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# Thomson problem: Wikis

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# Encyclopedia

Updated live from Wikipedia, last check: May 23, 2013 05:27 UTC (47 seconds ago)

The Thomson problem is that of determining the minimum (ground state) energy configuration of N classical electrons on the surface of a sphere (or 2-sphere S2). The electrons repel each other with a force given by Coulomb's law. This problem is named for J. J. Thomson, who posed it in 1904[1] as part of the development of the plum pudding model of the atom. In this model, the electrons formed spherical shells. This one hundred year old puzzle (Thomson wrote "I have not as yet succeeded in getting a general solution") has many important physical realizations including multi-electron bubbles and the surface ordering of liquid metal drops confined in Paul traps. The Thomson problem is solved exactly for a few particles on the sphere and the energy per particle is known for a large number of particles on the surface of the sphere. Numerical results provide the best known solutions for a wide range of particle numbers. The configurations that are found have a great variety of geometrical structure.

 "No fact discovered about the atom can be trivial, nor fail to accelerate the progress of physical science, for the greater part of natural philosophy is the outcome of the structure and mechanism of the atom." —Sir J. J. Thomson[2]

One can also ask for ground states of particles interacting with arbitrary potentials: this is the generalized Thomson problem. The generalized Thomson problem arises, for example, in determining the arrangements of the protein subunits which comprise the shells of spherical viruses. The "particles" in this application are clusters of protein subunits arranged on a shell. Other realizations include regular arrangements of colloid particles in colloidosomes, proposed for encapsulation of active ingredients such as drugs, nutrients or living cells, fullerene patterns of carbon atoms, and VSEPR Theory. An example with long range logarithmic interactions is provided by the Abrikosov vortices which would form at low temperatures in a superconducting metal shell with a large monopole at the center.

The Thomson problem is of outstanding mathematical interest not only for the asymptotics (large N behavior) of the growth of minimum energy configurations but also for the characteristics of the minimum energy configurations themselves.

## Best known solutions

In the following table N is the number of charges on a sphere of radius 1. E1 is the energy (any pair of charges with distance r have potential energy 1/r). The symmetry type is given in Schönflies notation (see Point groups in three dimensions). ri are the positions of the charges. Most symmetry types require the sum (and thus the electric dipole moment) to be zero.

It is customary to also consider the polyhedron formed by the convex hull of the charges. Thus vi is the number of vertices where the given number of edges meet. e is the total number of edges and f3 and f4 are the number of triangle and quadrilateral faces. θ1 is the smallest angle between any two charges.

N E1 Symmetry $\left| \sum \mathbf{r}_{i} \right|$ v3 v4 v5 v6 v7 v8 e f3 f4 θ1 Equivalent Polyhedron
2 0.500000000 $D_{\infty h}$ 0 - - - - - - - - - 180.000°
3 1.732050808 D3h 0 - - - - - - - - - 120.000°
4 3.674234614 Td 0 4 0 0 0 0 0 6 4 0 109.471° tetrahedron
5 6.474691495 D3h 0 2 3 0 0 0 0 9 6 0 90.000° triangular dipyramid
6 9.985281374 Oh 0 0 6 0 0 0 0 12 8 0 90.000° octahedron
7 14.452997414 D5h 0 0 5 2 0 0 0 15 10 0 72.000° pentagonal dipyramid
8 19.675287861 D4d 0 0 8 0 0 0 0 16 8 2 71.694° square antiprism
9 25.759986531 D3h 0 0 3 6 0 0 0 21 14 0 61.190° triaugmented triangular prism
10 32.716949460 D4d 0 0 2 8 0 0 0 24 16 0 64.996° gyroelongated square dipyramid
11 40.596450510 C2v 0.013219635 0 2 8 1 0 0 27 18 0 58.540°
12 49.165253058 Ih 0 0 0 12 0 0 0 30 20 0 63.435° icosahedron
13 58.853230612 C2v 0.008820367 0 1 10 2 0 0 33 22 0 52.317°
14 69.306363297 D6d 0 0 0 12 2 0 0 36 24 0 52.866° gyroelongated hexagonal dipyramid
15 80.670244114 D3 0 0 0 12 3 0 0 39 26 0 49.225°
16 92.911655300 Td 0 0 0 12 4 0 0 42 28 0 48.936°
17 106.050404829 D5h 0 0 0 12 5 0 0 45 30 0 50.108°
18 120.084467447 D4d 0 0 2 8 8 0 0 48 32 0 47.534°
19 135.089467557 C2v 0.000135163 0 0 14 5 0 0 50 32 1 44.910°
20 150.881568334 D3h 0 0 0 12 8 0 0 54 36 0 46.093°
21 167.641622399 C2v 0.001406124 0 1 10 10 0 0 57 38 0 44.321°
22 185.287536149 Td 0 0 0 12 10 0 0 60 40 0 43.302°
23 203.930190663 D3 0 0 0 12 11 0 0 63 42 0 41.481°
24 223.347074052 O 0 0 0 24 0 0 0 60 32 6 42.065° snub cube
25 243.812760299 Cs 0.001021305 0 0 14 11 0 0 68 44 1 39.610°
26 265.133326317 C2 0.001919065 0 0 12 14 0 0 72 48 0 38.842°
27 287.302615033 D5h 0 0 0 12 15 0 0 75 50 0 39.940°
28 310.491542358 T 0 0 0 12 16 0 0 78 52 0 37.824°
29 334.634439920 D3 0 0 0 12 17 0 0 81 54 0 36.391°
30 359.603945904 D2 0 0 0 12 18 0 0 84 56 0 36.942°
31 385.530838063 C3v 0.003204712 0 0 12 19 0 0 87 58 0 36.373°
32 412.261274651 Ih 0 0 0 12 20 0 0 90 60 0 37.377°
33 440.204057448 Cs 0.004356481 0 0 15 17 1 0 92 60 1 33.700°
34 468.904853281 D2 0 0 0 12 22 0 0 96 64 0 33.273°
35 498.569872491 C2 0.000419208 0 0 12 23 0 0 99 66 0 33.100°
36 529.122408375 D2 0 0 0 12 24 0 0 102 68 0 33.229°
37 560.618887731 D5h 0 0 0 12 25 0 0 105 70 0 32.332°
38 593.038503566 D6d 0 0 0 12 26 0 0 108 72 0 33.236°
39 626.389009017 D3h 0 0 0 12 27 0 0 111 74 0 32.053°
40 660.675278835 Td 0 0 0 12 28 0 0 114 76 0 31.916°
41 695.916744342 D3h 0 0 0 12 29 0 0 117 78 0 31.528°
42 732.078107544 D5h 0 0 0 12 30 0 0 120 80 0 31.245°
43 769.190846459 C2v 0.000399668 0 0 12 31 0 0 123 82 0 30.867°
44 807.174263085 Oh 0 0 0 24 20 0 0 120 72 6 31.258°
45 846.188401061 D3 0 0 0 12 33 0 0 129 86 0 30.207°
46 886.167113639 T 0 0 0 12 34 0 0 132 88 0 29.790°
47 927.059270680 Cs 0.002482914 0 0 14 33 0 0 134 88 1 28.787°
48 968.713455344 O 0 0 0 24 24 0 0 132 80 6 29.690°
49 1011.557182654 C3 0.001529341 0 0 12 37 0 0 141 94 0 28.387°
50 1055.182314726 D6d 0 0 0 12 38 0 0 144 96 0 29.231°
51 1099.819290319 D3 0 0 0 12 39 0 0 147 98 0 28.165°
52 1145.418964319 C3 0.000457327 0 0 12 40 0 0 150 100 0 27.670°
53 1191.922290416 C2v 0.000278469 0 0 18 35 0 0 150 96 3 27.137°
54 1239.361474729 C2 0.000137870 0 0 12 42 0 0 156 104 0 27.030°
55 1287.772720783 C2 0.000391696 0 0 12 43 0 0 159 106 0 26.615°
56 1337.094945276 D2 0 0 0 12 44 0 0 162 108 0 26.683°
57 1387.383229253 D3 0 0 0 12 45 0 0 165 110 0 26.702°
58 1438.618250640 D2 0 0 0 12 46 0 0 168 112 0 26.155°
59 1490.773335279 C2 0.000154286 0 0 14 43 2 0 171 114 0 26.170°
60 1543.830400976 D3 0 0 0 12 48 0 0 174 116 0 25.958°
61 1597.941830199 C1 0.001091717 0 0 12 49 0 0 177 118 0 25.392°
62 1652.909409898 D5 0 0 0 12 50 0 0 180 120 0 25.880°
63 1708.879681503 D3 0 0 0 12 51 0 0 183 122 0 25.257°
64 1765.802577927 D2 0 0 0 12 52 0 0 186 124 0 24.920°
65 1823.667960264 C2 0.000399515 0 0 12 53 0 0 189 126 0 24.527°
66 1882.441525304 C2 0.000776245 0 0 12 54 0 0 192 128 0 24.765°
67 1942.122700406 D5 0 0 0 12 55 0 0 195 130 0 24.727°
68 2002.874701749 D2 0 0 0 12 56 0 0 198 132 0 24.433°
69 2064.533483235 D3 0 0 0 12 57 0 0 201 134 0 24.137°
70 2127.100901551 D2d 0 0 0 12 50 0 0 200 128 4 24.291°
71 2190.649906425 C2 0.001256769 0 0 14 55 2 0 207 138 0 23.803°
72 2255.001190975 I 0 0 0 12 60 0 0 210 140 0 24.492°
73 2320.633883745 C2 0.001572959 0 0 12 61 0 0 213 142 0 22.810°
74 2387.072981838 C2 0.000641539 0 0 12 62 0 0 216 144 0 22.966°
75 2454.369689040 D3 0 0 0 12 63 0 0 219 146 0 22.736°
76 2522.674871841 C2 0.000943474 0 0 12 64 0 0 222 148 0 22.886°
77 2522.674871841 D5 0 0 0 12 65 0 0 225 150 0 23.286°
78 2662.046474566 Th 0 0 0 12 66 0 0 228 152 0 23.426°
79 2733.248357479 Cs 0.000702921 0 0 12 63 1 0 230 152 1 22.636°
80 2805.355875981 D4d 0 0 0 16 64 0 0 232 152 2 22.778°
81 2878.522829664 C2 0.000194289 0 0 12 69 0 0 237 158 0 21.892°
82 2952.569675286 D2 0 0 0 12 70 0 0 240 160 0 22.206°
83 3027.528488921 C2 0.000339815 0 0 14 67 2 0 243 162 0 21.646°
84 3103.465124431 C2 0.000401973 0 0 12 72 0 0 246 164 0 21.513°
85 3180.361442939 C2 0.000416581 0 0 12 73 0 0 249 166 0 21.498°
86 3258.211605713 C2 0.001378932 0 0 12 74 0 0 252 168 0 21.522°
87 3337.000750014 C2 0.000754863 0 0 12 75 0 0 255 170 0 21.456°
88 3416.720196758 D2 0 0 0 12 76 0 0 258 172 0 21.486°
89 3497.439018625 C2 0.000070891 0 0 12 77 0 0 261 174 0 21.182°
90 3579.091222723 D3 0 0 0 12 78 0 0 264 176 0 21.230°
91 3661.713699320 C2 0.000033221 0 0 12 79 0 0 267 178 0 21.105°
92 3745.291636241 D2 0 0 0 12 80 0 0 270 180 0 21.026°
93 3829.844338421 C2 0.000213246 0 0 12 81 0 0 273 182 0 20.751°
94 3915.309269620 D2 0 0 0 12 82 0 0 276 184 0 20.952°
95 4001.771675565 C2 0.000116638 0 0 12 83 0 0 279 186 0 20.711°
96 4001.771675565 C2 0.000036310 0 0 12 84 0 0 282 188 0 20.687°
97 4177.533599622 C2 0.000096437 0 0 12 85 0 0 285 190 0 20.450°
98 4266.822464156 C2 0.000112916 0 0 12 86 0 0 288 192 0 20.422°
99 4357.139163132 C2 0.000156508 0 0 12 87 0 0 291 194 0 20.284°
100 4448.350634331 T 0 0 0 12 88 0 0 294 196 0 20.297°
101 4540.590051694 D3 0 0 0 12 89 0 0 297 198 0 20.011°
102 4633.736565899 D3 0 0 0 12 90 0 0 300 200 0 20.040°
103 4727.836616833 C2 0.000201245 0 0 12 91 0 0 303 202 0 19.907°
104 4822.876522746 D6 0 0 0 12 92 0 0 306 204 0 19.957°
105 4919.000637616 D3 0 0 0 12 93 0 0 309 206 0 19.842°
106 5015.984595705 D2 0 0 0 12 94 0 0 312 208 0 19.658°
107 5113.953547724 C2 0.000064137 0 0 12 95 0 0 315 210 0 19.327°
108 5212.813507831 C2 0.000432525 0 0 12 96 0 0 318 212 0 19.327°
109 5312.735079920 C2 0.000647299 0 0 14 93 2 0 321 214 0 19.103°
110 5413.549294192 D6 0 0 0 12 98 0 0 324 216 0 19.476°
111 5515.293214587 D3 0 0 0 12 99 0 0 327 218 0 19.255°
112 5618.044882327 D5 0 0 0 12 100 0 0 330 220 0 19.351°
113 5721.824978027 D3 0 0 0 12 101 0 0 333 222 0 18.978°
114 5826.521572163 C2 0.000149772 0 0 12 102 0 0 336 224 0 18.836°
115 5932.181285777 C3 0.000049972 0 0 12 103 0 0 339 226 0 18.458°
116 6038.815593579 C2 0.000259726 0 0 12 104 0 0 342 228 0 18.386°
117 6146.342446579 C2 0.000127609 0 0 12 105 0 0 345 230 0 18.566°
118 6254.877027790 C2 0.000332475 0 0 12 106 0 0 348 232 0 18.455°
119 6364.347317479 C2 0.000685590 0 0 12 107 0 0 351 234 0 18.336°
120 6474.756324980 Cs 0.001373062 0 0 12 108 0 0 354 236 0 18.418°
121 6586.121949584 C3 0.000838863 0 0 12 109 0 0 357 238 0 18.199°
122 6698.374499261 Ih 0 0 0 12 110 0 0 360 240 0 18.612°
123 6811.827228174 C2v 0.001939754 0 0 14 107 2 0 363 242 0 17.840°
124 6926.169974193 D2 0 0 0 12 112 0 0 366 244 0 18.111°
125 7041.473264023 C2 0.000088274 0 0 12 113 0 0 369 246 0 17.867°
126 7157.669224867 D4 0 0 2 16 100 8 0 372 248 0 17.920°
127 7274.819504675 D5 0 0 0 12 115 0 0 375 250 0 17.877°
128 7393.007443068 C2 0.000054132 0 0 12 116 0 0 378 252 0 17.814°
129 7512.107319268 C2 0.000030099 0 0 12 117 0 0 381 254 0 17.743°
130 7632.167378912 C2 0.000025622 0 0 12 118 0 0 384 256 0 17.683°
131 7753.205166941 C2 0.000305133 0 0 12 119 0 0 387 258 0 17.511°
132 7875.045342797 I 0 0 0 12 120 0 0 390 260 0 17.958°
133 7998.179212898 C3 0.000591438 0 0 12 121 0 0 393 262 0 17.133°
134 8122.089721194 C2 0.000470268 0 0 12 122 0 0 396 264 0 17.214°
135 8246.909486992 D3 0 0 0 12 123 0 0 399 266 0 17.431°
136 8372.743302539 T 0 0 0 12 124 0 0 402 268 0 17.485°
137 8499.534494782 D5 0 0 0 12 125 0 0 405 270 0 17.560°
138 8627.406389880 C2 0.000473576 0 0 12 126 0 0 408 272 0 16.924°
139 8756.227056057 C2 0.000404228 0 0 12 127 0 0 411 274 0 16.673°
140 8885.980609041 C1 0.000630351 0 0 13 126 1 0 414 276 0 16.773°
141 9016.615349190 C2v 0.000376365 0 0 14 126 0 1 417 278 0 16.962°
142 9148.271579993 C2 0.000550138 0 0 12 130 0 0 420 280 0 16.840°
143 9280.839851192 C2 0.000255449 0 0 12 131 0 0 423 282 0 16.782°
144 9414.371794460 D2 0 0 0 12 132 0 0 426 284 0 16.953°
145 9548.928837232 Cs 0.000094938 0 0 12 133 0 0 429 286 0 16.841°
146 9684.381825575 D2 0 0 0 12 134 0 0 432 288 0 16.905°
147 9820.932378373 C2 0.000636651 0 0 12 135 0 0 435 290 0 16.458°
148 9958.406004270 C2 0.000203701 0 0 12 136 0 0 438 292 0 16.627°
149 10096.859907397 C1 0.000638186 0 0 14 133 2 0 441 294 0 16.344°
150 10236.196436701 T 0 0 0 12 138 0 0 444 296 0 16.405°
151 10376.571469275 C2 0.000153836 0 0 12 139 0 0 447 298 0 16.163°
152 10517.867592878 D2 0 0 0 12 140 0 0 450 300 0 16.117°
153 10660.082748237 D3 0 0 0 12 141 0 0 453 302 0 16.390°
154 10803.372421141 C2 0.000735800 0 0 12 142 0 0 456 304 0 16.078°
155 10947.574692279 C2 0.000603670 0 0 12 143 0 0 459 306 0 15.990°
156 11092.798311456 C2 0.000508534 0 0 12 144 0 0 462 308 0 15.822°
157 11238.903041156 C2 0.000357679 0 0 12 145 0 0 465 310 0 15.948°
158 11385.990186197 C2 0.000921918 0 0 12 146 0 0 468 312 0 15.987°
159 11534.023960956 C2 0.000381457 0 0 12 147 0 0 471 314 0 15.960°
160 11683.054805549 D2 0 0 0 12 148 0 0 474 316 0 15.961°
161 11833.084739465 C2 0.000056447 0 0 12 149 0 0 477 318 0 15.810°
162 11984.050335814 D3 0 0 0 12 150 0 0 480 320 0 15.813°
163 12136.013053220 C2 0.000120798 0 0 12 151 0 0 483 322 0 15.675°
164 12288.930105320 D2 0 0 0 12 152 0 0 486 324 0 15.655°
165 12442.804451373 C2 0.000091119 0 0 12 153 0 0 489 326 0 15.651°
166 12597.649071323 D2d 0 0 0 16 146 4 0 492 328 0 15.607°
167 12753.469429750 C2 0.000097382 0 0 12 155 0 0 495 330 0 15.600°
168 12910.212672268 D3 0 0 0 12 156 0 0 498 332 0 15.655°
169 13068.006451127 Cs 0.000068102 0 0 13 155 1 0 501 334 0 15.537°
170 13226.681078541 D2d 0 0 0 12 158 0 0 504 336 0 15.569°
171 13386.355930717 D3 0 0 0 12 159 0 0 507 338 0 15.497°
172 13547.018108787 C2v 0.000547291 0 0 14 156 2 0 510 340 0 15.292°
173 13708.635243034 Cs 0.000286544 0 0 12 161 0 0 513 342 0 15.225°
174 13871.187092292 D2 0 0 0 12 162 0 0 516 344 0 15.366°
175 14034.781306929 C2 0.000026686 0 0 12 163 0 0 519 346 0 15.252°
176 14199.354775632 C1 0.000283978 0 0 12 164 0 0 522 348 0 15.101°
177 14364.837545298 D5 0 0 0 12 165 0 0 525 350 0 15.269°
178 14531.309552587 D2 0 0 0 12 166 0 0 528 352 0 15.145°
179 14698.754594220 C1 0.000125113 0 0 13 165 1 0 531 354 0 14.968°
180 14867.099927525 D2 0 0 0 12 168 0 0 534 356 0 15.067°
181 15036.467239769 C2 0.000304193 0 0 12 169 0 0 537 358 0 15.002°
182 15206.730610906 D5 0 0 0 12 170 0 0 540 360 0 15.155°
183 15378.166571028 C1 0.000467899 0 0 12 171 0 0 543 362 0 14.747°
184 15550.421450311 T 0 0 0 12 172 0 0 546 364 0 14.932°
185 15723.720074072 C2 0.000389762 0 0 12 173 0 0 549 366 0 14.775°
186 15897.897437048 C1 0.000389762 0 0 12 174 0 0 552 368 0 14.739°
187 16072.975186320 D5 0 0 0 12 175 0 0 555 370 0 14.848°
188 16249.222678879 D2 0 0 0 12 176 0 0 558 372 0 14.740°
189 16426.371938862 C2 0.000020732 0 0 12 177 0 0 561 374 0 14.671°
190 16604.428338501 C3 0.000586804 0 0 12 178 0 0 564 376 0 14.501°
191 16783.452219362 C1 0.001129202 0 0 13 177 1 0 567 378 0 14.195°
192 16963.338386460 I 0 0 0 12 180 0 0 570 380 0 14.819°
193 17144.564740880 C2 0.000985192 0 0 12 181 0 0 573 382 0 14.144°
194 17326.616136471 C1 0.000322358 0 0 12 182 0 0 576 384 0 14.350°
195 17509.489303930 D3 0 0 0 12 183 0 0 579 386 0 14.375°
196 17693.460548082 C2 0.000315907 0 0 12 184 0 0 582 388 0 14.251°
197 17878.340162571 D5 0 0 0 12 185 0 0 585 390 0 14.147°
198 18064.262177195 C2 0.000011149 0 0 12 186 0 0 588 392 0 14.237°
199 18251.082495640 C1 0.000534779 0 0 12 187 0 0 591 394 0 14.153°
200 18438.842717530 D2 0 0 0 12 188 0 0 594 396 0 14.222°
201 18627.591226244 C1 0.001048859 0 0 13 187 1 0 597 398 0 13.830°
202 18817.204718262 D5 0 0 0 12 190 0 0 600 400 0 14.189°
203 19007.981204580 Cs 0.000600343 0 0 12 191 0 0 603 402 0 13.977°
204 19199.540775603 Th 0 0 0 12 192 0 0 606 404 0 14.291°
212 20768.053085964 I 0 0 0 12 200 0 0 630 420 0 14.118°
214 21169.910410375 T 0 0 0 12 202 0 0 636 424 0 13.771°
217 21779.856080418 D5 0 0 0 12 205 0 0 645 430 0 13.902°
232 24961.252318934 T 0 0 0 12 220 0 0 690 460 0 13.260°
256 30506.687515847 T 0 0 0 12 244 0 0 762 508 0 12.572°
257 30749.941417346 D5 0 0 0 12 245 0 0 765 510 0 12.672°
272 34515.193292681 Ih 0 0 0 12 260 0 0 810 540 0 12.335°
282 37147.294418462 I 0 0 0 12 270 0 0 840 560 0 12.166°
292 39877.008012909 D5 0 0 0 12 280 0 0 870 580 0 11.857°
306 43862.569780797 Th 0 0 0 12 294 0 0 912 608 0 11.628°
312 45629.313804002 C2 0.000306163 0 0 12 300 0 0 930 620 0 11.299°
315 46525.825643432 D3 0 0 0 12 303 0 0 939 626 0 11.337°
317 47128.310344520 D5 0 0 0 12 305 0 0 945 630 0 11.423°
318 47431.056020043 D3 0 0 0 12 306 0 0 948 632 0 11.219°
334 52407.728127822 T 0 0 0 12 322 0 0 996 664 0 11.058°
348 56967.472454334 Th 0 0 0 12 336 0 0 1038 692 0 10.721°
357 59999.922939598 D5 0 0 0 12 345 0 0 1065 710 0 10.728°
358 60341.830924588 T 0 0 0 12 346 0 0 1068 712 0 10.647°
372 65230.027122557 I 0 0 0 12 360 0 0 1110 740 0 10.531°
382 68839.426839215 D5 0 0 0 12 370 0 0 1140 760 0 10.379°
390 71797.035335953 Th 0 0 0 12 378 0 0 1164 776 0 10.222°
392 72546.258370889 I 0 0 0 12 380 0 0 1170 780 0 10.278°
400 75582.448512213 T 0 0 0 12 388 0 0 1194 796 0 10.068°

## References

1. ^ J. J. Thomson, "On the Structure of the Atom: an Investigation of the Stability and Periods of Oscillation of a number of Corpuscles arranged at equal intervals around the Circumference of a Circle; with Application of the Results to the Theory of Atomic Structure", Philosophical Magazine Series 6, Volume 7, Number 39, pp. 237--265, March 1904
2. ^ Sir J.J. Thomson, The Romanes Lecture, 1914 (The Atomic Theory)

## Notes

• T. Erber and G. M. Hockney, "Complex Systems: Equilibrium Configurations of N Equal Charges on a Sphere $(2\leq N\leq 112)$", Advances in Chemical Physics, Volume 98, pp. 495--594, 1997.