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## G = C19×C3⋊C8order 456 = 23·3·19

### Direct product of C19 and C3⋊C8

Aliases: C19×C3⋊C8, C3⋊C152, C573C8, C6.C76, C76.4S3, C228.6C2, C12.2C38, C114.3C4, C38.2Dic3, C4.2(S3×C19), C2.(Dic3×C19), SmallGroup(456,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C19×C3⋊C8
 Chief series C1 — C3 — C6 — C12 — C228 — C19×C3⋊C8
 Lower central C3 — C19×C3⋊C8
 Upper central C1 — C76

Generators and relations for C19×C3⋊C8
G = < a,b,c | a19=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C19×C3⋊C8
Regular action on 456 points
Generators in S456
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19)(20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38)(39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57)(58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76)(77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95)(96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114)(115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133)(134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152)(153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171)(172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190)(191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209)(210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228)(229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247)(248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266)(267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285)(286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304)(305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323)(324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342)(343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361)(362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380)(381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399)(400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418)(419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437)(438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456)
(1 422 207)(2 423 208)(3 424 209)(4 425 191)(5 426 192)(6 427 193)(7 428 194)(8 429 195)(9 430 196)(10 431 197)(11 432 198)(12 433 199)(13 434 200)(14 435 201)(15 436 202)(16 437 203)(17 419 204)(18 420 205)(19 421 206)(20 168 304)(21 169 286)(22 170 287)(23 171 288)(24 153 289)(25 154 290)(26 155 291)(27 156 292)(28 157 293)(29 158 294)(30 159 295)(31 160 296)(32 161 297)(33 162 298)(34 163 299)(35 164 300)(36 165 301)(37 166 302)(38 167 303)(39 330 221)(40 331 222)(41 332 223)(42 333 224)(43 334 225)(44 335 226)(45 336 227)(46 337 228)(47 338 210)(48 339 211)(49 340 212)(50 341 213)(51 342 214)(52 324 215)(53 325 216)(54 326 217)(55 327 218)(56 328 219)(57 329 220)(58 377 381)(59 378 382)(60 379 383)(61 380 384)(62 362 385)(63 363 386)(64 364 387)(65 365 388)(66 366 389)(67 367 390)(68 368 391)(69 369 392)(70 370 393)(71 371 394)(72 372 395)(73 373 396)(74 374 397)(75 375 398)(76 376 399)(77 278 242)(78 279 243)(79 280 244)(80 281 245)(81 282 246)(82 283 247)(83 284 229)(84 285 230)(85 267 231)(86 268 232)(87 269 233)(88 270 234)(89 271 235)(90 272 236)(91 273 237)(92 274 238)(93 275 239)(94 276 240)(95 277 241)(96 177 320)(97 178 321)(98 179 322)(99 180 323)(100 181 305)(101 182 306)(102 183 307)(103 184 308)(104 185 309)(105 186 310)(106 187 311)(107 188 312)(108 189 313)(109 190 314)(110 172 315)(111 173 316)(112 174 317)(113 175 318)(114 176 319)(115 151 406)(116 152 407)(117 134 408)(118 135 409)(119 136 410)(120 137 411)(121 138 412)(122 139 413)(123 140 414)(124 141 415)(125 142 416)(126 143 417)(127 144 418)(128 145 400)(129 146 401)(130 147 402)(131 148 403)(132 149 404)(133 150 405)(248 439 343)(249 440 344)(250 441 345)(251 442 346)(252 443 347)(253 444 348)(254 445 349)(255 446 350)(256 447 351)(257 448 352)(258 449 353)(259 450 354)(260 451 355)(261 452 356)(262 453 357)(263 454 358)(264 455 359)(265 456 360)(266 438 361)
(1 445 414 38 80 227 306 365)(2 446 415 20 81 228 307 366)(3 447 416 21 82 210 308 367)(4 448 417 22 83 211 309 368)(5 449 418 23 84 212 310 369)(6 450 400 24 85 213 311 370)(7 451 401 25 86 214 312 371)(8 452 402 26 87 215 313 372)(9 453 403 27 88 216 314 373)(10 454 404 28 89 217 315 374)(11 455 405 29 90 218 316 375)(12 456 406 30 91 219 317 376)(13 438 407 31 92 220 318 377)(14 439 408 32 93 221 319 378)(15 440 409 33 94 222 320 379)(16 441 410 34 95 223 321 380)(17 442 411 35 77 224 322 362)(18 443 412 36 78 225 323 363)(19 444 413 37 79 226 305 364)(39 176 382 201 343 134 161 239)(40 177 383 202 344 135 162 240)(41 178 384 203 345 136 163 241)(42 179 385 204 346 137 164 242)(43 180 386 205 347 138 165 243)(44 181 387 206 348 139 166 244)(45 182 388 207 349 140 167 245)(46 183 389 208 350 141 168 246)(47 184 390 209 351 142 169 247)(48 185 391 191 352 143 170 229)(49 186 392 192 353 144 171 230)(50 187 393 193 354 145 153 231)(51 188 394 194 355 146 154 232)(52 189 395 195 356 147 155 233)(53 190 396 196 357 148 156 234)(54 172 397 197 358 149 157 235)(55 173 398 198 359 150 158 236)(56 174 399 199 360 151 159 237)(57 175 381 200 361 152 160 238)(58 434 266 116 296 274 329 113)(59 435 248 117 297 275 330 114)(60 436 249 118 298 276 331 96)(61 437 250 119 299 277 332 97)(62 419 251 120 300 278 333 98)(63 420 252 121 301 279 334 99)(64 421 253 122 302 280 335 100)(65 422 254 123 303 281 336 101)(66 423 255 124 304 282 337 102)(67 424 256 125 286 283 338 103)(68 425 257 126 287 284 339 104)(69 426 258 127 288 285 340 105)(70 427 259 128 289 267 341 106)(71 428 260 129 290 268 342 107)(72 429 261 130 291 269 324 108)(73 430 262 131 292 270 325 109)(74 431 263 132 293 271 326 110)(75 432 264 133 294 272 327 111)(76 433 265 115 295 273 328 112)

G:=sub<Sym(456)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57)(58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76)(77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95)(96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114)(115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133)(134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171)(172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190)(191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209)(210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228)(229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247)(248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266)(267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285)(286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304)(305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323)(324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342)(343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361)(362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380)(381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399)(400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418)(419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437)(438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456), (1,422,207)(2,423,208)(3,424,209)(4,425,191)(5,426,192)(6,427,193)(7,428,194)(8,429,195)(9,430,196)(10,431,197)(11,432,198)(12,433,199)(13,434,200)(14,435,201)(15,436,202)(16,437,203)(17,419,204)(18,420,205)(19,421,206)(20,168,304)(21,169,286)(22,170,287)(23,171,288)(24,153,289)(25,154,290)(26,155,291)(27,156,292)(28,157,293)(29,158,294)(30,159,295)(31,160,296)(32,161,297)(33,162,298)(34,163,299)(35,164,300)(36,165,301)(37,166,302)(38,167,303)(39,330,221)(40,331,222)(41,332,223)(42,333,224)(43,334,225)(44,335,226)(45,336,227)(46,337,228)(47,338,210)(48,339,211)(49,340,212)(50,341,213)(51,342,214)(52,324,215)(53,325,216)(54,326,217)(55,327,218)(56,328,219)(57,329,220)(58,377,381)(59,378,382)(60,379,383)(61,380,384)(62,362,385)(63,363,386)(64,364,387)(65,365,388)(66,366,389)(67,367,390)(68,368,391)(69,369,392)(70,370,393)(71,371,394)(72,372,395)(73,373,396)(74,374,397)(75,375,398)(76,376,399)(77,278,242)(78,279,243)(79,280,244)(80,281,245)(81,282,246)(82,283,247)(83,284,229)(84,285,230)(85,267,231)(86,268,232)(87,269,233)(88,270,234)(89,271,235)(90,272,236)(91,273,237)(92,274,238)(93,275,239)(94,276,240)(95,277,241)(96,177,320)(97,178,321)(98,179,322)(99,180,323)(100,181,305)(101,182,306)(102,183,307)(103,184,308)(104,185,309)(105,186,310)(106,187,311)(107,188,312)(108,189,313)(109,190,314)(110,172,315)(111,173,316)(112,174,317)(113,175,318)(114,176,319)(115,151,406)(116,152,407)(117,134,408)(118,135,409)(119,136,410)(120,137,411)(121,138,412)(122,139,413)(123,140,414)(124,141,415)(125,142,416)(126,143,417)(127,144,418)(128,145,400)(129,146,401)(130,147,402)(131,148,403)(132,149,404)(133,150,405)(248,439,343)(249,440,344)(250,441,345)(251,442,346)(252,443,347)(253,444,348)(254,445,349)(255,446,350)(256,447,351)(257,448,352)(258,449,353)(259,450,354)(260,451,355)(261,452,356)(262,453,357)(263,454,358)(264,455,359)(265,456,360)(266,438,361), (1,445,414,38,80,227,306,365)(2,446,415,20,81,228,307,366)(3,447,416,21,82,210,308,367)(4,448,417,22,83,211,309,368)(5,449,418,23,84,212,310,369)(6,450,400,24,85,213,311,370)(7,451,401,25,86,214,312,371)(8,452,402,26,87,215,313,372)(9,453,403,27,88,216,314,373)(10,454,404,28,89,217,315,374)(11,455,405,29,90,218,316,375)(12,456,406,30,91,219,317,376)(13,438,407,31,92,220,318,377)(14,439,408,32,93,221,319,378)(15,440,409,33,94,222,320,379)(16,441,410,34,95,223,321,380)(17,442,411,35,77,224,322,362)(18,443,412,36,78,225,323,363)(19,444,413,37,79,226,305,364)(39,176,382,201,343,134,161,239)(40,177,383,202,344,135,162,240)(41,178,384,203,345,136,163,241)(42,179,385,204,346,137,164,242)(43,180,386,205,347,138,165,243)(44,181,387,206,348,139,166,244)(45,182,388,207,349,140,167,245)(46,183,389,208,350,141,168,246)(47,184,390,209,351,142,169,247)(48,185,391,191,352,143,170,229)(49,186,392,192,353,144,171,230)(50,187,393,193,354,145,153,231)(51,188,394,194,355,146,154,232)(52,189,395,195,356,147,155,233)(53,190,396,196,357,148,156,234)(54,172,397,197,358,149,157,235)(55,173,398,198,359,150,158,236)(56,174,399,199,360,151,159,237)(57,175,381,200,361,152,160,238)(58,434,266,116,296,274,329,113)(59,435,248,117,297,275,330,114)(60,436,249,118,298,276,331,96)(61,437,250,119,299,277,332,97)(62,419,251,120,300,278,333,98)(63,420,252,121,301,279,334,99)(64,421,253,122,302,280,335,100)(65,422,254,123,303,281,336,101)(66,423,255,124,304,282,337,102)(67,424,256,125,286,283,338,103)(68,425,257,126,287,284,339,104)(69,426,258,127,288,285,340,105)(70,427,259,128,289,267,341,106)(71,428,260,129,290,268,342,107)(72,429,261,130,291,269,324,108)(73,430,262,131,292,270,325,109)(74,431,263,132,293,271,326,110)(75,432,264,133,294,272,327,111)(76,433,265,115,295,273,328,112)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57)(58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76)(77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95)(96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114)(115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133)(134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171)(172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190)(191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209)(210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228)(229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247)(248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266)(267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285)(286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304)(305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323)(324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342)(343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361)(362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380)(381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399)(400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418)(419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437)(438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456), (1,422,207)(2,423,208)(3,424,209)(4,425,191)(5,426,192)(6,427,193)(7,428,194)(8,429,195)(9,430,196)(10,431,197)(11,432,198)(12,433,199)(13,434,200)(14,435,201)(15,436,202)(16,437,203)(17,419,204)(18,420,205)(19,421,206)(20,168,304)(21,169,286)(22,170,287)(23,171,288)(24,153,289)(25,154,290)(26,155,291)(27,156,292)(28,157,293)(29,158,294)(30,159,295)(31,160,296)(32,161,297)(33,162,298)(34,163,299)(35,164,300)(36,165,301)(37,166,302)(38,167,303)(39,330,221)(40,331,222)(41,332,223)(42,333,224)(43,334,225)(44,335,226)(45,336,227)(46,337,228)(47,338,210)(48,339,211)(49,340,212)(50,341,213)(51,342,214)(52,324,215)(53,325,216)(54,326,217)(55,327,218)(56,328,219)(57,329,220)(58,377,381)(59,378,382)(60,379,383)(61,380,384)(62,362,385)(63,363,386)(64,364,387)(65,365,388)(66,366,389)(67,367,390)(68,368,391)(69,369,392)(70,370,393)(71,371,394)(72,372,395)(73,373,396)(74,374,397)(75,375,398)(76,376,399)(77,278,242)(78,279,243)(79,280,244)(80,281,245)(81,282,246)(82,283,247)(83,284,229)(84,285,230)(85,267,231)(86,268,232)(87,269,233)(88,270,234)(89,271,235)(90,272,236)(91,273,237)(92,274,238)(93,275,239)(94,276,240)(95,277,241)(96,177,320)(97,178,321)(98,179,322)(99,180,323)(100,181,305)(101,182,306)(102,183,307)(103,184,308)(104,185,309)(105,186,310)(106,187,311)(107,188,312)(108,189,313)(109,190,314)(110,172,315)(111,173,316)(112,174,317)(113,175,318)(114,176,319)(115,151,406)(116,152,407)(117,134,408)(118,135,409)(119,136,410)(120,137,411)(121,138,412)(122,139,413)(123,140,414)(124,141,415)(125,142,416)(126,143,417)(127,144,418)(128,145,400)(129,146,401)(130,147,402)(131,148,403)(132,149,404)(133,150,405)(248,439,343)(249,440,344)(250,441,345)(251,442,346)(252,443,347)(253,444,348)(254,445,349)(255,446,350)(256,447,351)(257,448,352)(258,449,353)(259,450,354)(260,451,355)(261,452,356)(262,453,357)(263,454,358)(264,455,359)(265,456,360)(266,438,361), (1,445,414,38,80,227,306,365)(2,446,415,20,81,228,307,366)(3,447,416,21,82,210,308,367)(4,448,417,22,83,211,309,368)(5,449,418,23,84,212,310,369)(6,450,400,24,85,213,311,370)(7,451,401,25,86,214,312,371)(8,452,402,26,87,215,313,372)(9,453,403,27,88,216,314,373)(10,454,404,28,89,217,315,374)(11,455,405,29,90,218,316,375)(12,456,406,30,91,219,317,376)(13,438,407,31,92,220,318,377)(14,439,408,32,93,221,319,378)(15,440,409,33,94,222,320,379)(16,441,410,34,95,223,321,380)(17,442,411,35,77,224,322,362)(18,443,412,36,78,225,323,363)(19,444,413,37,79,226,305,364)(39,176,382,201,343,134,161,239)(40,177,383,202,344,135,162,240)(41,178,384,203,345,136,163,241)(42,179,385,204,346,137,164,242)(43,180,386,205,347,138,165,243)(44,181,387,206,348,139,166,244)(45,182,388,207,349,140,167,245)(46,183,389,208,350,141,168,246)(47,184,390,209,351,142,169,247)(48,185,391,191,352,143,170,229)(49,186,392,192,353,144,171,230)(50,187,393,193,354,145,153,231)(51,188,394,194,355,146,154,232)(52,189,395,195,356,147,155,233)(53,190,396,196,357,148,156,234)(54,172,397,197,358,149,157,235)(55,173,398,198,359,150,158,236)(56,174,399,199,360,151,159,237)(57,175,381,200,361,152,160,238)(58,434,266,116,296,274,329,113)(59,435,248,117,297,275,330,114)(60,436,249,118,298,276,331,96)(61,437,250,119,299,277,332,97)(62,419,251,120,300,278,333,98)(63,420,252,121,301,279,334,99)(64,421,253,122,302,280,335,100)(65,422,254,123,303,281,336,101)(66,423,255,124,304,282,337,102)(67,424,256,125,286,283,338,103)(68,425,257,126,287,284,339,104)(69,426,258,127,288,285,340,105)(70,427,259,128,289,267,341,106)(71,428,260,129,290,268,342,107)(72,429,261,130,291,269,324,108)(73,430,262,131,292,270,325,109)(74,431,263,132,293,271,326,110)(75,432,264,133,294,272,327,111)(76,433,265,115,295,273,328,112) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19),(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38),(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57),(58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76),(77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95),(96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114),(115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133),(134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152),(153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171),(172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190),(191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209),(210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228),(229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247),(248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266),(267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285),(286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304),(305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323),(324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342),(343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361),(362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380),(381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399),(400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418),(419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437),(438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456)], [(1,422,207),(2,423,208),(3,424,209),(4,425,191),(5,426,192),(6,427,193),(7,428,194),(8,429,195),(9,430,196),(10,431,197),(11,432,198),(12,433,199),(13,434,200),(14,435,201),(15,436,202),(16,437,203),(17,419,204),(18,420,205),(19,421,206),(20,168,304),(21,169,286),(22,170,287),(23,171,288),(24,153,289),(25,154,290),(26,155,291),(27,156,292),(28,157,293),(29,158,294),(30,159,295),(31,160,296),(32,161,297),(33,162,298),(34,163,299),(35,164,300),(36,165,301),(37,166,302),(38,167,303),(39,330,221),(40,331,222),(41,332,223),(42,333,224),(43,334,225),(44,335,226),(45,336,227),(46,337,228),(47,338,210),(48,339,211),(49,340,212),(50,341,213),(51,342,214),(52,324,215),(53,325,216),(54,326,217),(55,327,218),(56,328,219),(57,329,220),(58,377,381),(59,378,382),(60,379,383),(61,380,384),(62,362,385),(63,363,386),(64,364,387),(65,365,388),(66,366,389),(67,367,390),(68,368,391),(69,369,392),(70,370,393),(71,371,394),(72,372,395),(73,373,396),(74,374,397),(75,375,398),(76,376,399),(77,278,242),(78,279,243),(79,280,244),(80,281,245),(81,282,246),(82,283,247),(83,284,229),(84,285,230),(85,267,231),(86,268,232),(87,269,233),(88,270,234),(89,271,235),(90,272,236),(91,273,237),(92,274,238),(93,275,239),(94,276,240),(95,277,241),(96,177,320),(97,178,321),(98,179,322),(99,180,323),(100,181,305),(101,182,306),(102,183,307),(103,184,308),(104,185,309),(105,186,310),(106,187,311),(107,188,312),(108,189,313),(109,190,314),(110,172,315),(111,173,316),(112,174,317),(113,175,318),(114,176,319),(115,151,406),(116,152,407),(117,134,408),(118,135,409),(119,136,410),(120,137,411),(121,138,412),(122,139,413),(123,140,414),(124,141,415),(125,142,416),(126,143,417),(127,144,418),(128,145,400),(129,146,401),(130,147,402),(131,148,403),(132,149,404),(133,150,405),(248,439,343),(249,440,344),(250,441,345),(251,442,346),(252,443,347),(253,444,348),(254,445,349),(255,446,350),(256,447,351),(257,448,352),(258,449,353),(259,450,354),(260,451,355),(261,452,356),(262,453,357),(263,454,358),(264,455,359),(265,456,360),(266,438,361)], [(1,445,414,38,80,227,306,365),(2,446,415,20,81,228,307,366),(3,447,416,21,82,210,308,367),(4,448,417,22,83,211,309,368),(5,449,418,23,84,212,310,369),(6,450,400,24,85,213,311,370),(7,451,401,25,86,214,312,371),(8,452,402,26,87,215,313,372),(9,453,403,27,88,216,314,373),(10,454,404,28,89,217,315,374),(11,455,405,29,90,218,316,375),(12,456,406,30,91,219,317,376),(13,438,407,31,92,220,318,377),(14,439,408,32,93,221,319,378),(15,440,409,33,94,222,320,379),(16,441,410,34,95,223,321,380),(17,442,411,35,77,224,322,362),(18,443,412,36,78,225,323,363),(19,444,413,37,79,226,305,364),(39,176,382,201,343,134,161,239),(40,177,383,202,344,135,162,240),(41,178,384,203,345,136,163,241),(42,179,385,204,346,137,164,242),(43,180,386,205,347,138,165,243),(44,181,387,206,348,139,166,244),(45,182,388,207,349,140,167,245),(46,183,389,208,350,141,168,246),(47,184,390,209,351,142,169,247),(48,185,391,191,352,143,170,229),(49,186,392,192,353,144,171,230),(50,187,393,193,354,145,153,231),(51,188,394,194,355,146,154,232),(52,189,395,195,356,147,155,233),(53,190,396,196,357,148,156,234),(54,172,397,197,358,149,157,235),(55,173,398,198,359,150,158,236),(56,174,399,199,360,151,159,237),(57,175,381,200,361,152,160,238),(58,434,266,116,296,274,329,113),(59,435,248,117,297,275,330,114),(60,436,249,118,298,276,331,96),(61,437,250,119,299,277,332,97),(62,419,251,120,300,278,333,98),(63,420,252,121,301,279,334,99),(64,421,253,122,302,280,335,100),(65,422,254,123,303,281,336,101),(66,423,255,124,304,282,337,102),(67,424,256,125,286,283,338,103),(68,425,257,126,287,284,339,104),(69,426,258,127,288,285,340,105),(70,427,259,128,289,267,341,106),(71,428,260,129,290,268,342,107),(72,429,261,130,291,269,324,108),(73,430,262,131,292,270,325,109),(74,431,263,132,293,271,326,110),(75,432,264,133,294,272,327,111),(76,433,265,115,295,273,328,112)])

228 conjugacy classes

 class 1 2 3 4A 4B 6 8A 8B 8C 8D 12A 12B 19A ··· 19R 38A ··· 38R 57A ··· 57R 76A ··· 76AJ 114A ··· 114R 152A ··· 152BT 228A ··· 228AJ order 1 2 3 4 4 6 8 8 8 8 12 12 19 ··· 19 38 ··· 38 57 ··· 57 76 ··· 76 114 ··· 114 152 ··· 152 228 ··· 228 size 1 1 2 1 1 2 3 3 3 3 2 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 3 ··· 3 2 ··· 2

228 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + - image C1 C2 C4 C8 C19 C38 C76 C152 S3 Dic3 C3⋊C8 S3×C19 Dic3×C19 C19×C3⋊C8 kernel C19×C3⋊C8 C228 C114 C57 C3⋊C8 C12 C6 C3 C76 C38 C19 C4 C2 C1 # reps 1 1 2 4 18 18 36 72 1 1 2 18 18 36

Matrix representation of C19×C3⋊C8 in GL2(𝔽457) generated by

 185 0 0 185
,
 456 456 1 0
,
 222 246 24 235
G:=sub<GL(2,GF(457))| [185,0,0,185],[456,1,456,0],[222,24,246,235] >;

C19×C3⋊C8 in GAP, Magma, Sage, TeX

C_{19}\times C_3\rtimes C_8
% in TeX

G:=Group("C19xC3:C8");
// GroupNames label

G:=SmallGroup(456,3);
// by ID

G=gap.SmallGroup(456,3);
# by ID

G:=PCGroup([5,-2,-19,-2,-2,-3,190,42,7604]);
// Polycyclic

G:=Group<a,b,c|a^19=b^3=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

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text:
Algebraic Properties of Real Numbers

# Algebraic Properties of Real Numbers

It is important to first review the Axioms of the Field of Real Numbers page before reading forward. We will be proving many (rather simple) theorems using these axioms and subsequent results that we prove. Note that the proofs given below are not necessarily the only way to prove each theorem. Some properties can be proved in a variety of ways just by using the axioms.

We will also note that in the later proofs on this page, we will use earlier proofs to verify these properties.

 Theorem 1: If $a$ and $b$ are real numbers such that $a + b = a$, then $b = 0$.
• Proof:
(1)
\begin{align} a + b = a \\ a + b + (-a) = a + (-a) \\ a + (-a) + b \overset{A2} = a + (-a) \\ (a + (-a)) + b \overset{A2} = (a + (-a)) \\ (0) + b \overset{A4} = (0) \\ b \overset{A3} = 0 \quad \blacksquare \end{align}
 Theorem 2: If $a$ and $b$ are real numbers such that $a \cdot b = a$, then $b = 1$.
• Proof:
(2)
\begin{align} a\cdot b = a \\ a^{-1}(ab) = a^{-1}a \\ (a^{-1}a)b \overset{M2} = a^{-1}a \\ (1)b \overset{M4} = 1 \\ b \overset{M3} = 1 \quad \blacksquare \end{align}
 Theorem 3: If $a$ is a real number then $a \cdot 0 = 0$.
• Proof:
(3)
\begin{align} a \cdot 0 = 0 \\ a + a \cdot 0 = a + 0 \\ a(1 + 0) \overset{D} = a + 0 \\ a(1) = a + 0 \\ a(1) \overset{A3} = a \\ a \overset{M3} = a \quad \blacksquare \end{align}
• This implies that $a \cdot 0 = 0$ since we have shown that $a + (a \cdot 0) = a + 0$.
 Theorem 4: If $a$ and $b$ are real numbers where $b \neq 0$, then if $a \cdot b = 1$, then $b = a^{-1}$.
• Proof:
(4)
\begin{align} b = b 1 \cdot b \overset{M3} = b \\ (a \cdot a^{-1}) \cdot b \overset{A4} = b \\ (a^{-1}) \cdot a \cdot b \overset{M1} = b \\ a^{-1}\cdot (1) = b \\ a^{-1} \overset{M3} = b \quad \blacksquare \end{align}
 Theorem 5: If $a$ and $b$ are real numbers and $a \cdot b = 0$ then $a = 0$ or $b = 0$ or both $a, b = 0$.
• Proof: Without loss of generality, assume that $b \neq 0$. Then we get that:
(5)
\begin{align} a \cdot b = 0 \\ a^{-1} \cdot a \cdot b = a^{-1} \cdot 0 \\ a^{-1} \cdot a \cdot b \overset{Theorem\:3}= 0 \\ (a^{-1} \cdot a) \cdot b \overset{M2} = 0 \\ 1 \cdot b \overset{M4} = 0 \\ b \overset{M3} = 0 \quad \blacksquare \end{align}
 Theorem 6: If $a$ and $b$ are real numbers such that $a + b = 0$ then $b = -a$.
• Proof:
(6)
\begin{align} a + b = 0 \\ (-a) + a + b = (-a) + 0 \\ ((-a) + a) + b \overset{A2} = (-a) + 0 \\ 0 + b \overset{A4} = (-a) + 0 \\ b \overset{A3} = (-a) + 0 \\ b \overset{A3} = -a \quad \blacksquare \end{align}
 Theorem 7: If $a$ is a real number, then $(-1)a = -a$.
• Proof:
(7)
\begin{align} (-1) \cdot a = -a \\ a + (-1) \cdot a = a + (-a) \\ 1a + (-1) \cdot a \overset{M3} = a + (-a) \\ (1 + (-1)) \cdot a \overset{D} = a + (-a) \\ (0) \cdot a = a + (-a) \\ 0 \overset{Theorem\:3}= a + (-a) \\ 0 \overset{A4} = 0 \end{align}
• Therefore $(-1)a = -a$ since $a + (-1)a = a + (-a)$. $\blacksquare$
 Theorem 8: If $a$ is a real number then $-(-a) = a$.
• Proof:
(8)
\begin{align} -(-a) = a \\ (-a) - (-a) = (-a) + a \\ (-a) - (-a) \overset{A4} 0 \\ a + (-a) - (-a) = a + 0 \\ (a + (-a)) - (-a) \overset{A2} = a + 0 \\ 0 - (-a) \overset{A4} = a + 0 \\ -(-a) \overset{A3} = a \quad \blacksquare \end{align}
 Theorem 9: $(-1) \cdot (-1) = 1$.
• Proof:
(9)
\begin{align} (-1) \cdot (-1) = 1 \\ a \cdot (-1) \cdot (-1) = a \cdot 1 \\ (-1) \cdot a \cdot (-1) \overset{M1} = a \cdot 1 \\ (-a)(-1) \overset{Theorem \: 7} = a \cdot 1 \\ (-1)(-a) \overset{M1} = a \cdot 1 \\ -(-a) \overset{Theorem \: 7} = a \cdot 1 \\ -(-a) \overset{M3} = a \end{align}
• We showed in Theorem 8 that $-(-a) = a$, therefore, $(-1)(-1) = 1$ since $a \cdot (-1) \cdot (-1) = a \cdot 1$. $\blacksquare$
 Theorem 10: If $a$ and $b$ are real numbers then $-(a + b) = (-a) + (-b)$.
• Proof:
(10)
\begin{align} -(a + b) = (-a) + (-b) \\ (-1) \cdot (a + b) \overset{Theorem \: 7} = (-a) + (-b) \\ (-1) \cdot a + (-1) \cdot b \overset{D} = (-a) + (-b) \\ (-a) + (-b) \overset{Theorem \: 7} = (-a) + (-b) \quad \blacksquare \end{align}
 Theorem 11: If $a$ and $b$ are real numbers then $(-a) \cdot (-b) = a \cdot b$.
• Proof:
(11)
\begin{align} (-a) \cdot (-b) = a \cdot b \\ (-1) \cdot (-a) \cdot (-1) \cdot (b) \overset{Theorem \:7} = a \cdot b \\ (-1) \cdot (-1) \cdot (a) \cdot b \overset{M1} = a \cdot b \\ 1 \cdot a \cdot b \overset{Theorem \: 9} = a \cdot b \\ a \cdot b \overset{M3} = a \cdot b \quad \blacksquare \end{align}
 Theorem 12: If $a$ and $b$ are real numbers such that $b \neq 0$ then $-(a \cdot b^{-1}) = (-a) \cdot b^{-1}$.
• Proof:
(12)
\begin{align} -(a \cdot b^{-1}) = (-a) \cdot b^{-1} \\ (-1) \cdot (a \cdot b^{-1} \overset{Theorem \: 7} = (-a) \cdot b^{-1} \\ ((-1) \cdot a) \cdot b^{-1} \overset{M2} = (-a) \cdot b^{-1} \\ (-a) \cdot b^{-1} \overset{Theorem \: 7} = (-a) \cdot b^{-1} \quad \blacksquare \end{align}

format: plain


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A Tic Tac Toe game developed by joke

# JokeTTT: a Tic Tac Toe game developed by joke

JokeTTT is a library providing tic tac toe structures useful to develop games based on several kind of players. It was developed just for fun and to learn Python and some concepts of machine learning.

## Project setup

The project has been tested only with python3 on Ubuntu Linux. If you have python3 installed in your machine, just install the package with the usual pip command.

To avoid the usual problems with messy Python configurations (python 2 vs. 3, packages to install, etc.), conda is used for development.

For those that wants to do the same and does not know conda, this is a a quick reference:

If you have conda installed, enter the project directory (the one with the environment.yml file) and enter the following command:

conda env create

This shall be done only the one time. After this the jokettt conda environment is created. It can be activated with the command:

conda activate jokettt

To update the conda environment every time that the environment.yml is changed, enter the command:

conda env update

## Demo programs

For example of simple applications that uses the jokettt classes, see the jokettt_demo repository.

## Project details

Uploaded source
Uploaded py3


text:
# Conjecture on bernoulli numbers and binomial coefficients

Crossposted from

https://math.stackexchange.com/questions/4116414/conjecture-on-bernoulli-numbers-and-binomial-coefficients

In playing around with some formulas, I have come up with the following conjecture. I have checked it for a lot of cases, and have good reason to believe it to be true. If anyone could help, I would sincerely appreciate it, and I would be happy to include as co-author.

Some notation:

• The exponent $$R(i)$$ means the formal raising by $$2$$s for $$i$$ terms. For example, $$(k-7)^{R(3)} = (k-7)(k-5)(k-3)$$.

• The exponent $$D(i)$$ means the formal descending by $$2$$s for $$i$$ terms. For example, $$k^{D(4)} = (k)(k-2)(k-4)(k-6)$$.

• $$k^{D(1)}:=k$$, $$k^{D(0)}:=1$$ and $$k^{D(-1)}:=\frac{1}{k+2}$$.

• !! denotes the double factorial, for example $$9!! = 1.3.5.7.9$$

• $$B_i$$ denotes the ith Bernoulli number, with the convention that $$B_1 = 1/2$$.

In what follows, $$M$$ is a positive odd integer bigger than or equal to $$3$$, $$k$$ is a formal variable. For a given odd $$M\geq 3$$, the following is a polynomial in $$k$$. My conjecture is that it has roots $$1,3,5,.., M-2$$.

$$\sum\limits_{i=1}^{\frac{M+1}{2}}\binom{\frac{M+1}{2}}{i}\left[(k-(M-2))^{R(i)}+(-1)^{i+1}(2i-1)!!\right]B_{\frac{M+1}{2}-i}\left[k^{D(\frac{M+1}{2}-i-1)}\right]$$

For example, when $$M=3$$, we get

$$\binom{2}{1}[(k-1)^{R(1)}+1!!]B_{1}k^{D(0)}+\binom{2}{2}[(k-1)^{R(2)}-3!!]B_{0}[k^{D(-1)}]$$

$$=2[k]B_1+[(k-1)(k+1)-3]\frac{B_0}{k+2},$$ which has a root of $$1$$.

When $$M=5$$ we get

$$\binom{3}{1}[(k-3)^{R(1)}+1!!]B_2[k^{D(1)}]+\binom{3}{2}[(k-3)^{R(2)}-3!!]B_1[k^{D(0)}]+\binom{3}{3}[(k-3)^{R(3)}+5!!]B_0[k^{D(-1)}]$$

$$=3[k-3+1]B_2[k]+3[(k-3)(k-1)-1.3]B_1+[(k-3)(k-1)(k+1)+1.3.5]\frac{B_0}{k+2}.$$

Putting $$k=1$$ gives

$$-3B_2-9B_1+5B_0 = 0,$$

putting $$k=3$$ gives

$$9B_2-9B_1+3B_0 = 0.$$

As another examples, when $$M=9$$ we get

$$\binom{5}{1}[(k-7)+1]B_4[(k)(k-2)(k-4)] + \binom{5}{2}[(k-7)(k-5)-1.3]B_3[k(k-2)] + \binom{5}{3}[(k-7)(k-5)(k-3)+1.3.5]B_2[k] + \binom{5}{4}[(k-7)(k-5)(k-3)(k-1)-1.3.5.7]B_1 + \binom{5}{5}[(k-7)(k-5)(k-3)(k-1)(k+1)+1.3.5.7.9]\frac{B_0}{k+2}$$

Putting $$k=7$$, using that $$B_3=0$$ and factoring out 1.3.5.7, we get

$$\binom{5}{1}B_4+\binom{5}{3}B_2-\binom{5}{4}B_1+\binom{5}{5}B_0 = -\frac{5}{30}+\frac{10}{6}-\frac{5}{2}+1 = 0$$

In special cases the conjecture boils down to well known identities involving Bernoulli numbers and binomial coefficients. I would appreciate an elementary proof, but really anything will do. Perhaps someone into combinatorics or generating functions can help? Thanks again.

• You should definitely put the whole question into the text here, and not make people visit the other site to see it. – Sam Hopkins May 4 at 22:11
• @Fox you can hit 'edit' on the other question and copy and paste the source (then cancel the edit), rather than re-type the thing. – David Roberts May 4 at 22:54
• @David, Sam: thanks! – Fox Mulder May 5 at 0:41
• Strengthening a conjecture usually makes it easier to prove, so it is worth noting that empirically the given polynomial for $M = 2m+1$ appears to be $(m+1) (k - 1)^{D(m)}$ (and, FWIW, this holds for $M=1$ too). – Peter Taylor May 7 at 16:52

UPDATE. My earlier answer was incorrect due to miscalculation. Below I give a proof of the conjecture.

First, we notice that $$k^{R(i)} = (k/2+i-1)_i\cdot 2^i = \binom{k/2+i-1}i\cdot i!\cdot 2^i$$. Similarly, $$k^{D(i)} = (k/2)_i\cdot 2^i = \binom{k/2}i\cdot i!\cdot 2^i$$ and $$(2i-1)!!=\frac{(2i)!}{i!2^i}=\binom{2i}{i}\cdot \frac{i!}{2^i}$$. Then the proposed conjecture follows from the following polynomial (in $$x$$) identity: $$\sum_{i=0}^m \big(\binom{x+\frac12-m+i}i + (-1)^{i+1}\binom{2i}{i} \frac{1}{4^i}\big) B^+_{m-i} \binom{x+1}{m-i} = (x+1) \binom{x-\frac12}{m-1}$$ by taking $$m = \frac{M+1}2$$ and $$x=\frac{k}2$$. Noticing that $$\binom{x+\frac12-m+i}i = (-1)^i \binom{m-x-\frac32}i$$ and $$B^+_{m-i} = (-1)^{m-i} B^-_{m-i}$$, we rewrite the above identity as $$(\star)\qquad \sum_{i=0}^m \big(\binom{m-x-\frac32}i - \binom{2i}{i} \frac{1}{4^i}\big) B^-_{m-i} \binom{x+1}{m-i} = (-1)^m (x+1) \binom{x-\frac12}{m-1}.$$

Proof. Since for a fixed $$m$$ the l.h.s. and r.h.s. of $$(\star)$$ are polynomials in $$x$$, it is enough to prove $$(\star)$$ for $$x$$ being a nonnegative integer. We notice that $$\binom{m-x-\frac32}i - \binom{2i}{i}\frac{1}{4^i} = [z^i]\ \big( (1+z)^{m-x-\frac32} - (1-z)^{-\frac12}\big)$$ while $$B^-_{m-i} \binom{x+1}{m-i} = [z^{m-i}]\ z^{x+1}{\cal B}_{x+1}(\frac1z),$$ where $${\cal B}_{x+1}(t)$$ is the $$(x+1)$$-st Bernoulli polynomial and $$[z^n]$$ is the operator extracting the coefficient of $$z^n$$.

It follows that the l.h.s. of $$(\star)$$ equals $$[z^m]\ \big( (1+z)^{m-x-\frac32} - (1-z)^{-\frac12}\big) z^{x+1}{\cal B}_{x+1}(\frac1z).$$

Using Lagrange–Bürmann formula, we conclude that $$[z^m]\ (1+z)^{m-x-\frac32} z^{x+1}{\cal B}_{x+1}(\frac1z) = [z^m]\ (1-z)^{-\frac12} z^{x+1} {\cal B}_{x+1}(\frac1z-1).$$ Then, by the properties of Bernoulli polynomials, $${\cal B}_{x+1}(\frac1z-1) = {\cal B}_{x+1}(\frac1z) - (x+1) (\frac1z-1)^x.$$

So, the Bernoulli polynomials in the l.h.s. of $$(\star)$$ cancel out, and it reduces to $$-(x+1)\cdot [z^m]\ z (1-z)^{x-\frac12} = (-1)^m (x+1) \binom{x-\frac12}{m-1}.$$ QED

• Hi Max, thanks for your answer, maybe I'm doing something incorrect, but it seems like when M=9, k=7 is indeed an integer root, I put the calculation in the body of the question at the end, it was too long for a comment. Thanks again for looking into this, greatly appreciated! – Fox Mulder May 7 at 13:58
• @FoxMulder: You seem to be correct, and there was a typo in my code causing miscalculation. I'm taking my answer down for now. – Max Alekseyev May 7 at 16:12
• @FoxMulder: I've added a proof of your conjecture. – Max Alekseyev May 8 at 16:05
• Thanks Max, greatly appreciated! If it's ok with you, I'll email you to chat a bit about this and the wider problem, thanks again! – Fox Mulder May 9 at 1:52
• Sure, you are welcome to email me to maxal@gwu.edu – Max Alekseyev May 9 at 2:24


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# On Monday at The Capsule Corporation, the ratio of the number

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On Monday at The Capsule Corporation, the ratio of the number  [#permalink]

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13 Jan 2019, 06:43
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On Monday at The Capsule Corporation, the ratio of the number of employees who had meetings to the number of employees who did not have meetings was 2 to 3. On Tuesday at The Capsule Corporation, the ratio of the number of employees who had meetings to the number of employees who did not have meetings was 3 to 4. Which of the following could be the number of employees at the Capsule Corporation? (Assume the number of employees who showed up at work on Monday is same as the number that showed up on Tuesday.
A)5
B)10
C)15
D)25
E)35

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Re: On Monday at The Capsule Corporation, the ratio of the number  [#permalink]

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13 Jan 2019, 07:11
Abhi077 wrote:
On Monday at The Capsule Corporation, the ratio of the number of employees who had meetings to the number of employees who did not have meetings was 2 to 3. On Tuesday at The Capsule Corporation, the ratio of the number of employees who had meetings to the number of employees who did not have meetings was 3 to 4. Which of the following could be the number of employees at the Capsule Corporation? (Assume the number of employees who showed up at work on Monday is same as the number that showed up on Tuesday.
A)5
B)10
C)15
D)25
E)35

Mon -> $$\frac{Meeting}{No Meeting}$$ = $$\frac{2}{3}$$

Tue-> $$\frac{Meeting}{No Meeting}$$ = $$\frac{3}{4}$$

Number of employees is same, this means it should be LCM of 5 and 7

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Re: On Monday at The Capsule Corporation, the ratio of the number &nbs [#permalink] 13 Jan 2019, 07:11
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# On Monday at The Capsule Corporation, the ratio of the number

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# Do modern GPUs contain barrel shifters?

I'm aware that most modern GPUs, although designed for floating point, are more or less equivalent in integer performance these days, with a few caveats like the lack of a fused multiply add. I'm not sure how this applies to shift operations though. I'm doing Marching Cubes on GPU, initially writing out a 32-bit packed position for each surface cube then unpacking these in a later pass to the actual vertices in that cube, like this :

ivec3 unpackedPos = ivec3( packedPos >> 20 & 0x3FF,
packedPos >> 10 & 0x3FF,
packedPos & 0x3FF);

It just occurred to me to wonder if shader units have barrel shifters in them these days? Am I doing 2 shifts here or 30?

EDIT>> I'm an idiot... Thanks for the answers guys, useful to know, but I've been going about this all wrong. I should just be using the RGB10_A2UI texture format then packing/ unpacking with a single image load/store instruction instead of messing around with bitshifts myself.

RE_EDIT>> Or not... This method apparently works on red boxes but not on green ones, so it's back to bit-shifts.

• 24 bit shifters are used in single precision floating point to align mantissas, so the compiler might generate a few, but I don't think you'll see 30. Mar 5 '16 at 18:34


text:
By Topic

# IEEE Transactions on Industrial Electronics and Control Instrumentation

## Filter Results

Displaying Results 1 - 25 of 25
• ### Comparison of a Local Linearization Algorithm with Standard Numerical Integration Methods for Real-Time Simulation

Publication Year: 1980, Page(s):129 - 132
| | PDF (835 KB)

The local linearization algorithm is presented as a possible numerical integration scheme to be used in real-time simulation. A second-order nonlinear example problem is solved using different methods. The local linearization approach is shown to require less computing time and give significant improvement in accuracy over the classical second-order integration methods. View full abstract»

• ### Design and Application of a Microprocessor PID Predictor Controller

Publication Year: 1980, Page(s):133 - 137
Cited by:  Papers (20)  |  Patents (2)
| | PDF (1131 KB)

A digital PID predictor controller is designed and implemented in a microprocessor. The general purpose system controller described permits the use of integer arithmetic for efficient operation with minimum throughput delay time. The stability of the system is investigated and a normalized chart for the selection of the control coefficients for a class of applications is developed. The controller ... View full abstract»

• ### Motor Thermal Protection by Continuous Monitoring of Winding Resistance

Publication Year: 1980, Page(s):137 - 141
Cited by:  Papers (47)  |  Patents (1)
| | PDF (1230 KB)

Simple practical techniques which sense ac line current afford protection of motor windings for many applications. A more complete protection system is described which continuously monitors average winding resistance and hence temperature by means of an easily applied dc injection technique. No additional motor connections are required. View full abstract»

• ### An Application of Distributed Computer Control of Railroad

Publication Year: 1980, Page(s):141 - 146
Cited by:  Patents (1)
| | PDF (1474 KB)

There are three generations of railroad control: 1) early electromechanical control; 2) central computer control; and 3) distributed computer control. In the second-generation, tracks were divided into numbers of track sections and applications had been aimed toward detecting individual track section as to whether or not there is a train on the track and dispatching necessary information to trains... View full abstract»

• ### Stability Analysis of Induction Motor Drives Using Phase-Locked Loop Control System

Publication Year: 1980, Page(s):147 - 155
Cited by:  Papers (27)  |  Patents (1)
| | PDF (1810 KB)

This paper presents the analysis of an induction motor drive using a phase-locked loop speed control system. A linear continuous model of the drive is developed and the system response is analysed using conventional root locus techniques. Various compensating filters'' and feedback signals are considered and the need for addition of derivative feedback is shown. A sampled data model is used to s... View full abstract»

• ### Application of RF Circuit Design Principles to Distributed Power Converters

Publication Year: 1980, Page(s):156 - 164
Cited by:  Papers (70)  |  Patents (9)
| | PDF (5875 KB)

The application of RF circuit design principles to high-frequency power converters is described. Compared to conventional converter design, emphasis is placed on obtaining sinusoidal-type waveforms (instead of rectangular-type waveforms) to minimize device switching time requirements and alleviate transforming requirements. A 25-W 48-V to 5-V dc-to-dc converter design using a 5-MHz switching frequ... View full abstract»

• ### Stability Analysis of a Thyristor DC-DC Converter

Publication Year: 1980, Page(s):165 - 168
Cited by:  Papers (1)
| | PDF (691 KB)

Stability of a dc-dc converter where the average value of the output voltage is maintained constant is investigated. Simplified equations are developed for the average value of the output during sampling periods. The converter is modeled as a chopper pulsewidth modulated (CPWM) system and stability conditions are derived. By simulation, the accuracy of the results is verified. View full abstract»

• ### A Microprocessor-Based Current Controller for SCR-DC Motor Drives

Publication Year: 1980, Page(s):169 - 176
Cited by:  Papers (10)  |  Patents (1)
| | PDF (1838 KB)

A new current controller for a three-phase full-wave silicon controlled rectifier (SCR) bridge, supplying a separately excited armature-controlled dc motor driving an inertial load is developed. The controller operates in two modes, one applicable when the motor armature current is continuous, the other when it is discontinuous. The current controller is then used in speed or position control of a... View full abstract»

• ### Comparison of Multiple-Output DC-DC Converters Using Cross Regulation

Publication Year: 1980, Page(s):176 - 189
Cited by:  Papers (37)  |  Patents (2)
| | PDF (2955 KB)

A new multiple-output dc-dc converter is analyzed, in which the cross regulation is performed by the energy-storage reactor as well as the transformer. The steady-state and the dynamic characteristics of this converter are compared both theoretically and experimentally with those of the conventional one composed of the step-up-down type circuit. As the results of comparative analysis, it is reveal... View full abstract»

• ### An On-Line System for Identifying the Angular Orientation of a Class of Engineering Components

Publication Year: 1980, Page(s):189 - 196
| | PDF (1995 KB)

This paper describes the principle of operation and the development of a system for on-line identification of the angular orientation of a class of flat'' objects-the class of objects whose angular orientation can be uniquely determined by examining the features of the exterior profile or silhouette. The laboratory system described herein makes use of a digitized image of the component produced ... View full abstract»

• ### A Microprocessor-Based PLL Speed Control System Converter-Fed Synchronous Motor

Publication Year: 1980, Page(s):196 - 201
Cited by:  Papers (10)  |  Patents (1)
| | PDF (1367 KB)

A microprocessor-based digital phase-locked loop (PLL) speed control system of a self-controlled synchronous motor fed by a current source inverter is presented. The hardware and software structures of the system are shown in detail. Sophisticated methods for synthesizing PLL control systems are proposed. Experimental results demonstrate the validity of the proposed methods. The minimum maintenanc... View full abstract»

• ### Harmonic Reduction in Inverters by Use of Sinusoidal Pulsewidth Modulation

Publication Year: 1980, Page(s):201 - 207
Cited by:  Papers (15)
| | PDF (1531 KB)

This paper presents a new technique for harmonic reduction in inverters using sinusoidal pulsewidth modulation. In this inverter, the output wave with sinusoidal pulsewidth distribution can be kept in maximum degree of symmetry, and so it offers excellent harmonic control in comparison with any other modulation technique. Detailed analysis shows that the harmonics lower than the ninth (or the thir... View full abstract»

• ### The Analysis of Control Systems with Distributed Lag

Publication Year: 1980, Page(s):207 - 210
Cited by:  Papers (2)
| | PDF (945 KB)

This is an extension of an earlier paper [1] dealing with the stability analysis of distributed-lag systems. Quite a few systems have been analyzed based on the dual phase-comparison technique [2]. A brief discussion of a recent paper [3] has also been included. View full abstract»

• ### Power Factor of AC Controllers for Inductive Loads

Publication Year: 1980, Page(s):210 - 212
Cited by:  Papers (22)
| | PDF (759 KB)

A general relationship for the input power factor of ac controllers with inductive loads is derived in this paper. Its simplified form is given for the phase angle, integral cycle, and symmetrical pulsewidth modulated controllers. The latter may or may not have a freewheeling path for the load. View full abstract»

• ### Analog Signal Processing for Tachometer, Accelerometer, and Torque-Speed Curve Plotting of Motors by Use of Silicon Solar Cells

Publication Year: 1980, Page(s):213 - 218
Cited by:  Papers (2)
| | PDF (1411 KB)

An optical tachometer is developed by the analog processing of the signals obtained from the silicon solar cell displacement sensors. Differentiating the speed signal, satisfactorily smooth acceleration signal is obtained by simple filtering. The torque-speed curve plotting for small motors by the application of this technique and the effects of the filtering on the plot are described. View full abstract»

• ### Microprocessor-Based System for Processing Redundant Instrumentation Signals

Publication Year: 1980, Page(s):218 - 222
Cited by:  Papers (4)
| | PDF (1251 KB)

A microprocessor-based system for processing redundant instrumentation signals is proposed. The prototype of the system has been produced using general-purpose microprocessors. The system can provide a complex trip function in place of a conventional hardwired simple 2 out of 3 trip circuit. Self testing and fail safe features are included in the system. The feasibility of the system is assessed b... View full abstract»

• ### Microprocessor Control of DC 3-Phase Thyristor Inverter Crcuits

Publication Year: 1980, Page(s):223 - 228
Cited by:  Papers (1)  |  Patents (2)
| | PDF (3319 KB)

The paper describes a microprocessor-orientated 50-kW direct current to three-phase variable frequency thyristor inverter which is suitable for induction machine speed and/or torque control. The microprocessor in real time analyzes speed and torque requirements, current and rotor speed feedback signals and calculates output frequency requirements. The microprocessor directly controls the main thyr... View full abstract»

• ### Microprocessor Control of Position or Speed of an SCR DC Motor Drive

Publication Year: 1980, Page(s):228 - 234
Cited by:  Papers (10)
| | PDF (1415 KB)

In silicon-controlled rectifier (SCR) dc motor drive control, it is desirable for a microprocessor to perform both the control law computations and the logical functions of SCR firing. However, because of their limited real-time capabilities, present-day microprocessors cannot accommodate excessively lengthy control algorithms. This paper presents a method, suitable for microprocessor implementati... View full abstract»

• ### A Performance Study of Control Systems with Dead Time

Publication Year: 1980, Page(s):234 - 241
Cited by:  Papers (15)
| | PDF (1410 KB)

This paper discusses the performance of the Smith Dead Time (SDT) controller for systems with appreciable dead time. Tuning, modeling accuracy, and the effects of parameter variations and measurement noise are all studied using a simulated second-order process with dead time. The corresponding performance of the conventional proportional plus integral (PI) controller is used as a reference for com... View full abstract»

• ### A Programmable Firing Controller

Publication Year: 1980, Page(s):242 - 246
Cited by:  Papers (4)
| | PDF (1186 KB)

A digital programmable firing controller to drive solid-state power modulators is presented. The advantages over analog or hybrid systems are outlined. The simplicity, expandability, reliability, and repeatability without any channel crosstalk is stressed. Programming on line while maintaining strict waveshape symmetry is realized. View full abstract»

• ### Vibrating Fluid Densimeters: A Solution to the Viscosity Problem

Publication Year: 1980, Page(s):247 - 253
Cited by:  Papers (3)  |  Patents (11)
| | PDF (1842 KB)

The natural frequency of a resonator immerged in a fluid is density dependent. A major disadvantage of densimeters based on this principle is their poor accuracy for viscous fluids. We solve this difficulty by making an automatic control of the physical phase at which the resonator oscillates. This is done by a modulation technique which, additionally, gives the quality factor and, therefore, the ... View full abstract»

• ### Letters to the Editor

Publication Year: 1980, Page(s):253 - 258
| PDF (2829 KB)
• ### Contributors

Publication Year: 1980, Page(s):258 - 262
| PDF (12393 KB)
• ### For essential new data in electrotechnology, look to the source.

Publication Year: 1980, Page(s): 263
| PDF (1351 KB)
• ### Our IEEE periodicals have only one competitor as an information source in electrotechnology [advertisement]

Publication Year: 1980, Page(s): 264
| PDF (1361 KB)

## Aims & Scope

This Transactions ceased production in 1981. The current retitled publication is IEEE Transactions on Industrial Electronics.

Full Aims & Scope


text:
2022-12-26

A)Subtraction
B)Multiplication
C)Division

Neenuegopddd

Expert

The correct answer is b Multiplication
Example:
3+3+3+3=12
Also, 3×4=12
Here, 3 added 4 times repeatedly is 12 and also 3 multiplied by 4 is also 12.
Hence, multiplication is nothing but repeated addition.

Do you have a similar question?


text:
in - Maple Help

MultiSet/member

test if an expression is a member of a MultiSet

 Calling Sequence member( x, M ) x in M

Parameters

 M - MultiSet; a MultiSet x - anything, an expression

Description

 • member( x, M ) returns true if x is a member of the MultiSet M, false otherwise
 • Note: member( x, M ) is equivalent to evalb( x in M ) and also equivalent to evalb( M[x] <> 0 ).
 • Note: As a MultiSet is unordered, attempting to use the three-argument form of the member command, as in member( x, M, 'k' ) will result in an error.

Examples

 > $M≔\mathrm{MultiSet}\left(a=2,b=5\right)$
 ${M}{≔}\left\{\left[{a}{,}{2}\right]{,}\left[{b}{,}{5}\right]\right\}$ (1)
 > $\mathrm{member}\left(b,M\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{member}\left(c,M\right)$
 ${\mathrm{false}}$ (3)
 > $\mathrm{evalb}\left(a\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}M\right)$
 ${\mathrm{true}}$ (4)
 > $\mathrm{member}\left(a,M,'k'\right)$

Compatibility

 • The MultiSet/member command was introduced in Maple 2016.


text:
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An induction coil is generally used to
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If $y=a^{x}$ passes through $(2 ; 25)$, calculate the value of $a$.
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Measurements are collected from 100 subjects from each of two sources. The data yield the following frequency histograms. The number above each rectangle is its height.
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A rectangle with length $L$ and width $W$ has a perimeter $P$ given by $P=L+W+L+W$. Simplify this expression.
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A photograph with length $L$ and width $W$ is placed in a mat of width 8 centimeters $(\mathrm{cm})$. The area of the mat (in square centimeters, or $\mathrm{cm}^{2}$ ) is found to be $A=(L+16)(W+16)-L \cdot W .$  A photograph with length $L$ and width $W$ is placed in a mat of width 8 centimeters $(\mathrm{cm})$. The area of the mat (in square centimeters, or ...
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The standard normal curve has a total area to be under one, and it is symmetric around zero. True or False?
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The sides of a rectangle are positive whole numbers. If its area is $A$ and its perimeter is $P$, which of the following numbers cannot equal $A+P ?$
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# Understanding the Result of Camera Calibration and its Units

Hi,

I am quite new to camera calibration.

I have 35 images, based on the following functions :

findChessboardCorners() cornerSubPix() drawChessboardCorners() calibrateCamera()

I now have

1. Camera Matrix - 3x3 dimension
2. Distortion Coefficient - 1x5 dimension
3. Rotation Vector - 35x3 dimension
4. Translation Vector - 35x3 dimension

Questions :

1. camera matrix

 [ 2421.70421334667, 0, 2138.107444179007;
0, 2429.244140012221, 1136.772764836856;
0, 0, 1]

In the above , what are the units of fx, fy , Cx and Cy ?

1. Distortion Coefficient
[-0.2877455633196141,
0.2968104208806526, -0.001127291269463514, -0.002010576239298727, -0.3457992571078199]

Do we have any Units for K1, K2, P1, P2, K3 ?

1. Rotation Vector
[-0.1019788539700399; 0.3918509002001774; 0.0734955967493143]
[-0.1165422449849887; 0.5017699087572564; 0.4257284491781231]
[-0.1233545118567792; 0.5922964766688477; 1.001270611741778]
[-0.5253276710841507; -0.01653828323651414; -1.386034124708322]
so on ... for all 35 images

What about the unit of Rotation Matrix? How should I understand it? What does this values mean?

1. Translation Vector
[152.8348988935562, -168.8318694822206; 951.100836109663]
[191.0103136141713; -207.2053481732149; 963.7313891311128]
[307.0885175766131; -212.1491832662802; 945.4874998124949]
[221.391205120318; -22.64308775049023; 910.7108013142725]
so on ... for all 35 images

What about the unit of Translation Vector? How should I understand it ? What does its values mean?

I also tries the camera calibration app in Matlab, I set the same image but it does not show the same result?

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Sort by » oldest newest most voted

Summarizing:

1. Camera matrix:
This is the inner orientation of the camera that can be calculated when calibrating the camera. A camera lens may have a fixed focal length of 35 mm but this is often not accurate enough for further calculations so when calibrating with a chessboard you calculate these values more accurate.
So fx and fy are the focal lengths expressed in pixels.
Cx and Cy describe the coordinates of the so called principal point that should be in the center of the image. It is e.g. not in the center of the image if you cropped the image, what you should never do when calibrating.
fx, fy, cx, cy are given in Pixels in Computer Vision ( and openCV) but e.g. in Photogrammetry you often use mm

2. Distortion Coefficient:
As the name already says, the parameters show the distortions of the image. It can also be calculated when calibrating the camera with a chessboard.
There are up to 6 parameters for radial distortion (k1-k6) and two parameters for tangential distortion (p1-p2). Look at the images in the link.
k1-k6, p1-p2 don't have units.

3. Rotation vector and translation vector:
As far as I understood for every image you take of the chessboard the translation and rotation is estimated from the model coordinate space (object points estimated from inner orientation) to the world coordinate space (real object points in a coordinate system you defined).
Rotation can be e.g. given in Angles (-Pi, Pi) or (0, 2*Pi) or in quaternions.
Translation can be given in km, m, dm.... (depends how you specified your values).

More detailed information is given e.g. in Szeliski - Computer Vision Algorithms and Applications (http://szeliski.org/Book/drafts/Szeli...)

more

Official site

GitHub

Wiki

Documentation


text:
Question:

# How do you calculate the moles of a unknown substance given in grams?

## Convert grams of solute into moles of solute. Measure the amount of solution. The solution will be in liters before the final...

Tags:
Chemistry
Solutions
Physical chemistry
Colloidal chemistry
Melanocytic nevus
Apparent molar property
Osmolarity

9


text:
## Calculus 8th Edition

See sec.1-8, Intermediate Value Theorem. If f is continuous on [a,b], N any number between f(a) and f(b), f(a) $\neq$ f(b). Then, there exists a c in [a,b] such that f(c)=N. ------------- $\pi\approx 3.14,\qquad 3 < \pi < 4$ $f(-1) < \pi < f(1)$. Apply the Intermediate Value Theorem with $a=-1,\ b=1,\ N=\pi$. The statement is true.


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# Math Help - trignometry

1. ## trignometry

solve the eqn:

1/log(base 4) (x+1)/(x+2) < 1/log(base 4) (x+3)

regret me
the topic is LOGARITHMS

2. Hello, grgrsanjay!

I have a very long solution . . . and it may be incomplete.
Someone check it carefully . . . please!

$\text{Solve: }\;\dfrac{1}{\log_4\!\left(\frac{x+1}{x+2}\right)} \;<\; \dfrac{1}{\log_4(x+3)}\;\;[1]$

We begin with: . $1 \:<\:2$

Add $x\!:\;\;x+1 \:<\:x+2\;\;[2]$

Domain: $x \,>\,-1 \quad\Rightarrow\quad x+1 \,>\,0 \quad\Rightarrow\quad x+2 \,>\,0$

Divide [2] by positive $x+2\!:$ . $\dfrac{x+1}{x+2} \:<\:1$

Take logs, base 4: . $\log_4\!\left(\dfrac{x+1}{x+2}\right)\:<\:\log_4(1 ) \:=\:0$

That is: . $\log_4\!\left(\dfrac{x+1}{x+2}\right)$ .is negative.

Multiply [1] by negative $\log_4\!\left(\frac{x+1}{x+2}\right)\!:$ . $1 \;>\;\dfrac{\log_4\!\left(\frac{x+1}{x+2}\right)}{ \log_4(x+3)}\;\;[3]$

Since $x \,>\, -1$, then $x+3 \,>\,2 \quad\Rightarrow\quad \log_4(x+3) \:>\:0$

Multiply [3] by positive $\log_4(x+3)\!:$ . $\log_4(x+3) \;>\;\log_4\!\left(\dfrac{x+1}{x+2}\right)$

Take anti-logs: . $x+3\;>\;\dfrac{x+1}{x+2} \quad\Rightarrow\quad x^2 + 4x + 5 \;>\;0$

This is an up-opening parabola with its vertex at (-2, 1).
. . Hence, it is always above the $x$-axis.

The inequality is true for all $x$ in the domain: . $x \:>\:-1$


text:
# 21: Linear Variational Theory

Recap of Lecture 20

Last lecture continued the discussion of variational method approach to approximate the solutions of systems that we cannot analytically solve the Schrodinger equation. The system that motivates us is multi-electron atoms, and we focused on the He atom primarily. This approach requires postulating a trial wavefunction and then calculating the energy of that function as a function of the parameters that describe the trial wavefunction. Then we can minimize the energy as a function of these parameters and the closer the wavefunction "looks" like the true wavefunction (that we do not know), the closer the trail energy matches the true energy (however, the trial energy is ALWAYS higher in energy than the true energy). We went over several example trial wavefunctions for the He atom showing the more complex wavefunctions give better results than the simple ones (including the "Ignorance is Bliss" approximation with an average effective charge). We then transition into the Heisenberg's matrix representation of Quantum mechanics which was the segway to the linear variational method, which addresses trial functions that are a linear combination of a basis functions. We will continue that discussed next lecture.

## Overview (again) of Variational Method Approximation

We can always construct a variational energy for a trial wavefunction given a specific Hamilitonian

$E_{trial} = \dfrac{\langle \psi_{trial}| \hat{H} | \psi_{trial} \rangle }{\langle \psi_{trial}| \psi_{trial} \rangle} \ge E_{true}\label{7.3.1b}$

The variational energy is an upper bound to the true ground state energy of a given molecule. The general approach of this method consists in choosing a "trial wavefunction" depending on one or more parameters, and finding the values of these parameters for which the expectation value of the energy is the lowest possible.

Simplified algorithmic flowchart of the Variational Method approximation

Example $$\PageIndex{1}$$

Using the variational method approximation, find the ground state energy of a particle in a box using this trial function:

$| \phi \rangle = N\cos\left(\dfrac{\pi x}{L}\right) \nonumber$

How does is it compare to the true ground state energy?

Solution

The problem asks that we apply variational methods approximation to our trial wavefunction.

$E_{\phi} = \dfrac {\langle \phi | \hat{H} | \phi \rangle} { \langle \phi | \phi \rangle} \ge E_o \nonumber$

Let's look at the denominator:

$\langle \phi | \phi \rangle = 1 = \int_{0}^{L}N^2\cos^2\Big(\dfrac{\pi x}{L}\Big)\nonumber$

Performing this integral and solving for N yields

$N = \sqrt{\dfrac{2}{L}}\nonumber$

Now let's look at the numerator:

The Hamiltonian for a particle in a one dimensional box is $$\hat{H} = \dfrac{-\hbar^2}{2m}\dfrac{d^2}{dx^2}$$

\begin{align*} \langle \phi | \hat{H} | \phi \rangle &= \langle N\cos\Big(\dfrac{\pi x}{L}\Big) \Big| \dfrac{-\hbar^2}{2m}\dfrac{d^2}{dx^2} \Big| N\cos\Big(\dfrac{\pi x}{L}\Big) \rangle\nonumber \\ &= \int_{0}^{L}N\cos\Big(\dfrac{\pi x}{L}\Big)\dfrac{-\hbar^2}{2m}\dfrac{d^2}{dx^2}N\cos\Big(\dfrac{\pi x}{L}\Big)dx\nonumber \\ &= \dfrac{N^2\pi^2 \hbar^2}{2mL^2}\int_{0}^{L}\cos^2\Big(\dfrac{\pi x}{L}\Big) dx\end{align*}

where $$N = \sqrt{\dfrac{2}{L}}$$. The above equation after the integral becomes

$\dfrac{\pi^2 \hbar^2}{mL^3}\Big(\dfrac{L}{2}\Big)\nonumber$

Now the variational energy for this trail wavefunction is

$E_\phi = \dfrac{\pi^2 \hbar^2}{2mL^2}\nonumber$

This is equal to the ground state energy of the particle in a box that we calculated from the Schrodinger equation using

$\psi = \sqrt{\dfrac{2}{L}}\sin(\dfrac{n\pi x}{L})\nonumber$

This is not a good trail wavefucntion since we are not varying any parameter, and moreover it is the eigenstate of energy. So the guess (trail wavefunction) is accurate... this almost never happens.

## Let's look at a Different Type of a Trial Wavefunction

Let's expand $$\phi$$ into a linear combination of basis functions:

$| \psi_{trial} \rangle = \sum_j^N a_j | \phi_j \rangle \label{Ex1}$

and similarly

$\langle \psi_{trial} | = \sum_j^N a_j^* \langle \phi_j | \label{Ex2}$

In these cases, one says that a 'linear variational' calculation is being performed.

Basis Functions

The set of functions {$$\phi_j$$} are called the 'linear variational' basis functions and are usually selected:

• to obey all of the boundary conditions that the exact state $$| \psi _{trial} \rangle$$ obeys,
• to be functions of the the same coordinates as $$| \psi _{trial} \rangle$$, and
• to be of the same symmetry as $$| \psi _{trial} \rangle$$.

Beyond these conditions, the {$$\phi_j$$} are nothing more than members of a set of functions that are convenient to deal with (e.g., convenient to evaluate Hamiltonian terms elements $$\langle \phi_i|H|\phi_j \rangle$$ that can, in principle, be made complete if more and more such functions are included in the expansion in Equations $$\ref{Ex1}$$ and $$\ref{Ex2}$$ (i.e., increase $$N$$).

Examples

For example, For the Hydrogen atom wavefunctions, $$\phi$$ could be expanded into a linear combination of Scalable Gaussian functions

$|\phi_{trial} \rangle = \sum_{j=1}^{N} c_{j} e^{-\alpha_{j} r^2}\label{6B}$

or for $$H_2$$, (this is call the Liner Combination of Atomic Orbitals approximation discussed in detail in later sections)

$| \phi_{trial} \rangle = c_1 \psi_{1s_A} + c_2 \psi_{1s_B}$

The goal is to solve for the set of all $$c$$ values that minimize the energy $$E_{trial}$$.

$E_{trial} = \dfrac{ \langle \psi _{trial}| \hat {H} | \psi _{trial} \rangle}{\langle \psi _{trial} | \psi _{trial} \rangle} \label{7.1.8}$

Substituting Equation \ref{Ex2} into Equation \ref{7.1.8} involves addressing the numerator and denominator individually. For the numerator, the integral can be expanded thusly:

\begin{align} \langle\psi_{trial} |H| \psi_{trial} \rangle &= \sum_{i}^{N} \sum_{j} ^{N}a_i^{*} a_j \langle \phi_i|H|\phi_j \rangle. \\[5pt] &= \sum_{i,\,j} ^{N,\,N}a_i^{*} a_j \langle \phi_i|H|\phi_j \rangle. \label{MatrixElement}\end{align}

We often rewrite the following integral in Equation \ref{MatrixElement} (as a function of the basis elements, not the trial wavefunction) as

$H_{ij} = \langle \phi_i|H|\phi_j \rangle$

So the numerator of the right side of Equation \ref{7.1.8} becomes

$\langle\psi_{trial} |H| \psi_{trial} \rangle = \sum_{i,\,j} ^{N,\,N}a_i^{*} a_j H_{ij} \label{numerator}$

Similarly, the denominator of the right side of Equation \ref{7.1.8} can be expanded

$\langle \psi_{trial}|\psi_{trial} \rangle = \sum_{i,\,j} ^{N,\,N}a_i^{*} a_j \langle \phi_i | \phi_j \rangle \label{overlap}$

We often rewrite the following integral in Equation \ref{overlap} (as a function of the basis elements, not the trial wavefunction) as

$S_{ij} = \langle \phi_i|\phi_j \rangle$

where $$S_{ij}$$ are overlap integrals between the different {$$\phi_j$$} functions. So Equation \ref{overlap} is

$\langle \psi_{trial}|\psi_{trial} \rangle = \sum_{i,\,j} ^{N,\,N}a_i^{*} a_j S_{ij} \label{denominator}$

Orthonormality

There is no explicit rule that the {$$\phi_j$$} functions have to be orthogonal and normal functions, although they often are selected that was for convenience. Therefore, a priori, $$S_{ij}$$ does not have to be $$\delta_{ij}$$.

Substituting Equations \ref{numerator} and \ref{denominator} into the variational energy formula (Equation \ref{7.1.8}) results in

$E_{trial} = \dfrac{ \displaystyle \sum_{i,\,j} ^{N,\,N}a_i^{*} a_j H_{ij} }{ \displaystyle \sum_{i,\,j} ^{N,\,N}a_i^{*} a_j S_{ij} } \label{Var}$

## Minimizing the Variational Energy

The optimum coefficients are found by searching for minima in the variational energy landscape spanned by varying the $$\{a_i\}$$ coefficients (Figure $$\PageIndex{1}$$).

We want to minimize the variation energy with respect to the linear coefficients $$\{a_i\}$$, which requires that

$\dfrac{\partial E_{trial}}{\partial a_i}= 0$

for all $$i$$.

For the "normal" variational method discussed last time, the variation energy can be a nonlinear function of one or more parameters. Minimizing the variation energy therefore may require linear or non-linear regression. However, the linearatiy of the trial function and the nature of the variational energy allows for more simplistic linear regression.

The expression for variational energy (Equation \ref{Var}) can be rearranged

$E_{trial} \sum_{i,\,j} ^{N,\,N} a_i^*a_j S_{ij} = \sum_{i,\,j} ^{N,\,N} a_i^* a_j H_{ij} \label{7.2.9}$

Differentiating both sides of Equation $$\ref{7.2.9}$$ for the $$k^{th}$$ coefficient gives,

$\underbrace{ \dfrac{\partial E_{trial}}{\partial a_k} \sum_{i,\,j} ^{N,\,N} a_i^*a_j S_{ij}+ E_{trial} \sum_i \sum_j \left[ \dfrac{ \partial a_i^*}{\partial a_k} a_j + \dfrac {\partial a_j}{\partial c_k} a_i^* \right ]S_{ij} }_{\text{product rule}}= \sum_{i,\,j} ^{N,\,N} \left [ \dfrac{\partial a_i^*}{\partial a_k} a_j + \dfrac{ \partial a_j}{\partial a_k}a_i^* \right] H_{ij} \label{7.2.10}$

Since the coefficients are linearly independent (Equation \ref{Ex1})

$\dfrac{\partial a_i^*}{ \partial a_k} = \delta_{ik}$

and

$S_{ij}^* = S_{ji}$

and also since the Hamiltonian is a Hermetian Operator

$H_{ij}^* =H_{ji}$

then Equation $$\ref{7.2.10}$$ simplifies to

$\dfrac{\partial E_{trial}}{\partial a_k} \sum_i \sum_j a_i^*a_j S_{ij}+ 2E_{trial} \sum_i S_{ik} = 2 \sum_i a_i H_{ik} \label{7.2.11}$

Hermitian Operators

Hermitian operators are operators that satisfy the general formula

$\langle \phi_i | \hat{A} | \phi_j \rangle = \langle \phi_j | \hat{A} | \phi_i \rangle \label{Herm1}$

If that condition is met, then $$\hat{A}$$ is a Hermitian operator. For any operator that generates a real eigenvalue (e.g., observables), then that operator is Hermitian. The Hamiltonian $$\hat{H}$$ meets the condition and a Hermitian operator. Equation \ref{Herm1} can be rewriten as

$A_{ij} =A_{ji}$

where

$A_{ij} = \langle \phi_i | \hat{A} | \phi_j \rangle$

and

$A_{ji} = \langle \phi_j | \hat{A} | \phi_i \rangle$

Therefore, when applied to the Hamiltonian operator

$\boxed{H_{ij}^* =H_{ji}.}$

At the minimum variational energy, when

$\dfrac{\partial E_{trial}}{\partial a_k} = 0$

then Equation $$\ref{7.2.11}$$ gives

${\sum _i^N a_i (H_{ik}–E_{trial} S_{ik}) = 0} \label{7.2.12}$

for all $$k$$. The equations in $$\ref{7.2.12}$$ are call the Secular Equations.

(left) Simplified algorithmic flowchart of the "normal" Variational Method approximation vs. (right) Simplified algorithmic flowchart of the linear Variational Method approximation. Noticed the similarities and differences between two.

## Secular Determinant

From the secular equations with an orthonormal functions (Equation \ref{7.2.12}), we have $$k$$ simultaneous secular equations in $$k$$ unknowns. These equations can also be written in matrix notation, and for a non-trivial solution (i.e. $$c_i \neq 0$$ for all $$i$$), the determinant of the secular matrix must be equal to zero.

${ | H–E_{trial}S| = 0} \label{7.2.13}$

For a 3x3 system (i.e., expanded to three terms) Equation \ref{7.2.13} looks like:

$\begin{vmatrix} H_{11}-E_{trial}S_{11}&H_{12}-E_{trial}S_{12} \\ H_{12}-E_{trial}S_{12}&H_{22}-E_{trial}S_{22}\end{vmatrix}=0$

when the determinate is expanded, it will give a polynomial that will have N roots (solutions of E_{trials}).

Determinants OVerview

• The determinant is a real number, it is not a matrix.
• The determinant can be a negative number.
• It is not associated with absolute value at all except that they both use vertical lines.
• The determinant only exists for square matrices (2×2, 3×3, ... n×n). The determinant of a 1×1 matrix is that single value in the determinant.
• The inverse of a matrix will exist only if the determinant is not zero.

Determinants can be expanded using Minors and Cofactors

The method is called expansion using minors and cofactors. Before we can use them, we need to define them. It is the product of the elements on the main diagonal minus the product of the elements off the main diagonal.

In the case of a 2 × 2 matrix, the specific formula for the determinant is

{\displaystyle {\begin{aligned}|A|={\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc.\end{aligned}}}

Similarly, suppose we have a 3 × 3 matrix A, and we want the specific formula for its determinant |A|:

{\displaystyle {\begin{aligned}|A|={\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}&=a\,{\begin{vmatrix}e&f\\h&i\end{vmatrix}}-b\,{\begin{vmatrix}d&f\\g&i\end{vmatrix}}+c\,{\begin{vmatrix}d&e\\g&h\end{vmatrix}}\\&=aei+bfg+cdh-ceg-bdi-afh.\end{aligned}}}

To solve this determinate in Equation \ref{7.2.13}, it should be expanded to generate a polynomial (a characteristic equation) that can be directly solved with linear methods (i.e., find the roots  - different $$E_{trail}$$ values that satisfy the secular equations).

Example $$\PageIndex{1}$$: Optimizing a two-term basis set

If $$|\psi_{trial} \rangle$$ is a linear combination of two functions. In math terms,

$|\psi_{trial} \rangle= \sum_{n=1}^{N=2} a_n |f_n\rangle = a_1 |\phi_1 \rangle + a_2 | \phi_2 \rangle$

then the secular determinant (Equation $$\ref{7.2.13}$$), in matrix formulation would look like this

$\begin{vmatrix} H_{11}-E_{trial}S_{11}&H_{12}-E_{trial}S_{12} \\ H_{12}-E_{trial}S_{12}&H_{22}-E_{trial}S_{22}\end{vmatrix}=0$

Solution

Solving the secular equations is done by finding $$E_{trial}$$ and putting the value into the expansion of the secular determinant

$a_1^2 H_{11} + 2a_1 a_2 H_{12}+ a_2^2 H_{22}=0$

and

$a_1(H_{12} - E_{trial}S_{12}) + a_2(H_{22} - E_{trial}S_{22}) = 0$

Equation $$\ref{7.2.13}$$ can be solved to obtain the energies $$E$$. When arranged in order of increasing energy, these provide approximations to the energies of the first $$k$$ states (each having an energy higher than the true energy of the state by virtue of the variation theorem). To find the energies of a larger number of states we simply use a greater number of basis functions $$\{\phi_i\}$$ in the trial wavefunction (Example \ref{Ex1}). To obtain the approximate wavefunction for a particular state, we substitute the appropriate energy into the secular equations and solve for the coefficients $$a_i$$.

Using this method it is possible to find all the coefficients $$a_1 \ldots a_k$$ in terms of one coefficient; normalizing the wavefunction provides the absolute values for the coefficients.

Example: Polynomial Basis set for Approximating the Wavefunction of a Particle in an Infinite Potential Well

The potential well with infinite barriers is defined:

$V(x) = \infty \;for | x| > |L|$

and

$V(x) = 0 \;for \; |x| \leq |L|$

and it forces the wave function to vanish at the boundaries of the well at $$x=\pm a$$. The exact solution for this problem is known and treated previously. Here we discuss a linear variational approach to be compared with the exact solution. We take $$a=1$$ and use natural units such that $$\hbar^2/2m=1$$.

As the all variational methods problems with a basis set, the trial wavefunction is expanded

$| \varphi \rangle = \sum_n f_n(x)$

As basis functions we take simple polynomials that vanish on the boundaries of the well:

$\psi_n(x)=x^n(x-1)(x+1)$

with $$n=0,1,2,3...$$

The reason for choosing this particular form of basis functions is that the relevant matrix elements can easily be calculated analytically.

Solution

We start we the overlap matrix:

$S_{mn}=\langle \psi_n\vert\psi_m \rangle = \int_{-1}^1 \psi_n(x) \psi_m(x) dx.$

Working out the integrals, one obtains

$S_{mn}=\dfrac{2}{n+m+5} - \dfrac{4}{n+m+3} + \dfrac{2}{n+m+1}$

for when $$n+m$$ even, and zero otherwise.

We need to calculate the Hamiltonian matrix elements:

$H_{mn}=\langle \psi_n \vert p^2 \vert \psi_m \rangle = \int_{-1}^1 \psi_n(x) \left(-\frac{d^2}{dx^2} \right) \psi_m(x) dx$

$= -8 \left[ \dfrac{1-m-n-2mn} {(m+n+3)(m+n+1)(m+n-1)} \right]$

for when $$n+m$$ even, and zero otherwise.

## Linear Combination of Trial Wavefunctions (e.g., Gaussian functions)

In electron calculations, the trial function approximation can be constructed in terms of the ground-state wavefunctions (remember that even the radial component of higher orbitals decay exponentially when far from the nucleus:

$\phi = \sum_{j=1}^{N} c_j e^{-\alpha_j r}\label{10A}$

An alternative to this trial wavefunction is the combination of Gaussian functions:

$\phi = \sum_{j=1}^{N} c_j e^{-\alpha_j r^2}\label{10}$

We know the energy of the hydrogen atom, but using a set of $$N$$ Gaussian functions gives...

$$N$$ $$E_{min}$$ in Hartree units
1 -0.42441
2 -0.48581
3 -0.49697
: :
5 -0.49976
: :
16 -0.49998

A quick note on units - $$1 Hartree = 2 R_{\infty}$$

These sets of Gaussian functions are used in computer programs for molecular quantum mechanic calculations, because the integrals are much simpler than using the one-electron Hydrogen atomic functions. Besides, for multi-electron atoms, the H-atom functions are not as accurate.

Overview of Variational Methods

1. The hyrogen atom is the only atom with an exact solution.
2. Hydrogen wave functions are used as the approximation for atomic wave functions in multielectron atoms for atomic wavefunctions in multielectron atoms.
3. The variational principle states that any wave function we choose that satisfies the Schrödinger equation will give an energy greater than the true energy of the system.
4. The variation method provides a general prescription for improving on any wave function with a parameter by minimizing that function with respect to the parameter minimizing that function with respect to the parameter