])s 1 f 418.791 y 63.532 (and)s 5 f 85.797 (@)s 6 f 420.591 y 92.168 (t)s 1 f 418.791 y 96.393 (\050)s 5 f 101.06 (\277)s 107.664 (\301)s 1 f 114.814 ([)s 5 f 118.147 (\256)s 1 f 125.868 (]\051,)s 140.346 (and)s 162.611 (the)s 182.209 (second)s 219.874 (deriv)s 245.574 (ativ)s 265.574 (es)s 278.572 (terms)s 310.936 (is)s 321.934 (simply)s 9 f 409.791 y 359.265 (P)s 6 f 422.391 y 371.932 (\256;\257)s 5 f 418.791 y 389.494 (A)s 1 f 398.494 ([)s 5 f 401.827 (\256)s 409.548 (;)s 414.881 (\257)s 1 f 422.302 (])s 7 f 425.635 (r)s 4 f 414.436 y 435.635 (2)s 6 f 422.189 y 435.635 (\256\257)s 5 f 418.791 y 448.347 (\273)s 4 f 414.436 y 454.149 ([)s 6 f 456.866 (\256)s 4 f 463.104 (])s 1 f 418.791 y 466.421 (.)s 433.191 y 0 (Summarizing,)s 76.538 (the)s 97.609 (\257rst)s 122.78 (of)s 136.85 (the)s 157.921 (determining)s 225.025 (equations)s 279.162 (\0501.10\051)s 314.233 (is)s 326.704 (no)s 339.037 (w,)s 355.542 (in)s 369.946 (blo)s 386.279 (c)s 391.279 (k)s 402.017 (notation)s 450.421 (and)s 447.591 y 0 (with)s 27.333 (no)s 44 (sum)s 69.4 (on)s 5 f 86.067 (\256)s 1 f 93.788 (,)s 5 f 478.53 y 80.024 (@)s 6 f 480.33 y 86.395 (t)s 5 f 478.53 y 90.62 (\273)s 4 f 473.575 y 96.422 ([)s 6 f 99.139 (\256)s 4 f 105.377 (])s 1 f 478.53 y 115.361 (+)s 7 f 131.36 (f)s 5 f 137.36 (\301)s 1 f 144.511 ([)s 5 f 147.844 (\256)s 1 f 155.565 (])s 5 f 158.898 (;)s 164.231 (\273)s 4 f 473.575 y 170.033 ([)s 6 f 172.75 (\256)s 4 f 178.989 (])s 7 f 478.53 y 182.305 (g)s 6 f 480.33 y 188.305 (\256)s 1 f 478.53 y 201.81 (+)s 9 f 467.129 y 218.883 (X)s 6 f 492.963 y 217.81 (\257)s 8 f 223.683 (6)s 4 f 223.683 (=)s 6 f 231.05 (\256)s 1 f 478.53 y 237.289 (\050)s 5 f 241.955 (\301)s 1 f 249.105 ([)s 5 f 252.439 (\257)s 1 f 259.859 (])s 7 f 265.859 (\242)s 271.859 (r)s 6 f 480.33 y 281.859 (\257)s 1 f 478.53 y 288.332 (\051)s 5 f 292.999 (\273)s 4 f 473.575 y 298.801 ([)s 6 f 301.518 (\256)s 4 f 307.756 (])s 1 f 478.53 y 315.073 (+)s 7 f 513.196 y 82.691 (\241)s 5 f 98.691 (@)s 6 f 514.996 y 105.061 (t)s 1 f 513.196 y 109.287 (\050)s 5 f 113.953 (\277)s 120.557 (\301)s 1 f 127.707 ([)s 5 f 131.041 (\256)s 1 f 138.762 (]\051)s 153.428 (+)s 9 f 501.796 y 169.428 (X)s 6 f 527.63 y 170.955 (\257)s 176.828 (;\260)s 5 f 513.196 y 188.761 (A)s 1 f 197.761 ([)s 5 f 201.095 (\257)s 208.516 (;)s 213.849 (\260)s 1 f 220.728 (])s 7 f 224.062 (r)s 4 f 508.242 y 234.062 (2)s 6 f 516.163 y 234.062 (\257)s 239.935 (\260)s 5 f 513.196 y 246.092 (\273)s 4 f 508.242 y 251.894 ([)s 6 f 254.611 (\256)s 4 f 260.849 (])s 1 f 549.063 y 91.357 (=)s 9 f 535.743 y 108.024 (h)s 5 f 549.063 y 113.691 (@)s 6 f 550.863 y 120.061 (t)s 5 f 549.063 y 124.286 (\273)s 4 f 544.108 y 130.089 ([)s 6 f 132.805 (\256)s 4 f 139.044 (])s 1 f 549.063 y 149.027 (+)s 7 f 165.027 (f)s 5 f 171.027 (\301)s 1 f 178.177 ([)s 5 f 181.51 (\256)s 1 f 189.231 (])s 5 f 192.565 (;)s 197.898 (\273)s 4 f 544.108 y 203.7 ([)s 6 f 206.417 (\256)s 4 f 212.655 (])s 7 f 549.063 y 215.972 (g)s 6 f 550.863 y 221.972 (\256)s 7 f 549.063 y 231.477 (\241)s 5 f 247.477 (@)s 6 f 550.863 y 253.848 (t)s 1 f 549.063 y 258.073 (\050)s 5 f 262.739 (\277)s 269.344 (\301)s 1 f 276.494 ([)s 5 f 279.827 (\256)s 1 f 287.548 (]\051)s 302.214 (+)s 5 f 318.214 (A)s 1 f 327.214 ([)s 5 f 330.548 (\256)s 338.269 (;)s 343.602 (\256)s 1 f 351.323 (])s 5 f 354.656 (\273)s 4 f 544.108 y 360.458 ([)s 6 f 363.175 (\256)s 4 f 369.413 (])s 9 f 535.743 y 372.73 (i)s 1 f 549.063 y 380.397 (+)s 574.263 y 94.691 (+)s 9 f 562.863 y 111.763 (X)s 6 f 588.697 y 110.691 (\257)s 8 f 116.564 (6)s 4 f 116.564 (=)s 6 f 123.93 (\256)s 1 f 574.263 y 130.169 (\050)s 5 f 134.835 (\301)s 1 f 141.985 ([)s 5 f 145.319 (\257)s 1 f 152.74 (])s 7 f 158.74 (\242)s 164.74 (r)s 6 f 576.063 y 174.74 (\257)s 1 f 574.263 y 181.213 (\051)s 5 f 185.879 (\273)s 4 f 569.309 y 191.681 ([)s 6 f 194.398 (\256)s 4 f 200.637 (])s 1 f 574.263 y 210.62 (+)s 9 f 562.863 y 243.94 (X)s 4 f 589.163 y 226.62 (\050)s 6 f 230.37 (\257)s 236.243 (;\260)s 4 f 244.65 (\051)s 8 f 248.4 (6)s 4 f 248.4 (=\050)s 6 f 259.517 (\256;\256)s 4 f 274.844 (\051)s 5 f 574.263 y 280.594 (A)s 1 f 289.594 ([)s 5 f 292.927 (\257)s 300.348 (;)s 305.681 (\260)s 1 f 312.561 (])s 7 f 315.894 (r)s 4 f 569.309 y 325.894 (2)s 6 f 577.23 y 325.894 (\257)s 331.767 (\260)s 5 f 574.263 y 337.924 (\273)s 4 f 569.309 y 343.726 ([)s 6 f 346.443 (\256)s 4 f 352.682 (])s 1 f 574.263 y 363.332 (=)s 379.998 (0)s 5 f 385.998 (:)s 1 f 532.196 y 439.087 (\0502)s 5 f 449.754 (:)s 1 f 453.087 (29\051)s 616.802 y 0 (Note)s 29 (that)s 55 (the)s 75.667 (square)s 113.434 (brac)s 135.8 (k)s 141.8 (et)s 155.8 (corresp)s 193.6 (onds)s 221.667 (to)s 236.334 (the)s 257 (equation)s 306 (for)s 324.367 (the)s 345.034 (one-particle)s 410.401 (pro)s 428.101 (cess.)s 643.202 y 0 (Let)s 21.5 (us)s 36.9 (no)s 49.233 (w)s 61.9 (consider)s 108.667 (the)s 129.333 (second)s 168.067 (of)s 181.734 (\0501.10\051.)s 221.067 (W)s 232.4 (e)s 241.734 (will)s 264.4 (use)s 285.134 (the)s 305.8 (same)s 335.867 (blo)s 352.201 (c)s 357.201 (k)s 367.534 (notation.)s 672.002 y 228.877 (11)s end endp %%Page: 14 14 %%BeginPageSetup initializepage (degasperis; page: 14 of 26)setjob %%EndPageSetup gS 0 0 552 730 rC 0 0 :M f1_12 sf ( )S 4 0 :M f8_12 sf .071 .007( )J f4_12 sf .158 .016( )J 15 0 :M f5_12 sf .138 .014( )J 24 0 :M f6_12 sf ( )S f9_12 sf 11.923 1.192( )J 50 0 :M f7_12 sf ( )S 0 0 :M initclip TeXdict begin 1 10 1200 1000 div mul /CMR10 findfont fdef 4 7 1200 1000 div mul /CMR7 findfont fdef 5 10 1200 1000 div mul /CMMI10 findfont fdef 6 7 1200 1000 div mul /CMMI7 findfont fdef 7 10 1200 1000 div mul /CMSY10 findfont fdef 8 7 1200 1000 div mul /CMSY7 findfont fdef 9 10 1200 1000 div mul /CMEX10 findfont fdef 42 41 translate 72 72.27 div dup scale 1 f 12 y 0 (W)s 11.333 (e)s 20.667 (will)s 43.333 (write)s 5 f 74.033 (B)s 1 f 87.738 (in)s 101.738 (blo)s 118.071 (c)s 123.071 (k)s 133.404 (form)s 161.771 (as)s 5 f 55.532 y 141.896 (B)s 1 f 158.933 (=)s 9 f 27.812 y 175.6 (0)s 48.932 y 175.6 (B)s 56.612 y 175.6 (@)s 5 f 37.199 y 190.242 (R)s 1 f 199.446 ([1)s 5 f 208.779 (;)s 1 f 214.112 (1])s 5 f 237.587 (:)s 242.921 (:)s 248.254 (:)s 265.729 (R)s 1 f 274.933 ([1)s 5 f 284.267 (;)s 289.6 (N)s 1 f 300.55 (])s 49.865 y 205.177 (.)s 54.665 y 205.177 (.)s 59.465 y 205.177 (.)s 51.065 y 238.254 (.)s 54.665 y 242.921 (.)s 58.265 y 247.587 (.)s 49.865 y 283.139 (.)s 54.665 y 283.139 (.)s 59.465 y 283.139 (.)s 5 f 73.865 y 188.1 (R)s 1 f 197.304 ([)s 5 f 200.638 (N)s 210.921 (;)s 1 f 216.254 (1])s 5 f 237.587 (:)s 242.921 (:)s 248.254 (:)s 263.587 (R)s 1 f 272.792 ([)s 5 f 276.125 (N)s 286.408 (;)s 291.742 (N)s 1 f 302.692 (])s 9 f 27.812 y 308.025 (1)s 48.932 y 308.025 (C)s 56.612 y 308.025 (A)s 1 f 55.532 y 324.525 (;)s 439.087 (\0502)s 5 f 449.754 (:)s 1 f 453.087 (30\051)s 103.159 y 0 (notice)s 35.872 (that)s 5 f 62.41 (R)s 1 f 71.614 ([)s 5 f 74.948 (\256)s 82.669 (;)s 88.002 (\256)s 1 f 95.723 (])s 103.595 (are)s 124.166 (an)s 136.5 (tisymmetric)s 203.471 (matrices,)s 255.578 (while)s 287.45 (in)s 301.988 (general)s 5 f 343.893 (R)s 1 f 353.097 ([)s 5 f 356.431 (\256)s 364.152 (;)s 369.485 (\257)s 1 f 376.906 (])s 384.47 (=)s 5 f 398.034 (R)s 6 f 98.804 y 407.238 (T)s 1 f 103.159 y 414.797 ([)s 5 f 418.131 (\257)s 425.551 (;)s 430.885 (\256)s 1 f 438.606 (].)s 452.221 (F)s 459.054 (or)s 117.559 y 0 (later)s 27.928 (reference,)s 81.577 (the)s 102.138 (submatrix)s 5 f 159.132 (R)s 1 f 168.337 ([1)s 5 f 177.67 (;)s 1 f 183.003 (1])s 196.231 (will)s 218.792 (also)s 242.753 (b)s 249.753 (e)s 258.981 (denoted)s 304.209 (as)s 5 f 318.836 (M)s 1 f 331.786 (.)s 340.418 (The)s 364.979 (matrix)s 5 f 403.907 (\276)s 411.194 (B)s 1 f 424.793 (will)s 447.354 (th)s 458.354 (us)s 131.959 y 0 (ha)s 12.333 (v)s 18.333 (e)s 27.667 (blo)s 44 (c)s 49 (k)s 59.333 (comp)s 87.667 (onen)s 112 (ts)s 148.025 y 171.054 (\050)s 5 f 175.72 (\276)s 183.008 (B)s 1 f 192.712 (\051)s 4 f 141.792 y 197.379 ([)s 6 f 200.095 (\256)s 4 f 206.334 (])s 152.112 y 197.379 ([)s 6 f 200.095 (\257)s 4 f 205.968 (])s 1 f 148.025 y 216.984 (=)s 5 f 233.65 (\275)s 1 f 239.855 ([)s 5 f 243.188 (\256)s 1 f 250.909 (])s 5 f 256.242 (B)s 1 f 265.946 ([1)s 5 f 275.28 (;)s 280.613 (\257)s 1 f 288.034 (])s 5 f 295.367 (:)s 1 f 439.087 (\0502)s 5 f 449.754 (:)s 1 f 453.087 (31\051)s 187.018 y 0 (The)s 24.667 (term)s 7 f 53.367 (f)s 5 f 59.367 (\276)s 6 f 188.817 y 66.224 (k)s 5 f 187.018 y 72.128 (;)s 77.461 (\273)s 7 f 83.263 (g)s 6 f 182.663 y 89.263 (i)s 7 f 187.018 y 96.592 (\264)s 5 f 109.258 (\276)s 6 f 181.251 y 116.546 (j)s 190.637 y 116.115 (k)s 5 f 187.018 y 122.019 (@)s 6 f 188.817 y 128.39 (j)s 5 f 187.018 y 133.447 (\273)s 6 f 182.663 y 139.249 (i)s 7 f 187.018 y 145.91 (\241)s 5 f 157.91 (\273)s 6 f 182.663 y 163.712 (j)s 5 f 187.018 y 168.769 (@)s 6 f 188.817 y 175.14 (j)s 5 f 187.018 y 180.196 (\276)s 6 f 182.663 y 187.483 (i)s 190.416 y 187.053 (k)s 1 f 187.018 y 196.957 (is)s 209.024 (written,)s 254.39 (in)s 268.39 (blo)s 284.724 (c)s 289.724 (k)s 300.057 (notation,)s 9 f 206.449 y 141.768 (\263)s 5 f 219.769 y 148.935 (\276)s 4 f 213.536 y 156.222 ([)s 6 f 158.939 (\260)s 4 f 164.496 (])s 223.856 y 155.791 ([)s 6 f 158.508 (\257)s 4 f 164.381 (])s 7 f 219.769 y 170.479 (\242)s 176.479 (r)s 6 f 221.569 y 186.479 (\260)s 9 f 206.449 y 192.637 (\264)s 5 f 219.769 y 201.803 (\273)s 4 f 214.814 y 207.605 ([)s 6 f 210.322 (\256)s 4 f 216.56 (])s 7 f 219.769 y 226.544 (\241)s 9 f 206.449 y 242.544 (\263)s 5 f 219.769 y 249.71 (\273)s 4 f 214.814 y 255.512 ([)s 6 f 258.229 (\260)s 4 f 263.786 (])s 7 f 219.769 y 269.77 (\242)s 275.77 (r)s 6 f 221.569 y 285.77 (\260)s 9 f 206.449 y 291.927 (\264)s 5 f 219.769 y 301.094 (\276)s 4 f 213.536 y 308.381 ([)s 6 f 311.098 (\256)s 4 f 317.336 (])s 223.856 y 307.951 ([)s 6 f 310.667 (\257)s 4 f 316.54 (])s 1 f 219.769 y 324.653 (;)s 439.087 (\0502)s 5 f 449.754 (:)s 1 f 453.087 (32\051)s 255.434 y 0 (b)s 6.333 (y)s 16.275 (observing)s 69.651 (that)s 5 f 95.26 (\276)s 4 f 249.201 y 102.547 ([)s 6 f 105.264 (\256)s 4 f 111.503 (])s 259.521 y 102.117 ([)s 6 f 104.833 (\257)s 4 f 110.706 (])s 1 f 255.434 y 118.152 (=)s 5 f 130.819 (\275)s 1 f 137.023 ([)s 5 f 140.357 (\256)s 1 f 148.077 (])s 5 f 151.411 (\261)s 6 f 257.234 y 156.744 (\257)s 162.617 (;)s 4 f 165.467 (1)s 1 f 255.434 y 170.85 (,)s 177.871 (w)s 186.204 (e)s 195.146 (get)s 214.755 (that)s 240.364 (this)s 263.373 (is)s 275.048 (trivially)s 320.024 (zero)s 344.999 (for)s 5 f 362.975 (\257)s 373.729 (\076)s 1 f 386.395 (1,)s 399.416 (and)s 422.358 (for)s 5 f 440.333 (\257)s 1 f 451.087 (=)s 463.754 (1)s 273.476 y 0 (it)s 12 (is)s 24.067 (easily)s 57.133 (written)s 99.167 (as)s 113.9 (\050)s 5 f 118.567 (\275)s 1 f 124.771 ([)s 5 f 128.104 (\260)s 1 f 134.984 (])s 7 f 140.984 (\242)s 146.984 (r)s 6 f 275.276 y 156.984 (\260)s 1 f 273.476 y 163.141 (\051)s 5 f 167.808 (\273)s 4 f 269.121 y 173.61 ([)s 6 f 176.326 (\256)s 4 f 182.565 (])s 7 f 273.476 y 188.548 (\241)s 1 f 200.548 (\050)s 5 f 205.215 (\273)s 4 f 269.121 y 211.017 ([)s 6 f 213.734 (\260)s 4 f 219.291 (])s 7 f 273.476 y 225.274 (\242)s 231.274 (r)s 6 f 275.276 y 241.274 (\260)s 1 f 273.476 y 247.432 (\051)s 5 f 252.098 (\275)s 1 f 258.302 ([)s 5 f 261.636 (\256)s 1 f 269.357 (].)s 299.876 y 0 (As)s 7 f 18.228 (r)s 6 f 301.676 y 28.228 (\260)s 5 f 299.876 y 34.386 (\275)s 1 f 40.59 ([)s 5 f 43.923 (\256)s 1 f 51.644 (])s 59.135 (=)s 72.627 (0)s 83.122 (for)s 5 f 101.984 (\260)s 7 f 113.021 (6)s 1 f 113.021 (=)s 5 f 126.513 (\256)s 1 f 134.233 (,)s 142.185 (this)s 166.08 (can)s 188.575 (also)s 213.137 (b)s 220.137 (e)s 229.965 (rewritten,)s 285.984 (in)s 300.479 (the)s 321.641 (same)s 352.202 (notation)s 400.697 (as)s 415.926 (ab)s 428.926 (o)s 434.592 (v)s 440.592 (e)s 450.421 (and)s 314.276 y 0 (with)s 27.333 (no)s 44 (sum)s 69.4 (on)s 5 f 86.067 (\256)s 1 f 93.788 (,)s 101.121 (as)s 7 f 330.409 y 141.886 (f)s 5 f 147.886 (\275)s 1 f 154.091 ([)s 5 f 157.424 (\256)s 1 f 165.145 (])s 5 f 168.478 (;)s 173.811 (\273)s 4 f 325.454 y 179.613 ([)s 6 f 182.33 (\256)s 4 f 188.569 (])s 7 f 330.409 y 191.885 (g)s 6 f 332.209 y 197.885 (\256)s 1 f 330.409 y 211.391 (+)s 9 f 319.009 y 228.305 (X)s 6 f 344.842 y 227.39 (\260)s 8 f 232.948 (6)s 4 f 232.948 (=)s 6 f 240.314 (\256)s 1 f 330.409 y 248.553 (\050)s 5 f 253.22 (\275)s 1 f 259.424 ([)s 5 f 262.757 (\260)s 1 f 269.637 (])s 7 f 275.636 (\242)s 281.636 (r)s 6 f 332.209 y 291.636 (\260)s 1 f 330.409 y 297.794 (\051)s 5 f 302.46 (\273)s 4 f 325.454 y 308.263 ([)s 6 f 310.979 (\256)s 4 f 317.218 (])s 1 f 330.409 y 324.534 (;)s 439.087 (\0502)s 5 f 449.754 (:)s 1 f 453.087 (33\051)s 379.069 y 0 (The)s 24.667 (terms)s 5 f 58.1 (\277)s 64.704 (@)s 6 f 380.869 y 71.075 (t)s 5 f 379.069 y 75.3 (\276)s 6 f 374.715 y 82.588 (i)s 382.468 y 82.157 (k)s 1 f 379.069 y 92.061 (and)s 115.395 (\050)s 5 f 120.061 (@)s 6 f 380.869 y 126.432 (t)s 5 f 379.069 y 130.657 (\277)s 1 f 137.261 (\051)s 5 f 141.928 (\276)s 6 f 374.715 y 149.216 (i)s 382.468 y 148.785 (k)s 1 f 379.069 y 158.689 (also)s 182.756 (v)s 188.423 (anish)s 219.823 (for)s 5 f 238.189 (k)s 248.148 (\076)s 260.814 (n)s 1 f 272.017 (and)s 295.35 (are)s 315.384 (easily)s 348.45 (written)s 390.484 (for)s 5 f 408.851 (k)s 7 f 418.809 (\267)s 5 f 431.475 (n)s 1 f 438.678 (.)s 405.469 y 0 (Let)s 21.11 (us)s 36.119 (summarize)s 95.829 (our)s 116.805 (discussion)s 172.614 (for)s 190.591 (the)s 210.867 (second)s 249.21 (of)s 262.486 (\0501.10\051.)s 301.689 (F)s 308.522 (or)s 5 f 322.832 (k)s 332.79 (\076)s 345.457 (n)s 1 f 352.66 (,)s 359.681 (this)s 382.69 (reduces,)s 428.478 (in)s 442.087 (blo)s 458.421 (c)s 463.421 (k)s 419.869 y 0 (notation,)s 51.333 (to)s 435.936 y 155.304 (\050)s 5 f 159.97 (\276)s 167.258 (B)s 1 f 176.962 (\051)s 4 f 429.703 y 181.629 ([)s 6 f 184.345 (\256)s 4 f 190.584 (])s 440.023 y 181.629 ([)s 6 f 184.345 (\257)s 4 f 190.218 (])s 1 f 435.936 y 201.234 (=)s 5 f 217.9 (\275)s 1 f 224.105 ([)s 5 f 227.438 (\256)s 1 f 235.159 (])s 5 f 242.492 (R)s 1 f 251.696 ([1)s 5 f 261.03 (;)s 266.363 (\257)s 1 f 273.784 (])s 284.45 (=)s 301.117 (0)s 5 f 311.117 (:)s 1 f 439.087 (\0502)s 5 f 449.754 (:)s 1 f 453.087 (34\051)s 460.603 y 0 (If)s 5 f 12.433 (\275)s 1 f 23.07 (is)s 35.57 (of)s 49.67 (rank)s 5 f 77.803 (n)s 1 f 85.005 (,)s 92.88 (this)s 116.713 (implies)s 5 f 157.879 (R)s 1 f 167.084 ([1)s 5 f 176.417 (;)s 181.75 (\257)s 1 f 189.171 (])s 196.559 (=)s 209.948 (0)s 220.381 (for)s 5 f 239.18 (\257)s 7 f 250.656 (6)s 1 f 250.656 (=)s 264.044 (1.)s 280.01 (Due)s 305.61 (to)s 320.709 (the)s 341.809 (an)s 354.142 (tisymmetry)s 418.676 (of)s 5 f 432.775 (B)s 1 f 442.479 (,)s 450.354 (this)s 475.003 y 0 (means)s 36.718 (that)s 62.702 (the)s 83.353 (only)s 109.67 (nonzero)s 154.354 (blo)s 170.688 (c)s 175.688 (k)s 186.005 (submatrix)s 243.089 (are)s 263.107 (the)s 283.758 (diagonal)s 331.742 (ones,)s 5 f 361.796 (R)s 1 f 371 ([)s 5 f 374.333 (\256)s 382.054 (;)s 387.388 (\256)s 1 f 395.108 (].)s 407.103 (Notice)s 444.754 (ho)s 457.087 (w-)s 489.403 y 0 (ev)s 11.333 (er)s 25.045 (that)s 50.724 (for)s 5 f 68.769 (\256)s 7 f 79.823 (6)s 1 f 79.823 (=)s 92.49 (1)s 102.168 (these)s 132.58 (represen)s 175.713 (ts)s 188.792 (in)s 198.458 (ternal)s 232.837 (transformations)s 319.049 (of)s 332.394 (the)s 352.739 (\134ghost\042)s 5 f 396.484 (n)s 1 f 403.687 (-dimensional)s 503.803 y 0 (Wiener)s 42.182 (pro)s 59.882 (cesses)s 94.564 (\050)s 5 f 99.231 (v)s 6 f 499.449 y 105.478 (nk)s 4 f 116.714 (+1)s 5 f 503.803 y 129.464 (;)s 134.797 (:::;)s 150.131 (v)s 6 f 499.449 y 156.378 (nk)s 4 f 167.614 (+)s 6 f 174.981 (n)s 1 f 503.803 y 181.513 (\051)s 190.662 (\050)s 5 f 195.328 (k)s 7 f 206.09 (\270)s 1 f 219.56 (1\051)s 234.708 (whic)s 258.375 (h)s 269.524 (pla)s 285.19 (y)s 296.006 (no)s 313.154 (role)s 337.003 (in)s 351.485 (the)s 372.634 (problem;)s 423.39 (as)s 438.605 (in)s 453.087 (the)s 5 f 518.203 y 0 (N)s 1 f 14.283 (=)s 26.95 (2)s 36.698 (case,)s 65.229 (these)s 95.71 (w)s 104.044 (ould)s 130.458 (giv)s 145.792 (e)s 154.873 (spurious)s 202.121 (symmetries,)s 268.419 (mapp)s 298.086 (ed)s 313.834 (to)s 328.248 (the)s 348.663 (iden)s 370.329 (tit)s 382.663 (y)s 392.744 (in)s 406.492 (terms)s 439.673 (of)s 453.087 (the)s 532.603 y 0 (original)s 5 f 43.973 (N)s 1 f 54.923 (-particle)s 102.897 (problem,)s 153.688 (and)s 177.628 (should)s 216.301 (b)s 223.301 (e)s 233.241 (disregarded.)s 303.861 (Th)s 318.861 (us)s 334.868 (the)s 356.141 (only)s 383.081 (nonzero)s 428.387 (relev)s 452.754 (an)s 465.087 (t)s 547.003 y 0 (submatrix)s 57.1 (in)s 5 f 71.1 (B)s 1 f 84.804 (is)s 96.871 (the)s 5 f 117.538 (R)s 1 f 126.742 ([1)s 5 f 136.075 (;)s 1 f 141.409 (1])s 154.075 (=)s 5 f 166.742 (M)s 1 f 183.692 (blo)s 200.025 (c)s 205.025 (k.)s 573.403 y 0 (F)s 6.833 (or)s 5 f 21.533 (k)s 7 f 31.492 (\267)s 5 f 44.158 (n)s 1 f 55.361 (the)s 76.028 (second)s 114.761 (of)s 128.428 (\0501.10\051)s 163.095 (is)s 175.161 (written,)s 220.528 (in)s 234.528 (blo)s 250.861 (c)s 255.861 (k)s 266.195 (notation,)s 317.528 (as)s 9 f 592.215 y 74.102 (h)s 7 f 605.535 y 79.769 (f)s 5 f 85.769 (\275)s 1 f 91.973 ([)s 5 f 95.306 (\256)s 1 f 103.027 (])s 5 f 106.36 (;)s 111.694 (\273)s 4 f 600.581 y 117.496 ([)s 6 f 120.213 (\256)s 4 f 126.451 (])s 7 f 605.535 y 129.768 (g)s 6 f 607.335 y 135.768 (\256)s 7 f 605.535 y 149.273 (\241)s 5 f 165.273 (\277)s 171.877 (@)s 6 f 607.335 y 178.248 (t)s 5 f 605.535 y 182.473 (\275)s 1 f 188.677 ([)s 5 f 192.01 (\256)s 1 f 199.731 (])s 7 f 209.731 (\241)s 1 f 221.731 (\0501)s 5 f 232.398 (=)s 1 f 238.398 (2\051\050)s 5 f 253.731 (@)s 6 f 607.335 y 260.102 (t)s 5 f 605.535 y 264.327 (\277)s 1 f 270.931 (\051)s 5 f 275.598 (\275)s 1 f 281.802 ([)s 5 f 285.135 (\256)s 1 f 292.856 (])s 7 f 302.856 (\241)s 5 f 318.856 (\275)s 1 f 325.06 ([)s 5 f 328.394 (\256)s 1 f 336.115 (])s 5 f 339.448 (R)s 1 f 348.652 ([1)s 5 f 357.985 (;)s 1 f 363.319 (1])s 9 f 592.215 y 376.652 (i)s 1 f 605.535 y 388.319 (+)s 630.735 y 78.769 (+)s 9 f 619.335 y 95.683 (X)s 6 f 645.169 y 94.769 (\260)s 8 f 100.326 (6)s 4 f 100.326 (=)s 6 f 107.693 (\256)s 1 f 630.735 y 115.931 (\050)s 5 f 120.598 (\275)s 1 f 126.802 ([)s 5 f 130.135 (\260)s 1 f 137.015 (])s 7 f 143.015 (\242)s 149.015 (r)s 6 f 632.535 y 159.015 (\260)s 1 f 630.735 y 165.172 (\051)s 5 f 169.839 (\273)s 4 f 625.781 y 175.641 ([)s 6 f 178.357 (\256)s 4 f 184.596 (])s 1 f 630.735 y 195.246 (=)s 211.912 (0)s 5 f 221.912 (:)s 1 f 622.869 y 439.087 (\0502)s 5 f 449.754 (:)s 1 f 453.087 (35\051)s 672.002 y 228.877 (12)s end endp %%Page: 15 15 %%BeginPageSetup initializepage (degasperis; page: 15 of 26)setjob %%EndPageSetup gS 0 0 552 730 rC 0 0 :M f0_12 sf ( )S 4 0 :M f1_12 sf ( )S f2_12 sf ( )S f8_12 sf ( )S 14 0 :M f4_12 sf .136 .014( )J 22 0 :M f5_12 sf .138 .014( )J 31 0 :M f6_12 sf ( )S f9_12 sf 11.923 1.192( )J 57 0 :M f7_12 sf ( )S f10_12 sf 9.389 .939( )J 0 0 :M initclip TeXdict begin 0 10 1200 1000 div mul /CMBX10 findfont fdef 1 10 1200 1000 div mul /CMR10 findfont fdef 2 10 1200 1000 div mul /CMTI10 findfont fdef 4 7 1200 1000 div mul /CMR7 findfont fdef 5 10 1200 1000 div mul /CMMI10 findfont fdef 6 7 1200 1000 div mul /CMMI7 findfont fdef 7 10 1200 1000 div mul /CMSY10 findfont fdef 8 7 1200 1000 div mul /CMSY7 findfont fdef 9 10 1200 1000 div mul /CMEX10 findfont fdef 10 7 1200 1000 div mul /CMBX7 findfont fdef 42 41 translate 72 72.27 div dup scale 1 f 12 y 0 (Again,)s 38.333 (the)s 59 (square)s 96.767 (brac)s 119.134 (k)s 125.134 (et)s 139.134 (corresp)s 176.934 (onds)s 205 (to)s 219.667 (the)s 240.334 (equation)s 289.334 (for)s 307.701 (the)s 328.367 (one-particle)s 393.734 (pro)s 411.434 (cess.)s 38.4 y 0 (W)s 11.333 (e)s 20.948 (will)s 43.896 (no)s 56.229 (w)s 69.177 (summarize)s 129.558 (our)s 151.206 (results)s 189.654 (for)s 208.302 (the)s 5 f 229.25 (N)s 1 f 240.2 (-particle)s 287.848 (pro)s 305.548 (cess)s 329.962 (de\257ned)s 371.577 (b)s 377.91 (y)s 388.525 (an)s 405.473 (Ito)s 424.754 (equation)s 52.8 y 0 (in)s 14 (the)s 34.667 (same)s 64.733 (w)s 73.067 (a)s 78.733 (y)s 89.067 (as)s 103.8 (w)s 112.133 (e)s 121.467 (did)s 142.134 (for)s 160.5 (the)s 181.167 (t)s 185.5 (w)s 193.834 (o-particle)s 247.2 (pro)s 264.9 (cess.)s 0 f 79.2 y 0 (Prop)s 30.067 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115.149 (6)s 4 f 115.149 (=)s 6 f 122.516 (\256)s 1 f 146.301 y 128.754 (\050)s 5 f 133.421 (\301)s 1 f 140.571 ([)s 5 f 143.904 (\257)s 1 f 151.325 (])s 7 f 157.325 (\242)s 163.325 (r)s 6 f 148.101 y 173.325 (\257)s 1 f 146.301 y 179.798 (\051)s 5 f 184.465 (\273)s 4 f 141.347 y 190.267 ([)s 6 f 192.984 (\256)s 4 f 199.222 (])s 1 f 146.301 y 209.499 (+)s 9 f 134.901 y 243.112 (X)s 4 f 161.201 y 225.792 (\050)s 6 f 229.542 (\257)s 235.415 (;\260)s 4 f 243.822 (\051)s 8 f 247.572 (6)s 4 f 247.572 (=\050)s 6 f 258.689 (\256;\256)s 4 f 274.016 (\051)s 5 f 146.301 y 279.766 (A)s 1 f 288.766 ([)s 5 f 292.099 (\257)s 299.52 (;)s 304.853 (\260)s 1 f 311.733 (])s 7 f 315.066 (r)s 4 f 141.347 y 325.066 (2)s 6 f 149.268 y 325.066 (\257)s 330.939 (\260)s 5 f 146.301 y 337.096 (\273)s 4 f 141.347 y 342.899 ([)s 6 f 345.615 (\256)s 4 f 351.854 (])s 1 f 146.301 y 362.797 (=)s 379.757 (0)s 9 f 169.781 y 79.436 (h)s 7 f 183.102 y 85.103 (f)s 5 f 91.103 (\275)s 1 f 97.307 ([)s 5 f 100.641 (\256)s 1 f 108.361 (])s 5 f 111.695 (;)s 117.028 (\273)s 4 f 178.147 y 122.83 ([)s 6 f 125.547 (\256)s 4 f 131.785 (])s 7 f 183.102 y 135.102 (g)s 4 f 185.268 y 141.102 ([)s 5 f 183.102 y 144.419 (\256)s 1 f 152.14 (])s 7 f 162.433 (\241)s 5 f 178.726 (\277)s 185.33 (@)s 6 f 184.902 y 191.701 (t)s 5 f 183.102 y 195.926 (\275)s 1 f 202.13 ([)s 5 f 205.464 (\256)s 1 f 213.185 (])s 7 f 223.478 (\241)s 1 f 235.478 (\0501)s 5 f 246.144 (=)s 1 f 252.144 (2\051\050)s 5 f 267.478 (@)s 6 f 184.902 y 273.849 (t)s 5 f 183.102 y 278.074 (\277)s 1 f 284.678 (\051)s 5 f 289.345 (\275)s 1 f 295.549 ([)s 5 f 298.882 (\256)s 1 f 306.603 (])s 7 f 316.896 (\241)s 5 f 333.189 (\275)s 1 f 339.394 ([)s 5 f 342.727 (\256)s 1 f 350.448 (])s 5 f 353.781 (M)s 9 f 169.781 y 371.024 (i)s 1 f 183.102 y 382.984 (+)s 208.302 y 92.983 (+)s 9 f 196.902 y 110.191 (X)s 6 f 222.735 y 109.276 (\260)s 8 f 114.833 (6)s 4 f 114.833 (=)s 6 f 122.2 (\256)s 1 f 208.302 y 130.439 (\050)s 5 f 135.105 (\275)s 1 f 141.31 ([)s 5 f 144.643 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5 f 158.283 (;)s 163.617 (\264)s 7 f 170.005 (g)s 10 f 375.408 y 176.005 (x)s 7 f 373.608 y 185.007 (\241)s 5 f 201.007 (@)s 6 f 375.408 y 207.378 (t)s 1 f 373.608 y 211.603 ([)s 5 f 214.936 (\277)s 221.54 (\301)s 1 f 228.69 (\050)s 0 f 233.357 (x)s 5 f 240.64 (;)s 245.974 (t)s 1 f 250.307 (\051])s 264.974 (+)s 5 f 280.974 (A)s 1 f 289.974 (\050)s 0 f 294.64 (x)s 5 f 301.924 (;)s 0 f 307.257 (x)s 1 f 314.54 (\051)s 7 f 319.207 (r)s 4 f 368.654 y 329.207 (2)s 10 f 376.575 y 329.207 (xx)s 5 f 373.608 y 341.277 (\264)s 7 f 391.608 y 79.199 (f)s 5 f 85.199 (\275)s 1 f 91.403 (\050)s 0 f 96.069 (x)s 5 f 103.353 (;)s 108.686 (t)s 1 f 113.019 (\051)s 5 f 117.686 (;)s 123.019 (\264)s 7 f 129.408 (g)s 10 f 393.408 y 135.408 (x)s 7 f 391.608 y 148.41 (\241)s 5 f 164.41 (\277)s 171.014 (@)s 6 f 393.408 y 177.385 (t)s 5 f 391.608 y 181.61 (\275)s 1 f 187.814 (\050)s 0 f 192.481 (x)s 5 f 199.764 (;)s 205.097 (t)s 1 f 209.431 (\051)s 7 f 220.764 (\241)s 1 f 236.764 (\0501)s 5 f 247.431 (=)s 1 f 253.431 (2\051\050)s 5 f 268.764 (@)s 6 f 393.408 y 275.135 (t)s 5 f 391.608 y 279.36 (\277)s 1 f 285.964 (\051)s 5 f 290.631 (\275)s 1 f 296.835 (\050)s 0 f 301.502 (x)s 5 f 308.785 (;)s 314.118 (t)s 1 f 318.451 (\051)s 7 f 325.785 (\241)s 5 f 337.785 (\275)s 1 f 343.989 (\050)s 0 f 348.656 (x)s 5 f 355.939 (;)s 361.272 (t)s 1 f 365.606 (\051)s 5 f 370.272 (M)s 1 f 387.222 (;)s 382.524 y 439.087 (\0502)s 5 f 449.754 (:)s 1 f 453.087 (37\051)s 415.111 y 0 (these)s 30.733 (coincide)s 76.733 (with)s 104.067 (\0501.10\051.)s 7 f 143.4 (\246)s 1 f 441.511 y 0 (By)s 20.372 (setting)s 5 f 61.31 (\264)s 1 f 73.597 (=)s 5 f 88.827 (\264)s 1 f 95.216 (\050)s 0 f 99.883 (x)s 5 f 107.166 (;)s 112.499 (t)s 1 f 116.833 (\051,)s 130.756 (instead)s 173.694 (than)s 203.233 (the)s 225.438 (general)s 268.343 (situation)s 5 f 319.948 (\264)s 1 f 332.235 (=)s 5 f 347.465 (\264)s 1 f 353.854 (\050)s 0 f 358.521 (z)s 5 f 364.654 (;)s 369.987 (t)s 1 f 374.321 (\051,)s 388.244 (in)s 403.782 (the)s 425.987 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365.257 (a)s 5 f 374.431 (N)s 1 f 385.38 (-particle)s 431.921 (pro)s 449.621 (cess)s 537.511 y 0 (whic)s 23.667 (h)s 33.417 (are)s 2 f 52.534 (not)s 1 f 72.484 (symmetries)s 134.735 (of)s 147.485 (the)s 167.235 (OPP)s 195.985 (de\257ned)s 236.402 (b)s 242.735 (y)s 252.152 (the)s 271.902 (same)s 301.052 (Ito)s 319.136 (equation,)s 370.736 (as)s 384.553 (w)s 392.887 (e)s 401.304 (already)s 442.754 (knew)s 551.911 y 0 (from)s 28.367 (the)s 49.033 (discussion)s 105.234 (of)s 118.9 (the)s 139.567 (t)s 143.9 (w)s 152.234 (o-particle)s 205.6 (pro)s 223.3 (cess.)s 0 f 590.357 y 0 (3.)s 16.867 (Examples)s 79.293 (\173)s 90.793 (I:)s 104.459 (W-symmetries.)s 1 f 628.802 y 0 (In)s 16.384 (this)s 41.168 (and)s 65.885 (the)s 87.936 (follo)s 109.936 (wing)s 139.986 (sections,)s 189.85 (w)s 198.183 (e)s 208.9 (will)s 232.951 (illustrate)s 285.101 (the)s 307.152 (results)s 346.702 (of)s 361.753 (previous)s 410.904 (sections)s 457.088 (b)s 463.421 (y)s 643.202 y 0 (some)s 29.273 (simple)s 65.879 (examples.)s 122.014 (W)s 133.348 (e)s 141.887 (start)s 169.86 (b)s 176.193 (y)s 185.733 (considering,)s 251.198 (in)s 264.404 (this)s 287.01 (section,)s 329.775 (examples)s 380.715 (of)s 5 f 393.587 (W)s 1 f 406.587 (-symmetries)s 672.002 y 228.877 (13)s end endp %%Page: 16 16 %%BeginPageSetup initializepage (degasperis; page: 16 of 26)setjob %%EndPageSetup gS 0 0 552 730 rC 0 0 :M f0_12 sf ( )S f1_12 sf ( )S 7 0 :M f3_12 sf .09 .009( )J f8_12 sf .06 .006( )J 17 0 :M f4_12 sf .136 .014( )J 25 0 :M f5_12 sf .138 .014( )J 34 0 :M f6_12 sf ( )S f9_12 sf 11.923 1.192( )J 59 0 :M f7_12 sf ( )S 0 0 :M initclip TeXdict begin 0 10 1200 1000 div mul /CMBX10 findfont fdef 1 10 1200 1000 div mul /CMR10 findfont fdef 3 10 1200 1000 div mul /CMTT10 findfont fdef 4 7 1200 1000 div mul /CMR7 findfont fdef 5 10 1200 1000 div mul /CMMI10 findfont fdef 6 7 1200 1000 div mul /CMMI7 findfont fdef 7 10 1200 1000 div mul /CMSY10 findfont fdef 8 7 1200 1000 div mul /CMSY7 findfont fdef 9 10 1200 1000 div mul /CMEX10 findfont fdef 42 41 translate 72 72.27 div dup scale 1 f 12 y 0 (for)s 19.296 (some)s 50.293 (\050OPP)s 85.555 (de\257ned)s 127.818 (b)s 134.151 (y\051)s 150.081 (Ito)s 170.011 (equations;)s 228.472 (these)s 260.135 (are)s 281.098 (just)s 305.761 (the)s 327.357 (m)s 337.024 (ultidimensional)s 422.021 (examples)s 26.4 y 0 (considered)s 59.496 (in)s 74.226 ([4],)s 95.138 (so)s 110.6 (that)s 137.33 (the)s 158.726 (symmetries)s 222.622 (not)s 244.685 (in)s 254.352 (v)s 260.352 (olving)s 296.748 (transformations)s 384.011 (of)s 398.407 (the)s 0 f 419.804 (w)s 1 f 434.691 (can)s 457.421 (b)s 464.421 (e)s 40.8 y 0 (read)s 26.7 (from)s 55.067 (there.)s 3 f 67.2 y 0 (Example)s 50.4 (1.)s 1 f 68.333 (Let)s 89.833 (us)s 105.233 (consider)s 152 (the)s 172.666 (t)s 177 (w)s 185.333 (o-dimensional)s 261.4 (Ito)s 280.4 (equation)s 96.622 y 151.159 (d)s 5 f 157.826 (x)s 1 f 172.017 (=)s 5 f 188.684 (y)s 1 f 196.998 (d)s 5 f 203.664 (t)s 1 f 117.668 y 151.703 (d)s 5 f 158.37 (y)s 1 f 172.017 (=)s 7 f 188.684 (\241)s 5 f 198.017 (k)s 4 f 112.713 y 204.642 (2)s 5 f 117.668 y 210.026 (y)s 1 f 218.34 (d)s 5 f 225.006 (t)s 1 f 236.006 (+)s 7 f 106.582 y 252.006 (p)s 18.008 .48 262.006 106.582 r 1 f 117.668 y 262.006 (2)s 5 f 268.006 (k)s 4 f 114.201 y 274.631 (2)s 1 f 117.668 y 282.014 (d)s 5 f 288.681 (w)s 1 f 297.595 (\050)s 5 f 302.262 (t)s 1 f 306.595 (\051)s 315.262 (;)s 107.145 y 445.087 (\0503)s 5 f 455.754 (:)s 1 f 459.087 (1\051)s 147.424 y 0 (this)s 23.4 (is)s 35.467 (example)s 82.467 (4)s 92.467 (of)s 106.134 ([4].)s 127.467 (The)s 152.134 (v)s 158.134 (ector)s 5 f 188.167 (\273)s 1 f 197.969 (and)s 221.302 (the)s 241.969 (an)s 254.302 (tisymmetric)s 320.736 (matrix)s 5 f 359.769 (B)s 1 f 373.474 (will)s 396.14 (b)s 403.14 (e)s 412.474 (written)s 454.507 (as)s 5 f 184.38 y 145.022 (\273)s 1 f 158.157 (=)s 9 f 167.459 y 174.824 (\265)s 5 f 177.18 y 185.657 (\256)s 191.58 y 185.807 (\257)s 9 f 167.459 y 195.378 (\266)s 5 f 184.38 y 214.211 (;)s 227.545 (B)s 1 f 244.582 (=)s 9 f 167.459 y 261.249 (\265)s 1 f 177.18 y 276.324 (0)s 5 f 298.99 (b)s 7 f 191.58 y 272.082 (\241)s 5 f 281.415 (b)s 1 f 298.565 (0)s 9 f 167.459 y 306.565 (\266)s 5 f 184.38 y 321.399 (:)s 1 f 445.087 (\0503)s 5 f 455.754 (:)s 1 f 459.087 (2\051)s 220.669 y 0 (If)s 12.963 (w)s 21.296 (e)s 31.593 (lo)s 41.26 (ok)s 58.556 (at)s 74.186 (the)s 95.816 (second)s 135.512 (set)s 155.208 (of)s 169.838 (determining)s 237.501 (equations)s 292.198 (and)s 316.494 (single)s 350.858 (out)s 373.154 (the)s 394.784 (one)s 417.747 (for)s 5 f 437.077 (i)s 1 f 446.149 (=)s 460.421 (2,)s 5 f 235.069 y 0 (k)s 1 f 11.636 (=)s 25.981 (1)s 36.988 (w)s 45.321 (e)s 55.661 (ha)s 67.994 (v)s 73.994 (e)s 84.335 (immediately)s 153.675 (that)s 5 f 180.682 (b)s 1 f 190.843 (=)s 205.187 (0,)s 219.779 (i.e.)s 243.466 (in)s 258.473 (this)s 282.88 (case)s 309.287 (w)s 317.62 (e)s 327.96 (ha)s 340.293 (v)s 346.293 (e)s 356.634 (no)s 374.307 (new)s 399.981 (symmetry)s 457.087 (b)s 463.421 (y)s 249.469 y 0 (allo)s 18.333 (wing)s 47 (transformation)s 128.8 (of)s 142.467 (the)s 0 f 163.134 (w)s 1 f 173.292 (.)s 7 f 181.959 (\246)s 3 f 275.869 y 0 (Example)s 50.4 (2.)s 1 f 68.298 (W)s 79.631 (e)s 88.86 (consider)s 135.522 (no)s 147.855 (w)s 160.417 (the)s 180.979 (equation,)s 233.228 (again)s 265.124 (t)s 269.457 (w)s 277.79 (o-dimensional,)s 357.106 (giv)s 372.44 (en)s 388.335 (as)s 402.964 (example)s 449.859 (5)s 459.754 (in)s 290.269 y 0 ([4];)s 20 (this)s 43.4 (is)s 305.624 y 166.168 (d)s 5 f 172.835 (x)s 1 f 187.027 (=)s 203.693 (\050)s 5 f 208.36 (a)s 4 f 307.424 y 214.703 (1)s 5 f 305.624 y 220.086 (=x)s 1 f 232.945 (\051)s 239.611 (d)s 5 f 246.278 (t)s 1 f 255.278 (+)s 269.278 (d)s 5 f 275.945 (w)s 4 f 307.424 y 284.536 (1)s 1 f 305.624 y 289.919 (\050)s 5 f 294.586 (t)s 1 f 298.919 (\051)s 323.624 y 166.713 (d)s 5 f 173.38 (y)s 1 f 187.027 (=)s 5 f 203.693 (a)s 4 f 325.424 y 210.036 (2)s 1 f 323.624 y 217.42 (d)s 5 f 224.086 (t)s 1 f 233.086 (+)s 247.086 (d)s 5 f 253.753 (w)s 4 f 325.424 y 262.344 (2)s 1 f 323.624 y 267.727 (\050)s 5 f 272.394 (t)s 1 f 276.727 (\051)s 5 f 285.394 (:)s 1 f 314.624 y 445.087 (\0503)s 5 f 455.754 (:)s 1 f 459.087 (3\051)s 347.38 y 0 (W)s 11.333 (e)s 21.386 (use)s 42.839 (the)s 64.226 (same)s 95.012 (notation)s 143.732 (\0503.2\051)s 173.118 (as)s 188.571 (ab)s 201.571 (o)s 207.238 (v)s 213.238 (e)s 223.291 (for)s 5 f 242.377 (\273)s 1 f 252.899 (and)s 5 f 276.952 (B)s 1 f 286.656 (,)s 294.889 (and)s 318.942 (pass)s 345.795 (directly)s 390.215 (to)s 405.601 (consider)s 453.087 (the)s 361.78 y 0 (second)s 38.733 (set)s 57.467 (of)s 71.133 (determining)s 137.834 (equations)s 191.567 (\0501.10\051.)s 230.9 (As)s 248.634 (no)s 260.967 (w)s 5 f 273.634 (\276)s 1 f 284.255 (=)s 5 f 296.921 (I)s 1 f 303.138 (,)s 310.471 (these)s 341.205 (read)s 367.905 (simply)s 5 f 391.536 y 169.65 (@)s 6 f 393.336 y 176.021 (k)s 5 f 391.536 y 181.925 (\273)s 6 f 386.581 y 187.728 (i)s 7 f 391.536 y 194.389 (\241)s 1 f 206.389 (\050)s 5 f 211.056 (\277)s 6 f 393.336 y 216.302 (t)s 5 f 391.536 y 220.527 (=)s 1 f 226.527 (2\051)s 5 f 237.193 (\261)s 6 f 386.581 y 242.981 (i)s 394.502 y 242.527 (k)s 7 f 391.536 y 251.098 (\241)s 5 f 263.098 (B)s 6 f 386.581 y 272.802 (i)s 394.502 y 272.2 (k)s 1 f 391.536 y 281.437 (=)s 294.104 (0)s 445.087 (\0503)s 5 f 455.754 (:)s 1 f 459.087 (4\051)s 421.291 y 0 (and)s 23.333 (therefore)s 73.733 (w)s 82.067 (e)s 91.4 (ha)s 103.733 (v)s 109.733 (e,)s 122.4 (writing)s 5 f 163.767 (\277)s 6 f 423.091 y 169.013 (t)s 1 f 421.291 y 176.571 (=)s 189.238 (2)s 5 f 195.238 (\265)s 1 f 201.204 (\050)s 5 f 205.871 (t)s 1 f 210.204 (\051,)s 9 f 443.463 y 162.455 (8)s 454.264 y 162.455 (\076)s 457.864 y 162.455 (\074)s 479.464 y 162.455 (\076)s 483.064 y 162.455 (:)s 5 f 449.847 y 175.121 (\256)s 6 f 451.647 y 182.798 (x)s 1 f 449.847 y 192.173 (=)s 5 f 204.839 (\265)s 1 f 230.434 (\050)s 5 f 235.1 (i;)s 244.568 (k)s 1 f 254.526 (=)s 267.193 (1)s 5 f 273.193 (;)s 1 f 278.526 (1\051)s 5 f 464.247 y 175.121 (\256)s 6 f 466.047 y 182.798 (y)s 1 f 464.247 y 191.899 (=)s 5 f 204.566 (b)s 1 f 230.434 (\050)s 5 f 235.1 (i;)s 244.568 (k)s 1 f 254.526 (=)s 267.193 (1)s 5 f 273.193 (;)s 1 f 278.526 (2\051)s 5 f 478.647 y 175.121 (\257)s 6 f 480.447 y 181.909 (x)s 1 f 478.647 y 191.284 (=)s 7 f 203.95 (\241)s 5 f 213.284 (b)s 1 f 230.434 (\050)s 5 f 235.1 (i;)s 244.568 (k)s 1 f 254.526 (=)s 267.193 (2)s 5 f 273.193 (;)s 1 f 278.526 (1\051)s 5 f 493.047 y 175.121 (\257)s 6 f 494.847 y 181.909 (y)s 1 f 493.047 y 191.01 (=)s 5 f 203.677 (\265)s 1 f 230.434 (\050)s 5 f 235.1 (i;)s 244.568 (k)s 1 f 254.526 (=)s 267.193 (2)s 5 f 273.193 (;)s 1 f 278.526 (2\051)s 297.193 (.)s 471.664 y 445.087 (\0503)s 5 f 455.754 (:)s 1 f 459.087 (5\051)s 521.369 y 0 (These)s 34.733 (mean)s 66.733 (w)s 75.067 (e)s 84.4 (can)s 106.4 (write)s 5 f 533.725 y 170.954 (\256)s 1 f 186.008 (=)s 5 f 202.674 (\265)s 208.641 (x)s 1 f 218.166 (+)s 5 f 230.166 (by)s 1 f 244.296 (+)s 5 f 256.296 (\256)s 4 f 535.525 y 263.973 (0)s 1 f 533.725 y 269.356 (\050)s 5 f 274.023 (t)s 1 f 278.356 (\051)s 5 f 551.725 y 171.254 (\257)s 1 f 186.008 (=)s 7 f 202.674 (\241)s 5 f 212.008 (bx)s 1 f 226.683 (+)s 5 f 238.682 (\265)s 244.649 (y)s 1 f 253.63 (+)s 5 f 265.63 (\257)s 4 f 553.525 y 272.417 (0)s 1 f 551.725 y 277.801 (\050)s 5 f 282.467 (t)s 1 f 286.801 (\051)s 5 f 295.467 (:)s 1 f 542.725 y 445.087 (\0503)s 5 f 455.754 (:)s 1 f 459.087 (6\051)s 575.481 y 0 (The)s 24.667 (\257rst)s 49.433 (set)s 68.167 (of)s 81.833 (determining)s 148.534 (equations)s 202.267 (\0501.10\051)s 236.934 (reads)s 268.367 (in)s 282.367 (this)s 305.767 (case)s 5 f 610.888 y 89.06 (\256)s 6 f 612.688 y 96.736 (t)s 1 f 610.888 y 103.628 (+)s 9 f 597.568 y 115.628 (h)s 5 f 602.77 y 122.734 (a)s 4 f 604.57 y 129.077 (1)s 11.726 .48 122.734 608.128 r 5 f 619.119 y 125.168 (x)s 610.888 y 135.901 (\256)s 6 f 612.688 y 143.577 (x)s 1 f 610.888 y 152.285 (+)s 5 f 164.285 (a)s 4 f 612.688 y 170.628 (2)s 5 f 610.888 y 176.012 (\256)s 6 f 612.688 y 183.688 (y)s 1 f 610.888 y 192.123 (+)s 5 f 602.77 y 205.82 (a)s 4 f 604.57 y 212.164 (1)s 12.242 .48 205.563 608.128 r 5 f 619.119 y 205.563 (x)s 4 f 615.652 y 212.421 (2)s 5 f 610.888 y 219.244 (\256)s 9 f 597.568 y 226.965 (i)s 7 f 610.888 y 235.299 (\241)s 5 f 602.77 y 248.739 (a)s 4 f 604.57 y 255.082 (1)s 11.726 .48 248.739 608.128 r 5 f 619.119 y 251.173 (x)s 610.888 y 261.905 (\277)s 6 f 612.688 y 267.151 (t)s 1 f 610.888 y 274.042 (+)s 602.77 y 287.482 (1)s 6 .48 287.482 608.128 r 619.119 y 287.482 (2)s 610.888 y 296.922 (\050)s 5 f 301.589 (\256)s 6 f 612.688 y 309.265 (xx)s 1 f 610.888 y 323.415 (+)s 5 f 335.415 (\256)s 6 f 612.688 y 343.092 (y)s 348.26 (y)s 1 f 610.888 y 354.028 (\051)s 362.028 (=)s 374.695 (0)s 5 f 639.771 y 89.06 (\257)s 6 f 641.571 y 95.847 (t)s 1 f 639.771 y 102.739 (+)s 9 f 626.45 y 114.739 (h)s 5 f 631.652 y 121.845 (a)s 4 f 633.452 y 128.188 (1)s 11.726 .48 121.845 637.011 r 5 f 648.002 y 124.279 (x)s 639.771 y 135.012 (\257)s 6 f 641.571 y 141.799 (x)s 1 f 639.771 y 150.507 (+)s 5 f 162.507 (a)s 4 f 641.571 y 168.851 (2)s 5 f 639.771 y 174.234 (\257)s 6 f 641.571 y 181.021 (y)s 9 f 626.45 y 186.789 (i)s 7 f 639.771 y 195.123 (\241)s 5 f 207.123 (a)s 4 f 641.571 y 213.466 (2)s 5 f 639.771 y 218.849 (\277)s 6 f 641.571 y 224.095 (t)s 1 f 639.771 y 230.987 (+)s 631.652 y 244.427 (1)s 6 .48 244.427 637.011 r 648.002 y 244.427 (2)s 639.771 y 253.867 (\050)s 5 f 258.533 (\257)s 6 f 641.571 y 265.321 (xx)s 1 f 639.771 y 279.471 (+)s 5 f 291.471 (\257)s 6 f 641.571 y 298.258 (y)s 303.426 (y)s 1 f 639.771 y 309.194 (\051)s 317.194 (=)s 329.861 (0)s 5 f 339.861 (;)s 1 f 624.519 y 441.721 (\0503)s 5 f 452.387 (:)s 1 f 455.721 (7)s 8 f 620.164 y 461.721 (0)s 1 f 624.519 y 465.087 (\051)s 672.002 y 228.877 (14)s end endp %%Page: 17 17 %%BeginPageSetup initializepage (degasperis; page: 17 of 26)setjob %%EndPageSetup gS 0 0 552 730 rC 0 0 :M f0_12 sf ( )S f1_12 sf ( )S 7 0 :M f3_12 sf .09 .009( )J f8_12 sf .06 .006( )J 17 0 :M f4_12 sf .136 .014( )J 25 0 :M f5_12 sf .138 .014( )J 34 0 :M f6_12 sf ( )S f9_12 sf 11.923 1.192( )J 59 0 :M f7_12 sf ( )S 0 0 :M initclip TeXdict begin 0 10 1200 1000 div mul /CMBX10 findfont fdef 1 10 1200 1000 div mul /CMR10 findfont fdef 3 10 1200 1000 div mul /CMTT10 findfont fdef 4 7 1200 1000 div mul /CMR7 findfont fdef 5 10 1200 1000 div mul /CMMI10 findfont fdef 6 7 1200 1000 div mul /CMMI7 findfont fdef 7 10 1200 1000 div mul /CMSY10 findfont fdef 8 7 1200 1000 div mul /CMSY7 findfont fdef 9 10 1200 1000 div mul /CMEX10 findfont fdef 42 41 translate 72 72.27 div dup scale 1 f 12 y 0 (whic)s 23.667 (h)s 34.333 (with)s 61.667 (\0503.6\051)s 90.333 (reduce)s 128.367 (to)s 36.692 y 137.888 (1)s 6 .48 137.888 42.05 r 53.042 y 137.888 (2)s 5 f 44.81 y 145.328 (\277)s 6 f 46.61 y 150.574 (tt)s 5 f 44.81 y 158.424 (x)s 1 f 167.949 (+)s 5 f 179.949 (\256)s 4 f 46.61 y 187.625 (0)s 8 f 39.856 y 193.009 (0)s 1 f 44.81 y 199.042 (+)s 5 f 211.042 (ba)s 4 f 46.61 y 222.535 (2)s 1 f 44.81 y 230.585 (+)s 5 f 36.692 y 244.025 (a)s 4 f 38.492 y 250.368 (1)s 5 f 36.692 y 255.751 (by)s 23.19 .48 244.025 42.05 r 53.042 y 249.499 (x)s 4 f 49.575 y 256.358 (2)s 1 f 44.81 y 271.322 (+)s 5 f 36.692 y 284.762 (a)s 4 f 38.492 y 291.105 (1)s 5 f 36.692 y 296.488 (\256)s 4 f 38.492 y 304.165 (0)s 24.786 .48 284.762 42.05 r 5 f 53.042 y 291.034 (x)s 4 f 49.575 y 297.892 (2)s 1 f 44.81 y 314.321 (=)s 326.988 (0)s 66.175 y 137.888 (1)s 6 .48 137.888 71.533 r 82.525 y 137.888 (2)s 5 f 74.293 y 145.328 (\277)s 6 f 76.093 y 150.574 (tt)s 5 f 74.293 y 158.424 (y)s 1 f 167.405 (+)s 5 f 179.405 (\257)s 4 f 76.093 y 186.192 (0)s 8 f 68.926 y 191.575 (0)s 7 f 74.293 y 197.609 (\241)s 5 f 66.175 y 211.049 (a)s 4 f 67.975 y 217.392 (1)s 5 f 66.175 y 222.775 (b)s 16.876 .48 211.049 71.533 r 82.525 y 216.058 (x)s 1 f 74.293 y 232.032 (+)s 5 f 66.175 y 245.472 (a)s 4 f 67.975 y 251.815 (2)s 11.726 .48 245.472 71.533 r 1 f 82.525 y 248.335 (2)s 5 f 74.293 y 258.638 (\277)s 6 f 76.093 y 263.884 (t)s 7 f 74.293 y 270.775 (\241)s 5 f 282.775 (a)s 4 f 76.093 y 289.119 (2)s 5 f 74.293 y 294.502 (\277)s 6 f 76.093 y 299.748 (t)s 1 f 74.293 y 307.306 (=)s 319.973 (0)s 5 f 329.973 (:)s 1 f 58.442 y 438.954 (\0503)s 5 f 449.621 (:)s 1 f 452.954 (7)s 8 f 54.087 y 458.954 (00)s 1 f 58.442 y 465.087 (\051)s 104.217 y 0 (These)s 34.733 (imply)s 63.4 (,)s 70.733 (as)s 85.467 (the)s 106.133 (unkno)s 138.134 (wn)s 157.467 (functions)s 209.2 (dep)s 228.2 (end)s 250.867 (on)s 5 f 267.534 (t)s 1 f 275.867 (alone,)s 5 f 130.776 y 85.479 (\277)s 6 f 132.576 y 90.725 (tt)s 1 f 130.776 y 101.908 (=)s 114.575 (0)s 5 f 124.575 (;)s 133.908 (\256)s 4 f 132.576 y 141.585 (0)s 8 f 125.821 y 146.968 (0)s 1 f 130.776 y 153.668 (=)s 7 f 166.335 (\241)s 5 f 175.668 (ba)s 4 f 132.576 y 187.161 (2)s 5 f 130.776 y 196.544 (;)s 205.878 (\257)s 4 f 132.576 y 212.665 (0)s 8 f 125.409 y 218.049 (0)s 1 f 130.776 y 224.749 (=)s 5 f 237.415 (a)s 4 f 132.576 y 243.758 (2)s 5 f 130.776 y 249.142 (\277)s 6 f 132.576 y 254.387 (t)s 5 f 130.776 y 258.612 (=)s 1 f 264.612 (2)s 5 f 274.612 (;)s 1 f 277.946 (\0503)s 5 f 288.612 (:)s 1 f 291.946 (8)s 8 f 125.821 y 297.946 (0)s 1 f 130.776 y 301.313 (\051)s 5 f 148.776 y 277.946 (a)s 4 f 150.576 y 284.289 (1)s 5 f 148.776 y 289.672 (\256)s 4 f 150.576 y 297.349 (0)s 1 f 148.776 y 306.065 (=)s 318.732 (0)s 5 f 328.732 (;)s 338.065 (a)s 4 f 150.576 y 344.408 (1)s 5 f 148.776 y 349.792 (b)s 1 f 358.275 (=)s 370.941 (0)s 5 f 380.941 (;)s 1 f 438.954 (\0503)s 5 f 449.621 (:)s 1 f 452.954 (8)s 8 f 144.421 y 458.954 (00)s 1 f 148.776 y 465.087 (\051)s 175.335 y 0 (and)s 23.514 (for)s 5 f 42.061 (a)s 4 f 177.135 y 48.404 (1)s 7 f 175.335 y 57.422 (6)s 1 f 57.422 (=)s 70.389 (0)s 80.57 (w)s 88.903 (e)s 98.417 (also)s 122.664 (ha)s 134.997 (v)s 140.997 (e)s 5 f 150.511 (\256)s 4 f 177.135 y 158.188 (0)s 1 f 175.335 y 167.205 (=)s 180.173 (0,)s 5 f 193.732 (b)s 1 f 202.516 (=)s 215.483 (0.)s 230.692 (Th)s 245.692 (us,)s 264.651 (unless)s 5 f 300.298 (a)s 4 f 177.135 y 306.641 (1)s 1 f 175.335 y 315.658 (=)s 328.626 (0,)s 342.185 (w)s 350.518 (e)s 360.032 (do)s 376.879 (not)s 398.393 (ha)s 410.726 (v)s 416.726 (e)s 426.24 (an)s 438.574 (y)s 449.088 (new)s 189.735 y 0 (symmetry)s 56.1 (with)s 83.434 (resp)s 105.2 (ect)s 124.534 (to)s 139.2 (the)s 159.867 (case)s 185.267 (analyzed)s 234.934 (in)s 248.934 ([4].)s 216.135 y 0 (In)s 16.575 (the)s 38.817 (case)s 5 f 65.792 (a)s 4 f 217.935 y 72.135 (1)s 1 f 216.135 y 83.476 (=)s 98.768 (0)s 110.343 (w)s 118.676 (e)s 129.585 (are)s 151.193 (left)s 173.768 (with)s 202.676 (\0503.8\047\051)s 236.251 (alone,)s 272.887 (as)s 289.195 (\0503.8\042\051)s 325.437 (are)s 347.045 (automatically)s 424.287 (satis\257ed.)s 230.535 y 0 (W)s 11.333 (riting)s 5 f 44.001 (\277)s 1 f 53.939 (=)s 66.606 (2)s 5 f 72.606 (c)s 4 f 232.335 y 77.799 (1)s 5 f 230.535 y 83.182 (t)s 1 f 90.118 (+)s 5 f 102.054 (c)s 4 f 232.335 y 107.247 (2)s 1 f 230.535 y 116.599 (w)s 124.932 (e)s 134.233 (obtain)s 171.535 (immediately)s 5 f 239.836 (\256)s 4 f 232.335 y 247.513 (0)s 1 f 230.535 y 256.229 (=)s 7 f 268.896 (\241)s 5 f 278.229 (a)s 4 f 232.335 y 284.573 (2)s 5 f 230.535 y 289.956 (bt)s 1 f 302.042 (+)s 5 f 313.978 (c)s 4 f 232.335 y 319.171 (3)s 1 f 230.535 y 328.523 (and)s 5 f 351.824 (\257)s 4 f 232.335 y 358.612 (0)s 1 f 230.535 y 367.328 (=)s 5 f 379.995 (c)s 4 f 232.335 y 385.188 (1)s 5 f 230.535 y 390.571 (a)s 4 f 232.335 y 396.914 (2)s 5 f 230.535 y 402.298 (t)s 1 f 409.234 (+)s 5 f 421.17 (c)s 4 f 232.335 y 426.363 (4)s 1 f 230.535 y 431.746 (,)s 439.054 (where)s 5 f 244.935 y 0 (b)s 1 f 9.15 (and)s 32.483 (the)s 5 f 53.15 (c)s 6 f 246.735 y 58.343 (i)s 1 f 244.935 y 62.338 (\047s)s 74.405 (are)s 94.438 (arbitrary)s 145.538 (real)s 168.905 (constan)s 208.639 (ts.)s 271.335 y 0 (Th)s 15 (us,)s 36.021 (for)s 5 f 56.218 (a)s 4 f 273.135 y 62.561 (1)s 1 f 271.335 y 74.328 (=)s 90.044 (0)s 101.874 (w)s 110.208 (e)s 121.371 (ha)s 133.704 (v)s 139.704 (e)s 150.868 (\257v)s 163.535 (e)s 174.698 (symmetries;)s 243.943 (four)s 270.807 (of)s 286.303 (these)s 318.867 (\173)s 330.697 (asso)s 352.497 (cited)s 383.66 (to)s 400.157 (the)s 422.654 (arbitrary)s 285.735 y 0 (constan)s 39.733 (ts)s 5 f 52.74 (c)s 6 f 287.535 y 57.933 (i)s 1 f 285.735 y 65.535 (\173)s 75.142 (do)s 91.415 (not)s 112.356 (in)s 122.022 (v)s 128.022 (olv)s 143.356 (e)s 152.296 (the)s 0 f 172.569 (w)s 1 f 186.334 (v)s 192.334 (ector)s 221.975 (and)s 244.915 (are)s 264.555 (giv)s 279.888 (en)s 295.495 (b)s 301.828 (y)s 5 f 311.768 (X)s 4 f 287.535 y 321.71 (1)s 1 f 285.735 y 330.427 (=)s 343.093 (2)s 5 f 349.093 (t@)s 6 f 287.535 y 359.798 (t)s 1 f 285.735 y 365.903 (+)s 5 f 377.116 (x@)s 6 f 287.535 y 390.346 (x)s 1 f 285.735 y 398.267 (+)s 409.481 (\050)s 5 f 414.148 (y)s 1 f 422.342 (+)s 5 f 433.555 (a)s 4 f 287.535 y 439.898 (2)s 5 f 285.735 y 445.282 (t)s 1 f 449.615 (\051)s 5 f 454.282 (@)s 6 f 287.535 y 460.653 (y)s 1 f 285.735 y 466.421 (,)s 5 f 300.135 y 0 (X)s 4 f 301.935 y 9.942 (2)s 1 f 300.135 y 18.658 (=)s 5 f 31.325 (@)s 6 f 301.935 y 37.696 (t)s 1 f 300.135 y 41.921 (,)s 5 f 48.882 (X)s 4 f 301.935 y 58.823 (3)s 1 f 300.135 y 67.54 (=)s 5 f 80.207 (@)s 6 f 301.935 y 86.578 (x)s 1 f 300.135 y 92.619 (,)s 5 f 99.58 (X)s 4 f 301.935 y 109.522 (4)s 1 f 300.135 y 118.239 (=)s 5 f 130.905 (@)s 6 f 301.935 y 137.276 (y)s 1 f 300.135 y 143.044 (;)s 150.067 (the)s 170.268 (\257fth)s 195.47 (one)s 217.005 (is)s 228.606 (a)s 238.14 (new)s 262.342 (symmetry)s 317.976 (with)s 344.844 (a)s 354.379 (nonzero)s 5 f 398.614 (B)s 1 f 411.853 (matrix)s 450.421 (and)s 314.535 y 0 (is)s 12.067 (giv)s 27.4 (en)s 43.4 (b)s 49.733 (y)s 5 f 334.519 y 108.562 (X)s 4 f 336.319 y 118.504 (5)s 1 f 334.519 y 131.221 (=)s 147.887 (\050)s 5 f 152.554 (a)s 4 f 336.319 y 158.897 (2)s 5 f 334.519 y 164.28 (t)s 7 f 171.28 (\241)s 5 f 183.28 (y)s 1 f 189.594 (\051)s 5 f 326.401 y 197.13 (@)s 13.896 .48 195.701 331.759 r 342.751 y 195.701 (@)s 202.738 (x)s 1 f 334.519 y 213.703 (+)s 5 f 225.703 (x)s 326.401 y 235.158 (@)s 13.351 .48 234.002 331.759 r 342.751 y 234.002 (@)s 241.039 (y)s 1 f 334.519 y 251.46 (+)s 5 f 263.459 (w)s 4 f 336.319 y 272.05 (2)s 5 f 326.401 y 283.861 (@)s 21.012 .48 278.874 331.759 r 342.751 y 278.874 (@)s 285.911 (w)s 4 f 344.551 y 294.502 (1)s 7 f 334.519 y 303.992 (\241)s 5 f 315.992 (w)s 4 f 336.319 y 324.583 (1)s 5 f 326.401 y 336.394 (@)s 21.012 .48 331.407 331.759 r 342.751 y 331.407 (@)s 338.444 (w)s 4 f 344.551 y 347.035 (2)s 5 f 334.519 y 357.858 (:)s 7 f 359.021 y 0 (\246)s 3 f 385.421 y 0 (Example)s 50.4 (3.)s 1 f 70.522 (W)s 81.855 (e)s 91.918 (will)s 115.315 (no)s 127.648 (w)s 141.044 (consider)s 188.541 (the)s 209.937 (next)s 237.667 (example)s 285.397 (in)s 300.126 ([4],)s 321.039 (i.e.)s 343.894 (the)s 365.291 (t)s 369.624 (w)s 377.957 (o-dimensional)s 454.754 (Ito)s 399.821 y 0 (system)s 413.555 y 155.407 (d)s 5 f 162.074 (x)s 1 f 176.265 (=)s 192.932 (cos)s 208.998 (\050)s 5 f 213.665 (t)s 1 f 217.998 (\051d)s 5 f 229.332 (w)s 4 f 415.355 y 237.923 (1)s 7 f 413.555 y 245.973 (\241)s 1 f 257.973 (sin)s 272.706 (\050)s 5 f 277.373 (t)s 1 f 281.706 (\051d)s 5 f 293.04 (w)s 4 f 415.355 y 301.631 (2)s 1 f 431.555 y 155.951 (d)s 5 f 162.618 (y)s 1 f 176.265 (=)s 192.932 (sin)s 207.665 (\050)s 5 f 212.332 (t)s 1 f 216.665 (\051d)s 5 f 227.999 (w)s 4 f 433.355 y 236.59 (1)s 1 f 431.555 y 244.639 (+)s 256.639 (cos)s 272.706 (\050)s 5 f 277.373 (t)s 1 f 281.706 (\051d)s 5 f 293.04 (w)s 4 f 433.355 y 301.631 (2)s 5 f 431.555 y 311.014 (:)s 1 f 422.555 y 445.087 (\0503)s 5 f 455.754 (:)s 1 f 459.087 (9\051)s 453.359 y 0 (W)s 11.333 (e)s 20.667 (will)s 43.333 (use)s 64.067 (again)s 96.067 (the)s 116.733 (same)s 146.8 (notation)s 194.8 (\0503.2\051)s 223.467 (for)s 5 f 241.834 (\273)s 1 f 251.636 (and)s 5 f 274.969 (B)s 1 f 284.673 (.)s 479.759 y 0 (Notice)s 37.667 (that)s 63.667 (no)s 76 (w)s 5 f 88.667 (f)s 1 f 99.167 (=)s 111.833 (0,)s 125.167 (so)s 139.9 (that)s 165.9 (the)s 186.567 (\257rst)s 211.334 (set)s 230.067 (of)s 243.734 (determining)s 310.434 (equations)s 364.167 (giv)s 379.501 (es)s 393.567 (simply)s 5 f 506.318 y 150.994 (\256)s 6 f 508.118 y 158.67 (xx)s 1 f 506.318 y 172.82 (+)s 5 f 184.82 (\256)s 6 f 508.118 y 192.497 (y)s 197.665 (y)s 1 f 506.318 y 206.766 (=)s 219.433 (0)s 5 f 229.433 (;)s 238.766 (\257)s 6 f 508.118 y 245.553 (xx)s 1 f 506.318 y 259.703 (+)s 5 f 271.703 (\257)s 6 f 508.118 y 278.491 (y)s 283.659 (y)s 1 f 506.318 y 292.76 (=)s 305.427 (0)s 5 f 315.427 (:)s 1 f 439.087 (\0503)s 5 f 449.754 (:)s 1 f 453.087 (10\051)s 532.877 y 0 (As)s 17.733 (for)s 36.1 (the)s 56.767 (second)s 95.5 (set,)s 117.567 (w)s 125.9 (e)s 135.233 (get)s 9 f 551.185 y 13.917 (8)s 561.985 y 13.917 (\076)s 565.585 y 13.917 (\074)s 587.186 y 13.917 (\076)s 590.786 y 13.917 (:)s 1 f 557.569 y 26.584 (cos)s 42.651 (\050)s 5 f 47.317 (t)s 1 f 51.651 (\051)s 5 f 56.317 (\256)s 6 f 559.369 y 63.994 (x)s 1 f 557.569 y 72.702 (+)s 84.702 (sin)s 99.435 (\050)s 5 f 104.102 (t)s 1 f 108.435 (\051)s 5 f 113.102 (\256)s 6 f 559.369 y 120.778 (y)s 1 f 557.569 y 129.213 (+)s 141.213 (sin)s 155.946 (\050)s 5 f 160.613 (t)s 1 f 164.946 (\051)s 5 f 169.613 (\277)s 7 f 178.884 (\241)s 1 f 190.884 (cos)s 206.95 (\050)s 5 f 211.617 (t)s 1 f 215.951 (\051\050)s 5 f 225.284 (\277)s 6 f 559.369 y 230.53 (t)s 5 f 557.569 y 234.755 (=)s 1 f 240.755 (2\051)s 7 f 254.088 (\241)s 5 f 266.088 (b)s 1 f 273.238 (sin)s 287.971 (\050)s 5 f 292.638 (t)s 1 f 296.971 (\051)s 304.971 (=)s 317.638 (0)s 348.305 (\050)s 5 f 352.971 (i;)s 362.439 (k)s 1 f 372.397 (=)s 385.064 (1)s 5 f 391.064 (;)s 1 f 396.397 (1\051)s 411.064 (\050e11\051)s 7 f 571.969 y 26.584 (\241)s 1 f 37.917 (sin)s 52.651 (\050)s 5 f 57.317 (t)s 1 f 61.651 (\051)s 5 f 66.317 (\256)s 6 f 573.769 y 73.994 (x)s 1 f 571.969 y 82.702 (+)s 94.702 (cos)s 110.769 (\050)s 5 f 115.435 (t)s 1 f 119.769 (\051)s 5 f 124.435 (\256)s 6 f 573.769 y 132.112 (y)s 7 f 571.969 y 140.546 (\241)s 1 f 152.546 (cos)s 168.613 (\050)s 5 f 173.28 (t)s 1 f 177.613 (\051)s 5 f 182.28 (\277)s 1 f 191.55 (+)s 203.55 (sin)s 218.284 (\050)s 5 f 222.95 (t)s 1 f 227.284 (\051\050)s 5 f 236.617 (\277)s 6 f 573.769 y 241.863 (t)s 5 f 571.969 y 246.088 (=)s 1 f 252.088 (2\051)s 7 f 265.421 (\241)s 5 f 277.421 (b)s 1 f 284.571 (cos)s 300.638 (\050)s 5 f 305.305 (t)s 1 f 309.638 (\051)s 317.638 (=)s 330.305 (0)s 348.305 (\050)s 5 f 352.971 (i;)s 362.439 (k)s 1 f 372.397 (=)s 385.064 (1)s 5 f 391.064 (;)s 1 f 396.397 (2\051)s 411.064 (\050e12\051)s 586.369 y 26.584 (cos)s 42.651 (\050)s 5 f 47.317 (t)s 1 f 51.651 (\051)s 5 f 56.317 (\257)s 6 f 588.169 y 63.105 (x)s 1 f 586.369 y 71.813 (+)s 83.813 (sin)s 98.546 (\050)s 5 f 103.213 (t)s 1 f 107.546 (\051)s 5 f 112.213 (\257)s 6 f 588.169 y 119.001 (y)s 1 f 586.369 y 127.435 (+)s 139.435 (cos)s 155.502 (\050)s 5 f 160.169 (t)s 1 f 164.502 (\051)s 5 f 169.169 (\277)s 7 f 178.439 (\241)s 1 f 190.439 (sin)s 205.173 (\050)s 5 f 209.839 (t)s 1 f 214.173 (\051\050)s 5 f 223.506 (\277)s 6 f 588.169 y 228.752 (t)s 5 f 586.369 y 232.977 (=)s 1 f 238.977 (2\051)s 252.31 (+)s 5 f 264.31 (b)s 1 f 271.46 (cos)s 287.527 (\050)s 5 f 292.193 (t)s 1 f 296.527 (\051)s 304.527 (=)s 317.193 (0)s 348.305 (\050)s 5 f 352.971 (i;)s 362.439 (k)s 1 f 372.397 (=)s 385.064 (2)s 5 f 391.064 (;)s 1 f 396.397 (1\051)s 411.064 (\050e21\051)s 7 f 600.769 y 26.584 (\241)s 1 f 37.917 (sin)s 52.651 (\050)s 5 f 57.317 (t)s 1 f 61.651 (\051)s 5 f 66.317 (\257)s 6 f 602.569 y 73.105 (x)s 1 f 600.769 y 81.813 (+)s 93.813 (cos)s 109.88 (\050)s 5 f 114.546 (t)s 1 f 118.88 (\051)s 5 f 123.546 (\257)s 6 f 602.569 y 130.334 (y)s 7 f 600.769 y 138.769 (\241)s 1 f 150.768 (sin)s 165.502 (\050)s 5 f 170.169 (t)s 1 f 174.502 (\051)s 5 f 179.169 (\277)s 7 f 188.439 (\241)s 1 f 200.439 (cos)s 216.506 (\050)s 5 f 221.173 (t)s 1 f 225.506 (\051\050)s 5 f 234.839 (\277)s 6 f 602.569 y 240.085 (t)s 5 f 600.769 y 244.31 (=)s 1 f 250.31 (2\051)s 7 f 263.644 (\241)s 5 f 275.643 (b)s 1 f 282.793 (sin)s 297.527 (\050)s 5 f 302.193 (t)s 1 f 306.527 (\051)s 314.527 (=)s 327.193 (0)s 348.305 (\050)s 5 f 352.971 (i;)s 362.439 (k)s 1 f 372.397 (=)s 385.064 (2)s 5 f 391.064 (;)s 1 f 396.397 (2\051)s 411.064 (\050e22\051)s 445.73 (.)s 614.402 y 439.087 (\0503)s 5 f 449.754 (:)s 1 f 453.087 (11\051)s 628.802 y 0 (W)s 11.333 (e)s 21.219 (will)s 44.437 (denote)s 83.656 (b)s 89.989 (y)s 5 f 100.874 (@)s 6 f 630.602 y 107.245 (x)s 1 f 628.802 y 113.287 (\050)s 5 f 117.953 (e)s 1 f 123.541 (11\051)s 144.76 (the)s 165.978 (equation)s 215.53 (obtained)s 265.415 (di\256eren)s 304.115 (tiating)s 343.334 (\050)s 5 f 348.001 (e)s 1 f 353.588 (11\051)s 374.807 (with)s 402.692 (resp)s 424.459 (ect)s 444.344 (to)s 5 f 459.563 (x)s 1 f 466.421 (,)s 643.202 y 0 (and)s 23.333 (so)s 38.067 (on)s 54.733 (for)s 73.1 (similar)s 112.534 (expressions.)s 672.002 y 228.877 (15)s end endp %%Page: 18 18 %%BeginPageSetup initializepage (degasperis; page: 18 of 26)setjob %%EndPageSetup gS 0 0 552 730 rC 0 0 :M f1_12 sf ( )S f3_12 sf .055 .006( )J f8_12 sf ( )S 13 0 :M f4_12 sf .136 .014( )J 21 0 :M f5_12 sf .138 .014( )J 30 0 :M f6_12 sf ( )S f7_12 sf ( )S 0 0 :M initclip TeXdict begin 1 10 1200 1000 div mul /CMR10 findfont fdef 3 10 1200 1000 div mul /CMTT10 findfont fdef 4 7 1200 1000 div mul /CMR7 findfont fdef 5 10 1200 1000 div mul /CMMI10 findfont fdef 6 7 1200 1000 div mul /CMMI7 findfont fdef 7 10 1200 1000 div mul /CMSY10 findfont fdef 9 10 1200 1000 div mul /CMEX10 findfont fdef 42 41 translate 72 72.27 div dup scale 1 f 12 y 0 (By)s 18.833 (considering)s 81.6 (sin)s 96.334 (\050)s 5 f 101 (t)s 1 f 105.334 (\051)s 5 f 110 (@)s 6 f 13.8 y 116.371 (x)s 1 f 12 y 122.413 (\050)s 5 f 127.079 (e)s 1 f 132.667 (11\051)s 152 (+)s 164 (cos)s 180.067 (\050)s 5 f 184.734 (t)s 1 f 189.067 (\051)s 5 f 193.734 (@)s 6 f 13.8 y 200.104 (y)s 1 f 12 y 205.873 (\050)s 5 f 210.539 (e)s 1 f 216.127 (11\051,)s 240.127 (w)s 248.46 (e)s 257.793 (obtain)s 36.406 y 143.815 (sin)s 158.548 (\050)s 5 f 163.215 (t)s 1 f 167.548 (\051)s 174.215 (cos)s 190.281 (\050)s 5 f 194.948 (t)s 1 f 199.281 (\051[)s 5 f 207.281 (\256)s 6 f 38.206 y 214.958 (xx)s 1 f 36.406 y 229.108 (+)s 5 f 241.108 (\256)s 6 f 38.206 y 248.784 (y)s 253.952 (y)s 1 f 36.406 y 259.72 (])s 265.72 (+)s 5 f 277.72 (\256)s 6 f 38.206 y 285.397 (xy)s 1 f 36.406 y 299.94 (=)s 312.606 (0)s 322.606 (;)s 439.087 (\0503)s 5 f 449.754 (:)s 1 f 453.087 (12\051)s 60.813 y 0 (recalling)s 50.324 (\0503.10\051,)s 91.187 (this)s 116.877 (means)s 155.901 (that)s 5 f 184.191 (\256)s 6 f 62.613 y 191.868 (xy)s 1 f 60.813 y 210.228 (=)s 226.712 (0.)s 248.25 (Similarly)s 294.283 (,)s 304.48 (b)s 310.813 (y)s 323.437 (considering)s 388.494 (sin)s 403.227 (\050)s 5 f 407.894 (t)s 1 f 412.227 (\051)s 5 f 416.894 (@)s 6 f 62.613 y 423.265 (x)s 1 f 60.813 y 429.306 (\050)s 5 f 433.973 (e)s 1 f 439.561 (21\051)s 460.421 (+)s 75.213 y 0 (cos)s 16.067 (\050)s 5 f 20.733 (t)s 1 f 25.067 (\051)s 5 f 29.733 (@)s 6 f 77.013 y 36.104 (y)s 1 f 75.213 y 41.872 (\050)s 5 f 46.539 (e)s 1 f 52.127 (21\051)s 74.324 (w)s 82.657 (e)s 93.52 (obtain)s 5 f 132.384 (\257)s 6 f 77.013 y 139.172 (xy)s 1 f 75.213 y 156.265 (=)s 171.482 (0.)s 190.74 (By)s 211.103 (considering)s 275.4 (sin)s 290.134 (\050)s 5 f 294.8 (t)s 1 f 299.134 (\051)s 5 f 303.8 (@)s 6 f 77.013 y 310.171 (y)s 1 f 75.213 y 315.939 (\050)s 5 f 320.606 (e)s 1 f 326.193 (11\051)s 346.547 (+)s 359.567 (cos)s 375.634 (\050)s 5 f 380.3 (t)s 1 f 384.634 (\051)s 5 f 389.3 (@)s 6 f 77.013 y 395.671 (y)s 1 f 75.213 y 401.439 (\050)s 5 f 406.106 (e)s 1 f 411.694 (12\051)s 433.89 (w)s 442.224 (e)s 453.087 (ob-)s 89.613 y 0 (tain)s 5 f 24.808 (\256)s 6 f 91.413 y 32.484 (y)s 37.652 (y)s 1 f 89.613 y 46.989 (=)s 59.891 (0,)s 73.401 (and)s 96.876 (th)s 107.876 (us)s 123.417 (b)s 129.75 (y)s 140.225 (\0503.10\051)s 5 f 175.033 (\256)s 6 f 91.413 y 182.709 (xx)s 1 f 89.613 y 197.761 (=)s 210.663 (0)s 220.805 (as)s 235.679 (w)s 244.012 (ell.)s 265.103 (Similarly)s 311.136 (,)s 318.646 (sin)s 333.38 (\050)s 5 f 338.046 (t)s 1 f 342.38 (\051)s 5 f 347.046 (@)s 6 f 91.413 y 353.417 (y)s 1 f 89.613 y 359.185 (\050)s 5 f 363.852 (e)s 1 f 369.439 (21\051)s 388.867 (+)s 400.961 (cos)s 417.028 (\050)s 5 f 421.694 (t)s 1 f 426.028 (\051)s 5 f 430.694 (@)s 6 f 91.413 y 437.065 (y)s 1 f 89.613 y 442.833 (\050)s 5 f 447.5 (e)s 1 f 453.087 (22\051)s 104.013 y 0 (yields)s 5 f 33.733 (\257)s 6 f 105.813 y 40.521 (xx)s 1 f 104.013 y 55.337 (=)s 5 f 68.004 (\257)s 6 f 105.813 y 74.792 (y)s 79.96 (y)s 1 f 104.013 y 89.061 (=)s 101.728 (0.)s 116.394 (Th)s 131.394 (us,)s 5 f 150.128 (\256)s 1 f 161.849 (and)s 5 f 185.182 (\257)s 1 f 196.603 (are)s 216.636 (at)s 231.303 (most)s 260.703 (linear)s 294.07 (in)s 5 f 308.07 (x)s 1 f 318.928 (and)s 5 f 342.261 (y)s 1 f 348.575 (.)s 130.413 y 0 (W)s 11.333 (e)s 20.667 (use)s 41.4 (again)s 73.4 (com)s 94.4 (binations)s 146.467 (of)s 160.134 (the)s 180.8 (equations)s 234.534 (giv)s 249.867 (en)s 265.867 (in)s 279.867 (\0503.11\051:)s 9 f 144.619 y 44.54 (8)s 155.419 y 44.54 (\076)s 159.02 y 44.54 (\076)s 162.62 y 44.54 (\074)s 184.22 y 44.54 (\076)s 187.82 y 44.54 (\076)s 191.42 y 44.54 (:)s 1 f 155.55 y 57.206 (sin)s 71.94 (\050)s 5 f 76.606 (t)s 1 f 80.94 (\051\050)s 5 f 90.273 (e)s 1 f 95.861 (11\051)s 115.194 (+)s 127.194 (cos)s 143.261 (\050)s 5 f 147.927 (t)s 1 f 152.261 (\051\050)s 5 f 161.594 (e)s 1 f 167.181 (12\051)s 187.181 (:)s 5 f 201.848 (\256)s 6 f 157.35 y 209.524 (y)s 1 f 155.55 y 217.959 (+)s 5 f 229.959 (\277)s 1 f 236.563 ([sin)s 4 f 150.503 y 254.63 (2)s 1 f 155.55 y 260.013 (\050)s 5 f 264.68 (t)s 1 f 269.013 (\051)s 7 f 276.347 (\241)s 1 f 288.347 (cos)s 4 f 151.195 y 304.413 (2)s 1 f 155.55 y 309.797 (\050)s 5 f 314.463 (t)s 1 f 318.797 (\051])s 7 f 329.463 (\241)s 5 f 341.463 (b)s 1 f 349.947 (=)s 362.613 (0)s 169.95 y 57.206 (sin)s 71.94 (\050)s 5 f 76.606 (t)s 1 f 80.94 (\051\050)s 5 f 90.273 (e)s 1 f 95.861 (21\051)s 115.194 (+)s 127.194 (cos)s 143.261 (\050)s 5 f 147.927 (t)s 1 f 152.261 (\051\050)s 5 f 161.594 (e)s 1 f 167.181 (22\051)s 187.181 (:)s 5 f 201.848 (\257)s 6 f 171.75 y 208.636 (y)s 7 f 169.95 y 217.07 (\241)s 5 f 229.07 (\277)s 6 f 171.75 y 234.316 (t)s 5 f 169.95 y 238.541 (=)s 1 f 244.541 (2)s 253.874 (=)s 266.541 (0)s 184.35 y 57.206 (cos)s 73.273 (\050)s 5 f 77.94 (t)s 1 f 82.273 (\051\050)s 5 f 91.606 (e)s 1 f 97.194 (11\051)s 7 f 116.527 (\241)s 1 f 128.527 (sin)s 143.261 (\050)s 5 f 147.927 (t)s 1 f 152.261 (\051\050)s 5 f 161.594 (e)s 1 f 167.181 (12\051)s 187.181 (:)s 5 f 201.848 (\256)s 6 f 186.15 y 209.524 (x)s 1 f 184.35 y 218.233 (+)s 230.233 (2)s 238.233 (sin)s 252.966 (\050)s 5 f 257.633 (t)s 1 f 261.966 (\051)s 268.633 (cos)s 284.699 (\050)s 5 f 289.366 (t)s 1 f 293.699 (\051)s 5 f 298.366 (\277)s 7 f 307.637 (\241)s 1 f 319.637 (\050)s 5 f 324.303 (\277)s 6 f 186.15 y 329.549 (t)s 5 f 184.35 y 333.774 (=)s 1 f 339.774 (2\051)s 353.774 (=)s 366.441 (0)s 198.75 y 57.206 (cos)s 73.273 (\050)s 5 f 77.94 (t)s 1 f 82.273 (\051\050)s 5 f 91.606 (e)s 1 f 97.194 (21\051)s 7 f 116.527 (\241)s 1 f 128.527 (sin)s 143.261 (\050)s 5 f 147.927 (t)s 1 f 152.261 (\051\050)s 5 f 161.594 (e)s 1 f 167.181 (22\051)s 187.181 (:)s 5 f 201.848 (\257)s 6 f 200.55 y 208.636 (x)s 1 f 198.75 y 217.344 (+)s 5 f 229.344 (\277)s 1 f 238.615 (+)s 5 f 250.615 (b)s 1 f 259.098 (=)s 271.764 (0)s 384.441 (.)s 176.42 y 439.087 (\0503)s 5 f 449.754 (:)s 1 f 453.087 (13\051)s 233.76 y 0 (W)s 11.333 (e)s 20.667 (obtain)s 58 (in)s 72 (this)s 95.4 (w)s 103.733 (a)s 109.4 (y)s 5 f 257.5 y 108.086 (\256)s 1 f 123.14 (=)s 139.807 ([)s 5 f 143.14 (\277)s 6 f 259.3 y 148.386 (t)s 5 f 257.5 y 152.611 (=)s 1 f 158.611 (2)s 7 f 167.278 (\241)s 1 f 179.278 (sin)s 194.011 (\0502)s 5 f 204.678 (t)s 1 f 209.011 (\051)s 5 f 213.678 (\277)s 1 f 220.282 (])s 5 f 223.615 (x)s 1 f 233.14 (+)s 245.14 ([)s 5 f 248.474 (b)s 1 f 256.29 (+)s 268.29 (cos)s 284.357 (\0502)s 5 f 295.023 (t)s 1 f 299.357 (\051)s 5 f 304.023 (\277)s 1 f 310.628 (])s 5 f 313.961 (y)s 1 f 322.942 (+)s 5 f 334.941 (\256)s 4 f 259.3 y 342.618 (0)s 1 f 257.5 y 348.001 (\050)s 5 f 352.668 (t)s 1 f 357.001 (\051)s 5 f 275.5 y 108.386 (\257)s 1 f 123.14 (=)s 7 f 139.807 (\241)s 1 f 149.14 ([)s 5 f 152.474 (b)s 1 f 160.29 (+)s 5 f 172.29 (\277)s 1 f 178.894 (])s 5 f 182.228 (x)s 1 f 191.753 (+)s 203.753 ([)s 5 f 207.086 (\277)s 6 f 277.3 y 212.332 (t)s 5 f 275.5 y 216.557 (=)s 1 f 222.557 (2])s 5 f 231.89 (y)s 1 f 240.871 (+)s 5 f 252.871 (\257)s 4 f 277.3 y 259.658 (0)s 1 f 275.5 y 265.041 (\050)s 5 f 269.708 (t)s 1 f 274.041 (\051)s 282.708 (;)s 266.5 y 439.087 (\0503)s 5 f 449.754 (:)s 1 f 453.087 (14\051)s 299.906 y 0 (one)s 22 (can)s 44 (c)s 49 (hec)s 66 (k,)s 79.667 (e.g.)s 103 (b)s 109.333 (y)s 119.667 (substituting)s 186.467 (in)s 200.467 (\0503.11\051,)s 238.467 (that)s 264.467 (this)s 287.867 (is)s 299.934 (the)s 320.601 (most)s 350.001 (general)s 391.367 (solution.)s 326.306 y 0 (By)s 19.741 (setting)s 5 f 60.049 (\277)s 1 f 71.498 (=)s 5 f 85.678 (\256)s 4 f 328.106 y 93.354 (0)s 1 f 326.306 y 103.583 (=)s 5 f 117.762 (\257)s 4 f 328.106 y 124.55 (0)s 1 f 326.306 y 134.779 (=)s 148.958 (0,)s 163.426 (w)s 171.759 (e)s 182 (obtain)s 220.241 (the)s 241.815 (new)s 267.389 (\050with)s 300.297 (resp)s 322.064 (ect)s 342.304 (to)s 357.879 ([4]\051)s 380.12 (symmetry)s 431.22 (,)s 439.687 (corre-)s 340.706 y 0 (sp)s 11.733 (onding)s 51.067 (to)s 5 f 65.733 (\256)s 1 f 76.788 (=)s 5 f 89.454 (by)s 1 f 104.918 (and)s 5 f 128.251 (\257)s 1 f 139.006 (=)s 7 f 151.672 (\241)s 5 f 161.006 (bx)s 1 f 177.014 (with)s 5 f 204.347 (b)s 1 f 213.497 (arbitrary)s 259.597 (,)s 266.931 (i.e.)s 287.597 (giv)s 302.931 (en)s 318.931 (b)s 325.264 (y)s 5 f 371.646 y 103.501 (X)s 1 f 121.717 (=)s 9 f 354.726 y 138.384 (\265)s 5 f 371.646 y 147.217 (y)s 363.528 y 158.4 (@)s 13.896 .48 154.971 368.886 r 379.877 y 154.971 (@)s 162.009 (x)s 7 f 371.646 y 174.973 (\241)s 5 f 188.973 (x)s 363.528 y 200.429 (@)s 13.351 .48 197.272 368.886 r 379.877 y 197.272 (@)s 204.309 (y)s 9 f 354.726 y 212.063 (\266)s 1 f 371.646 y 227.563 (+)s 9 f 354.726 y 243.563 (\265)s 5 f 371.646 y 252.396 (w)s 4 f 366.691 y 261.31 (2)s 5 f 363.528 y 275.282 (@)s 21.335 .48 268.134 368.886 r 379.877 y 268.134 (@)s 275.171 (w)s 4 f 376.411 y 284.085 (1)s 7 f 371.646 y 295.575 (\241)s 5 f 309.575 (w)s 4 f 366.691 y 318.489 (1)s 5 f 363.528 y 332.461 (@)s 21.335 .48 325.312 368.886 r 379.877 y 325.312 (@)s 332.35 (w)s 4 f 376.411 y 341.263 (2)s 9 f 354.726 y 348.087 (\266)s 5 f 371.646 y 362.92 (:)s 1 f 439.087 (\0503)s 5 f 449.754 (:)s 1 f 453.087 (15\051)s 415.253 y 0 (Note)s 29.944 (that)s 56.888 (this)s 81.232 (is)s 94.242 (nothing)s 139.186 (else)s 162.864 (than)s 191.808 (a)s 202.751 (sim)s 220.485 (ultaneous)s 275.495 (\050and)s 304.439 (iden)s 326.106 (tical\051)s 358.383 (rotation)s 405.361 (in)s 420.305 (the)s 441.915 (\050)s 5 f 446.582 (x;)s 458.774 (y)s 1 f 465.087 (\051)s 429.653 y 0 (and)s 23.333 (in)s 37.333 (the)s 58 (\050)s 5 f 62.667 (w)s 4 f 431.453 y 71.258 (1)s 5 f 429.653 y 76.641 (;)s 81.974 (w)s 4 f 431.453 y 90.565 (2)s 1 f 429.653 y 95.949 (\051)s 104.615 (planes.)s 7 f 146.016 (\246)s 3 f 456.053 y 0 (Example)s 50.4 (4.)s 1 f 72.687 (W)s 84.021 (e)s 94.805 (will)s 118.924 (no)s 131.257 (w)s 145.375 (consider)s 5 f 193.593 (n)s 1 f 206.248 (uncoupled)s 265.033 (equal)s 298.151 (\134Langevin)s 357.436 (harmonic)s 411.587 (oscillators\042)s 470.453 y 0 (sub)s 18.733 (ject)s 43.314 (to)s 59.562 (indep)s 88.562 (enden)s 118.896 (t)s 129.143 (sto)s 144.877 (c)s 149.877 (hastic)s 186.191 (noises)s 222.572 (of)s 237.82 (p)s 244.82 (ossibly)s 285.534 (di\256eren)s 324.234 (t)s 334.482 (strength.)s 391.325 (This)s 420.306 (system)s 461.687 (is)s 484.853 y 0 (written)s 42.033 (as)s 499.253 y 122.892 (d)s 5 f 129.559 (x)s 6 f 494.298 y 136.417 (i)s 1 f 499.253 y 147.746 (=)s 7 f 164.412 (\241)s 5 f 173.746 (x)s 6 f 494.298 y 180.604 (i)s 1 f 499.253 y 186.599 (d)s 5 f 193.266 (t)s 1 f 202.266 (+)s 7 f 489.64 y 216.266 (p)s 15.62 .48 226.266 489.64 r 1 f 499.253 y 226.266 (2)s 5 f 232.266 (s)s 6 f 501.052 y 237.891 (i)s 1 f 499.253 y 243.886 (d)s 5 f 250.553 (w)s 6 f 494.298 y 259.466 (i)s 1 f 499.253 y 275.462 (\050no)s 296.795 (sum)s 322.195 (on)s 338.862 (i\051)s 439.087 (\0503)s 5 f 449.754 (:)s 1 f 453.087 (16\051)s 520.1 y 0 (and)s 22.816 (is)s 34.366 (considered)s 92.616 (e.g.)s 115.777 (in)s 129.26 ([5];)s 148.916 (ob)s 161.249 (viously)s 5 f 201.466 (s)s 6 f 521.9 y 207.091 (i)s 7 f 520.1 y 214.419 (\270)s 1 f 227.086 (0,)s 240.005 (and)s 262.822 (w)s 271.155 (e)s 279.972 (assume)s 320.921 (here)s 346.438 (they)s 372.921 (are)s 392.438 (strictly)s 433.021 (greater)s 534.5 y 0 (than)s 28 (zero.)s 560.9 y 0 (The)s 24.667 (\257rst)s 49.433 (set)s 68.167 (of)s 81.833 (determining)s 148.534 (equations)s 202.267 (\0501.10\051)s 236.934 (read)s 263.634 (no)s 275.967 (w)s 5 f 585.306 y 145.814 (\273)s 6 f 580.352 y 151.616 (i)s 588.273 y 151.064 (t)s 1 f 585.306 y 158.277 (+)s 5 f 170.277 (\273)s 6 f 580.352 y 176.08 (i)s 7 f 585.306 y 182.741 (\241)s 5 f 194.741 (x)s 6 f 580.352 y 201.6 (j)s 5 f 585.306 y 206.656 (\273)s 6 f 580.352 y 212.458 (i)s 588.273 y 211.906 (j)s 1 f 585.306 y 219.629 (+)s 5 f 231.629 (\277)s 6 f 587.106 y 236.875 (t)s 5 f 585.306 y 241.1 (x)s 6 f 580.352 y 247.958 (i)s 1 f 585.306 y 254.62 (+)s 5 f 266.62 (s)s 6 f 587.106 y 272.245 (k)s 5 f 585.306 y 278.149 (\273)s 6 f 580.352 y 283.951 (i)s 588.273 y 283.399 (k)s 288.703 (k)s 1 f 585.306 y 297.94 (=)s 310.607 (0)s 5 f 320.607 (;)s 1 f 439.087 (\0503)s 5 f 449.754 (:)s 1 f 453.087 (17\051)s 609.713 y 0 (while)s 31.333 (the)s 52 (second)s 90.733 (set)s 109.467 (is)s 121.533 (\050no)s 142.867 (sum)s 168.267 (on)s 5 f 184.934 (i;)s 194.401 (k)s 1 f 201.026 (\051)s 5 f 639.771 y 170.92 (s)s 6 f 641.571 y 176.545 (k)s 5 f 639.771 y 182.45 (\273)s 6 f 634.816 y 188.252 (i)s 642.737 y 187.7 (k)s 1 f 639.771 y 196.937 (=)s 5 f 209.604 (s)s 6 f 641.571 y 215.229 (i)s 5 f 639.771 y 219.224 (B)s 6 f 634.816 y 228.928 (i)s 642.737 y 228.326 (k)s 1 f 639.771 y 236.897 (+)s 631.653 y 250.337 (1)s 6 .48 250.337 637.011 r 648.002 y 250.337 (2)s 5 f 639.771 y 257.777 (s)s 6 f 641.571 y 263.402 (i)s 5 f 639.771 y 267.397 (\277)s 6 f 641.571 y 272.643 (t)s 5 f 639.771 y 276.868 (\261)s 6 f 641.571 y 282.201 (ik)s 5 f 639.771 y 295.5 (:)s 1 f 439.087 (\0503)s 5 f 449.754 (:)s 1 f 453.087 (18\051)s 672.002 y 228.877 (16)s end endp %%Page: 19 19 %%BeginPageSetup initializepage (degasperis; page: 19 of 26)setjob %%EndPageSetup gS 0 0 552 730 rC 0 0 :M f0_12 sf ( )S 4 0 :M f1_12 sf ( )S f3_12 sf .055 .006( )J f8_12 sf ( )S 17 0 :M f4_12 sf .136 .014( )J 25 0 :M f5_12 sf .138 .014( )J 34 0 :M f6_12 sf ( )S f9_12 sf 11.923 1.192( )J 0 0 :M initclip TeXdict begin 0 10 1200 1000 div mul /CMBX10 findfont fdef 1 10 1200 1000 div mul /CMR10 findfont fdef 3 10 1200 1000 div mul /CMTT10 findfont fdef 4 7 1200 1000 div mul /CMR7 findfont fdef 5 10 1200 1000 div mul /CMMI10 findfont fdef 6 7 1200 1000 div mul /CMMI7 findfont fdef 7 10 1200 1000 div mul /CMSY10 findfont fdef 8 7 1200 1000 div mul /CMSY7 findfont fdef 42 41 translate 72 72.27 div dup scale 1 f 12 y 0 (The)s 24.667 (latter)s 57.367 (giv)s 72.7 (es)s 86.767 (at)s 101.433 (once)s 5 f 26.4 y 175.347 (\273)s 6 f 21.445 y 181.149 (i)s 1 f 26.4 y 192.477 (=)s 5 f 209.144 (C)s 6 f 21.445 y 218.579 (i)s 29.367 y 217.721 (j)s 1 f 26.4 y 222.777 (\050)s 5 f 227.444 (t)s 1 f 231.777 (\051)s 5 f 236.444 (x)s 6 f 21.445 y 243.302 (j)s 1 f 26.4 y 253.025 (+)s 5 f 267.025 (\256)s 6 f 21.445 y 274.745 (i)s 1 f 26.4 y 278.741 (\050)s 5 f 283.407 (t)s 1 f 287.741 (\051)s 439.087 (\0503)s 5 f 449.754 (:)s 1 f 453.087 (19\051)s 48.869 y 0 (where)s 34.7 (the)s 55.367 (matrix)s 5 f 94.4 (C)s 1 f 107.835 (is)s 119.902 (giv)s 135.235 (en)s 151.235 (b)s 157.569 (y)s 167.902 (\050again)s 204.569 (no)s 221.235 (sum)s 246.635 (on)s 5 f 263.302 (i;)s 272.77 (k)s 1 f 279.395 (\051)s 5 f 82.576 y 171.135 (C)s 6 f 77.621 y 180.57 (i)s 85.542 y 179.711 (k)s 1 f 82.576 y 192.949 (=)s 74.458 y 211.056 (1)s 6 .48 211.056 79.816 r 90.807 y 211.056 (2)s 5 f 82.576 y 218.496 (\277)s 6 f 84.376 y 223.741 (t)s 5 f 82.576 y 227.966 (\261)s 6 f 77.621 y 233.754 (i)s 85.542 y 233.3 (k)s 1 f 82.576 y 245.871 (+)s 5 f 74.458 y 264.265 (s)s 6 f 76.258 y 269.89 (i)s 11.529 .48 263.31 79.816 r 5 f 90.807 y 263.31 (s)s 6 f 92.607 y 268.935 (k)s 5 f 82.576 y 276.28 (B)s 6 f 77.621 y 285.984 (i)s 85.542 y 285.382 (k)s 5 f 82.576 y 295.286 (:)s 1 f 439.087 (\0503)s 5 f 449.754 (:)s 1 f 453.087 (20\051)s 116.462 y 0 (Inserting)s 50.433 (this)s 73.833 (in)s 83.5 (to)s 98.167 (\0503.17\051,)s 136.167 (the)s 156.834 (terms)s 190.267 (con)s 207.934 (taining)s 5 f 248.6 (B)s 1 f 262.305 (cancel)s 298.305 (out)s 319.638 (and)s 342.971 (w)s 351.305 (e)s 360.638 (obtain)s 5 f 144.518 y 171.118 (\277)s 6 f 146.318 y 176.364 (tt)s 1 f 144.518 y 187.548 (=)s 7 f 200.214 (\241)s 1 f 209.548 (2)s 5 f 215.548 (\277)s 6 f 146.318 y 220.793 (t)s 5 f 144.518 y 229.018 (;)s 238.352 (\256)s 6 f 139.563 y 246.073 (i)s 147.484 y 246.028 (t)s 1 f 144.518 y 253.586 (=)s 7 f 266.253 (\241)s 5 f 275.586 (\256)s 6 f 139.563 y 283.307 (i)s 1 f 144.518 y 295.302 (;)s 439.087 (\0503)s 5 f 449.754 (:)s 1 f 453.087 (21\051)s 172.573 y 0 (these)s 31.644 (are)s 52.587 (kno)s 71.254 (wn)s 91.498 (from)s 120.775 (example)s 168.685 (7)s 179.596 (of)s 194.173 ([4])s 211.75 (and)s 235.993 (giv)s 251.327 (e)s 261.57 (symmetries)s 325.647 (studied)s 368.625 (there.)s 406.722 (Setting)s 5 f 448.966 (\277)s 1 f 460.421 (=)s 5 f 186.973 y 0 (\256)s 6 f 182.618 y 7.721 (i)s 1 f 186.973 y 15.049 (=)s 27.716 (0)s 36.827 (to)s 50.605 (discard)s 91.149 (these)s 120.994 (kno)s 139.66 (wn)s 158.105 (symmetries,)s 223.894 (w)s 232.227 (e)s 240.672 (still)s 263.183 (ha)s 275.516 (v)s 281.516 (e)s 289.961 (the)s 309.739 (solution)s 354.25 (corresp)s 392.05 (onding)s 430.494 (to)s 5 f 444.272 (B)s 1 f 457.087 (an)s 201.373 y 0 (arbitrary)s 51.63 (\050real,)s 83.66 (an)s 95.993 (tisymmetric\051)s 167.624 (constan)s 207.357 (t)s 216.554 (matrix)s 256.118 (and)s 5 f 279.982 (\273)s 6 f 197.018 y 285.784 (i)s 1 f 201.373 y 293.996 (=)s 5 f 307.547 (C)s 6 f 197.018 y 316.982 (i)s 204.771 y 320.123 (k)s 5 f 201.373 y 326.028 (x)s 6 f 197.018 y 332.886 (k)s 1 f 201.373 y 338.79 (,)s 346.786 (where)s 5 f 382.017 (C)s 6 f 197.018 y 391.452 (i)s 204.771 y 394.593 (k)s 1 f 201.373 y 404.715 (=)s 418.265 (\050)s 5 f 422.932 (s)s 6 f 203.173 y 428.557 (i)s 5 f 201.373 y 432.552 (=s)s 6 f 203.173 y 444.177 (k)s 1 f 201.373 y 450.081 (\051)s 5 f 454.748 (B)s 6 f 197.018 y 464.452 (i)s 204.771 y 463.85 (k)s 1 f 215.773 y 0 (\050no)s 21.333 (sum)s 46.733 (on)s 5 f 63.4 (i;)s 72.868 (k)s 1 f 83.493 (here\051.)s 242.173 y 0 (The)s 25.162 (meaning)s 73.657 (of)s 87.818 (this)s 111.713 (result)s 145.642 (is)s 158.203 (ob)s 170.537 (vious:)s 207.26 (w)s 215.593 (e)s 225.421 (can)s 247.916 (act)s 268.411 (on)s 285.573 (this)s 309.468 (system)s 349.763 (b)s 356.096 (y)s 366.925 (an)s 384.086 (arbitrary)s 5 f 435.681 (S)s 443.731 (O)s 1 f 453.218 (\050)s 5 f 457.885 (n)s 1 f 465.087 (\051)s 256.573 y 0 (rotation)s 46.033 (in)s 60.033 (the)s 0 f 80.7 (w)s 1 f 94.858 (space,)s 130.258 (while)s 161.592 (op)s 174.592 (erating)s 215.292 (a)s 225.292 (related)s 265.325 (rotation)s 311.359 (in)s 325.359 (the)s 0 f 346.026 (x)s 1 f 357.309 (space.)s 282.973 y 0 (If)s 12.426 (w)s 20.76 (e)s 30.52 (assume)s 5 f 72.413 (s)s 4 f 284.773 y 78.038 (1)s 1 f 282.973 y 87.465 (=)s 5 f 100.843 (s)s 4 f 284.773 y 106.468 (2)s 1 f 282.973 y 115.895 (=)s 5 f 129.272 (:::)s 1 f 143.316 (=)s 5 f 156.694 (s)s 6 f 284.773 y 162.319 (n)s 1 f 282.973 y 172.895 (=)s 5 f 186.272 (s)s 1 f 191.897 (,)s 199.764 (w)s 208.097 (e)s 217.857 (ha)s 230.19 (v)s 236.19 (e)s 245.95 (indeed)s 5 f 284.376 (C)s 1 f 297.855 (=)s 5 f 311.233 (B)s 1 f 325.363 (and)s 349.123 (the)s 370.216 (rotations)s 421.41 (in)s 0 f 435.836 (w)s 1 f 450.421 (and)s 0 f 297.373 y 0 (x)s 1 f 12.441 (spaces)s 50.4 (do)s 68.225 (just)s 93.116 (coincide.)s 147.258 (F)s 154.091 (or)s 169.949 (general)s 5 f 212.474 (s)s 4 f 299.173 y 218.099 (1)s 5 f 297.373 y 223.482 (;)s 228.816 (:::;)s 244.149 (s)s 6 f 299.173 y 249.774 (n)s 1 f 297.373 y 256.306 (,)s 265.087 (the)s 286.912 (\050)s 5 f 291.579 (s)s 6 f 299.173 y 297.204 (i)s 5 f 297.373 y 301.199 (=s)s 6 f 299.173 y 312.824 (k)s 1 f 297.373 y 318.728 (\051)s 328.553 (factors)s 368.811 (relating)s 414.003 (the)s 5 f 435.828 (C)s 1 f 450.421 (and)s 5 f 311.773 y 0 (B)s 1 f 14.218 (matrices)s 62.833 (are)s 83.38 (also)s 107.961 (easily)s 141.542 (understo)s 187.309 (o)s 193.642 (d:)s 210.004 (w)s 218.338 (e)s 228.185 (could)s 260.699 (rescale)s 299.98 (eac)s 316.314 (h)s 327.495 (of)s 341.676 (the)s 5 f 362.857 (x)s 6 f 307.418 y 369.715 (i)s 1 f 311.773 y 378.224 (b)s 384.558 (y)s 395.405 (a)s 405.92 (factor)s 7 f 302.746 y 440.801 (p)s 15.62 .48 450.801 302.746 r 1 f 311.773 y 450.801 (2)s 5 f 456.801 (s)s 6 f 313.573 y 462.426 (i)s 1 f 311.773 y 466.421 (,)s 5 f 326.173 y 0 (x)s 6 f 321.818 y 6.858 (i)s 1 f 326.173 y 14.187 (=)s 7 f 317.146 y 26.853 (p)s 15.62 .48 36.853 317.146 r 1 f 326.173 y 36.853 (2)s 5 f 42.853 (s)s 6 f 327.973 y 48.478 (i)s 5 f 326.173 y 52.474 (y)s 6 f 321.818 y 58.787 (i)s 1 f 326.173 y 66.44 (\050no)s 87.431 (sum)s 112.488 (on)s 5 f 128.812 (i)s 1 f 132.947 (\051,)s 144.673 (arriving)s 189.397 (at)s 203.721 (the)s 224.045 (manifestly)s 281.769 (rotationally)s 346.46 (in)s 356.127 (v)s 361.794 (arian)s 388.16 (t)s 5 f 396.485 (n)s 1 f 403.687 (-dimensional)s 340.573 y 0 (Ito)s 19 (equation)s 68 (d)s 5 f 74.667 (y)s 6 f 336.218 y 80.981 (i)s 1 f 340.573 y 88.309 (=)s 7 f 100.976 (\241)s 5 f 110.309 (y)s 6 f 336.218 y 116.623 (i)s 1 f 340.573 y 120.618 (d)s 5 f 127.285 (t)s 1 f 134.285 (+)s 146.285 (d)s 5 f 152.951 (w)s 6 f 336.218 y 161.865 (i)s 1 f 340.573 y 165.86 (.)s 7 f 174.527 (\246)s 3 f 366.973 y 0 (Example)s 50.4 (5.)s 1 f 68.154 (Let)s 89.119 (us)s 103.983 (consider)s 150.215 (systems)s 194.213 (in)s 207.677 (whic)s 231.344 (h)s 241.475 (the)s 261.606 (drift)s 5 f 288.104 (f)s 1 f 295.271 (\050)s 5 f 299.938 (x;)s 312.129 (t)s 1 f 316.462 (\051)s 324.594 (is)s 336.125 (a)s 345.589 (time-indep)s 401.923 (enden)s 432.256 (t)s 440.387 (linear)s 381.373 y 0 (function)s 47 (of)s 60.667 (the)s 5 f 81.334 (x)s 1 f 92.192 (and)s 5 f 115.525 (\276)s 1 f 126.813 (is)s 138.879 (constan)s 178.613 (t)s 187.279 (and)s 210.613 (in)s 220.28 (v)s 226.28 (ertible,)s 266.98 (i.e.)s 409.428 y 169.731 (d)s 5 f 176.398 (x)s 6 f 404.473 y 183.256 (i)s 1 f 409.428 y 194.584 (=)s 5 f 211.251 (M)s 6 f 404.473 y 224.201 (i)s 412.395 y 222.893 (j)s 5 f 409.428 y 228.196 (x)s 6 f 404.473 y 235.054 (j)s 1 f 409.428 y 240.111 (d)s 5 f 246.777 (t)s 1 f 253.777 (+)s 5 f 265.777 (\276)s 6 f 404.473 y 273.065 (i)s 412.395 y 272.634 (k)s 1 f 409.428 y 278.538 (d)s 5 f 285.205 (w)s 6 f 404.473 y 294.119 (k)s 1 f 409.428 y 439.087 (\0503)s 5 f 449.754 (:)s 1 f 453.087 (22\051)s 437.483 y 0 (with)s 5 f 27.333 (M)s 1 f 44.283 (and)s 5 f 67.617 (\276)s 1 f 78.904 (constan)s 118.638 (t)s 127.304 (matrices.)s 463.883 y 0 (In)s 15 (this)s 38.4 (case)s 63.8 (the)s 84.467 (second)s 123.2 (of)s 136.867 (\0501.10\051)s 171.534 (is)s 183.6 (simply)s 5 f 491.939 y 158.4 (\276)s 6 f 486.172 y 165.688 (j)s 495.559 y 165.257 (k)s 5 f 491.939 y 171.161 (\273)s 6 f 486.984 y 176.963 (i)s 494.905 y 176.411 (j)s 7 f 491.939 y 184.134 (\241)s 1 f 196.134 (\050)s 5 f 200.801 (\277)s 6 f 493.739 y 206.047 (t)s 5 f 491.939 y 210.272 (=)s 1 f 216.272 (2\051)s 5 f 226.938 (\276)s 6 f 486.984 y 234.226 (i)s 494.905 y 233.795 (k)s 7 f 491.939 y 242.366 (\241)s 5 f 254.366 (\276)s 6 f 486.984 y 261.654 (i)s 494.905 y 261.223 (p)s 5 f 491.939 y 266.77 (B)s 6 f 486.172 y 276.474 (p)s 495.559 y 275.872 (k)s 1 f 491.939 y 285.354 (=)s 298.021 (0)s 308.021 (;)s 439.087 (\0503)s 5 f 449.754 (:)s 1 f 453.087 (23\051)s 521.488 y 0 (di\256eren)s 38.7 (tiating)s 77.367 (this)s 100.767 (in)s 5 f 114.767 (x)s 6 f 517.134 y 121.625 (m)s 1 f 521.488 y 134.741 (and)s 158.074 (m)s 167.741 (ultiplying)s 222.074 (from)s 250.441 (the)s 271.108 (left)s 292.108 (b)s 298.441 (y)s 308.774 (\050)s 5 f 313.441 (\276)s 6 f 517.134 y 320.728 (T)s 1 f 521.488 y 328.288 (\051)s 8 f 517.134 y 332.955 (\241)s 4 f 340.455 (1)s 1 f 521.488 y 349.838 (w)s 358.171 (e)s 367.505 (get)s 5 f 387.505 (\273)s 6 f 517.134 y 393.307 (i)s 524.612 y 392.755 (j)s 397.211 (m)s 1 f 521.488 y 409.66 (=)s 422.326 (0,)s 435.66 (i.e.)s 5 f 551.614 y 181.727 (\273)s 6 f 546.659 y 187.529 (i)s 1 f 551.614 y 194.857 (=)s 5 f 207.524 (L)s 6 f 546.659 y 215.691 (i)s 554.58 y 215.691 (j)s 1 f 551.614 y 220.747 (\050)s 5 f 225.414 (t)s 1 f 229.747 (\051)s 5 f 234.414 (x)s 6 f 546.659 y 241.272 (j)s 1 f 551.614 y 248.995 (+)s 5 f 260.995 (P)s 6 f 546.659 y 270.366 (i)s 1 f 551.614 y 274.361 (\050)s 5 f 279.027 (t)s 1 f 283.361 (\051)s 439.087 (\0503)s 5 f 449.754 (:)s 1 f 453.087 (24\051)s 579.669 y 0 (\050cf.)s 22.098 (example)s 68.391 (8)s 77.684 (in)s 90.977 ([4]\051.)s 116.742 (W)s 128.075 (e)s 136.702 (can)s 157.995 (th)s 168.995 (us)s 183.688 (rewrite)s 223.715 (the)s 243.675 (\257rst)s 267.735 (of)s 280.695 (\0501.10\051)s 314.654 (as)s 328.681 (a)s 337.974 (system)s 377.067 (of)s 390.027 (three)s 420.021 (equations)s 594.069 y 0 (in)s 14 (matrix)s 53.033 (form,)s 5 f 607.802 y 178.394 (@)s 6 f 609.602 y 184.765 (t)s 5 f 607.802 y 188.99 (L)s 1 f 199.823 (+)s 211.823 ([)s 5 f 215.156 (L;)s 228.656 (M)s 1 f 241.606 (])s 252.273 (=)s 5 f 268.939 (\277)s 6 f 609.602 y 274.185 (t)s 5 f 607.802 y 278.41 (M)s 625.802 y 178.394 (@)s 6 f 627.602 y 184.765 (t)s 5 f 625.802 y 188.99 (P)s 1 f 205.694 (=)s 5 f 222.36 (M)s 235.31 (P)s 643.802 y 178.394 (L\276)s 1 f 197.181 (=)s 5 f 209.848 (\276)s 217.135 (B)s 1 f 229.506 (+)s 241.506 (\050)s 5 f 246.173 (\277)s 6 f 645.602 y 251.419 (t)s 5 f 643.802 y 255.644 (=)s 1 f 261.644 (2\051)s 5 f 272.31 (\276)s 283.598 (:)s 1 f 625.802 y 439.087 (\0503)s 5 f 449.754 (:)s 1 f 453.087 (25\051)s 672.002 y 228.877 (17)s end endp %%Page: 20 20 %%BeginPageSetup initializepage (degasperis; page: 20 of 26)setjob %%EndPageSetup gS 0 0 552 730 rC 0 0 :M f0_12 sf ( )S 4 0 :M f1_12 sf ( )S f3_12 sf ( )S 14 0 :M f8_12 sf .1 .01( )J 18 0 :M f4_12 sf .116 .012( )J f5_12 sf .148 .015( )J 34 0 :M f6_12 sf ( )S f9_12 sf 11.923 1.192( )J 60 0 :M f7_12 sf ( )S 0 0 :M initclip TeXdict begin 0 10 1200 1000 div mul /CMBX10 findfont fdef 1 10 1200 1000 div mul /CMR10 findfont fdef 3 10 1200 1000 div mul /CMTT10 findfont fdef 4 7 1200 1000 div mul /CMR7 findfont fdef 5 10 1200 1000 div mul /CMMI10 findfont fdef 6 7 1200 1000 div mul /CMMI7 findfont fdef 7 10 1200 1000 div mul /CMSY10 findfont fdef 8 7 1200 1000 div mul /CMSY7 findfont fdef 9 10 1200 1000 div mul /CMEX10 findfont fdef 42 41 translate 72 72.27 div dup scale 1 f 12 y 0 (The)s 25.376 (latter)s 58.785 (one)s 81.494 (implies)s 5 f 122.936 (L)s 1 f 135.618 (=)s 5 f 149.466 (\276)s 156.754 (B)s 166.458 (\276)s 8 f 7.645 y 173.746 (\241)s 4 f 181.246 (1)s 1 f 12 y 189.768 (+)s 202.241 (\050)s 5 f 206.908 (\277)s 6 f 13.8 y 212.153 (t)s 5 f 12 y 216.378 (=)s 1 f 222.378 (2\051)s 5 f 233.045 (I)s 1 f 243.971 (and)s 268.013 (inserting)s 318.156 (this)s 342.265 (in)s 356.974 (the)s 378.35 (\257rst)s 403.825 (w)s 412.159 (e)s 422.201 (get,)s 446.421 (with)s 9 f 25.2 y 1.475 (f)s 5 f 28.233 y 0 (M)s 1 f 16.283 (:=)s 5 f 32.283 (\276)s 8 f 23.879 y 39.571 (\241)s 4 f 47.071 (1)s 5 f 28.233 y 52.454 (M)s 65.404 (\276)s 1 f 72.692 (,)s 5 f 46.508 y 150.35 (\277)s 6 f 48.308 y 155.596 (t)s 9 f 43.474 y 161.296 (f)s 5 f 46.508 y 159.821 (M)s 7 f 175.438 (\241)s 9 f 33.188 y 187.438 (h)s 43.474 y 194.579 (f)s 5 f 46.508 y 193.104 (M)s 206.054 (;)s 211.387 (B)s 9 f 33.188 y 221.092 (i)s 7 f 46.508 y 229.425 (\241)s 1 f 241.425 (\050)s 5 f 246.092 (\277)s 6 f 48.308 y 251.337 (tt)s 5 f 46.508 y 259.187 (=)s 1 f 265.187 (2)s 5 f 271.187 (I)s 1 f 277.404 (\051)s 289.404 (=)s 306.071 (0)s 5 f 316.071 (:)s 1 f 439.087 (\0503)s 5 f 449.754 (:)s 1 f 453.087 (26\051)s 73.849 y 0 (F)s 6.833 (or)s 5 f 21.58 (B)s 1 f 34.694 (=)s 47.438 (0)s 57.484 (w)s 65.817 (e)s 75.197 (get)s 95.243 (the)s 115.956 (symmetries)s 179.169 (discussed)s 231.415 (in)s 245.461 ([4],)s 265.519 (while)s 296.898 (for)s 5 f 315.311 (\277)s 1 f 325.326 (=)s 338.069 (0)s 348.115 (w)s 356.449 (e)s 365.828 (get)s 385.874 (new)s 410.587 (symmetries)s 90.082 y 0 (with)s 5 f 27.333 (B)s 1 f 41.038 (satisfying)s 94.504 ([)s 5 f 97.838 (B)s 107.542 (;)s 9 f 87.049 y 114.35 (f)s 5 f 90.082 y 112.875 (M)s 1 f 125.825 (])s 132.492 (=)s 145.158 (0.)s 7 f 159.825 (\246)s 3 f 116.482 y 0 (Example)s 50.4 (6.)s 1 f 69.387 (Consider)s 119.839 (the)s 140.857 (general)s 182.575 (case)s 208.327 (where)s 5 f 243.378 (\276)s 1 f 254.585 (=)s 5 f 267.837 (sI)s 1 f 284.03 (\050)s 5 f 288.697 (s)s 7 f 298.241 (6)s 1 f 298.241 (=)s 311.494 (0)s 321.845 (a)s 332.197 (real)s 355.915 (n)s 362.248 (um)s 378.581 (b)s 385.581 (er\051.)s 410.003 (The)s 435.021 (second)s 130.882 y 0 (set)s 18.733 (of)s 32.4 (\0501.10\051)s 67.067 (is)s 79.133 (no)s 91.467 (w)s 5 f 146.223 y 181.443 (\273)s 6 f 141.269 y 187.245 (i)s 149.19 y 186.693 (k)s 1 f 146.223 y 199.93 (=)s 5 f 216.597 (B)s 6 f 141.269 y 226.301 (i)s 149.19 y 225.699 (k)s 1 f 146.223 y 234.27 (+)s 246.27 (\050)s 5 f 250.936 (\277)s 6 f 148.023 y 256.182 (t)s 5 f 146.223 y 260.407 (=)s 1 f 266.407 (2\051)s 5 f 277.074 (\261)s 6 f 141.269 y 282.861 (i)s 149.19 y 282.407 (k)s 1 f 146.223 y 439.087 (\0503)s 5 f 449.754 (:)s 1 f 453.087 (27\051)s 169.964 y 0 (and)s 23.333 (the)s 44 (\257rst)s 68.767 (set)s 87.5 (reduces)s 130.267 (to)s 5 f 199.705 y 162.98 (@)s 6 f 201.505 y 169.351 (t)s 1 f 199.705 y 173.576 (\050)s 5 f 178.242 (\273)s 6 f 194.751 y 184.045 (i)s 7 f 199.705 y 190.706 (\241)s 5 f 202.706 (\277)s 209.31 (f)s 6 f 194.751 y 216.477 (i)s 1 f 199.705 y 220.472 (\051)s 227.806 (+)s 7 f 239.806 (f)s 5 f 245.806 (f)s 252.306 (;)s 257.639 (\273)s 7 f 263.441 (g)s 1 f 276.774 (=)s 293.441 (0)s 5 f 303.441 (:)s 1 f 439.087 (\0503)s 5 f 449.754 (:)s 1 f 453.087 (28\051)s 229.446 y 0 (W)s 11.333 (e)s 21.361 (notice)s 57.389 (that)s 84.083 (setting)s 5 f 124.177 (\277)s 1 f 135.272 (=)s 149.096 (0,)s 163.297 (w)s 171.63 (e)s 181.658 (get)s 5 f 202.352 (\273)s 6 f 225.092 y 208.154 (i)s 1 f 229.446 y 216.64 (=)s 5 f 230.464 (B)s 6 f 225.092 y 240.168 (i)s 232.845 y 239.566 (k)s 5 f 229.446 y 245.47 (x)s 6 f 225.092 y 252.328 (k)s 1 f 229.446 y 262.927 (and)s 286.954 (hence)s 5 f 320.982 (@)s 6 f 231.246 y 327.353 (t)s 5 f 229.446 y 331.578 (\273)s 6 f 225.092 y 337.38 (i)s 1 f 229.446 y 345.866 (=)s 359.689 (0;)s 374.064 (th)s 385.064 (us,)s 404.665 (for)s 423.726 (\0503.28\051)s 459.087 (to)s 243.846 y 0 (hold,)s 5 f 30 (B)s 1 f 43.704 (has)s 65.104 (to)s 79.771 (satisfy)s 7 f 117.238 (f)s 5 f 123.238 (B)s 132.942 (x;)s 145.134 (f)s 7 f 152.3 (g)s 1 f 161.634 (=)s 174.3 (0,)s 187.633 (i.e.)s 5 f 273.587 y 170.508 (B)s 6 f 268.633 y 180.212 (i)s 276.554 y 179.61 (j)s 5 f 273.587 y 184.666 (f)s 6 f 268.633 y 191.833 (j)s 7 f 273.587 y 199.556 (\241)s 5 f 211.556 (B)s 6 f 267.821 y 221.26 (j)s 277.207 y 220.658 (k)s 5 f 273.587 y 226.562 (x)s 6 f 268.633 y 233.42 (k)s 5 f 273.587 y 239.324 (@)s 6 f 275.387 y 245.695 (j)s 5 f 273.587 y 250.751 (f)s 6 f 268.633 y 257.918 (i)s 1 f 273.587 y 269.247 (=)s 285.913 (0)s 5 f 295.913 (:)s 1 f 439.087 (\0503)s 5 f 449.754 (:)s 1 f 453.087 (29\051)s 7 f 303.328 y 0 (\246)s 3 f 329.728 y 0 (Example)s 50.4 (7.)s 1 f 68.376 (As)s 86.123 (a)s 96.138 (\257nal)s 122.819 (and)s 146.166 (again)s 178.181 (v)s 184.181 (ery)s 204.562 (simple)s 241.976 (example,)s 292.327 (w)s 300.661 (e)s 310.008 (consider)s 356.789 (the)s 5 f 377.47 (n)s 1 f 384.673 (-dimensional)s 454.754 (Ito)s 344.128 y 0 (equation)s 359.469 y 154.115 (d)s 5 f 160.782 (x)s 6 f 354.515 y 167.64 (i)s 1 f 359.469 y 174.968 (=)s 7 f 187.635 (\241)s 1 f 196.968 (\0501)s 7 f 210.302 (\241)s 5 f 222.302 (\270)s 7 f 229.302 (jj)s 5 f 235.968 (x)s 7 f 242.827 (jj)s 4 f 354.515 y 249.493 (2)s 1 f 359.469 y 254.877 (\051)s 5 f 259.543 (x)s 6 f 354.515 y 266.402 (i)s 1 f 359.469 y 270.397 (d)s 5 f 277.064 (t)s 1 f 284.063 (+)s 296.063 (d)s 5 f 302.73 (w)s 6 f 354.515 y 311.644 (i)s 1 f 359.469 y 439.087 (\0503)s 5 f 449.754 (:)s 1 f 453.087 (30\051)s 383.211 y 0 (with)s 5 f 28.335 (\270)s 1 f 40.337 (a)s 51.339 (nonzero)s 97.042 (real)s 121.41 (constan)s 161.144 (t.)s 177.483 (This)s 205.885 (w)s 214.218 (as)s 229.954 (conidered)s 284.989 (as)s 300.725 (example)s 348.727 (10)s 365.729 (in)s 380.731 ([4],)s 401.983 (where)s 437.685 (it)s 450.687 (w)s 459.021 (as)s 397.611 y 0 (sho)s 17.067 (wn)s 36.152 (that)s 61.904 (the)s 82.323 (only)s 108.408 (symmetry)s 164.26 (\050not)s 190.013 (allo)s 208.346 (wing)s 236.765 (action)s 272.517 (on)s 0 f 288.935 (w)s 1 f 299.094 (\051)s 307.512 (w)s 315.846 (as)s 330.331 (giv)s 345.665 (en)s 361.417 (b)s 367.75 (y)s 377.835 (time)s 404.921 (translations.)s 424.011 y 0 (In)s 15.487 (the)s 36.64 (presen)s 69.74 (t)s 78.893 (case,)s 5 f 108.235 (\276)s 1 f 119.666 (=)s 5 f 133.144 (I)s 1 f 139.36 (,)s 147.302 (and)s 171.122 (th)s 182.122 (us)s 198.008 (the)s 219.161 (discussion)s 275.848 (of)s 290.001 (the)s 311.155 (previous)s 359.408 (example)s 406.894 (applies.)s 453.087 (W)s 464.421 (e)s 438.411 y 0 (c)s 5 (hec)s 22 (k)s 32.765 (that)s 59.196 (the)s 80.294 (system)s 120.525 (is)s 133.023 (rotationally)s 198.488 (in)s 208.155 (v)s 213.821 (arian)s 240.188 (t,)s 252.727 (i.e.)s 274.688 (w)s 283.021 (e)s 292.786 (consider)s 5 f 339.984 (\277)s 1 f 350.64 (=)s 364.025 (0.)s 379.986 (With)s 5 f 411.417 (\273)s 6 f 434.056 y 417.219 (i)s 1 f 438.411 y 425.267 (=)s 5 f 438.652 (B)s 6 f 434.056 y 448.356 (i)s 441.809 y 447.754 (k)s 5 f 438.411 y 453.658 (x)s 6 f 434.056 y 460.517 (k)s 1 f 438.411 y 466.421 (,)s 452.811 y 0 (w)s 8.333 (e)s 17.667 (ha)s 30 (v)s 36 (e)s 45.333 (in)s 59.333 (this)s 82.733 (case)s 7 f 482.552 y 78.796 (f)s 5 f 84.796 (\273)s 90.598 (;)s 95.931 (f)s 7 f 103.098 (g)s 6 f 477.597 y 109.098 (i)s 1 f 482.552 y 120.426 (=)s 7 f 137.093 (f)s 5 f 143.093 (B)s 152.797 (x;)s 1 f 164.989 (\0501)s 7 f 178.322 (\241)s 5 f 190.322 (\270)s 7 f 197.322 (jj)s 5 f 203.988 (x)s 7 f 210.847 (jj)s 4 f 477.597 y 217.514 (2)s 1 f 482.552 y 222.897 (\051)s 5 f 227.564 (x)s 7 f 234.422 (g)s 1 f 247.755 (=)s 7 f 264.422 (\241)s 1 f 273.755 (2)s 5 f 279.755 (\270x)s 6 f 477.597 y 293.613 (i)s 9 f 469.231 y 299.609 (\263)s 5 f 482.552 y 306.775 (x)s 6 f 477.597 y 313.634 (j)s 5 f 482.552 y 318.69 (B)s 6 f 476.785 y 328.394 (j)s 486.171 y 327.792 (k)s 5 f 482.552 y 333.696 (x)s 6 f 477.597 y 340.554 (k)s 9 f 469.231 y 346.459 (\264)s 1 f 482.552 y 360.959 (=)s 377.625 (0)s 5 f 387.625 (;)s 1 f 439.087 (\0503)s 5 f 449.754 (:)s 1 f 453.087 (31\051)s 515.893 y 0 (the)s 20.879 (last)s 43.825 (equalit)s 79.158 (y)s 89.704 (follo)s 111.704 (wing)s 140.583 (from)s 169.162 (the)s 190.041 (an)s 202.374 (tisymmetry)s 266.686 (of)s 5 f 280.565 (B)s 1 f 290.269 (.)s 299.573 (That)s 329.785 (is,)s 345.451 (w)s 353.784 (e)s 363.329 (conclude)s 412.875 (that)s 439.087 (\0503.30\051)s 530.293 y 0 (is)s 12.067 (indeed)s 50.067 (symmetric)s 108.5 (under)s 142.534 (sim)s 160.267 (ultaneous)s 214.334 (iden)s 236 (tical)s 262.667 (rotations)s 313.434 (of)s 327.101 (the)s 0 f 347.767 (x)s 1 f 359.051 (and)s 0 f 382.384 (w)s 1 f 396.542 (v)s 402.542 (ectors.)s 7 f 441.976 (\246)s 0 f 572.347 y 0 (4.)s 16.867 (Examples)s 79.293 (\173)s 90.793 (I)s 96.409 (I:)s 110.076 (symmetries)s 183.379 (of)s 199.096 (RDS.)s 1 f 614.402 y 0 (W)s 11.333 (e)s 21.487 (will)s 44.974 (no)s 57.308 (w)s 70.795 (illustrate)s 122.382 (the)s 143.869 (results)s 182.857 (of)s 197.344 (the)s 218.831 (previous)s 267.418 (section)s 308.305 (2,)s 322.664 (i.e.)s 345.793 (symmetries)s 409.78 (of)s 424.267 (RDS,)s 457.088 (b)s 463.421 (y)s 628.802 y 0 (some)s 29.214 (simple)s 65.762 (examples;)s 120.26 (these)s 150.141 (are)s 169.322 (again)s 200.47 (equations)s 253.351 (already)s 294.865 (considered)s 352.779 (in)s 365.927 ([4],)s 385.245 (so)s 399.126 (that)s 424.273 (the)s 444.087 (OPP)s 643.202 y 0 (symmetries)s 63.167 (are)s 83.2 (giv)s 98.534 (en)s 114.534 (there.)s 672.002 y 228.877 (18)s end endp %%Page: 21 21 %%BeginPageSetup initializepage (degasperis; page: 21 of 26)setjob %%EndPageSetup gS 0 0 552 730 rC 0 0 :M f1_12 sf ( )S f3_12 sf .055 .006( )J f8_12 sf ( )S 13 0 :M f4_12 sf .136 .014( )J 21 0 :M f5_12 sf .138 .014( )J 30 0 :M f6_12 sf ( )S f7_12 sf ( )S 0 0 :M initclip TeXdict begin 1 10 1200 1000 div mul /CMR10 findfont fdef 3 10 1200 1000 div mul /CMTT10 findfont fdef 4 7 1200 1000 div mul /CMR7 findfont fdef 5 10 1200 1000 div mul /CMMI10 findfont fdef 6 7 1200 1000 div mul /CMMI7 findfont fdef 7 10 1200 1000 div mul /CMSY10 findfont fdef 9 10 1200 1000 div mul /CMEX10 findfont fdef 42 41 translate 72 72.27 div dup scale 1 f 12 y 0 (As)s 18.482 (the)s 39.898 (main)s 70.647 (in)s 80.314 (terest)s 114.496 (here)s 141.279 (is)s 154.094 (in)s 168.844 (determining)s 236.293 (concrete)s 284.408 (examples)s 336.891 (of)s 351.307 (symmetries)s 415.223 (for)s 434.338 (a)s 445.087 (RDS)s 26.4 y 0 (de\257ned)s 41.01 (b)s 47.343 (y)s 57.353 (a)s 67.029 (giv)s 82.362 (en)s 98.039 (Ito)s 116.715 (equation)s 165.391 (whic)s 189.058 (h)s 199.401 (are)s 219.111 (not)s 240.12 (symmetries)s 302.963 (of)s 316.306 (the)s 336.649 (OPP)s 365.992 (de\257ned)s 407.001 (b)s 413.335 (y)s 423.344 (the)s 443.687 (same)s 40.8 y 0 (equation,)s 53.256 (and)s 77.328 (not)s 99.4 (in)s 114.138 (determining)s 181.576 (the)s 202.981 (most)s 233.119 (general)s 275.224 (symmetry)s 332.063 (of)s 346.468 (the)s 367.873 (giv)s 383.206 (en)s 399.944 (RDS,)s 432.683 (w)s 441.016 (e)s 451.087 (will)s 55.2 y 0 (use)s 20.733 (the)s 41.4 (ansatz)s 5 f 69.6 y 149.83 (\277)s 1 f 156.434 (\050)s 5 f 161.101 (t)s 1 f 165.434 (\051)s 177.434 (=)s 194.101 (0)s 208.101 (and)s 5 f 235.434 (@)s 6 f 71.4 y 241.805 (t)s 5 f 69.6 y 246.03 (\264)s 1 f 252.419 (\050)s 5 f 257.085 (x;)s 269.277 (y)s 275.591 (;)s 280.924 (t)s 1 f 285.258 (\051)s 297.258 (=)s 313.924 (0)s 445.087 (\0504)s 5 f 455.754 (:)s 1 f 459.087 (1\051)s 91.449 y 0 (to)s 15.725 (simplify)s 62.184 (the)s 83.909 (\173)s 94.968 (sev)s 111.034 (erely)s 141.126 (underdetermined)s 234.918 (\173)s 245.977 (determining)s 313.736 (equations)s 368.528 (\0502.20\051.)s 411.037 (F)s 417.87 (or)s 433.629 (ease)s 460.087 (of)s 105.849 y 0 (notation,)s 51.333 (w)s 59.667 (e)s 69 (will)s 91.667 (write)s 122.367 (simply)s 5 f 160.767 (\264)s 1 f 171.156 (for)s 5 f 189.523 (\264)s 1 f 195.911 (\050)s 5 f 200.578 (x;)s 212.77 (y)s 219.084 (;)s 224.417 (t)s 1 f 228.75 (\051.)s 3 f 132.249 y 0 (Example)s 50.4 (1.)s 1 f 69.446 (As)s 87.55 (a)s 97.921 (\257rst)s 123.059 (and)s 146.764 (v)s 152.764 (ery)s 173.501 (simple)s 211.272 (example,)s 262.07 (consider)s 309.207 (the)s 330.245 (trivial)s 366.316 (one-dimensional)s 454.754 (Ito)s 146.649 y 0 (equation)s 161.049 y 208.658 (d)s 5 f 215.324 (x)s 1 f 225.516 (=)s 238.183 (d)s 5 f 244.849 (w)s 1 f 257.763 (;)s 445.087 (\0504)s 5 f 455.754 (:)s 1 f 459.087 (2\051)s 182.897 y 0 (no)s 12.333 (w)s 5 f 24.575 (\301)s 1 f 31.725 (\050)s 5 f 36.392 (x;)s 48.583 (t)s 1 f 52.917 (\051)s 7 f 60.917 (\264)s 1 f 73.583 (0,)s 86.577 (and)s 5 f 109.485 (\275)s 1 f 115.689 (\050)s 5 f 120.356 (x;)s 132.548 (t)s 1 f 136.881 (\051)s 144.881 (=)s 157.547 (1.)s 172.073 (With)s 202.648 (these,)s 236.374 (and)s 259.283 (recalling)s 306.891 (that)s 5 f 332.466 (B)s 1 f 345.504 (=)s 358.17 (0)s 367.745 (for)s 385.687 (one-dimensional)s 197.297 y 0 (problems,)s 54.767 (\0502.20\051)s 89.434 (reduce)s 127.467 (to)s 5 f 223.957 y 146.033 (\264)s 6 f 225.757 y 151.992 (t)s 1 f 223.957 y 158.883 (+)s 170.883 (\0501)s 5 f 181.55 (=)s 1 f 187.55 (2\051)s 200.217 ([)s 5 f 203.55 (\264)s 6 f 225.757 y 209.508 (xx)s 1 f 223.957 y 223.658 (+)s 5 f 235.658 (\264)s 6 f 225.757 y 241.617 (y)s 246.785 (y)s 1 f 223.957 y 255.219 (+)s 267.219 (2)s 5 f 273.219 (\264)s 6 f 225.757 y 279.178 (xy)s 1 f 223.957 y 290.388 (])s 301.054 (=)s 317.721 (0)s 5 f 241.957 y 146.033 (\264)s 6 f 243.757 y 151.992 (x)s 1 f 241.957 y 160.7 (+)s 5 f 172.7 (\264)s 6 f 243.757 y 178.658 (y)s 7 f 241.957 y 187.093 (\241)s 5 f 199.093 (\277)s 6 f 243.757 y 204.339 (t)s 5 f 241.957 y 208.564 (=)s 1 f 214.564 (2)s 227.897 (=)s 244.564 (0)s 5 f 254.564 (:)s 1 f 232.957 y 445.087 (\0504)s 5 f 455.754 (:)s 1 f 459.087 (3\051)s 268.617 y 0 (It)s 13.764 (is)s 26.595 (no)s 38.928 (w)s 52.359 (immediate)s 111.79 (to)s 127.221 (notice)s 163.318 (that)s 5 f 190.083 (\264)s 1 f 196.472 (\050)s 5 f 201.138 (x;)s 213.33 (y)s 219.644 (;)s 224.977 (t)s 1 f 229.31 (\051)s 238.584 (=)s 5 f 252.524 (\256)s 1 f 260.245 (\050)s 5 f 264.911 (x)s 7 f 274.946 (\241)s 5 f 287.455 (y)s 1 f 293.769 (\051,)s 5 f 306.724 (\277)s 1 f 317.935 (=)s 331.875 (0,)s 346.164 (is)s 358.995 (a)s 369.759 (smo)s 390.826 (oth)s 412.923 (solution)s 459.087 (to)s 283.017 y 0 (\0504.3\051)s 28.584 (and)s 51.834 (th)s 62.834 (us)s 78.151 (a)s 88.068 (symmetry)s 144.085 (for)s 162.368 (the)s 182.952 (t)s 187.285 (w)s 195.619 (o-particle)s 248.902 (pro)s 266.602 (cesses)s 300.719 (de\257ned)s 341.97 (b)s 348.303 (y)s 358.553 (\0504.2\051,)s 390.487 (for)s 408.77 (an)s 421.104 (y)s 431.354 (smo)s 452.421 (oth)s 297.417 y 0 (function)s 5 f 48.385 (\256)s 1 f 56.105 (.)s 68.925 (Ob)s 84.592 (viously)s 120.325 (,)s 129.389 (suc)s 145.789 (h)s 157.84 (solutions)s 209.358 (can)s 232.743 (not)s 255.461 (b)s 262.461 (e)s 273.178 (in)s 282.845 (terpreted)s 336.296 (in)s 351.681 (terms)s 386.498 (of)s 401.55 (OPP)s 426.216 (,)s 434.934 (i.e.)s 459.754 (in)s 311.817 y 0 (particular)s 55.4 (they)s 82.4 (are)s 102.434 (not)s 123.767 (symmetries)s 186.934 (of)s 200.6 (the)s 221.267 (OPP)s 250.934 (de\257ned)s 292.267 (b)s 298.601 (y)s 308.934 (\0504.2\051.)s 7 f 342.267 (\246)s 3 f 338.217 y 0 (Example)s 50.4 (2.)s 1 f 68.333 (W)s 79.666 (e)s 88.999 (will)s 111.666 (no)s 124 (w)s 136.666 (consider)s 183.433 (the)s 204.1 (Ito)s 223.1 (equation)s 364.876 y 181.562 (d)s 5 f 188.229 (x)s 1 f 202.42 (=)s 219.087 (d)s 5 f 225.753 (t)s 1 f 232.753 (+)s 5 f 244.753 (x)s 1 f 251.612 (d)s 5 f 258.278 (w)s 1 f 267.192 (\050)s 5 f 271.859 (t)s 1 f 276.192 (\051)s 5 f 284.859 (;)s 1 f 445.087 (\0504)s 5 f 455.754 (:)s 1 f 459.087 (4\051)s 391.536 y 0 (i.e.)s 20.667 (the)s 41.333 (case)s 5 f 66.733 (\301)s 1 f 73.883 (\050)s 5 f 78.55 (x;)s 90.742 (t)s 1 f 95.075 (\051)s 103.075 (=)s 115.742 (1,)s 5 f 129.075 (\275)s 1 f 135.279 (\050)s 5 f 139.946 (x;)s 152.137 (t)s 1 f 156.471 (\051)s 164.471 (=)s 5 f 177.137 (x)s 1 f 183.996 (.)s 192.662 (The)s 217.329 (determining)s 284.029 (equations)s 337.763 (\0502.20\051)s 372.429 (are)s 392.463 (no)s 404.796 (w)s 5 f 418.364 y 87.076 (\264)s 6 f 420.164 y 93.034 (x)s 1 f 418.364 y 101.742 (+)s 5 f 113.742 (\264)s 6 f 420.164 y 119.701 (y)s 1 f 418.364 y 128.135 (+)s 5 f 140.135 (\264)s 6 f 420.164 y 146.094 (t)s 7 f 418.364 y 152.985 (\241)s 5 f 164.985 (\277)s 6 f 420.164 y 170.231 (t)s 1 f 418.364 y 177.123 (+)s 189.123 (\0501)s 5 f 199.789 (=)s 1 f 205.789 (2\051)s 9 f 408.644 y 218.456 (\243)s 5 f 418.364 y 223.456 (x)s 4 f 413.409 y 230.314 (2)s 5 f 418.364 y 235.698 (\264)s 6 f 420.164 y 241.656 (xx)s 1 f 418.364 y 255.806 (+)s 5 f 267.806 (y)s 4 f 413.409 y 274.12 (2)s 5 f 418.364 y 279.503 (\264)s 6 f 420.164 y 285.462 (xy)s 1 f 418.364 y 299.338 (+)s 311.338 (2)s 5 f 317.338 (xy)s 330.51 (\264)s 6 f 420.164 y 336.469 (xy)s 9 f 408.644 y 347.678 (\244)s 1 f 418.364 y 360.012 (=)s 376.678 (0)s 5 f 437.564 y 87.076 (x\264)s 6 f 439.364 y 99.892 (x)s 1 f 437.564 y 108.601 (+)s 5 f 120.601 (y)s 126.915 (\264)s 6 f 439.364 y 132.873 (y)s 1 f 437.564 y 145.974 (=)s 5 f 162.641 (\264)s 1 f 171.696 (+)s 183.696 (\0501)s 5 f 194.363 (=)s 1 f 200.363 (2\051)s 5 f 211.03 (x\277)s 6 f 439.364 y 223.134 (t)s 1 f 437.564 y 231.359 (;)s 427.88 y 445.087 (\0504)s 5 f 455.754 (:)s 1 f 459.087 (5\051)s 464.223 y 0 (with)s 28.9 (the)s 51.134 (ansatz)s 5 f 90.101 (\277)s 1 f 102.65 (=)s 117.928 (0)s 129.494 (the)s 151.728 (second)s 192.028 (of)s 207.262 (these)s 239.562 (yields)s 274.862 (immediately)s 5 f 344.763 (\264)s 1 f 357.096 (=)s 5 f 372.374 (a)s 1 f 378.717 (\050)s 5 f 383.384 (t)s 1 f 387.717 (\051)s 5 f 392.384 (x)s 1 f 402.954 (+)s 5 f 415.998 (b)s 1 f 421.148 (\050)s 5 f 425.815 (t)s 1 f 430.148 (\051)s 5 f 434.815 (y)s 1 f 441.129 (,)s 450.421 (and)s 478.623 y 0 (inserting)s 49.433 (this)s 72.833 (in)s 82.5 (to)s 97.167 (the)s 117.834 (\257rst)s 142.6 (of)s 156.267 (\0504.5\051)s 184.934 (w)s 193.267 (e)s 202.6 (get)s 5 f 222.6 (\264)s 1 f 232.322 (=)s 5 f 244.989 (c)s 1 f 250.182 (\050)s 5 f 254.849 (x)s 7 f 264.374 (\241)s 5 f 276.374 (y)s 1 f 282.688 (\051)s 291.354 (with)s 5 f 318.688 (c)s 1 f 327.881 (an)s 344.547 (arbitrary)s 395.648 (constan)s 435.381 (t.)s 505.023 y 0 (Again,)s 37.649 (this)s 60.192 (dep)s 79.192 (ends)s 105.736 (on)s 5 f 121.547 (y)s 1 f 131.005 (and)s 153.482 (th)s 164.482 (us)s 179.026 (cannot)s 217.503 (b)s 224.503 (e)s 232.98 (seen)s 258.191 (as)s 272.068 (a)s 281.212 (symmetry)s 336.456 (\050nor)s 361.633 (in)s 371.3 (terpreted)s 422.51 (in)s 435.654 (terms\051)s 519.423 y 0 (of)s 13.667 (the)s 34.333 (OPP)s 64 (de\257ned)s 105.334 (b)s 111.667 (y)s 122 (\0504.4\051.)s 7 f 155.334 (\246)s 3 f 545.823 y 0 (Example)s 50.4 (3.)s 1 f 68.333 (W)s 79.666 (e)s 88.999 (consider)s 135.766 (next)s 162.766 (the)s 183.433 (Ito)s 202.433 (equation)s 572.483 y 181.562 (d)s 5 f 188.229 (x)s 1 f 202.42 (=)s 5 f 219.087 (x)s 1 f 225.945 (d)s 5 f 232.612 (t)s 1 f 239.612 (+)s 251.612 (d)s 5 f 258.278 (w)s 1 f 267.192 (\050)s 5 f 271.859 (t)s 1 f 276.192 (\051)s 284.859 (;)s 445.087 (\0504)s 5 f 455.754 (:)s 1 f 459.087 (6\051)s 599.143 y 0 (the)s 20.667 (determining)s 87.367 (equations)s 141.1 (\0502.20\051)s 175.767 (for)s 194.134 (this)s 217.534 (are)s 5 f 625.802 y 86.892 (x\264)s 6 f 627.602 y 99.709 (x)s 1 f 625.802 y 108.417 (+)s 5 f 120.417 (y)s 126.731 (\264)s 6 f 627.602 y 132.689 (y)s 1 f 625.802 y 141.124 (+)s 5 f 153.124 (\264)s 6 f 627.602 y 159.082 (t)s 7 f 625.802 y 165.974 (\241)s 5 f 177.974 (\264)s 7 f 187.029 (\241)s 5 f 199.029 (x\277)s 6 f 627.602 y 211.133 (t)s 1 f 625.802 y 218.025 (+)s 230.025 (\0501)s 5 f 240.692 (=)s 1 f 246.692 (2\051)s 259.358 ([)s 5 f 262.692 (\264)s 6 f 627.602 y 268.65 (xx)s 1 f 625.802 y 282.8 (+)s 5 f 294.8 (\264)s 6 f 627.602 y 300.758 (y)s 305.926 (y)s 1 f 625.802 y 314.361 (+)s 326.361 (2)s 5 f 332.361 (\264)s 6 f 627.602 y 338.319 (xy)s 1 f 625.802 y 349.529 (])s 360.196 (=)s 376.862 (0)s 5 f 643.802 y 86.892 (\264)s 6 f 645.602 y 92.85 (x)s 1 f 643.802 y 101.558 (+)s 5 f 113.558 (\264)s 6 f 645.602 y 119.517 (y)s 1 f 643.802 y 132.618 (=)s 149.285 (\0501)s 5 f 159.951 (=)s 1 f 165.951 (2\051)s 5 f 176.618 (\277)s 6 f 645.602 y 181.864 (t)s 5 f 643.802 y 190.089 (:)s 1 f 634.802 y 445.087 (\0504)s 5 f 455.754 (:)s 1 f 459.087 (7\051)s 672.002 y 228.877 (19)s end endp %%Page: 22 22 %%BeginPageSetup initializepage (degasperis; page: 22 of 26)setjob %%EndPageSetup gS 0 0 552 730 rC 0 0 :M f1_12 sf ( )S f3_12 sf .055 .006( )J f8_12 sf ( )S 13 0 :M f4_12 sf .136 .014( )J 21 0 :M f5_12 sf .138 .014( )J 30 0 :M f6_12 sf ( )S f7_12 sf ( )S 0 0 :M initclip TeXdict begin 1 10 1200 1000 div mul /CMR10 findfont fdef 3 10 1200 1000 div mul /CMTT10 findfont fdef 4 7 1200 1000 div mul /CMR7 findfont fdef 5 10 1200 1000 div mul /CMMI10 findfont fdef 6 7 1200 1000 div mul /CMMI7 findfont fdef 7 10 1200 1000 div mul /CMSY10 findfont fdef 9 10 1200 1000 div mul /CMEX10 findfont fdef 42 41 translate 72 72.27 div dup scale 1 f 12 y 0 (With)s 30.165 (the)s 49.997 (ansatz)s 86.562 (\0504.1\051)s 114.394 (w)s 122.728 (e)s 131.226 (get)s 150.391 (from)s 177.923 (the)s 197.755 (second)s 235.654 (of)s 248.485 (these)s 5 f 278.384 (\264)s 1 f 288.106 (=)s 5 f 300.773 (\256)s 1 f 308.494 (\050)s 5 f 313.16 (x)s 7 f 321.016 (\241)s 5 f 331.346 (y)s 1 f 337.66 (\051)s 345.492 (w)s 357.323 (an)s 373.155 (arbitrary)s 423.421 (function.)s 26.4 y 0 (Inserting)s 50.622 (this)s 74.21 (in)s 88.399 (the)s 109.254 (\257rst)s 134.209 (of)s 148.064 (\0504.7\051)s 176.919 (w)s 185.253 (e)s 194.775 (get)s 214.963 (again)s 5 f 247.152 (\264)s 1 f 257.188 (=)s 5 f 270.169 (c)s 1 f 275.362 (\050)s 5 f 280.028 (x)s 7 f 289.679 (\241)s 5 f 301.804 (y)s 1 f 308.118 (\051)s 316.973 (with)s 5 f 344.495 (c)s 1 f 353.877 (an)s 370.732 (arbitrary)s 422.021 (constan)s 461.754 (t;)s 40.8 y 0 (and)s 23.594 (again,)s 59.254 (this)s 82.915 (dep)s 101.915 (ends)s 129.576 (on)s 5 f 146.503 (y)s 1 f 157.078 (and)s 180.672 (th)s 191.672 (us)s 207.333 (cannot)s 246.928 (b)s 253.928 (e)s 263.522 (seen)s 289.849 (as)s 304.844 (a)s 315.105 (symmetry)s 371.466 (\050nor)s 397.76 (in)s 407.427 (terpreted)s 459.754 (in)s 55.2 y 0 (terms\051)s 38.1 (of)s 51.767 (the)s 72.433 (OPP)s 102.1 (de\257ned)s 143.434 (b)s 149.767 (y)s 160.1 (\0504.6\051.)s 7 f 193.434 (\246)s 3 f 81.6 y 0 (Example)s 50.4 (4.)s 1 f 72.288 (As)s 91.34 (an)s 109.325 (example)s 157.643 (based)s 192.362 (on)s 210.347 (a)s 221.665 (t)s 225.999 (w)s 234.332 (o-dimensional)s 311.717 (Ito)s 332.036 (equation,)s 386.017 (w)s 394.351 (e)s 405.002 (consider)s 453.087 (the)s 96 y 0 (system)s 109.733 y 157.003 (d)s 5 f 163.67 (x)s 4 f 111.533 y 170.528 (1)s 1 f 109.733 y 183.245 (=)s 5 f 199.912 (x)s 4 f 111.533 y 206.77 (2)s 1 f 109.733 y 212.153 (d)s 5 f 218.82 (t)s 1 f 130.779 y 157.003 (d)s 5 f 163.67 (x)s 4 f 132.579 y 170.528 (2)s 1 f 130.779 y 183.245 (=)s 7 f 199.912 (\241)s 5 f 209.245 (k)s 4 f 125.824 y 215.87 (2)s 5 f 130.779 y 221.253 (x)s 4 f 132.579 y 228.112 (2)s 1 f 130.779 y 236.162 (+)s 7 f 119.693 y 248.162 (p)s 18.008 .48 258.162 119.693 r 1 f 130.779 y 258.162 (2)s 5 f 264.162 (k)s 4 f 127.312 y 270.787 (2)s 1 f 130.779 y 276.17 (d)s 5 f 282.837 (w)s 1 f 291.751 (\050)s 5 f 296.417 (t)s 1 f 300.751 (\051)s 5 f 309.417 (;)s 1 f 120.256 y 445.087 (\0504)s 5 f 455.754 (:)s 1 f 459.087 (8\051)s 149.906 y 0 (i.e.)s 20.667 (the)s 41.333 (system)s 81.133 (iden)s 102.8 (ti\257ed)s 133.467 (b)s 139.8 (y)s 5 f 178.475 y 132.404 (\301)s 1 f 142.888 (=)s 9 f 161.555 y 155.554 (\265)s 5 f 171.275 y 177.058 (x)s 4 f 173.075 y 183.917 (2)s 7 f 185.675 y 166.387 (\241)s 5 f 175.721 (k)s 4 f 181.32 y 182.346 (2)s 5 f 185.675 y 187.729 (x)s 4 f 187.475 y 194.588 (2)s 9 f 161.555 y 201.971 (\266)s 5 f 178.475 y 220.804 (;)s 234.137 (\275)s 1 f 243.675 (=)s 9 f 161.555 y 256.341 (\265)s 1 f 171.275 y 267.175 (0)s 296.179 (0)s 185.675 y 267.175 (0)s 7 f 175.175 y 285.175 (p)s 18.008 .48 295.175 175.175 r 1 f 185.675 y 295.175 (2)s 5 f 301.175 (k)s 4 f 182.209 y 307.8 (2)s 9 f 161.555 y 315.183 (\266)s 5 f 178.475 y 334.017 (:)s 1 f 445.087 (\0504)s 5 f 455.754 (:)s 1 f 459.087 (9\051)s 208.479 y 0 (W)s 11.333 (e)s 20.667 (will)s 43.333 (write)s 5 f 74.033 (\264)s 1 f 84.422 (as)s 99.156 (a)s 109.156 (t)s 113.489 (w)s 121.822 (o-v)s 137.822 (ector)s 167.856 (with)s 195.189 (comp)s 223.523 (onen)s 247.856 (ts)s 261.256 (\050)s 5 f 265.923 (\264)s 4 f 204.124 y 272.311 (1)s 5 f 208.479 y 277.695 (;)s 283.028 (\264)s 4 f 204.124 y 289.417 (2)s 1 f 208.479 y 294.8 (\051.)s 234.879 y 0 (Let)s 20.427 (us)s 34.753 (\257rst)s 58.446 (lo)s 68.113 (ok)s 83.373 (at)s 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75.426 (;)s 1 f 80.759 (2\051)s 95.426 (giv)s 110.759 (e,)s 123.426 (resp)s 145.193 (ectiv)s 169.859 (ely)s 183.859 (,)s 5 f 285.283 y 182.452 (@)s 189.49 (\264)s 4 f 280.928 y 195.878 (1)s 19.279 .48 182.217 290.641 r 5 f 301.632 y 182.217 (@)s 189.255 (x)s 4 f 303.432 y 196.113 (2)s 1 f 293.401 y 205.603 (+)s 5 f 285.283 y 219.043 (@)s 226.081 (\264)s 4 f 280.928 y 232.469 (1)s 18.81 .48 219.043 290.641 r 5 f 301.632 y 219.296 (@)s 226.333 (y)s 4 f 303.432 y 232.217 (2)s 1 f 293.401 y 246.626 (=)s 263.293 (0)s 5 f 318.534 y 182.452 (@)s 189.49 (\264)s 4 f 314.179 y 195.878 (2)s 19.279 .48 182.217 323.892 r 5 f 334.883 y 182.217 (@)s 189.255 (x)s 4 f 336.683 y 196.113 (2)s 1 f 326.652 y 205.603 (+)s 5 f 318.534 y 219.043 (@)s 226.081 (\264)s 4 f 314.179 y 232.469 (2)s 18.81 .48 219.043 323.892 r 5 f 334.883 y 219.296 (@)s 226.333 (y)s 4 f 336.683 y 232.217 (2)s 1 f 326.652 y 246.626 (=)s 318.534 y 264.733 (1)s 6 .48 264.733 323.892 r 334.883 y 264.733 (2)s 5 f 326.652 y 272.173 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4 f 378.888 y 288.946 (1)s 5 f 377.088 y 294.33 (;)s 299.663 (y)s 4 f 378.888 y 305.546 (1)s 5 f 377.088 y 310.93 (;)s 316.263 (z)s 1 f 322.371 (\051)s 5 f 331.038 (:)s 1 f 439.087 (\0504)s 5 f 449.754 (:)s 1 f 453.087 (11\051)s 411.124 y 0 (The)s 24.667 (\257rst)s 49.433 (set)s 68.167 (of)s 81.833 (equations)s 135.567 (\0502.20\051)s 170.234 (reads,)s 205 (for)s 223.367 (our)s 244.734 (Ito)s 263.734 (equation)s 312.734 (\0504.8\051,)s 5 f 440.845 y 67.249 (x)s 4 f 442.645 y 74.108 (2)s 5 f 432.727 y 81.166 (@)s 88.203 (\264)s 4 f 428.373 y 94.592 (1)s 19.279 .48 80.931 438.085 r 5 f 449.077 y 80.931 (@)s 87.969 (x)s 4 f 450.877 y 94.827 (1)s 7 f 440.845 y 104.317 (\241)s 5 f 116.317 (k)s 4 f 435.891 y 122.942 (2)s 5 f 440.845 y 128.325 (x)s 4 f 442.645 y 135.183 (2)s 5 f 432.727 y 142.242 (@)s 149.279 (\264)s 4 f 428.373 y 155.668 (1)s 19.279 .48 142.007 438.085 r 5 f 449.077 y 142.007 (@)s 149.044 (x)s 4 f 450.877 y 155.903 (2)s 7 f 440.845 y 165.393 (\241)s 5 f 177.393 (\264)s 4 f 435.891 y 183.781 (2)s 1 f 440.845 y 191.831 (+)s 5 f 203.831 (y)s 4 f 442.645 y 209.715 (2)s 5 f 432.727 y 216.538 (@)s 223.576 (\264)s 4 f 428.373 y 229.965 (1)s 18.81 .48 216.538 438.085 r 5 f 449.077 y 216.791 (@)s 223.828 (y)s 4 f 450.877 y 229.712 (1)s 7 f 440.845 y 239.454 (\241)s 5 f 251.454 (k)s 4 f 435.891 y 258.079 (2)s 5 f 440.845 y 263.463 (y)s 4 f 442.645 y 269.346 (2)s 5 f 432.727 y 276.17 (@)s 283.207 (\264)s 4 f 428.373 y 289.596 (1)s 18.81 .48 276.17 438.085 r 5 f 449.077 y 276.422 (@)s 283.46 (y)s 4 f 450.877 y 289.343 (2)s 1 f 440.845 y 299.086 (+)s 5 f 432.727 y 312.526 (@)s 319.563 (\264)s 4 f 428.373 y 325.952 (1)s 18.81 .48 312.526 438.085 r 5 f 449.077 y 316.245 (@)s 323.283 (t)s 7 f 440.845 y 335.442 (\241)s 5 f 347.442 (x)s 4 f 442.645 y 354.3 (2)s 5 f 440.845 y 359.684 (\277)s 6 f 442.645 y 364.93 (t)s 1 f 440.845 y 369.155 (+)s 474.096 y 85.916 (+)s 5 f 97.916 (k)s 4 f 469.142 y 104.541 (2)s 9 f 457.176 y 111.924 (\267)s 5 f 465.978 y 119.698 (@)s 4 f 461.623 y 126.735 (2)s 5 f 465.978 y 132.118 (\264)s 4 f 461.623 y 138.507 (1)s 24.193 .48 119.698 471.336 r 5 f 482.328 y 122.155 (@)s 129.192 (x)s 4 f 478.194 y 136.05 (2)s 485.528 y 136.05 (2)s 1 f 474.096 y 147.997 (+)s 5 f 465.978 y 161.437 (@)s 4 f 461.623 y 168.475 (2)s 5 f 465.978 y 173.858 (\264)s 4 f 461.623 y 180.247 (1)s 24.193 .48 161.437 471.336 r 5 f 482.328 y 164.166 (@)s 171.204 (y)s 4 f 478.194 y 177.518 (2)s 485.528 y 177.087 (2)s 1 f 474.096 y 189.737 (+)s 201.737 (2)s 5 f 465.978 y 215.872 (@)s 4 f 461.623 y 222.91 (2)s 5 f 465.978 y 228.293 (\264)s 4 f 461.623 y 234.682 (1)s 37.583 .48 209.177 471.336 r 5 f 482.328 y 209.177 (@)s 216.214 (x)s 4 f 484.128 y 223.073 (2)s 5 f 482.328 y 228.456 (@)s 235.494 (y)s 4 f 484.128 y 241.377 (2)s 9 f 457.176 y 248.2 (\270)s 1 f 474.096 y 261.867 (=)s 278.534 (0)s 5 f 508.214 y 67.249 (x)s 4 f 510.014 y 74.108 (2)s 5 f 500.096 y 81.166 (@)s 88.203 (\264)s 4 f 495.741 y 94.592 (2)s 19.279 .48 80.931 505.454 r 5 f 516.445 y 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508.214 y 311.094 (+)s 5 f 500.096 y 324.534 (@)s 331.572 (\264)s 4 f 495.741 y 337.961 (2)s 18.81 .48 324.534 505.454 r 5 f 516.445 y 328.254 (@)s 335.291 (t)s 1 f 508.214 y 347.451 (+)s 5 f 359.451 (k)s 4 f 503.259 y 366.076 (2)s 5 f 508.214 y 371.459 (x)s 4 f 510.014 y 378.317 (2)s 5 f 508.214 y 383.701 (\277)s 6 f 510.014 y 388.946 (t)s 1 f 508.214 y 393.171 (+)s 541.465 y 85.916 (+)s 5 f 97.916 (k)s 4 f 536.51 y 104.541 (2)s 9 f 524.544 y 111.924 (\267)s 5 f 533.346 y 119.698 (@)s 4 f 528.992 y 126.735 (2)s 5 f 533.346 y 132.118 (\264)s 4 f 528.992 y 138.507 (2)s 24.193 .48 119.698 538.705 r 5 f 549.696 y 122.155 (@)s 129.192 (x)s 4 f 551.496 y 136.05 (2)s 1 f 541.465 y 147.997 (+)s 5 f 533.346 y 161.437 (@)s 4 f 528.992 y 168.475 (2)s 5 f 533.346 y 173.858 (\264)s 4 f 528.992 y 180.247 (2)s 24.193 .48 161.437 538.705 r 5 f 549.696 y 164.382 (@)s 171.419 (y)s 4 f 551.496 y 177.303 (2)s 1 f 541.465 y 189.737 (+)s 201.737 (2)s 5 f 533.346 y 215.872 (@)s 4 f 528.992 y 222.91 (2)s 5 f 533.346 y 228.293 (\264)s 4 f 528.992 y 234.682 (2)s 37.583 .48 209.177 538.705 r 5 f 549.696 y 209.177 (@)s 216.214 (x)s 4 f 551.496 y 223.073 (2)s 5 f 549.696 y 228.456 (@)s 235.494 (y)s 4 f 551.496 y 241.377 (2)s 9 f 524.544 y 248.2 (\270)s 1 f 541.465 y 261.867 (=)s 278.534 (0)s 288.534 (;)s 490.912 y 439.087 (\0504)s 5 f 449.754 (:)s 1 f 453.087 (12\051)s 570.7 y 0 (note)s 25.603 (that)s 50.539 (with)s 76.809 (\0504.11\051)s 110.412 (the)s 130.015 (terms)s 162.384 (in)s 175.321 (square)s 212.024 (brac)s 234.39 (k)s 240.39 (ets)s 258.06 (v)s 263.727 (anish)s 294.063 (automatically)s 364.73 (,)s 371.212 (and)s 393.482 (w)s 401.815 (e)s 410.085 (are)s 429.054 (reduced)s 585.1 y 0 (to)s 5 f 604.187 y 130.405 (x)s 4 f 605.986 y 137.263 (2)s 5 f 596.068 y 144.321 (@)s 151.359 (\264)s 4 f 591.714 y 157.748 (1)s 19.279 .48 144.087 601.426 r 5 f 612.418 y 144.087 (@)s 151.124 (x)s 4 f 614.218 y 157.983 (1)s 1 f 604.187 y 171.472 (+)s 5 f 183.472 (y)s 4 f 605.986 y 189.356 (2)s 5 f 596.068 y 196.179 (@)s 203.217 (\264)s 4 f 591.714 y 209.606 (1)s 18.81 .48 196.179 601.426 r 5 f 612.418 y 196.432 (@)s 203.469 (y)s 4 f 614.218 y 209.353 (1)s 7 f 604.187 y 223.096 (\241)s 5 f 235.095 (k)s 4 f 599.232 y 241.72 (2)s 5 f 604.187 y 247.104 (z)s 596.068 y 254.652 (@)s 261.69 (\264)s 4 f 591.714 y 268.079 (1)s 18.81 .48 254.652 601.426 r 5 f 612.418 y 257.484 (@)s 264.522 (z)s 1 f 604.187 y 282.235 (=)s 5 f 298.902 (\264)s 4 f 599.232 y 305.291 (2)s 5 f 637.437 y 130.405 (x)s 4 f 639.237 y 137.263 (2)s 5 f 629.319 y 144.321 (@)s 151.359 (\264)s 4 f 624.965 y 157.748 (2)s 19.279 .48 144.087 634.677 r 5 f 645.669 y 144.087 (@)s 151.124 (x)s 4 f 647.469 y 157.983 (1)s 1 f 637.437 y 171.472 (+)s 5 f 183.472 (y)s 4 f 639.237 y 189.356 (2)s 5 f 629.319 y 196.179 (@)s 203.217 (\264)s 4 f 624.965 y 209.606 (2)s 18.81 .48 196.179 634.677 r 5 f 645.669 y 196.432 (@)s 203.469 (y)s 4 f 647.469 y 209.353 (1)s 7 f 637.437 y 223.096 (\241)s 5 f 235.095 (k)s 4 f 632.483 y 241.72 (2)s 5 f 637.437 y 247.104 (z)s 629.319 y 254.652 (@)s 261.69 (\264)s 4 f 624.965 y 268.079 (2)s 18.81 .48 254.652 634.677 r 5 f 645.669 y 257.484 (@)s 264.522 (z)s 1 f 637.437 y 282.235 (=)s 7 f 298.902 (\241)s 5 f 308.235 (k)s 4 f 632.483 y 314.86 (2)s 5 f 637.437 y 320.244 (\264)s 4 f 632.483 y 326.632 (2)s 5 f 637.437 y 336.016 (:)s 1 f 620.151 y 439.087 (\0504)s 5 f 449.754 (:)s 1 f 453.087 (13\051)s 672.002 y 228.877 (20)s end endp %%Page: 23 23 %%BeginPageSetup initializepage (degasperis; page: 23 of 26)setjob %%EndPageSetup gS 0 0 552 730 rC 0 0 :M f0_12 sf ( )S 4 0 :M f1_12 sf ( )S f3_12 sf ( )S 14 0 :M f8_12 sf .1 .01( )J 18 0 :M f4_12 sf .116 .012( )J f5_12 sf .148 .015( )J 34 0 :M f6_12 sf ( )S f7_12 sf ( )S 0 0 :M initclip TeXdict begin 0 10 1200 1000 div mul /CMBX10 findfont fdef 1 10 1200 1000 div mul /CMR10 findfont fdef 3 10 1200 1000 div mul /CMTT10 findfont fdef 4 7 1200 1000 div mul /CMR7 findfont fdef 5 10 1200 1000 div mul /CMMI10 findfont fdef 6 7 1200 1000 div mul /CMMI7 findfont fdef 7 10 1200 1000 div mul /CMSY10 findfont fdef 9 10 1200 1000 div mul /CMEX10 findfont fdef 42 41 translate 72 72.27 div dup scale 1 f 12 y 0 (These)s 34.733 (admit)s 69.4 (an)s 86.067 (in\257nit)s 117.067 (y)s 127.4 (of)s 141.067 (solutions)s 191.2 (in)s 205.2 (the)s 225.867 (form)s 5 f 38.216 y 169.571 (\264)s 1 f 183.293 (=)s 5 f 199.959 (h)s 1 f 206.873 (\050)s 5 f 211.54 (\263)s 1 f 217.675 (\051)s 7 f 225.009 (\242)s 9 f 21.296 y 231.009 (\265)s 5 f 31.016 y 241.842 (x)s 4 f 32.816 y 248.7 (1)s 7 f 31.016 y 256.75 (\241)s 5 f 268.75 (y)s 4 f 32.816 y 274.634 (1)s 1 f 45.416 y 257.929 (1)s 9 f 21.296 y 282.017 (\266)s 5 f 38.216 y 296.85 (;)s 1 f 439.087 (\0504)s 5 f 449.754 (:)s 1 f 453.087 (14\051)s 64.432 y 0 (where)s 5 f 34.7 (h)s 1 f 45.614 (is)s 57.681 (an)s 74.347 (arbitrary)s 125.447 (smo)s 146.514 (oth)s 167.848 (function)s 214.848 (of)s 5 f 84.115 y 155.63 (\263)s 1 f 165.099 (:=)s 5 f 185.099 (k)s 4 f 79.16 y 191.724 (2)s 1 f 84.115 y 197.107 (\050)s 5 f 201.774 (x)s 4 f 85.915 y 208.632 (1)s 7 f 84.115 y 216.682 (\241)s 5 f 228.682 (y)s 4 f 85.915 y 234.566 (1)s 1 f 84.115 y 239.949 (\051)s 247.282 (+)s 259.282 (\050)s 5 f 263.949 (x)s 4 f 85.915 y 270.807 (2)s 7 f 84.115 y 278.857 (\241)s 5 f 290.857 (y)s 4 f 85.915 y 296.74 (2)s 1 f 84.115 y 302.124 (\051)s 5 f 310.791 (:)s 1 f 439.087 (\0504)s 5 f 449.754 (:)s 1 f 453.087 (15\051)s 7 f 103.797 y 0 (\246)s 3 f 130.197 y 0 (Example)s 50.4 (5.)s 1 f 73.937 (W)s 85.27 (e)s 96.471 (will)s 121.006 (\257nally)s 159.207 (consider)s 207.842 (a)s 219.71 (simple)s 258.978 (but)s 282.846 (relev)s 307.213 (an)s 319.546 (t)s 330.081 (class)s 360.082 (of)s 5 f 375.617 (n)s 1 f 382.819 (-dimensional)s 454.754 (Ito)s 144.597 y 0 (equations,)s 57.067 (i.e.)s 77.733 (those)s 109.134 (with)s 136.467 (linear)s 169.834 (drift)s 5 f 196.867 (\301)s 1 f 208.017 (and)s 231.35 (constan)s 271.084 (t)s 279.75 (diagonal)s 327.751 (noise)s 357.817 (matrix)s 5 f 396.851 (\275)s 1 f 403.055 (:)s 164.28 y 147.186 (d)s 5 f 153.852 (x)s 6 f 159.325 y 160.711 (i)s 1 f 164.28 y 172.039 (=)s 5 f 188.706 (M)s 6 f 159.325 y 201.656 (i)s 167.247 y 204.347 (j)s 5 f 164.28 y 209.403 (x)s 6 f 159.325 y 216.262 (j)s 1 f 164.28 y 223.318 (d)s 5 f 229.985 (t)s 1 f 238.985 (+)s 7 f 153.194 y 252.985 (p)s 17.008 .48 262.985 153.194 r 1 f 164.28 y 262.985 (2)s 5 f 268.985 (s)s 4 f 160.813 y 274.61 (2)s 1 f 164.28 y 281.993 (d)s 5 f 288.66 (w)s 6 f 159.325 y 297.573 (i)s 1 f 164.28 y 301.569 (\050)s 5 f 306.235 (t)s 1 f 310.569 (\051)s 5 f 319.235 (:)s 1 f 439.087 (\0504)s 5 f 449.754 (:)s 1 f 453.087 (16\051)s 183.963 y 0 (Here)s 5 f 28.367 (M)s 1 f 45.317 (is)s 57.383 (a)s 67.383 (constan)s 107.117 (t)s 115.783 (\050)s 5 f 120.45 (n)s 7 f 130.32 (\243)s 5 f 142.319 (n)s 1 f 149.522 (\051)s 158.189 (matrix)s 197.222 (and)s 5 f 220.556 (s)s 1 f 230.181 (a)s 240.181 (real)s 263.548 (constan)s 303.281 (t.)s 210.363 y 0 (The)s 24.667 (equations)s 78.4 (\0502.20\051)s 113.067 (read)s 139.767 (in)s 153.767 (this)s 177.167 (case,)s 205.9 (and)s 229.234 (assuming)s 281.367 (directly)s 325.067 (\0504.1\051,)s 5 f 237.876 y 14.704 (M)s 6 f 232.11 y 27.654 (j)s 241.496 y 30.346 (k)s 5 f 237.876 y 36.25 (x)s 6 f 232.922 y 43.108 (k)s 5 f 229.758 y 51.218 (@)s 58.255 (\264)s 6 f 225.404 y 64.644 (i)s 18.952 .48 50.453 235.116 r 5 f 246.108 y 50.453 (@)s 57.49 (x)s 6 f 242.641 y 64.348 (j)s 7 f 237.876 y 77.511 (\241)s 5 f 89.511 (M)s 6 f 232.922 y 102.461 (i)s 240.843 y 105.153 (j)s 5 f 237.876 y 110.209 (\264)s 6 f 232.922 y 116.598 (j)s 1 f 237.876 y 128.321 (+)s 5 f 140.321 (M)s 6 f 232.11 y 153.271 (j)s 241.496 y 155.963 (k)s 5 f 237.876 y 161.867 (y)s 6 f 232.922 y 168.181 (k)s 5 f 229.758 y 176.018 (@)s 183.056 (\264)s 6 f 225.404 y 189.444 (i)s 18.408 .48 175.525 235.116 r 5 f 246.108 y 175.525 (@)s 182.562 (y)s 6 f 242.641 y 188.876 (j)s 1 f 237.876 y 202.039 (+)s 5 f 214.039 (s)s 9 f 220.956 y 221.664 (\267)s 5 f 229.758 y 236.987 (@)s 4 f 225.404 y 244.025 (2)s 5 f 229.758 y 249.408 (\264)s 6 f 225.404 y 255.797 (i)s 37.904 .48 229.438 235.116 r 5 f 246.108 y 229.438 (@)s 236.475 (x)s 6 f 242.641 y 243.333 (j)s 5 f 246.108 y 248.39 (@)s 255.427 (x)s 6 f 242.641 y 262.285 (j)s 1 f 237.876 y 273.448 (+)s 5 f 229.758 y 293.893 (@)s 4 f 225.404 y 300.931 (2)s 5 f 229.758 y 306.314 (\264)s 6 f 225.404 y 312.703 (i)s 36.815 .48 286.888 235.116 r 5 f 246.108 y 286.888 (@)s 293.926 (y)s 6 f 242.641 y 300.24 (j)s 5 f 246.108 y 305.296 (@)s 312.333 (y)s 6 f 242.641 y 318.647 (j)s 1 f 237.876 y 329.81 (+)s 341.81 (2)s 5 f 229.758 y 356.528 (@)s 4 f 225.404 y 363.565 (2)s 5 f 229.758 y 368.948 (\264)s 6 f 225.404 y 375.337 (i)s 37.36 .48 349.25 235.116 r 5 f 246.108 y 349.25 (@)s 356.288 (x)s 6 f 242.641 y 363.146 (j)s 5 f 246.108 y 368.202 (@)s 375.24 (y)s 6 f 242.641 y 381.554 (j)s 9 f 220.956 y 388.05 (\270)s 1 f 237.876 y 401.716 (=)s 418.383 (0)s 5 f 263.989 y 17.333 (@)s 24.371 (\264)s 6 f 259.635 y 30.76 (i)s 19.8 .48 16.144 269.348 r 5 f 280.339 y 16.144 (@)s 23.182 (x)s 6 f 276.872 y 30.04 (k)s 1 f 272.108 y 44.051 (+)s 5 f 263.989 y 58.408 (@)s 65.445 (\264)s 6 f 259.635 y 71.834 (i)s 19.256 .48 57.491 269.348 r 5 f 280.339 y 57.491 (@)s 64.528 (y)s 6 f 276.872 y 70.842 (k)s 1 f 272.108 y 85.52 (=)s 5 f 102.186 (B)s 6 f 267.153 y 111.89 (i)s 275.074 y 115.288 (k)s 1 f 272.108 y 125.193 (;)s 254.259 y 439.087 (\0504)s 5 f 449.754 (:)s 1 f 453.087 (17\051)s 299.068 y 0 (from)s 28.367 (the)s 49.033 (second)s 87.767 (of)s 101.434 (these)s 132.167 (w)s 140.5 (e)s 149.834 (ha)s 162.167 (v)s 168.167 (e,)s 180.834 (with)s 5 f 208.167 (z)s 6 f 294.713 y 214.275 (i)s 1 f 299.068 y 221.604 (:=)s 237.604 (\050)s 5 f 242.27 (x)s 6 f 294.713 y 249.129 (i)s 7 f 299.068 y 255.79 (\241)s 5 f 267.79 (y)s 6 f 294.713 y 274.104 (i)s 1 f 299.068 y 278.099 (\051,)s 5 f 318.751 y 155.877 (\264)s 6 f 313.796 y 162.266 (i)s 1 f 318.751 y 173.594 (=)s 5 f 190.261 (\256)s 6 f 313.796 y 197.982 (i)s 1 f 318.751 y 201.977 (\050)s 5 f 206.643 (z)s 4 f 313.796 y 212.752 (1)s 5 f 318.751 y 218.135 (;)s 223.468 (:::;)s 238.802 (z)s 6 f 313.796 y 244.91 (n)s 1 f 318.751 y 251.442 (\051)s 260.775 (+)s 5 f 274.775 (B)s 6 f 313.796 y 284.479 (i)s 321.717 y 287.877 (k)s 5 f 318.751 y 293.781 (x)s 6 f 313.796 y 300.64 (k)s 5 f 318.751 y 310.544 (:)s 1 f 439.087 (\0504)s 5 f 449.754 (:)s 1 f 453.087 (18\051)s 338.433 y 0 (The)s 24.667 (\257rst)s 49.433 (of)s 63.1 (\0504.17\051)s 97.767 (is)s 109.834 (then)s 5 f 365.947 y 112.41 (M)s 6 f 360.992 y 125.36 (i)s 368.914 y 128.052 (k)s 5 f 365.947 y 133.956 (z)s 6 f 360.992 y 140.065 (k)s 5 f 357.829 y 147.409 (@)s 154.446 (\256)s 6 f 353.474 y 162.167 (i)s 18.753 .48 147.409 363.187 r 5 f 374.178 y 147.684 (@)s 154.722 (z)s 6 f 370.712 y 160.83 (j)s 1 f 365.947 y 170.269 (+)s 5 f 182.269 (B)s 6 f 360.992 y 191.973 (i)s 368.914 y 195.371 (j)s 5 f 365.947 y 200.427 (M)s 6 f 360.18 y 213.377 (j)s 369.567 y 216.069 (k)s 5 f 365.947 y 221.973 (x)s 6 f 360.992 y 228.831 (k)s 7 f 365.947 y 237.402 (\241)s 5 f 249.402 (M)s 6 f 360.992 y 262.352 (i)s 368.914 y 265.044 (j)s 5 f 365.947 y 270.1 (\256)s 6 f 360.992 y 277.821 (j)s 7 f 365.947 y 285.544 (\241)s 5 f 297.544 (M)s 6 f 360.992 y 310.494 (i)s 368.914 y 313.185 (j)s 5 f 365.947 y 318.242 (B)s 6 f 360.18 y 327.946 (j)s 369.567 y 331.344 (k)s 5 f 365.947 y 337.248 (x)s 6 f 360.992 y 344.106 (k)s 1 f 365.947 y 354.011 (;)s 439.087 (\0504)s 5 f 449.754 (:)s 1 f 453.087 (19\051)s 388.994 y 0 (as)s 14.733 (b)s 21.733 (oth)s 5 f 43.067 (B)s 1 f 56.771 (and)s 5 f 80.104 (M)s 1 f 97.054 (cannot)s 136.388 (dep)s 155.388 (end)s 178.054 (on)s 5 f 194.721 (x)s 1 f 205.579 (or)s 5 f 220.279 (z)s 1 f 230.388 (v)s 236.055 (ariables,)s 283.488 (this)s 306.888 (splits)s 338.355 (in)s 348.021 (to)s 362.688 (t)s 367.021 (w)s 375.355 (o)s 385.355 (equations,)s 5 f 409.135 y 165.044 (B)s 6 f 404.18 y 174.748 (i)s 412.102 y 178.146 (j)s 5 f 409.135 y 183.202 (M)s 6 f 403.369 y 196.152 (j)s 412.755 y 198.844 (k)s 1 f 409.135 y 212.081 (=)s 5 f 228.748 (M)s 6 f 404.18 y 241.698 (i)s 412.102 y 244.389 (j)s 5 f 409.135 y 249.446 (B)s 6 f 403.369 y 259.15 (j)s 412.755 y 262.548 (k)s 5 f 436.566 y 165.044 (M)s 6 f 430.8 y 177.994 (j)s 440.186 y 180.685 (k)s 5 f 436.566 y 186.59 (z)s 6 f 431.611 y 192.698 (k)s 9 f 419.646 y 200.602 (\265)s 5 f 428.448 y 210.875 (@)s 217.913 (\256)s 6 f 424.093 y 225.634 (i)s 18.753 .48 210.875 433.806 r 5 f 444.798 y 211.151 (@)s 218.189 (z)s 6 f 441.331 y 224.297 (j)s 9 f 419.646 y 231.069 (\266)s 1 f 436.566 y 247.235 (=)s 5 f 263.902 (M)s 6 f 431.611 y 276.852 (i)s 439.533 y 279.544 (j)s 5 f 436.566 y 284.6 (\256)s 6 f 431.611 y 292.321 (j)s 5 f 436.566 y 301.377 (:)s 1 f 425.888 y 439.087 (\0504)s 5 f 449.754 (:)s 1 f 453.087 (20\051)s 476.637 y 0 (In)s 16.736 (particular,)s 77.639 (w)s 85.972 (e)s 97.041 (get)s 118.777 (solutions)s 170.646 (in)s 186.382 (the)s 208.785 (form)s 238.887 (\0504.18\051)s 275.29 (with)s 5 f 304.359 (\256)s 6 f 472.282 y 312.08 (i)s 1 f 476.637 y 316.075 (\050)s 0 f 320.742 (z)s 1 f 326.875 (\051)s 337.768 (=)s 5 f 353.327 (L)s 6 f 472.282 y 361.494 (i)s 480.035 y 365.494 (k)s 5 f 476.637 y 371.398 (z)s 6 f 472.282 y 377.507 (k)s 1 f 476.637 y 389.147 (for)s 409.249 (an)s 421.582 (y)s 433.652 (pair)s 460.087 (of)s 491.037 y 0 (matrices)s 5 f 48.1 (B)s 57.804 (;)s 63.138 (L)s 1 f 75.304 (whic)s 98.971 (h)s 109.638 (b)s 116.638 (oth)s 137.971 (comm)s 168.971 (ute)s 189.638 (with)s 5 f 216.971 (M)s 1 f 229.921 (,)s 510.72 y 169.325 ([)s 5 f 172.658 (B)s 182.363 (;)s 187.696 (M)s 1 f 200.646 (])s 211.313 (=)s 227.979 (0)s 241.312 (=)s 257.979 ([)s 5 f 261.312 (L;)s 274.812 (M)s 1 f 287.762 (])s 5 f 297.096 (:)s 1 f 439.087 (\0504)s 5 f 449.754 (:)s 1 f 453.087 (21\051)s 0 f 542.402 y 0 (Ac)s 16.183 (kno)s 37.65 (wledgemen)s 103.774 (ts.)s 1 f 127.012 (I)s 136.432 (w)s 144.765 (ould)s 172.519 (lik)s 185.185 (e)s 195.606 (to)s 211.359 (thank)s 246.78 (Niurk)s 276.147 (a)s 287.234 (Ro)s 302.401 (dr)s 312.434 (\266)s 313.767 (\263guez)s 345.521 (Quin)s 371.188 (tero,)s 400.58 (with)s 429 (whom)s 465.421 (I)s 556.802 y 0 (started)s 40.334 (the)s 60.569 (study)s 93.203 (of)s 106.437 (symmetries)s 169.172 (of)s 182.406 (sto)s 198.139 (c)s 203.139 (hastic)s 237.44 (equations)s 290.741 ([4].)s 311.93 (Our)s 336.198 (\257rst)s 360.532 (discussions,)s 424.453 (and)s 447.354 (th)s 458.354 (us)s 571.202 y 0 (ev)s 11.333 (en)s 23 (tually)s 57.002 (this)s 80.07 (w)s 88.403 (ork,)s 112.505 (w)s 120.838 (ere)s 139.873 (triggered)s 190.274 (b)s 196.608 (y)s 206.609 (in)s 216.276 (vitations)s 265.678 (of)s 279.013 (L.A.)s 305.848 (Ib)s 317.181 (ort)s 336.216 (in)s 349.884 (the)s 370.219 (Univ)s 395.219 (ersidad)s 436.321 (Carlos)s 585.602 y 0 (I)s 4.667 (I)s 9.333 (I)s 17.636 (de)s 33.605 (Madrid,)s 79.281 (and)s 102.583 (b)s 108.917 (y)s 119.219 (A.)s 135.522 (Sanc)s 159.855 (hez)s 181.158 (to)s 195.794 (presen)s 228.894 (t)s 237.53 (m)s 247.197 (y)s 257.499 (\173)s 267.469 (then)s 294.771 (v)s 300.771 (ery)s 321.107 (v)s 326.774 (ague)s 354.743 (\173)s 364.713 (ideas)s 394.749 (on)s 411.385 (the)s 432.021 (sub)s 450.754 (ject)s 600.002 y 0 (in)s 14.928 (his)s 34.589 (seminar)s 80.284 (at)s 95.878 (GISC;)s 133.223 (I)s 142.483 (w)s 150.816 (ould)s 178.411 (lik)s 191.077 (e)s 201.338 (to)s 216.933 (thank)s 252.194 (them)s 283.789 (again,)s 320.282 (as)s 335.943 (w)s 344.276 (ell)s 361.204 (as)s 376.865 (A.)s 394.126 (Degasp)s 432.36 (eris)s 455.388 (for)s 614.402 y 0 (his)s 18.662 (nev)s 36.662 (er)s 50.623 (failing)s 86.885 (hospitalit)s 135.952 (y)s 146.213 (in)s 160.142 (\134La)s 183.57 (Sapienza\042.)s 244.213 (Last)s 271.041 (but)s 292.97 (not)s 314.231 (least,)s 345.574 (thanks)s 384.569 (should)s 422.564 (go)s 438.493 (to)s 453.088 (the)s 628.802 y 0 (anon)s 25 (ymous)s 63.076 (referee)s 101.819 (of)s 115.828 ([4])s 132.837 (who)s 158.513 (urged)s 192.222 (us)s 207.965 (to)s 222.974 (study)s 256.383 (symmetry)s 312.826 (of)s 326.835 (random)s 371.211 (dynamical)s 429.221 (systems)s 643.202 y 0 (and)s 23.333 (not)s 44.667 (just)s 68.4 (of)s 82.067 (one-p)s 111.067 (oin)s 126.734 (t)s 135.4 (pro)s 153.1 (cesses.)s 672.002 y 228.877 (21)s end endp %%Page: 24 24 %%BeginPageSetup initializepage (degasperis; page: 24 of 26)setjob %%EndPageSetup gS 0 0 552 730 rC 0 0 :M f0_12 sf ( )S 4 0 :M f1_12 sf ( )S f2_12 sf ( )S f8_12 sf ( )S 14 0 :M f4_12 sf .136 .014( )J 22 0 :M f5_12 sf .138 .014( )J 31 0 :M f6_12 sf ( )S f7_12 sf ( )S 0 0 :M initclip TeXdict begin 0 10 1200 1000 div mul /CMBX10 findfont fdef 1 10 1200 1000 div mul /CMR10 findfont fdef 2 10 1200 1000 div mul /CMTI10 findfont fdef 4 7 1200 1000 div mul /CMR7 findfont fdef 5 10 1200 1000 div mul /CMMI10 findfont fdef 6 7 1200 1000 div mul /CMMI7 findfont fdef 7 10 1200 1000 div mul /CMSY10 findfont fdef 9 10 1200 1000 div mul /CMEX10 findfont fdef 42 41 translate 72 72.27 div dup scale 0 f 12 y 0 (App)s 26.15 (endix.)s 68.891 (Discrete)s 123.184 (symmetries)s 196.487 (of)s 212.204 (Ito)s 234.304 (equations.)s 1 f 54.654 y 0 (In)s 14.962 (this)s 38.323 (app)s 57.99 (endix)s 90.285 (w)s 98.618 (e)s 107.913 (brie\260)s 134.613 (y)s 144.908 (consider,)s 194.977 (for)s 213.306 (the)s 233.934 (sak)s 250.667 (e)s 259.962 (of)s 273.59 (completeness,)s 2 f 349.027 (discr)s 373.713 (ete)s 1 f 396.959 (symmetries)s 460.087 (of)s 69.054 y 0 (sto)s 15.733 (c)s 20.733 (hastic)s 55.536 (di\256eren)s 94.236 (tial)s 115.639 (equations.)s 174.247 (Similarly)s 225.35 (to)s 240.087 (what)s 270.156 (happ)s 296.489 (ens)s 317.292 (in)s 331.362 (the)s 352.098 (deterministic)s 424.267 (case,)s 453.087 (the)s 83.454 y 0 (resulting)s 49.568 (determining)s 116.403 (equations)s 170.272 (are)s 190.44 (in)s 204.575 (general)s 246.076 (to)s 257.076 (o)s 267.211 (di\261cult)s 311.346 (to)s 326.148 (b)s 333.148 (e)s 342.616 (attac)s 368.949 (k)s 374.949 (ed)s 391.084 (except)s 428.886 (for)s 447.387 (v)s 453.387 (ery)s 97.854 y 0 (simple)s 37.61 (classes)s 76.02 (of)s 89.896 (symmetries)s 153.273 (\050e.g.)s 181.903 (re\260)s 198.603 (ections,)s 242.265 (or)s 257.175 (ho)s 269.508 (w)s 277.842 (ev)s 289.175 (er)s 303.418 (linear)s 336.995 (ones\051.)s 373.691 (On)s 393.901 (the)s 414.777 (other)s 446.354 (side,)s 112.254 y 0 (they)s 27.448 (can)s 49.895 (b)s 56.895 (e)s 66.676 (used)s 5 f 94.524 (a)s 1 f 100.867 (\051)s 109.981 (to)s 125.096 (c)s 130.096 (hec)s 147.096 (k)s 157.877 (if)s 169.325 (a)s 179.772 (giv)s 195.106 (en)s 211.553 (discrete)s 256.101 (transformation)s 338.349 (is)s 350.863 (a)s 361.311 (symmetry)s 417.859 (of)s 431.973 (a)s 442.421 (giv)s 457.754 (en)s 126.654 y 0 (sto)s 15.733 (c)s 20.733 (hastic)s 56.245 (di\256eren)s 94.945 (tial)s 117.056 (equation;)s 170.557 (and)s 5 f 194.668 (b)s 1 f 199.818 (\051)s 209.263 (to)s 224.708 (determine)s 281.519 (the)s 302.964 (sto)s 318.698 (c)s 323.698 (hastic)s 359.209 (di\256eren)s 397.909 (tial)s 420.021 (equations)s 141.054 y 0 (whic)s 23.667 (h)s 34.063 (admit)s 68.459 (a)s 78.188 (giv)s 93.521 (en)s 109.251 (discrete)s 153.08 (transformation)s 234.61 (as)s 249.072 (a)s 258.802 (symmetry)s 309.902 (.)s 318.478 (See)s 339.541 ([3])s 355.937 (for)s 374.033 (a)s 383.762 (discussion)s 439.691 (of)s 453.087 (the)s 155.454 y 0 (deterministic)s 72.1 (case.)s 181.854 y 0 (W)s 11.333 (e)s 20.667 (consider)s 67.433 (again)s 99.433 (an)s 116.1 (Ito)s 135.1 (equation)s 184.1 (of)s 197.767 (the)s 218.434 (form)s 212.045 y 137.463 (d)s 5 f 144.13 (x)s 6 f 207.09 y 150.988 (i)s 1 f 212.045 y 162.317 (=)s 5 f 178.983 (f)s 6 f 207.09 y 186.15 (i)s 1 f 212.045 y 190.145 (\050)s 0 f 194.812 (x)s 5 f 202.095 (;)s 207.428 (t)s 1 f 211.762 (\051)s 218.428 (d)s 5 f 225.095 (t)s 1 f 236.095 (+)s 5 f 252.095 (\276)s 6 f 207.09 y 259.382 (i)s 215.011 y 262.952 (k)s 1 f 212.045 y 268.856 (\050)s 0 f 273.523 (x)s 5 f 280.806 (;)s 286.139 (t)s 1 f 290.473 (\051)s 297.139 (d)s 5 f 303.806 (w)s 6 f 207.09 y 312.72 (k)s 1 f 212.045 y 318.624 (\050)s 5 f 323.291 (t)s 1 f 327.624 (\051)s 442.087 (\050)s 5 f 446.754 (A:)s 1 f 459.087 (1\051)s 242.235 y 0 (with)s 5 f 27.844 (i;)s 37.312 (k)s 1 f 48.122 (=)s 61.64 (1)s 5 f 67.64 (;)s 72.974 (:::;)s 88.307 (n)s 1 f 95.51 (;)s 103.61 (w)s 111.943 (e)s 121.787 (will)s 144.965 (not)s 166.809 (consider)s 214.087 (here)s 240.632 (transformations)s 327.676 (acting)s 364.187 (on)s 381.365 (the)s 5 f 402.543 (t)s 1 f 411.387 (co)s 423.054 (ordinate,)s 256.635 y 0 (and)s 23.333 (will)s 46 (th)s 57 (us)s 72.4 (limit)s 101.067 (to)s 115.734 (consider)s 162.5 (the)s 183.167 (c)s 188.167 (hange)s 222.834 (of)s 236.5 (co)s 248.167 (ordinates)s 300.267 (in)s 314.267 (the)s 0 f 334.934 (x)s 1 f 346.217 (and)s 0 f 369.551 (w)s 1 f 383.709 (spaces)s 420.509 (giv)s 435.842 (en)s 451.842 (b)s 458.176 (y)s 5 f 286.826 y 137.822 (y)s 6 f 281.871 y 144.136 (i)s 1 f 286.826 y 151.464 (=)s 5 f 164.131 (\301)s 6 f 281.871 y 171.281 (i)s 1 f 286.826 y 175.276 (\050)s 5 f 179.943 (x;)s 192.134 (t)s 1 f 196.468 (\051)s 5 f 209.134 (;)s 222.467 (w)s 6 f 281.871 y 231.381 (k)s 1 f 286.826 y 237.286 (\050)s 5 f 241.952 (t)s 1 f 246.286 (\051)s 254.286 (=)s 5 f 266.952 (R)s 6 f 281.871 y 276.156 (k)s 289.793 y 280.064 (p)s 1 f 286.826 y 285.61 (\050)s 5 f 290.277 (t)s 1 f 294.61 (\051)s 5 f 299.277 (z)s 6 f 281.871 y 305.385 (p)s 1 f 286.826 y 310.932 (\050)s 5 f 315.599 (t)s 1 f 319.932 (\051)s 5 f 328.599 (:)s 1 f 442.087 (\050)s 5 f 446.754 (A:)s 1 f 459.087 (2\051)s 317.017 y 0 (As)s 16.847 (discussed)s 68.16 (in)s 81.274 (section)s 120.454 (1,)s 5 f 133.078 (R)s 1 f 145.395 (will)s 167.176 (b)s 174.176 (e)s 182.622 (assumed)s 229.869 (to)s 243.649 (b)s 250.649 (e)s 259.096 (a)s 268.209 (constan)s 307.943 (t)s 315.723 (orthogonal)s 374.87 (matrix,)s 5 f 416.527 (R)s 7 f 429.065 (2)s 5 f 440.398 (O)s 1 f 449.885 (\050)s 5 f 454.551 (n)s 1 f 461.754 (\051.)s 343.417 y 0 (With)s 31 (the)s 51.667 (Ito)s 70.667 (form)s 94.7 (ula,)s 118.034 (and)s 141.367 (denoting)s 190.7 (as)s 205.434 (usual)s 5 f 236.834 (A)s 1 f 249.167 (=)s 261.834 (\0501)s 5 f 272.5 (=)s 1 f 278.5 (2\051)s 5 f 289.167 (\276)s 296.455 (\276)s 6 f 339.062 y 303.742 (T)s 1 f 343.417 y 311.302 (,)s 318.635 (w)s 326.968 (e)s 336.302 (ha)s 348.635 (v)s 354.635 (e)s 363.968 (at)s 378.635 (once)s 381.438 y 62.536 (d)s 5 f 69.202 (y)s 6 f 376.484 y 75.516 (i)s 1 f 381.438 y 86.845 (=)s 9 f 364.518 y 103.511 (\267)s 5 f 373.32 y 111.669 (@)s 118.707 (\301)s 6 f 368.966 y 125.857 (i)s 18.952 .48 111.285 378.678 r 5 f 389.67 y 111.285 (@)s 118.322 (x)s 6 f 386.203 y 125.18 (j)s 5 f 381.438 y 131.677 (f)s 6 f 376.484 y 138.843 (j)s 1 f 381.438 y 148.566 (+)s 5 f 162.566 (A)s 6 f 376.484 y 171.566 (j)s 176.022 (k)s 5 f 373.32 y 190.96 (@)s 4 f 368.966 y 197.997 (2)s 5 f 373.32 y 203.381 (\301)s 6 f 368.966 y 210.531 (i)s 38.752 .48 183.367 378.678 r 5 f 389.67 y 183.367 (@)s 190.404 (x)s 6 f 386.203 y 197.262 (j)s 5 f 389.67 y 202.319 (@)s 209.356 (x)s 6 f 386.203 y 216.215 (k)s 1 f 381.438 y 228.225 (+)s 5 f 373.32 y 243.665 (@)s 250.703 (\301)s 6 f 368.966 y 257.853 (i)s 18.183 .48 243.665 378.678 r 5 f 389.67 y 247.071 (@)s 254.109 (t)s 9 f 364.518 y 263.288 (\270)s 1 f 381.438 y 273.621 (d)s 5 f 280.288 (t)s 1 f 291.288 (+)s 9 f 364.518 y 307.288 (\267)s 5 f 373.32 y 315.446 (@)s 322.483 (\301)s 6 f 368.966 y 329.633 (i)s 18.952 .48 315.061 378.678 r 5 f 389.67 y 315.061 (@)s 322.099 (x)s 6 f 386.203 y 328.957 (j)s 5 f 381.438 y 335.453 (\276)s 6 f 376.484 y 342.741 (j)s 384.405 y 346.31 (p)s 5 f 381.438 y 351.857 (R)s 6 f 375.672 y 361.061 (p)s 385.058 y 364.968 (k)s 9 f 364.518 y 370.873 (\270)s 1 f 381.438 y 381.206 (d)s 5 f 387.873 (z)s 6 f 376.484 y 393.981 (k)s 5 f 381.438 y 403.885 (:)s 1 f 442.087 (\050)s 5 f 446.754 (A:)s 1 f 459.087 (3\051)s 418.829 y 0 (Notice)s 37.667 (that)s 63.667 (here)s 5 f 89.7 (\301)s 1 f 100.183 (=)s 5 f 112.85 (\301)s 1 f 120 (\050)s 5 f 124.667 (x;)s 136.858 (t)s 1 f 141.192 (\051,)s 5 f 153.192 (\276)s 1 f 163.812 (=)s 5 f 176.479 (\276)s 1 f 183.767 (\050)s 5 f 188.433 (x;)s 200.625 (t)s 1 f 204.958 (\051,)s 216.958 (and)s 5 f 240.292 (R)s 1 f 253.496 (is)s 265.562 (constan)s 305.296 (t.)s 445.229 y 0 (Th)s 15 (us,)s 34.937 (requiring)s 86.967 (this)s 111.331 (is)s 124.36 (just)s 149.057 (\050A.1\051)s 181.687 (again,)s 218.224 (written)s 261.221 (in)s 276.184 (terms)s 310.581 (of)s 325.211 (\050)s 0 f 329.877 (y)s 5 f 337.352 (;)s 0 f 342.686 (z)s 1 f 348.819 (\051)s 358.449 (instead)s 400.812 (of)s 415.442 (\050)s 0 f 420.109 (x)s 5 f 427.392 (;)s 0 f 432.725 (w)s 1 f 442.883 (\051,)s 456.087 (w)s 464.421 (e)s 459.629 y 0 (obtain)s 37.333 (the)s 2 f 58 (determining)s 124.7 (e)s 129.606 (quations)s 177.446 (for)s 196.612 (discr)s 221.299 (ete)s 240.619 (symmetries)s 303.945 (of)s 318.052 (an)s 335.225 (Ito)s 354.265 (e)s 359.171 (quation)s 1 f 397.811 (:)s 5 f 488.866 y 42.509 (@)s 49.547 (\301)s 6 f 484.511 y 56.697 (i)s 1 f 488.866 y 60.692 (\050)s 5 f 65.358 (x;)s 77.55 (t)s 1 f 81.883 (\051)s 44.041 .48 42.509 494.224 r 5 f 505.216 y 55.054 (@)s 62.091 (x)s 6 f 501.749 y 68.949 (j)s 5 f 496.984 y 89.99 (f)s 6 f 492.03 y 97.157 (j)s 1 f 496.984 y 102.213 (\050)s 5 f 106.88 (x;)s 119.071 (t)s 1 f 123.405 (\051)s 132.738 (+)s 5 f 146.738 (A)s 6 f 492.03 y 155.738 (j)s 160.194 (k)s 1 f 496.984 y 166.098 (\050)s 5 f 170.765 (x;)s 182.957 (t)s 1 f 187.29 (\051)s 5 f 488.866 y 195.397 (@)s 4 f 484.511 y 202.434 (2)s 5 f 488.866 y 207.817 (\301)s 6 f 484.511 y 214.967 (i)s 1 f 488.866 y 218.963 (\050)s 5 f 223.629 (x;)s 235.821 (t)s 1 f 240.154 (\051)s 49.424 .48 195.397 494.224 r 5 f 505.216 y 200.733 (@)s 207.77 (x)s 6 f 501.749 y 214.628 (j)s 5 f 505.216 y 219.685 (@)s 226.722 (x)s 6 f 501.749 y 233.581 (k)s 1 f 496.984 y 250.927 (+)s 5 f 488.866 y 266.367 (@)s 273.405 (\301)s 6 f 484.511 y 280.555 (i)s 1 f 488.866 y 284.55 (\050)s 5 f 289.217 (x;)s 301.408 (t)s 1 f 305.742 (\051)s 44.041 .48 266.367 494.224 r 5 f 505.216 y 282.702 (@)s 289.74 (t)s 1 f 496.984 y 319.182 (=)s 5 f 335.848 (f)s 6 f 492.03 y 343.015 (i)s 1 f 496.984 y 349.01 (\050)s 5 f 353.677 (\301)s 1 f 360.827 (\050)s 5 f 365.493 (x;)s 377.685 (t)s 1 f 382.018 (\051)s 5 f 386.685 (;)s 392.018 (t)s 1 f 396.352 (\051)s 5 f 519.929 y 42.509 (@)s 49.547 (\301)s 6 f 515.574 y 56.697 (i)s 1 f 519.929 y 60.692 (\050)s 5 f 65.358 (x;)s 77.55 (t)s 1 f 81.883 (\051)s 44.041 .48 42.509 525.287 r 5 f 536.278 y 55.054 (@)s 62.091 (x)s 6 f 532.812 y 68.949 (j)s 5 f 528.047 y 89.99 (\276)s 6 f 523.092 y 97.278 (j)s 531.013 y 100.847 (p)s 1 f 528.047 y 106.394 (\050)s 5 f 111.06 (x;)s 123.252 (t)s 1 f 127.585 (\051)s 5 f 134.252 (R)s 6 f 522.28 y 143.456 (p)s 531.667 y 147.364 (k)s 1 f 528.047 y 160.601 (=)s 5 f 177.268 (\276)s 6 f 523.092 y 184.555 (i)s 531.013 y 188.125 (k)s 1 f 528.047 y 196.029 (\050)s 5 f 200.695 (\301)s 1 f 207.845 (\050)s 5 f 212.512 (x;)s 224.704 (t)s 1 f 229.037 (\051)s 5 f 233.704 (;)s 239.037 (t)s 1 f 243.37 (\051)s 5 f 254.037 (:)s 1 f 510.616 y 442.087 (\050)s 5 f 446.754 (A:)s 1 f 459.087 (4\051)s 0 f 573.602 y 0 (Remark.)s 1 f 62.679 (W)s 74.012 (e)s 85.233 (stress)s 120.02 (that)s 147.908 (w)s 156.241 (e)s 167.462 (ha)s 179.795 (v)s 185.795 (e)s 197.016 (actually)s 244.571 (considered)s 305.225 (symmetries)s 370.279 (of)s 385.833 (the)s 408.387 (one-particle)s 588.002 y 0 (pro)s 17.7 (cess)s 41.881 (de\257ned)s 83.261 (b)s 89.594 (y)s 99.975 (the)s 120.689 (Ito)s 139.736 (equation)s 188.783 (\050A.1\051;)s 223.854 (the)s 244.568 (extension)s 297.682 (to)s 5 f 312.396 (N)s 1 f 323.346 (-particle)s 370.76 (pro)s 388.46 (cesses)s 422.707 (w)s 431.04 (ould)s 457.754 (go)s 602.402 y 0 (along)s 32 (the)s 52.667 (lines)s 80.067 (of)s 93.734 (the)s 114.4 (discussion)s 170.6 (on)s 187.267 (section)s 227.334 (2.)s 7 f 242 (\246)s 1 f 628.802 y 0 (Let)s 22.179 (us)s 38.259 (no)s 50.592 (w)s 63.938 (brie\260)s 90.638 (y)s 101.651 (consider)s 149.097 (some)s 179.844 (v)s 185.844 (ery)s 206.89 (simple)s 244.969 (examples)s 297.382 (regarding)s 351.462 (discrete)s 396.241 (symmetries)s 460.087 (of)s 643.202 y 0 (Ito)s 19 (equations.)s 672.002 y 228.877 (22)s end endp %%Page: 25 25 %%BeginPageSetup initializepage (degasperis; page: 25 of 26)setjob %%EndPageSetup gS 0 0 552 730 rC 0 0 :M f1_12 sf ( )S f2_12 sf ( )S f3_12 sf ( )S 13 0 :M f8_12 sf .1 .01( )J 17 0 :M f4_12 sf .116 .012( )J f5_12 sf .148 .015( )J 34 0 :M f6_12 sf ( )S f9_12 sf 11.923 1.192( )J 0 0 :M initclip TeXdict begin 1 10 1200 1000 div mul /CMR10 findfont fdef 2 10 1200 1000 div mul /CMTI10 findfont fdef 3 10 1200 1000 div mul /CMTT10 findfont fdef 4 7 1200 1000 div mul /CMR7 findfont fdef 5 10 1200 1000 div mul /CMMI10 findfont fdef 6 7 1200 1000 div mul /CMMI7 findfont fdef 7 10 1200 1000 div mul /CMSY10 findfont fdef 8 7 1200 1000 div mul /CMSY7 findfont fdef 42 41 translate 72 72.27 div dup scale 3 f 12 y 0 (Example)s 50.4 (1.)s 1 f 70.361 (The)s 95.703 (simplest)s 143.179 (case)s 169.255 (of)s 183.598 (discrete)s 228.374 (transformation)s 310.85 (is)s 323.592 (pro)s 340.626 (vided)s 373.635 (b)s 379.968 (y)s 5 f 390.978 (\301)s 6 f 7.645 y 398.128 (i)s 1 f 12 y 402.123 (\050)s 5 f 406.789 (x;)s 418.981 (t)s 1 f 423.314 (\051)s 432.441 (=)s 7 f 446.234 (\241)s 5 f 455.567 (x)s 6 f 7.645 y 462.426 (i)s 1 f 12 y 466.421 (,)s 5 f 26.4 y 0 (R)s 1 f 12.537 (=)s 7 f 25.204 (\247)s 5 f 34.537 (I)s 1 f 40.754 (.)s 49.421 (In)s 64.421 (the)s 85.087 (case)s 5 f 110.487 (R)s 1 f 123.025 (=)s 5 f 135.692 (I)s 1 f 141.908 (,)s 149.241 (\050A.4\051)s 180.908 (reduce)s 218.942 (to)s 5 f 55.2 y 119.711 (f)s 6 f 50.245 y 126.878 (i)s 1 f 55.2 y 130.873 (\050)s 5 f 135.54 (x;)s 147.731 (t)s 1 f 152.065 (\051)s 160.065 (=)s 7 f 172.731 (\241)s 5 f 182.065 (f)s 6 f 50.245 y 189.231 (i)s 1 f 55.2 y 193.226 (\050)s 7 f 197.893 (\241)s 5 f 207.226 (x;)s 219.418 (t)s 1 f 223.751 (\051)s 5 f 232.418 (;)s 241.751 (\276)s 1 f 249.039 (\050)s 5 f 253.706 (x;)s 265.897 (t)s 1 f 270.231 (\051)s 278.23 (=)s 7 f 290.897 (\241)s 5 f 300.23 (\276)s 1 f 307.518 (\050)s 7 f 312.185 (\241)s 5 f 321.518 (x;)s 333.71 (t)s 1 f 338.043 (\051)s 5 f 346.71 (:)s 1 f 442.087 (\050)s 5 f 446.754 (A:)s 1 f 459.087 (5\051)s 84 y 0 (In)s 15 (the)s 35.667 (case)s 5 f 61.067 (\301)s 6 f 79.645 y 68.217 (i)s 1 f 84 y 72.212 (\050)s 5 f 76.879 (x;)s 89.07 (t)s 1 f 93.404 (\051)s 101.403 (=)s 7 f 114.07 (\241)s 5 f 123.403 (x)s 6 f 79.645 y 130.262 (i)s 1 f 84 y 134.257 (,)s 5 f 141.59 (R)s 1 f 154.128 (=)s 7 f 166.794 (\241)s 5 f 176.128 (I)s 1 f 182.344 (,)s 189.678 (w)s 198.011 (e)s 207.344 (get)s 227.344 (instead)s 5 f 112.8 y 124.378 (f)s 6 f 107.845 y 131.545 (i)s 1 f 112.8 y 135.54 (\050)s 5 f 140.206 (x;)s 152.398 (t)s 1 f 156.731 (\051)s 164.731 (=)s 7 f 177.398 (\241)s 5 f 186.731 (f)s 6 f 107.845 y 193.898 (i)s 1 f 112.8 y 197.893 (\050)s 7 f 202.56 (\241)s 5 f 211.893 (x;)s 224.085 (t)s 1 f 228.418 (\051)s 5 f 237.085 (;)s 246.418 (\276)s 1 f 253.706 (\050)s 5 f 258.372 (x;)s 270.564 (t)s 1 f 274.897 (\051)s 282.897 (=)s 5 f 295.564 (\276)s 1 f 302.851 (\050)s 7 f 307.518 (\241)s 5 f 316.851 (x;)s 329.043 (t)s 1 f 333.376 (\051)s 5 f 342.043 (:)s 1 f 442.087 (\050)s 5 f 446.754 (A:)s 1 f 459.087 (6\051)s 7 f 141.6 y 0 (\246)s 3 f 168 y 0 (Example)s 50.4 (2.)s 1 f 68.223 (W)s 79.556 (e)s 88.558 (can)s 110.228 (consider)s 156.664 (the)s 177 (case)s 202.069 (of)s 5 f 215.405 (n)s 1 f 226.277 (indep)s 255.277 (enden)s 285.611 (t)s 293.947 (\134Langevin)s 351.449 (oscillators\042,)s 416.685 (see)s 435.754 (\0503.16\051.)s 196.8 y 124.223 (d)s 5 f 130.89 (x)s 6 f 191.845 y 137.748 (i)s 1 f 196.8 y 149.077 (=)s 7 f 165.743 (\241)s 5 f 175.077 (a)s 4 f 198.967 y 181.42 (\050)s 6 f 185.17 (i)s 4 f 188.565 (\051)s 5 f 196.8 y 192.915 (x)s 6 f 191.845 y 199.773 (i)s 1 f 196.8 y 203.768 (d)s 5 f 210.435 (t)s 1 f 217.435 (+)s 5 f 229.435 (s)s 4 f 198.967 y 235.06 (\050)s 6 f 238.81 (i)s 4 f 242.205 (\051)s 1 f 196.8 y 246.555 (d)s 5 f 253.222 (w)s 6 f 191.845 y 262.136 (i)s 1 f 196.8 y 274.131 (\050no)s 295.464 (sum)s 320.864 (on)s 337.531 (i\051)s 442.087 (\050)s 5 f 446.754 (A:)s 1 f 459.087 (7\051)s 225.6 y 0 (with)s 5 f 27.535 (a)s 4 f 227.767 y 33.878 (\050)s 6 f 37.628 (i)s 4 f 41.023 (\051)s 5 f 225.6 y 45.373 (;)s 50.707 (s)s 4 f 227.767 y 56.332 (\050)s 6 f 60.082 (i)s 4 f 63.477 (\051)s 1 f 225.6 y 72.028 (p)s 79.028 (ositiv)s 107.095 (e)s 116.63 (constan)s 156.363 (ts.)s 175.035 (It)s 188.236 (is)s 200.505 (immediate)s 259.373 (to)s 274.241 (c)s 279.241 (hec)s 296.241 (k)s 306.776 (that)s 332.978 (\050A.6\051)s 364.846 (is)s 377.114 (satis\257ed.)s 428.519 (More)s 459.754 (in)s 240 y 0 (general,)s 44.7 (\050A.4\051)s 76.367 (read)s 103.067 (no)s 115.4 (w)s 128.067 (\050no)s 149.4 (sum)s 174.8 (o)s 180.467 (v)s 186.467 (er)s 5 f 200.5 (i)s 1 f 204.634 (,)s 211.968 (sum)s 237.368 (o)s 243.035 (v)s 249.035 (er)s 5 f 263.068 (j)s 1 f 272.696 (when)s 304.03 (rep)s 321.063 (eated\051)s 5 f 279.013 y 149.764 (a)s 4 f 281.18 y 156.107 (\050)s 6 f 159.857 (j)s 4 f 164.313 (\051)s 5 f 270.895 y 170.488 (@)s 177.525 (\301)s 6 f 266.54 y 184.675 (i)s 18.952 .48 170.103 276.253 r 5 f 287.245 y 170.103 (@)s 177.141 (x)s 6 f 289.045 y 183.999 (j)s 7 f 279.013 y 193.162 (\241)s 5 f 268.368 y 206.602 (s)s 4 f 264.013 y 212.227 (2)s 272.233 y 212.227 (\050)s 6 f 215.977 (j)s 4 f 220.433 (\051)s 18.181 .48 206.602 276.253 r 1 f 287.245 y 212.692 (2)s 5 f 270.895 y 234.832 (@)s 4 f 266.54 y 241.87 (2)s 5 f 270.895 y 247.253 (\301)s 6 f 266.54 y 254.403 (i)s 37.904 .48 227.663 276.253 r 5 f 287.245 y 227.663 (@)s 234.701 (x)s 6 f 283.778 y 241.559 (j)s 5 f 287.245 y 246.615 (@)s 253.653 (x)s 6 f 283.778 y 260.511 (j)s 1 f 279.013 y 274.34 (=)s 5 f 291.007 (a)s 4 f 281.18 y 297.35 (\050)s 6 f 301.1 (i)s 4 f 304.495 (\051)s 5 f 279.013 y 308.845 (\301)s 6 f 274.058 y 315.995 (i)s 5 f 313.509 y 149.764 (s)s 4 f 315.676 y 155.389 (\050)s 6 f 159.139 (j)s 4 f 163.595 (\051)s 5 f 305.391 y 169.77 (@)s 176.807 (\301)s 6 f 301.036 y 183.957 (i)s 18.952 .48 169.385 310.749 r 5 f 321.74 y 169.385 (@)s 176.422 (x)s 6 f 323.54 y 183.281 (j)s 5 f 313.509 y 189.777 (R)s 6 f 307.742 y 198.981 (j)s 317.129 y 202.889 (k)s 1 f 313.509 y 216.126 (=)s 5 f 232.793 (s)s 4 f 315.676 y 238.418 (\050)s 6 f 242.168 (i)s 4 f 245.563 (\051)s 5 f 313.509 y 249.913 (\261)s 6 f 308.554 y 255.7 (i)s 316.476 y 259.246 (k)s 5 f 313.509 y 269.15 (:)s 1 f 294.887 y 442.087 (\050)s 5 f 446.754 (A:)s 1 f 459.087 (8\051)s 7 f 346.107 y 0 (\246)s 3 f 372.507 y 0 (Example)s 50.4 (3.)s 1 f 68.876 (Let)s 90.557 (us)s 106.138 (consider)s 153.086 (the)s 173.934 (case)s 199.515 (of)s 2 f 213.363 (line)s 231.763 (ar)s 1 f 247.137 (symmetries)s 5 f 310.485 (\301)s 6 f 368.152 y 317.635 (i)s 1 f 372.507 y 321.63 (\050)s 5 f 326.297 (x;)s 338.488 (t)s 1 f 342.822 (\051)s 351.124 (=)s 5 f 364.092 (L)s 6 f 368.152 y 372.259 (i)s 375.474 y 376.259 (p)s 1 f 372.507 y 381.806 (\050)s 5 f 386.472 (t)s 1 f 390.806 (\051)s 5 f 395.472 (x)s 6 f 368.152 y 402.331 (p)s 1 f 372.507 y 407.877 (:)s 416.906 (no)s 429.24 (w)s 442.087 (\050A.4\051)s 386.907 y 0 (reduce)s 38.033 (to)s 5 f 398.62 y 145.734 (L)s 6 f 393.665 y 153.9 (i)s 401.587 y 157.9 (j)s 1 f 398.62 y 162.957 (\050)s 5 f 167.623 (t)s 1 f 171.957 (\051)s 5 f 176.623 (;)s 181.957 (f)s 6 f 393.665 y 189.123 (j)s 1 f 398.62 y 194.18 (\050)s 5 f 198.846 (x;)s 211.038 (t)s 1 f 215.371 (\051)s 227.371 (=)s 5 f 244.038 (f)s 6 f 393.665 y 251.205 (i)s 1 f 398.62 y 255.2 (\050)s 5 f 259.866 (Lx;)s 280.225 (t)s 1 f 284.558 (\051)s 5 f 418.533 y 145.734 (L)s 6 f 413.578 y 153.9 (i)s 421.5 y 157.9 (j)s 1 f 418.533 y 162.957 (\050)s 5 f 167.623 (t)s 1 f 171.957 (\051)s 5 f 178.623 (\276)s 6 f 413.578 y 185.911 (j)s 421.5 y 189.48 (p)s 1 f 418.533 y 195.027 (\050)s 5 f 199.694 (x;)s 211.885 (t)s 1 f 216.219 (\051)s 5 f 222.885 (R)s 6 f 412.766 y 232.09 (p)s 422.153 y 235.997 (k)s 1 f 418.533 y 249.234 (=)s 5 f 265.901 (\276)s 6 f 413.578 y 273.188 (i)s 421.5 y 276.758 (k)s 1 f 418.533 y 282.662 (\050)s 5 f 287.329 (Lx;)s 307.687 (t)s 1 f 312.02 (\051)s 5 f 320.687 (:)s 1 f 408.62 y 442.087 (\050)s 5 f 446.754 (A:)s 1 f 459.087 (9\051)s 442.646 y 0 (If)s 13.789 (w)s 22.122 (e)s 33.244 (further)s 75.432 (assume)s 5 f 118.688 (f)s 1 f 131.643 (is)s 145.498 (also)s 171.354 (linear)s 206.509 (in)s 5 f 222.297 (x)s 1 f 229.156 (,)s 5 f 238.725 (f)s 6 f 438.291 y 245.892 (i)s 1 f 442.646 y 249.887 (\050)s 5 f 254.553 (x;)s 266.745 (t)s 1 f 271.078 (\051)s 282.059 (=)s 5 f 297.707 (M)s 6 f 438.291 y 310.657 (i)s 445.769 y 313.348 (j)s 1 f 442.646 y 318.405 (\050)s 5 f 323.071 (t)s 1 f 327.405 (\051)s 5 f 332.071 (x)s 6 f 438.291 y 338.93 (j)s 1 f 442.646 y 343.986 (,)s 353.555 (these)s 386.077 (read)s 414.565 (in)s 430.354 (ob)s 442.687 (vious)s 457.046 y 0 (shorthand)s 56.767 (notation)s 471.446 y 144.738 ([)s 5 f 148.071 (L;)s 161.571 (M)s 1 f 174.521 (])s 181.188 (=)s 193.854 (0)s 5 f 207.854 (;)s 221.187 (L\276)s 1 f 236.642 (\050)s 5 f 241.308 (x)s 1 f 248.167 (\051)s 5 f 252.833 (R)s 1 f 265.371 (=)s 5 f 278.037 (\276)s 1 f 285.325 (\050)s 5 f 289.991 (Lx)s 1 f 305.017 (\051)s 321.683 (;)s 436.087 (\050)s 5 f 440.754 (A:)s 1 f 453.087 (10\051)s 494.246 y 0 (when)s 5 f 30.927 (\276)s 1 f 41.809 (dos)s 62.803 (not)s 83.73 (dep)s 102.731 (end)s 124.991 (on)s 5 f 141.252 (x)s 1 f 151.704 (and)s 174.632 (is)s 186.293 (in)s 195.959 (v)s 201.959 (ertible,)s 242.335 (w)s 250.668 (e)s 259.595 (get)s 279.189 (a)s 288.783 (linear)s 321.744 (symmetry)s 377.438 (from)s 405.399 (an)s 417.733 (y)s 5 f 427.66 (L)s 1 f 439.421 (whic)s 463.087 (h)s 508.646 y 0 (comm)s 31 (utes)s 56.4 (with)s 5 f 83.733 (M)s 1 f 96.683 (,)s 104.017 (with)s 5 f 131.35 (R)s 1 f 143.888 (=)s 5 f 156.554 (\276)s 8 f 504.291 y 163.842 (\241)s 4 f 171.342 (1)s 5 f 508.646 y 176.725 (L)s 8 f 504.291 y 184.892 (\241)s 4 f 192.392 (1)s 5 f 508.646 y 197.775 (\276)s 1 f 205.063 (.)s 7 f 213.729 (\246)s 3 f 535.046 y 0 (Example)s 50.4 (4.)s 1 f 69.953 (F)s 76.787 (or)s 92.027 (equations)s 146.3 (with)s 5 f 174.174 (\276)s 6 f 530.691 y 181.461 (i)s 538.444 y 185.031 (k)s 1 f 535.046 y 190.935 (\050)s 5 f 195.602 (x;)s 207.793 (t)s 1 f 212.127 (\051)s 221.027 (=)s 5 f 234.593 (s)s 4 f 536.846 y 240.218 (0)s 5 f 535.046 y 245.602 (\261)s 6 f 530.691 y 251.389 (i)s 538.444 y 254.935 (k)s 1 f 535.046 y 265.379 (\050with)s 5 f 297.92 (s)s 1 f 308.085 (a)s 318.625 (real)s 342.532 (constan)s 382.265 (t\051,)s 399.607 (the)s 420.814 (second)s 460.087 (of)s 549.446 y 0 (\050A.4\051)s 31.107 (alw)s 48.773 (a)s 54.44 (ys)s 68.947 (reduces)s 111.153 (to)s 125.26 (\050)s 5 f 129.926 (@)s 136.964 (\301)s 6 f 545.091 y 144.114 (i)s 5 f 549.446 y 148.109 (=@)s 161.146 (x)s 6 f 545.091 y 168.005 (j)s 1 f 549.446 y 173.061 (\051)s 181.061 (=)s 193.728 (\050)s 5 f 198.394 (R)s 8 f 545.091 y 207.598 (\241)s 4 f 215.098 (1)s 1 f 549.446 y 220.482 (\051)s 6 f 545.091 y 225.149 (i)s 552.569 y 229.149 (j)s 1 f 549.446 y 234.205 (.)s 242.685 (But)s 265.958 (this)s 288.798 (implies)s 5 f 328.971 (\301)s 1 f 339.561 (m)s 349.228 (ust)s 368.734 (b)s 375.734 (e)s 384.508 (a)s 393.947 (linear)s 426.754 (function)s 565.316 y 0 (of)s 13.291 (the)s 5 f 33.582 (x)s 1 f 40.44 (,)s 5 f 47.473 (\301)s 6 f 560.961 y 54.623 (i)s 1 f 565.316 y 58.618 (\050)s 5 f 63.285 (x;)s 75.477 (t)s 1 f 79.81 (\051)s 87.81 (=)s 5 f 100.477 (L)s 6 f 560.961 y 108.643 (i)s 568.439 y 112.643 (j)s 1 f 565.316 y 117.7 (\050)s 5 f 122.366 (t)s 1 f 126.7 (\051)s 5 f 131.366 (x)s 6 f 560.961 y 138.225 (j)s 1 f 565.316 y 146.905 (with)s 5 f 173.863 (L)s 1 f 185.654 (an)s 201.945 (orthogonal)s 261.603 (matrix)s 300.261 (\050)s 5 f 304.928 (L)s 1 f 316.427 (=)s 5 f 329.094 (R)s 8 f 560.961 y 338.298 (\241)s 4 f 345.798 (1)s 1 f 565.316 y 351.182 (\051.)s 364.39 (Th)s 379.39 (us)s 394.414 (in)s 408.039 (this)s 431.063 (case)s 456.087 (w)s 464.421 (e)s 581.186 y 0 (are)s 20.033 (alw)s 37.7 (a)s 43.367 (ys)s 58.433 (reduced)s 103.133 (to)s 117.8 (the)s 138.467 (simple)s 175.867 (equation)s 5 f 224.867 (L)s 6 f 576.831 y 233.034 (i)s 584.309 y 237.034 (j)s 5 f 581.186 y 242.09 (f)s 6 f 576.831 y 249.257 (j)s 1 f 581.186 y 254.313 (\050)s 5 f 258.98 (x;)s 271.171 (t)s 1 f 275.505 (\051)s 283.505 (=)s 5 f 296.171 (f)s 6 f 576.831 y 303.338 (i)s 1 f 581.186 y 307.333 (\050)s 5 f 312 (Lx;)s 332.358 (t)s 1 f 336.691 (\051,)s 5 f 348.691 (L)s 7 f 360.191 (2)s 5 f 371.525 (O)s 1 f 381.011 (\050)s 5 f 385.678 (n)s 1 f 392.881 (\051.)s 7 f 406.214 (\246)s 1 f 672.002 y 228.877 (23)s end endp %%Page: 26 26 %%BeginPageSetup initializepage (degasperis; page: 26 of 26)setjob %%EndPageSetup gS 0 0 552 730 rC 0 0 :M f0_12 sf ( )S 4 0 :M f1_12 sf ( )S f2_12 sf ( )S 0 0 :M initclip TeXdict begin 0 10 1200 1000 div mul /CMBX10 findfont fdef 1 10 1200 1000 div mul /CMR10 findfont fdef 2 10 1200 1000 div mul /CMTI10 findfont fdef 42 41 translate 72 72.27 div dup scale 0 f 12 y 0 (References.)s 1 f 52.8 y 0 ([1])s 16.667 (L.)s 31.5 (Arnold,)s 2 f 75.2 (R)s 83.34 (andom)s 122.593 (Dynamic)s 168.76 (al)s 182.253 (Systems)s 1 f 223.959 (,)s 231.293 (Springer)s 279.359 (\050Berlin\051)s 324.559 (1998)s 79.2 y 0 ([2])s 16.936 (L.)s 32.039 (Arnold)s 72.675 (and)s 96.278 (P)s 103.444 (.)s 111.047 (Imk)s 131.38 (eller,)s 161.084 (\134Normal)s 210.387 (forms)s 243.756 (for)s 262.392 (sto)s 278.125 (c)s 283.125 (hastic)s 318.128 (di\256eren)s 356.828 (tial)s 378.431 (equations\042,)s 2 f 441.834 (Pr)s 454.421 (ob.)s 93.6 y 0 (The)s 19.627 (ory)s 40.94 (and)s 64.246 (R)s 72.386 (el.)s 90.173 (Fields)s 0 f 125.312 (110)s 1 f 150.012 (\0501998\051,)s 190.679 (559-588)s 120 y 0 ([3])s 15.876 (G.)s 31.836 (Gaeta)s 66.462 (and)s 89.005 (M.A.)s 118.881 (Ro)s 134.048 (dr)s 144.081 (\266)s 145.415 (\263guez,)s 178.782 (\134Discrete)s 230.592 (symmetries)s 292.968 (of)s 305.844 (di\256eren)s 344.544 (tial)s 365.087 (equations\042,)s 2 f 427.522 (J.)s 441.068 (Phys.)s 134.4 y 0 (A:)s 16.893 (Math.)s 53.106 (Gen.)s 0 f 83.669 (29)s 1 f 101.469 (\0501996\051,)s 142.136 (859-880)s 160.8 y 0 ([4])s 16.19 (G.)s 32.464 (Gaeta)s 67.404 (and)s 90.261 (N.)s 106.117 (Ro)s 121.284 (dr)s 131.317 (\266)s 132.651 (\263guez)s 162.841 (Quin)s 188.508 (tero,)s 216.16 (\134Lie-p)s 249.327 (oin)s 264.993 (t)s 273.183 (symmetries)s 335.874 (and)s 358.73 (sto)s 374.464 (c)s 379.464 (hastic)s 413.721 (di\256eren)s 452.421 (tial)s 175.2 y 0 (equations\042,)s 2 f 63.067 (J.)s 77.34 (Phys.)s 111.546 (A:)s 128.44 (Math.)s 164.653 (Gen.)s 0 f 195.216 (32)s 1 f 213.016 (\0501999\051,)s 253.683 (8485-8505)s 201.6 y 0 ([5])s 16.178 (N.G.)s 44.774 (v)s 50.44 (an)s 66.619 (Kamp)s 98.952 (en,)s 2 f 117.895 (Sto)s 134.148 (chastic)s 173.872 (Pr)s 186.459 (o)s 191.979 (c)s 196.885 (esses)s 226.489 (in)s 240.76 (Physics)s 283.717 (and)s 306.575 (Chemistry)s 1 f 363.599 (\0502nd)s 391.111 (edition\051,)s 438.721 (North)s 216 y 0 (Holland)s 45 (\050Amsterdam\051)s 119.434 (1992)s 242.4 y 0 ([6])s 16.667 (P)s 23.833 (.J.)s 40.667 (Olv)s 59.333 (er,)s 2 f 76.7 (Applic)s 109.54 (ations)s 145.42 (of)s 159.526 (Lie)s 180.546 (gr)s 190.513 (oups)s 218.419 (to)s 232.832 (di\256er)s 259.972 (ential)s 293.399 (e)s 298.305 (quations)s 1 f 341.852 (,)s 349.185 (Springer)s 397.252 (\050Berlin\051)s 442.452 (1986)s 672.002 y 228.877 (24)s end endp %%Trailer end %%EOF ---------------0003100813818--